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$("#documenter > .docs-main").bind('click', function(ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass('visible')) { + sidebar.removeClass('visible'); + } + }); +}) + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function() { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css('max-width'), 10); + var L0 = e.width(); + if(L0 > L) { + var h0 = parseInt(e.css('font-size'), 10); + e.css('font-size', L * h0 / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on('orientationchange', resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function() { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if(typeof active !== 'undefined') { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +function set_theme(theme) { + var active = null; + var disabled = []; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + var themename = ss.ownerNode.getAttribute("data-theme-name"); + if(themename === null) continue; // ignore non-theme stylesheets + // Find the active theme + if(themename === theme) active = ss; + else disabled.push(ss); + } + if(active !== null) { + active.disabled = false; + if(active.ownerNode.getAttribute("data-theme-primary") === null) { + document.getElementsByTagName('html')[0].className = "theme--" + theme; + } else { + document.getElementsByTagName('html')[0].className = ""; + } + disabled.forEach(function(ss){ + ss.disabled = true; + }); + } + + // Store the theme in localStorage + if(typeof(window.localStorage) !== "undefined") { + window.localStorage.setItem("documenter-theme", theme); + } else { + console.error("Browser does not support window.localStorage"); + } +} + +// Theme picker setup +$(document).ready(function() { + // onchange callback + $('#documenter-themepicker').change(function themepick_callback(ev){ + var themename = $('#documenter-themepicker option:selected').attr('value'); + set_theme(themename); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if(typeof(window.localStorage) !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if(theme !== null) { + $('#documenter-themepicker option').each(function(i,e) { + e.selected = (e.value === theme); + }) + } else { + $('#documenter-themepicker option').each(function(i,e) { + e.selected = $("html").hasClass(`theme--${e.value}`); + }) + } + } +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function() { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if (typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === 'boolean' && DOCUMENTER_VERSION_SELECTOR_DISABLED) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function(x) { + target_href = version_selector_select.children("option:selected").get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if (typeof DOCUMENTER_CURRENT_VERSION !== 'undefined' && $('#version-selector > option').length == 0) { + var option = $(""); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== 'undefined') { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function(i,x){return x.text}); + DOC_VERSIONS.forEach(function(each) { + var version_url = documenterBaseURL + "/../" + each; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $(""); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if 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zovENb{1M=zLQ;rimn5^s3t|VL+@0MCQyG^3=%2E1^ZrfFkC5cxGI8jEAmc}g|9x*g zc&{-qPs_tmNzoZ5T8|nr59)D292sK9&n{i|$fpU`4JSKk-=Jv7@XwM2r7gCVsB?)zt)2S~>e zqL>mu_C;uwJ$q|F7P9wPkj|N=S@HpD@Mos89Lfxxvi{7trvpL3Ke~Y2Rv8-xS5Y0m z&QTviC34oWr^zLVZ1v2GQM>;`3MUJ4(L;Ivk_IHegxCYYc)pMli6ck;x?99INwTkL za$Q6jgLTF2Z7Og6KQGFBqzSzwf}Qw)tCbpAdrP2ezQOJ`>JV~<4mi@-Y$?e- + // MIT License + function parseUri (str) { + var o = parseUri.options, + m = o.parser[o.strictMode ? "strict" : "loose"].exec(str), + uri = {}, + i = 14; + + while (i--) uri[o.key[i]] = m[i] || ""; + + uri[o.q.name] = {}; + uri[o.key[12]].replace(o.q.parser, function ($0, $1, $2) { + if ($1) uri[o.q.name][$1] = $2; + }); + + return uri; + }; + parseUri.options = { + strictMode: false, + key: ["source","protocol","authority","userInfo","user","password","host","port","relative","path","directory","file","query","anchor"], + q: { + name: "queryKey", + parser: /(?:^|&)([^&=]*)=?([^&]*)/g + }, + parser: { + strict: /^(?:([^:\/?#]+):)?(?:\/\/((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?))?((((?:[^?#\/]*\/)*)([^?#]*))(?:\?([^#]*))?(?:#(.*))?)/, + loose: /^(?:(?![^:@]+:[^:@\/]*@)([^:\/?#.]+):)?(?:\/\/)?((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?)(((\/(?:[^?#](?![^?#\/]*\.[^?#\/.]+(?:[?#]|$)))*\/?)?([^?#\/]*))(?:\?([^#]*))?(?:#(.*))?)/ + } + }; + + $("#search-form").submit(function(e) { + e.preventDefault() + }) + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + lunr.stopWordFilter = lunr.generateStopWordFilter([ + 'a', + 'able', + 'about', + 'across', + 'after', + 'almost', + 'also', + 'am', + 'among', + 'an', + 'and', + 'are', + 'as', + 'at', + 'be', + 'because', + 'been', + 'but', + 'by', + 'can', + 'cannot', + 'could', + 'dear', + 'did', + 'does', + 'either', + 'ever', + 'every', + 'from', + 'got', + 'had', + 'has', + 'have', + 'he', + 'her', + 'hers', + 'him', + 'his', + 'how', + 'however', + 'i', + 'if', + 'into', + 'it', + 'its', + 'just', + 'least', + 'like', + 'likely', + 'may', + 'me', + 'might', + 'most', + 'must', + 'my', + 'neither', + 'no', + 'nor', + 'not', + 'of', + 'off', + 'often', + 'on', + 'or', + 'other', + 'our', + 'own', + 'rather', + 'said', + 'say', + 'says', + 'she', + 'should', + 'since', + 'so', + 'some', + 'than', + 'that', + 'the', + 'their', + 'them', + 'then', + 'there', + 'these', + 'they', + 'this', + 'tis', + 'to', + 'too', + 'twas', + 'us', + 'wants', + 'was', + 'we', + 'were', + 'what', + 'when', + 'who', + 'whom', + 'why', + 'will', + 'would', + 'yet', + 'you', + 'your' + ]) + + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!" + // would not find anything if searching for "add!", only for the entire qualification + lunr.tokenizer.separator = /[\s\-\.]+/ + + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + lunr.trimmer = function (token) { + return token.update(function (s) { + return s.replace(/^[^a-zA-Z0-9@!]+/, '').replace(/[^a-zA-Z0-9@!]+$/, '') + }) + } + + lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'juliaStopWordFilter') + lunr.Pipeline.registerFunction(lunr.trimmer, 'juliaTrimmer') + + var index = lunr(function () { + this.ref('location') + this.field('title',{boost: 100}) + this.field('text') + documenterSearchIndex['docs'].forEach(function(e) { + this.add(e) + }, this) + }) + var store = {} + + documenterSearchIndex['docs'].forEach(function(e) { + store[e.location] = {title: e.title, category: e.category, page: e.page} + }) + + $(function(){ + searchresults = $('#documenter-search-results'); + searchinfo = $('#documenter-search-info'); + searchbox = $('#documenter-search-query'); + searchform = $('.docs-search'); + sidebar = $('.docs-sidebar'); + function update_search(querystring) { + tokens = lunr.tokenizer(querystring) + results = index.query(function (q) { + tokens.forEach(function (t) { + q.term(t.toString(), { + fields: ["title"], + boost: 100, + usePipeline: true, + editDistance: 0, + wildcard: lunr.Query.wildcard.NONE + }) + q.term(t.toString(), { + fields: ["title"], + boost: 10, + usePipeline: true, + editDistance: 2, + wildcard: lunr.Query.wildcard.NONE + }) + q.term(t.toString(), { + fields: ["text"], + boost: 1, + usePipeline: true, + editDistance: 0, + wildcard: lunr.Query.wildcard.NONE + }) + }) + }) + searchinfo.text("Number of results: " + results.length) + searchresults.empty() + results.forEach(function(result) { + data = store[result.ref] + link = $(''+data.title+'') + link.attr('href', documenterBaseURL+'/'+result.ref) + if (data.category != "page"){ + cat = $('('+data.category+', '+data.page+')') + } else { + cat = $('('+data.category+')') + } + li = $('

  • ').append(link).append(" ").append(cat) + searchresults.append(li) + }) + } + + function update_search_box() { + querystring = searchbox.val() + update_search(querystring) + } + + searchbox.keyup(_.debounce(update_search_box, 250)) + searchbox.change(update_search_box) + + // Disable enter-key form submission for the searchbox on the search page + // and just re-run search rather than refresh the whole page. + searchform.keypress( + function(event){ + if (event.which == '13') { + if (sidebar.hasClass('visible')) { + sidebar.removeClass('visible'); + } + update_search_box(); + event.preventDefault(); + } + } + ); + + search_query_uri = parseUri(window.location).queryKey["q"] + if(search_query_uri !== undefined) { + search_query = decodeURIComponent(search_query_uri.replace(/\+/g, '%20')) + searchbox.val(search_query) + } + update_search_box(); + }) +}) + +}) diff --git a/v0.4.5/assets/shear.png b/v0.4.5/assets/shear.png new file mode 100644 index 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rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#ecf0f1}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#ecf0f1;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content 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kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content 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.button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#ecf0f1;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-primary,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink{background-color:#375a7f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:hover,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#335476;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-focused.docs-sourcelink{border-color:transparent;color:#fff}html.theme--documenter-dark 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form.docs-search>input.button{border-radius:290486px;padding-left:1em;padding-right:1em}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:0.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){border-radius:3px;font-size:.85em}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons 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h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content 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.pagination.is-rounded .pagination-previous,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination .pagination-previous,html.theme--documenter-dark .pagination.is-rounded .pagination-next,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination .pagination-next{padding-left:1em;padding-right:1em;border-radius:290486px}html.theme--documenter-dark .pagination.is-rounded .pagination-link,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination .pagination-link{border-radius:290486px}html.theme--documenter-dark .pagination,html.theme--documenter-dark .pagination-list{align-items:center;display:flex;justify-content:center;text-align:center}html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark 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    Edit on GitHub

    Function References

    FourierTools.conv — Function
    conv(u, v[, dims])

    Convolve u with v over dims dimensions with an FFT based method. Note, that this method introduces wrap-around artifacts without proper padding/windowing.

    Arguments

    • u is an array in real space.
    • v is the array to be convolved in real space as well.
    • Per default ntuple(+, min(N, M))) means that we perform the convolution over all dimensions of that array which has less dimensions. If dims is an array with integers, we perform convolution only over these dimensions. Eg. dims=[1,3] would perform the convolution over the first and third dimension. Second dimension is not convolved.

    If u and v are both a real valued array we use rfft and hence the output is real as well. If either u or v is complex we use fft and output is hence complex.

    Examples

    1D with FFT over all dimensions. We choose v to be a delta peak. Therefore convolution should act as identity.

    julia> u = [1 2 3 4 5]
+1×5 Array{Int64,2}:
+ 1  2  3  4  5
+julia> v = [0 0 1 0 0]
+1×5 Array{Int64,2}:
+ 0  0  1  0  0
+
+julia> conv(u, v)
+1×5 Matrix{Float64}:
+ 4.0  5.0  1.0  2.0  3.0

    2D with FFT with different dims arguments.

    julia> u = 1im .* [1 2 3; 4 5 6]
+2×3 Matrix{Complex{Int64}}:
+ 0+1im  0+2im  0+3im
+ 0+4im  0+5im  0+6im
+
+julia> v = [1im 0 0; 1im 0 0]
+2×3 Matrix{Complex{Int64}}:
+ 0+1im  0+0im  0+0im
+ 0+1im  0+0im  0+0im
+
+julia> conv(u, v)
+2×3 Matrix{ComplexF64}:
+ -5.0+0.0im  -7.0+0.0im  -9.0+0.0im
+ -5.0+0.0im  -7.0+0.0im  -9.0+0.0im
    source
    FourierTools.ccorr — Function
    ccorr(u, v[, dims]; centered=false)

    Calculates the cross-correlation between u and v along dims. centered=true moves the output of the cross-correlation to the Fourier center.

    If u and v are both a real valued array we use rfft and hence the output is real as well. If either u or v is complex we use fft and output is hence complex.

    Per default the correlation is performed along min(ndims(u), ndims(v)).

    julia> ccorr([1,1,0,0], [1,1,0,0], centered=true)
+4-element Vector{Float64}:
+ 0.0
+ 1.0
+ 2.0
+ 1.0
+
+julia> ccorr([1,1,0,0], [1,1,0,0])
+4-element Vector{Float64}:
+ 2.0
+ 1.0
+ 0.0
+ 1.0
+
+julia> ccorr([1im,0,0,0], [0,1im,0,0])
+4-element Vector{ComplexF64}:
+ 0.0 + 0.0im
+ 0.0 + 0.0im
+ 0.0 + 0.0im
+ 1.0 + 0.0im
+
+julia> ccorr([1im,0,0,0], [0,1im,0,0], centered=true)
+4-element Vector{ComplexF64}:
+ 0.0 + 0.0im
+ 1.0 + 0.0im
+ 0.0 + 0.0im
+ 0.0 + 0.0im
    source
    FourierTools.conv_psf — Function
    conv_psf(u, psf[, dims])

    conv_psf is a shorthand for conv(u,ifftshift(psf)). For examples see conv.

    source
    FourierTools.plan_conv — Function
    plan_conv(u, v [, dims]; kwargs...)

    Pre-plan an optimized convolution for arrays shaped like u and v (based on pre-plan FFT) along the given dimenions dims. dims = 1:ndims(u) per default. The 0 frequency of u must be located at the first entry.

    We return two arguments: The first one is v_ft (obtained by fft(v) or rfft(v)). The second return is the convolution function pconv. pconv itself has two arguments. pconv(u, v_ft=v_ft) where u is the object and v_ft the v_ft. This function achieves faster convolution than conv(u, u). Depending whether u is real or complex we do ffts or rffts Additionally, it is possible to provide flags=FFTW.MEASURE as kwargs to change the planning of the FFT.

    Examples

    julia> u = [1 2 3 4 5]
+1×5 Matrix{Int64}:
+ 1  2  3  4  5
+
+julia> v = [1 0 0 0 0]
+1×5 Matrix{Int64}:
+ 1  0  0  0  0
+
+julia> v_ft, pconv = plan_conv(u, v);
+
+julia> pconv(u, v_ft)
+1×5 Matrix{Float64}:
+ 1.0  2.0  3.0  4.0  5.0
+
+julia> pconv(u)
+1×5 Matrix{Float64}:
+ 1.0  2.0  3.0  4.0  5.0
    source
    FourierTools.plan_conv_psf — Function
    plan_conv_psf(u, psf [, dims]; kwargs...) where {T, N}

    plan_conv_psf is a shorthand for plan_conv(u, ifftshift(psf)). For examples see plan_conv.

    source
    FourierTools.plan_conv_psf_buffer — Function
    plan_conv_psf_buffer(u, psf [, dims]; kwargs...) where {T, N}

    plan_conv_psf_buffer is a shorthand for plan_conv_buffer(u, ifftshift(psf)). For examples see plan_conv.

    source
    diff --git a/v0.4.5/czt/index.html b/v0.4.5/czt/index.html new file mode 100644 index 0000000..8729774 --- /dev/null +++ b/v0.4.5/czt/index.html @@ -0,0 +1,74 @@ + +CZT · FourierTools.jl
    Edit on GitHub

    CZTs

    Chirp Z Transformations: Allows Fourier-transformation and at the same time zooming into the result, which is why it is also called the Zoomed-FFT algorithm. The algorithm is loosely based on a publication [Rabiner, Schafer, Rader, The Chirp z-Transform Algorithm, IEEE Trans AU 17(1969) p. 86]. It needs three FFTs to perform its work but one can be precalculated by using plan_czt. Variable zooms, transform dimensions, array center positions as well as output sizes are supported along wiht a low-level interface by specifingy a and w.

    FourierTools.czt — Function
    czt(xin, scale, dims=1:ndims(xin), dsize=size(xin,d); a=nothing, w=nothing, damp=ones(ndims(xin)), 
+    src_center=size(xin,d)÷2+1, dst_center=dsize÷2+1, remove_wrap=false, fft_flags=FFTW.ESTIMATE)

    Chirp z transform of the ND array xin The tuple scale defines the zoom factors in the Fourier domain. Each has to be bigger than one.

    See also: iczt, czt_1d

    The code is based on Rabiner, Schafer & Rader 1969, IEEE Trans. on Audio and Electroacoustics, 17,86-92

    Arguments:

    • xin: array to transform
    • scale: a tuple of factors (one for each dimension) to zoom into during the czt. Note that a factor of nothing (or 1.0) needs to be provided, if a dimension is not transformed.
    • dims: a tuple of dimensions over which to apply the czt.
    • dsize: a tuple specifying the destination size
    • a: defines the starting phase of the result CZT. This relates to the where the center of the destination array should be. The default is nothing which means it is calculated from the src_center argument.
    • w: defines the consecutive phases of the result array, i.e. the zoom. It is (default nothing) usually automatically calculated from the scaled and the damp argument. You only need to state it, if you want to use the low-level interface (e.g. for the Laplace transform).
    • damp: a multiplicative factor to apply as a damping coefficient to w.
    • src_center: position of the nominal central (zero-position) pixel in the source array. By default the Fourier-center size(src).÷2 .+1 is used.
    • dst_center: the center (zero-position) of the destination array. By default the Fourier-center size(dst).÷2 .+1 is used.
    • remove_wrap: if true, the positions that represent a wrap-around will be set to zero

    Example:

    julia> using IndexFunArrays
+
+julia> sz = (10,10);
+
+julia> xin = disc(sz,4)
+10×10 Matrix{Float64}:
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  1.0  1.0  1.0  1.0  1.0  0.0  0.0
+ 0.0  0.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  0.0
+ 0.0  0.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  0.0
+ 0.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0
+ 0.0  0.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  0.0
+ 0.0  0.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  0.0
+ 0.0  0.0  0.0  1.0  1.0  1.0  1.0  1.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0
+
+julia> xft = czt(xin,(1.2,1.3));
+
+julia> zoomed = real.(ift(xft))
+10×10 Matrix{Float64}:
+ -0.0197423    0.0233008  -0.0449251   0.00295724   0.205593  -0.166546   0.205593   0.00295724  -0.0449251   0.0233008
+  0.0239759   -0.028264    0.0541186  -0.0116475   -0.261294   0.312719  -0.261294  -0.0116475    0.0541186  -0.028264
+ -0.0569       0.0666104  -0.122277    0.140354     0.78259    1.34381    0.78259    0.140354    -0.122277    0.0666104
+  0.00540611  -0.0117886   0.0837357   1.30651      1.8283     1.50127    1.8283     1.30651      0.0837357  -0.0117886
+  0.11892     -0.147731    0.368046    1.76537      1.33218    1.66119    1.33218    1.76537      0.368046   -0.147731
+ -0.00389861   0.0145979   1.21842     1.52989      1.67375    1.543      1.67375    1.52989      1.21842     0.0145979
+  0.11892     -0.147731    0.368046    1.76537      1.33218    1.66119    1.33218    1.76537      0.368046   -0.147731
+  0.00540611  -0.0117886   0.0837357   1.30651      1.8283     1.50127    1.8283     1.30651      0.0837357  -0.0117886
+ -0.0569       0.0666104  -0.122277    0.140354     0.78259    1.34381    0.78259    0.140354    -0.122277    0.0666104
+  0.0239759   -0.028264    0.0541186  -0.0116475   -0.261294   0.312719  -0.261294  -0.0116475    0.0541186  -0.028264
    source
    FourierTools.plan_czt — Function
    plan_czt(xin, scale, dims, dsize=size(xin); a=nothing, w=nothing, damp=ones(ndims(xin)), 
+         src_center=size(xin).÷2 .+1, dst_center=dsize.÷2 .+1, remove_wrap=false, fft_flags=FFTW.ESTIMATE)

    creates a plan for an N-dimensional chirp z-transformation (CZT). The generated plan is then applied via muliplication. For details about the arguments, see czt().

    source
    FourierTools.iczt — Function
    iczt(xin ,scale, dims=1:length(size(xin)), dsize=size(xin,d); a=nothing, w=nothing, damp=1.0, 
+     src_center=size(xin,d)÷2+1, dst_center=dsize÷2+1, remove_wrap=false, fft_flags=FFTW.ESTIMATE)

    Inverse chirp z transform of the ND array xin The tuple scale defines the zoom factors in the Fourier domain. Each has to be bigger than one. The code is based on Rabiner, Schafer & Rader 1969, IEEE Trans. on Audio and Electroacoustics, 17,86-92

    Arguments:

    • xin: array to transform
    • scaled: factor to zoom into during the 1-dimensional czt.
    • d: single dimension to transform (as a tuple)
    • dsize: size of the destination array
    • a: defines the starting phase of the result CZT. This relates to the where the center of the destination array should be. The default is nothing which means it is calculated from the src_center argument.
    • w: defines the consecutive phases of the result array, i.e. the zoom. It is (default nothing) usually automatically calculated from the scaled and the damp argument. You only need to state it, if you want to use the low-level interface (e.g. for the Laplace transform).
    • damp: a multiplicative factor to apply as a damping coefficient to w.
    • src_center: position of the nominal central (zero-position) pixel in the source array. By default the Fourier-center size(src).÷2 .+1 is used.
    • dst_center: the center (zero-position) of the destination array. By default the Fourier-center size(dst).÷2 .+1 is used.
    • remove_wrap: if true, the positions that represent a wrap-around will be set to zero
    • pad_value: the value to pad wrapped data with.

    See also: czt, czt_1d

    Example

    
+julia> using IndexFunArrays
+
+julia> sz = (10,10);
+
+julia> xin = disc(sz,4)
+10×10 Matrix{Float64}:
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  1.0  1.0  1.0  1.0  1.0  0.0  0.0
+ 0.0  0.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  0.0
+ 0.0  0.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  0.0
+ 0.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0
+ 0.0  0.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  0.0
+ 0.0  0.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  0.0
+ 0.0  0.0  0.0  1.0  1.0  1.0  1.0  1.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0
+
+julia> xft = ft(xin);
+
+julia> iczt(xft,(1.2,1.3))
+10×10 Matrix{ComplexF64}:
+ 0.00648614+0.0213779im  0.0165456+0.0357733im  0.0389356+0.0482465im  -0.235491-0.156509im    …  0.178484-0.0730099im  -0.245418-5.88331e-5im  0.0471654-0.0814548im  0.0141525+0.0734892im
+  -0.104602-0.160481im   -0.163859-0.125535im    0.495205+0.135059im    0.660335+0.00736904im     0.764045-0.0497007im    0.67753+0.263814im      0.48095-0.0775406im  -0.159713-0.0637132im
+   0.139304+0.111348im    0.454973+0.106869im    0.970263-0.0249785im    1.25999-0.166495im        1.07328-0.0481437im    1.24013-0.14664im      0.986722-0.0414382im   0.450186+0.111656im
+  -0.035645-0.0311352im    1.03899-0.0589268im     1.1463-0.0940003im   0.790545+0.283668im       0.994255+0.134865im     0.80774-0.0124851im     1.13205+0.151519im     1.04314-0.130321im
+   0.292575+0.0853233im   0.929883+0.0687029im    1.06514-0.0649952im   0.989483-0.019913im        1.02311+0.018235im    0.979555-0.136654im      1.07337+0.0317868im    0.92749+0.0405597im
+    1.12254-0.0464723im    1.03467-0.0239316im    0.92709+0.0822984im     1.0521-0.0992709im   …  0.983655-0.0663123im     1.0521+0.0992709im     0.92709-0.0822984im    1.03467+0.0239316im
+   0.287928-0.0306724im    0.92749-0.0405597im    1.07337-0.0317868im   0.979555+0.136654im        1.01648+0.0597475im   0.989483+0.019913im      1.06514+0.0649952im   0.929883-0.0687029im
+ -0.0275957+0.169775im     1.04314+0.130321im     1.13205-0.151519im     0.80774+0.0124851im       1.00574+0.0629632im   0.790545-0.283668im       1.1463+0.0940003im    1.03899+0.0589268im
+   0.130009-0.120643im    0.450186-0.111656im    0.986722+0.0414382im    1.24013+0.14664im         1.06002+0.0348813im    1.25999+0.166495im     0.970263+0.0249785im   0.454973-0.106869im
+ -0.0965531+0.0404296im  -0.159713+0.0637132im    0.48095+0.0775406im    0.67753-0.263814im        0.77553-0.121603im    0.660335-0.00736904im   0.495205-0.135059im   -0.163859+0.125535im
    source
    FourierTools.czt_1d — Function
    czt_1d(xin , scaled , d; remove_wrap=false, pad_value=zero(eltype(xin)))

    Chirp z transform along a single direction d of an ND array xin. Note that the result type is defined by eltype(xin) and not by scales.

    The code is based on Rabiner, Schafer & Rader 1969, IEEE Trans. on Audio and Electroacoustics, 17,86-92

    Arguments:

    • xin: array to transform
    • scaled: factor to zoom into during the 1-dimensional czt.
    • d: single dimension to transform (as a tuple)
    • dsize: size of the destination array
    • a: defines the starting phase of the result CZT. This relates to the where the center of the destination array should be. The default is nothing which means it is calculated from the src_center argument.
    • w: defines the consecutive phases of the result array, i.e. the zoom. It is (default nothing) usually automatically calculated from the scaled and the damp argument. You only need to state it, if you want to use the low-level interface (e.g. for the Laplace transform).
    • damp: a multiplicative factor to apply as a damping coefficient to w.
    • src_center: position of the nominal central (zero-position) pixel in the source array. By default the F ourier-center size(src).÷2 .+1 is used.
    • dst_center: the center (zero-position) of the destination array. By default the Fourier-center size(dst).÷2 .+1 is used.
    • extra_phase: a phase ramp to apply to the final result relating to the srccenter. By default nothing which calculates this phase according to the `srccenter`.
    • global_phase: the initial phase of the destitation array. By default nothing which calculates this phase according to the centers.
    • remove_wrap: if true, the positions that represent a wrap-around will be set to zero
    • pad_value: the value to pad wrapped data with.
    source
    czt_1d(xin , plan::CZTPlan_1D)

    Chirp z transform along a single direction d of an ND array xin. Note that the result type is defined by eltype(xin) and not by scales. The plan can also be applied via multiplication with xin.

    The code is based on Rabiner, Schafer & Rader 1969, IEEE Trans. on Audio and Electroacoustics, 17,86-92

    Arguments

    `plan`:   A plan created via plan_czt_1d()
    source
    FourierTools.plan_czt_1d — Function
    plan_czt_1d(xin, scaled, d, dsize=size(xin,d); a=nothing, w=nothing, damp=1.0, src_center=(size(xin,d)+1)/2, 
+            dst_center=dsize÷2+1, remove_wrap=false, fft_flags=FFTW.ESTIMATE)

    creates a plan for an one-dimensional chirp z-transformation (CZT). The generated plan is then applied via muliplication. For details about the arguments, see czt_1d().

    source
    FourierTools.CZTPlan_1D — Type
    CZTPlan_1D{CT<:Complex, D<:Integer, AT<:AbstractArray{CT, D}, PT<:Number, PFFT<:AbstractFFTs.Plan, PIFFT<:AbstractFFTs.ScaledPlan}

    type used for the onedimensional plan of the chirp Z transformation (CZT). containing

    Members:

    `d`: dimension (only one!) to transform with this plan
+`pad_value`: the value to pad wrapped data with (zero is already handled by the `wd` term, if wanted). 
+`pad_ranges` :: tuple of two ranges of invalid positions, which can be replaced by pad values
+`aw`: factor to multiply input with
+`fft_fv`: fourier-transform (FFTW) of the convolutio kernel
+`wd`: factor to multiply the result of the convolution by
+`fftw_plan!`: plan for the forward FFTW of the convolution kernel
+`ifftw_plan!`: plan for the inverse FFTW of the convolution kernel
    source
    FourierTools.CZTPlan_ND — Type
    CZTPlan_ND{CT, D} # <: AbstractArray{T,D}

    type used for the onedimensional plan of the chirp Z transformation (CZT). containing

    Members:

    `plans`: vector of CZTPlan_1D for each of the directions of the ND array to transform
    source
    diff --git a/v0.4.5/fractional/index.html b/v0.4.5/fractional/index.html new file mode 100644 index 0000000..eebc722 --- /dev/null +++ b/v0.4.5/fractional/index.html @@ -0,0 +1,2 @@ + +Fractional Fourier Transform · FourierTools.jl
    Edit on GitHub

    Fractional Fourier Transforms

    Implementation of the Fast Fractional Fourier Transform based on Javier García, David Mas, and Rainer G. Dorsch, "Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm," Appl. Opt. 35, 7013-7018 (1996)

    FourierTools.frfft — Function
    frfft(arr, p; shift=false, method=:garcia)

    Calculates the fractional Fast Fourier transform (FRFFT) of the order p of arr. No dims argument is supported yet.

    If shift=false the FRFFT is calculated around the first entry. If shift=true the FRFFT is calculated aroound the center.

    Methods

    Several implementation exists. The following are implemented:

    • method=:garcia: A convolutional approach based on 2 FFTs. See García, J., Mas, D., & Dorsch, R. G. (1996). Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm. Applied Optics, 35(35), 7013. doi:10.1364/ao.35.007013
    source
    diff --git a/v0.4.5/helpers/index.html b/v0.4.5/helpers/index.html new file mode 100644 index 0000000..e80c7bb --- /dev/null +++ b/v0.4.5/helpers/index.html @@ -0,0 +1,69 @@ + +FFT Helpers · FourierTools.jl
    Edit on GitHub

    FFT helpers

    FourierTools.rffts — Function
    rffts(A [, dims])

    Calculates a rfft(A, dims) and then shift the frequencies to the center. dims[1] is not shifted, because there is no negative and positive frequency. The shift is done with ShiftedArrays and therefore doesn't allocate memory.

    See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

    source
    FourierTools.irffts — Function
    irffts(A, d, [, dims])

    Calculates a irfft(A, d, dims) and then shift the frequencies back to the corner. dims[1] is not shifted, because there is no negative and positive frequency. The shift is done with ShiftedArrays and therefore doesn't allocate memory.

    See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

    source
    FourierTools.rfftshift_view — Function
    rfftshift_view(A, dims)

    Shifts the frequencies to the center expect for dims[1] because there os no negative and positive frequency.

    source
    FourierTools.irfftshift_view — Function
    irfftshift_view(A, dims)

    Shifts the frequencies back to the corner except for dims[1] because there os no negative and positive frequency.

    source
    FourierTools.ft — Function
    ft(A [, dims])

    Digital Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(fft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with both coordinate systems in real and Fourier-space being centered at position CtrFT == size÷2+1

    The following identities are true:

    julia> sz = (5,5)
+(5, 5)
+
+julia> ft(ones(sz)) ≈ prod(sz) .* δ(sz)
+true
+
+julia> ft(δ(sz)) ≈ ones(sz)
+true

    See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

    source
    FourierTools.ift — Function
    ift(A [, dims])

    Digital inverse Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(ifft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with both coordinate systems in real and Fourier-space being centered at position CtrFT == size÷2+1

    The following identities are true:

    julia> sz = (5,6,7)
+(5, 6, 7)
+
+julia> ift(ones(sz)) ≈ δ(sz)
+true
+
+julia> ift(δ(sz)) ≈ ones(sz) ./ prod(sz)
+true

    See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

    source
    FourierTools.rft — Function
    rft(A [, dims])

    Digital real-valued Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(rfft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with the coordinate systems in real space centered at CtrFT == size÷2+1 and in (half) Fourier-space being centered at CtrRFT == setindex(size÷2 +1,1,1).

    The following identities are true:

    julia> sz = (6,6)
+(6, 6)
+
+julia> rft(δ(sz)) ≈ ones(rft_size(sz))
+true

    See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

    source
    FourierTools.irft — Function
    irft(A, d, [, dims])

    Digital real-valued inverse Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(irfft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with the coordinate systems in real space centered at CtrFT == size÷2+1 and in (half) Fourier-space being centered at CtrRFT == setindex(size÷2 +1,1,1). Note that the size d of the first transform direction [1] is a required argument.

    The following identities are true:

    julia> sz = (6,6)
+(6, 6)
+
+julia> irft(ones(rft_size(sz)),sz[1]) ≈ δ(sz)
+true

    See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

    source

    FFT Utils

    FourierTools.fftpos — Function
    fftpos(L, N, around=CenterFirst::Center)

    Construct a range from -L/2 to L/2 around around

    However, we ensure that those positions are in a way which they are useful for FFT operations. This means, that depending on the center a small offset is subtracted.

    See NDTools.Center for all center options. You need to load using NDTools to access all center options.

    Examples

    julia> collect(fftpos(1,4))
+4-element Vector{Float64}:
+ 0.0
+ 0.2916666666666667
+ 0.5833333333333334
+ 0.875
+
+julia> collect(fftpos(1,5))
+5-element Vector{Float64}:
+ 0.0
+ 0.225
+ 0.45
+ 0.675
+ 0.9
+
+julia> using NDTools
+
+julia> collect(fftpos(1,4, CenterFirst))
+4-element Vector{Float64}:
+ 0.0
+ 0.2916666666666667
+ 0.5833333333333334
+ 0.875
+
+julia> collect(fftpos(1,4, CenterFT))
+4-element Vector{Float64}:
+ -0.5833333333333333
+ -0.29166666666666663
+  3.70074341541719e-17
+  0.2916666666666667
+
+julia> collect(fftpos(1,4, CenterMiddle))
+4-element Vector{Float64}:
+ -0.4375
+ -0.14583333333333334
+  0.14583333333333334
+  0.4375
    source
    fftpos(l, N, around)

    Another fftpos method where the range is constructed around around. around is here a number indicating the index position around the range is constructed

    source
    FourierTools.rft_size — Function
    rft_size(sz::NTuple{Int})

    Returns the size of an rft or rfft performed on the data x, without performing the rfft. sz: corresponding real space size to obtain the rft size for

    source
    rft_size(arr)

    Returns the size of an rft or rfft performed on the data x, without performing the rfft.

    arr: array to optain the corresponding rft size for

    source
    FourierTools.rfft_size — Function
    rfft_size(size, dims)

    Returns the size rfft would return if applied to a real array. size is the input size to rfft and dims the dimensions the rfft transforms over. Actually we only would need first(dims).

    julia> using FFTW
+
+julia> rfft((ones((4,3,2))), (2,3)) |> size
+(4, 2, 2)
+
+julia> FourierTools.rfft_size((4,3,2), (2, 3))
+(4, 2, 2)
    source
    FourierTools.ft_center_diff — Function
    ft_center_diff(s [, dims])

    Calculates how much each dimension must be shifted that the center frequency is at the Fourier center. This if for a normal fft

    source
    FourierTools.rft_center_diff — Function
    rft_center_diff(s [, dims])

    Calculates how much each dimension must be shifted that the center frequency is at the Fourier center. This is for rfft. The dims[1] must be therefore not shifted!

    source
    FourierTools.center_pos — Function
    center_pos(x)

    Calculate the position of the center frequency. Size of the array is x

    Examples

    julia> FourierTools.center_pos(3)
+2
+julia> FourierTools.center_pos(4)
+3
    source

    FFT Helpers 2D

    FourierTools.fftshift2d — Function
    fftshift2d(mat::AbstractArray{T, N}) where {T, N}

    Short-hand for fftshift(mat, (1,2)). See fftshift for details.

    source
    FourierTools.ifftshift2d — Function
    ifftshift2d(mat::AbstractArray{T, N}) where {T, N}

    Short-hand for ifftshift(mat, (1,2)). See ifftshift for details.

    source
    FourierTools.fftshift2d_view — Function
    fftshift2d_view(mat::AbstractArray{T, N}) where {T, N}

    Short-hand for fftshift_view(mat, (1,2)). See fftshift_view for details.

    source
    FourierTools.ifftshift2d_view — Function
    ifftshift2d_view(mat::AbstractArray{T, N}) where {T, N}

    Short-hand for ifftshift_view(mat, (1,2)) performing only a 2D inverse ft. See ifft for details.

    source
    diff --git a/v0.4.5/index.html b/v0.4.5/index.html new file mode 100644 index 0000000..0807621 --- /dev/null +++ b/v0.4.5/index.html @@ -0,0 +1,2 @@ + +FourierTools.jl · FourierTools.jl
    Edit on GitHub

    FourierTools.jl

    This package contains various functions that are useful for working with and in Fourier space.

    Install FourierTools.jl via the package manager. Currently with

    julia> add FourierTools

    Check out the sub pages for different applications.

    diff --git a/v0.4.5/nfft/index.html b/v0.4.5/nfft/index.html new file mode 100644 index 0000000..7f5df8e --- /dev/null +++ b/v0.4.5/nfft/index.html @@ -0,0 +1,20 @@ + +NFFT · FourierTools.jl
    Edit on GitHub

    NFFT helpers

    Based on NFFT.jl we provide some convenient helper functions:

    FourierTools.nfft_nd — Function
    nfft_nd(src, dst_coords, dst_size=nothing; is_in_pixels=false, is_local_shift=false)

    performs an n-dimensional non-uniform FFT on grids with a regular topology. In comparison to the nfft() routine, which this computed is based on, this version does not require any reshape operations. See plan_nfft_nd for details on the arguments and usage examples. Note that the input can be Real valued and will be automatically converted to Complex.

    # A Zoomed transform in 3D
+julia> nfft_nd(rand(10,12,12), (t)-> (0.8*t[1], 0.7*t[2], 0.6*t[3]))
    source
    FourierTools.plan_nfft_nd — Function
    plan_nfft_nd(src, dst_coords; is_in_pixels=false, is_local_shift=false, pad_value=nothing, reltol=1e-9)

    Plans an n-dimensional non-uniform FFT on grids with a regular topology. In comparison to the nfft() routine, which this computed is based on, this version does not require any reshape operations.

    Arguments

    • src: source array
    • dst_coords: array of destination coordinates. This can be either an Array of Tuple or an Array with the last dimension ofsizelength(size(dst_coords))-1` referring to the destination coordinates where the FFT needs to be computed. Alternatively also a function mapping a tuple (of source index positions) to a tuple (of destination index positions). In the recommended mode, the indices are normalized to to Fouier frequency range (roughly speaking from -0.5:0.5).
    • is_in_pixels: A Boolean flag indicating whether dstcoords refers to coordinates in pixels or (default) in relative frequency positions. If `isin_pixels=true` is selected, destination coordinates (1-based indexing) as typical for array indexing is assumed and internally converted to relative positions.
    • is_local_shift: A Boolean controlling wether dst_coords refers to the destination coordinates or the relative distance from standard grid coordinates (size determind from dst_coordinates).
    • pad_value: if supplied, values outside the valid pixel range (roughly -0.5:0.5) are replaced by this complex-valued pad value.
    • is_adjoint: if true this plan is based on the adjoint rather than the ordinary plan
    • reltol: The numerical precision to which the results are computed. This is passed to the nfft routine. Worse precision is faster.
    # Lets try a 2D rotation:
+julia> using TestImages, NDTools, View5D, IndexFunArrays, FourierTools
+
+julia> img = Float64.(testimage("resolution"));
+
+# define a rotation operation
+julia> rot_alpha(a, t) = (cosd(a)*t[1] - sind(a)*t[2], sind(a)*t[1]+cosd(a)*t[2])
+
+julia> new_pos = rot_alpha.(10.0, idx(img, scale=ScaFT))
+
+julia> f = ift(img)
+
+julia> p = plan_nfft_nd(f, new_pos; is_local_shift=false, is_in_pixels=false)
+
+julia> g = real.(p * f)
+
+#display the result
+julia> @ve img, g
    source
    plan_nfft_nd(src::AbstractArray{T,D}, dst_fkt::Function, dst_size=size(src); is_in_pixels=false, is_adjoint=false, kwargs...)

    Plans an n-dimensional non-uniform FFT on grids with a regular topology defined via the function dst_fkt.

    Arguments

    • src: source array
    • dst_fkt: a function mapping a tuple (of source index positions) to a tuple (of destination index positions). In the recommended mode, the indices are normalized to to Fouier frequency range (roughly speaking from -0.5:0.5). If the named argument is_in_pixels is provided, the function is expected to act on one-index based pixel coordinates. This option is particularly interesting in combination with the argument is_loca_shift.
    • dst_size: this argument is only used for functions. If you require a different result size for dst_coords being a function, state it here.By defaul (dst_size=nothing) the destination size will be inferred form the argument new_pos or assumed to be size(src).
    • is_in_pixels: A Boolean flag indicating whether dstcoords refers to coordinates in pixels or (default) in relative frequency positions. If `isin_pixels=true` is selected, destination coordinates (1-based indexing) as typical for array indexing is assumed and internally converted to relative positions.
    • is_adjoint: if true this plan is based on the adjoint rather than the ordinary plan

    For other arguments and examples see the array-version of plan_nfft above.

    source
    diff --git a/v0.4.5/resampling/index.html b/v0.4.5/resampling/index.html new file mode 100644 index 0000000..392c301 --- /dev/null +++ b/v0.4.5/resampling/index.html @@ -0,0 +1,111 @@ + +Resampling (sinc Interpolation) · FourierTools.jl
    Edit on GitHub

    Resampling

    To sinc interpolate a signal, it is possible to zero pad a signal in Fourier space and to do an inverse Fourier transform effectively evaluating the Fourier series at more samples. If the signal was initially band-limited, sinc interpolation leads to smoother, perfectly interpolated signals

    Sinc interpolation

    Below you can find a simple example for up sampling using resample. Furthermore, there is an image interpolation Pluto.jl notebook in the examples folder. We can see that the interpolated signal matches the higher sampled signal well.

     begin
+	N_low = 128
+	x_min = 0.0
+	x_max = 16π
+	
+	xs_low = range(x_min, x_max, length=N_low+1)[1:N_low]
+	xs_high = range(x_min, x_max, length=5000)[1:end-1]
+	f(x) = sin(0.5*x) + cos(x) + cos(2 * x) + sin(0.25*x)
+	arr_low = f.(xs_low)
+	arr_high = f.(xs_high)
+end
+
+begin
+	N = 1000
+	xs_interp = range(x_min, x_max, length=N+1)[1:N]
+	arr_interp = resample(arr_low, N)
+end
+
+begin
+	scatter(xs_low, arr_low, legend=:bottomleft, markersize=2, label="Low sampling")
+	plot!(xs_interp, arr_interp, label="FFT based sinc interpolation", linestyle=:dash)
+	plot!(xs_high, arr_high, linestyle=:dashdotdot, label="High sampling")
+end

    Downsampling

    32 samples in the downsampled signal should be sufficient for Nyquist sampling. And as we can see, the downsampled signal still matches the original one.

    begin
+	N_ds = 32
+	xs_ds = range(x_min, x_max, length=N_ds+1)[1:N_ds]
+	arr_ds = resample(arr_high, N_ds)
+end
+
+begin
+	scatter(xs_low, arr_low, legend=:bottomleft, markersize=2, label="Low sampling")
+	plot!(xs_interp, arr_interp, label="FFT based sinc interpolation", linestyle=:dash)
+	plot!(xs_ds, arr_ds, label="resampled array", linestyle=:dot)	
+end

    Image Upsampling

    Having a Nyquist sampled image, it is possible to perform a sinc interpolation and creating visually much nicer images. However, the information content does not change between both images. The full Pluto notebook is here. The right image is the upsampled version of the left one.

    There are fast versions (upsample2) for upsampling by a factor of 2.

    Function References

    FourierTools.resample — Function
    resample(arr, new_size [, normalize=true])

    Calculates the sinc interpolation of an arr on a new array size new_size. It is a re-evaluation of the Fourier series at new grid points. new_size can be arbitrary. Each dimension is then independently either up or downsampled.

    This method is based on FFTs and therefore implicitly assumes periodic boundaries and a finite frequency support.

    normalize=true by default multiplies by an appropriate factor so that the array size is included in the scaling. This results in an array having roughly the same mean intensity.

    Basic Principle

    If size(new_size)[i] > size(arr)[i], we apply zero padding in Fourier space.

    If size(new_size)[i] < size(arr)[i], we cut out a centered part of the Fourier spectrum.

    We apply some tricks at the boundary to increase accuracy of highest frequencies.

    For real arrays we use rfft based operations, for complex one we use fft based ones.

    Examples

    sinc interpolation of 2 datapoints result in an approximation of cosine.

    julia> resample([2.0, 0.0], (6,))
+6-element Vector{Float64}:
+ 2.0
+ 1.5
+ 0.5
+ 0.0
+ 0.5
+ 1.5
+
+julia> resample([2.0, 0.0], (6,)) ≈ 1 .+ cos.(2π .* (0:5)./6)
+true
    source
    FourierTools.resample_nfft — Function
    resample_nfft(img, new_pos, dst_size=nothing; pixel_coords=false, is_local_shift=false, is_src_coords=true, reltol=1e-9)

    resamples an ND-array to a set of new positions new_pos measured in either in pixels (pixel_coords=true) or relative (Fourier-) image coordinates (pixel_coords=false). new_pos can be

    • an array of Tuples specifying the zoom along each direction
    • an N+1 dimensional array (for N-dimensional input data img) of destination positions, the last dimension enumerating the respective destination coordinate dimension.
    • a function accepting a coordinate Tuple and yielding a destination position Tuple.

    resample_nfft can perform a large range of possible resamplings. Note that the default setting is is_src_coords=true which means that the source coordinates of each destination position have to be specified. This has the advantage that the result has usually less artefacts, but the positions may be more less convenient to specify.

    Arguments

    • img: the image to apply resampling to
    • new_pos`: specifies the resampling. See description above.
    • dst_size: this argument optionally defines the output size. If you require a different result size for new_pos being a function or with is_src_coords=true, state it here. By default (dst_size=nothing) the destination size will be inferred form the argument new_pos or assumed to be size(img).
    • is_local_shift: specifies, whether the resampling coordinates refer to a relative shift or absoluter coordinates
    • is_in_pixels: specifies whether the coordinates (or relative distances) are given in pixel pitch units (is_in_pixels=true) or in units relative to the array sizes (Fourier convention)
    • is_src_coords: specifies, whether the resampling positions refer to sampling at source (is_src_coords=true) or destination coordinates
    • reltol: will be used as an argument to the nfft function spedifying the relative precision to calculate to

    See also: resample, resample_czt

    Examples

    julia> using FourierTools, TestImages, NDTools, View5D, IndexFunArrays
+
+julia> a = Float32.(testimage("resolution"));
+
+julia> b = resample_nfft(a, t -> (2.5f0 *sign(t[1])*t[1]^2, t[2]*(0.5f0+t[1]))); # a complicated deformation
+
+julia> sz = size(a);
+
+# stacking only the displacement along the last dimension:
+julia> new_pos = cat(xx(sz,scale=ScaFT), zeros(sz), dims=3);
+
+julia> c = resample_nfft(a, new_pos, is_local_shift=true); # stretch along x using an array
+
+julia> new_pos = cat(.-xx(sz,scale=ScaFT)./2, zeros(sz), dims=3);
+
+julia> c2 = resample_nfft(a, new_pos, is_local_shift=true, is_src_coords=false); # stretch along x using an array
+
+# Notice the difference in brightness between c and c2
+julia> @ve a b c c2 # visualize distortion and x-shrinks. 
+
+# Lets try a 2D rotation:
+# define a rotation operation
+julia> rot_alpha(a, t) = (cosd(a)*t[1] + sind(a)*t[2], -sind(a)*t[1]+cosd(a)*t[2])
+
+# positions as an array of tuples
+julia> new_pos = rot_alpha.(10.0, idx(a, scale=ScaFT))
+
+# lets do the resampling, this time by specifying the destination coordinates:
+julia> d = resample_nfft(a, new_pos, is_src_coords=false);
+
+#display the result
+julia> @ve a d
+
+#how about a spiral deformation?
+julia> new_pos = rot_alpha.(rr(a), idx(a, scale=ScaFT))
+
+julia> e = resample_nfft(a, new_pos);
+
+julia> f = resample_nfft(a, new_pos, is_src_coords=false);
+
+# observe the artefacts generated by undersampling in the destination grid
+julia> @ve a e f
    source
    FourierTools.resample_czt — Function
    resample_czt(arr, rel_zoom; shear=nothing, shear_dim=nothing, fix_nyquist=false, new_size = size(arr), 
+             do_damp=false, rel_pad=0.2, remove_wrap=true)

    resamples the image with fixed factors or a list of separable functions using the chirp z transform algorithm. The data is first padded by a relative amount rel_pad which is needed to avoid wrap-around problems. As opposed to resample(), this routine allows for arbitrary non-integer zoom factors. It is reasonably fast but only allows a stretch (via rel_zoom) and a shift (via shear in pixels) per line or column.

    Note that each entry of the tuple in rel_zoom or shear describes the zoom or shear to apply to all other dimensions individually per entry along this dimension number.

    Arguments:

    • arr: array to resample
    • rel_zoom: factors to zoom as a tuple or a tuple of functions defining the zooms
    • shear: a tuple of shears or a tuple of shear functions defining the shears
    • shear_dim: which dimension to shear
    • fix_nyquist: defines whether to apply fix_nyquist when using the applyshiftstrength! function.
    • do_damp: applies a padding and damping outside the region to zoom, to avoid artefacts
    • rel_pad: amount of padding to apply, if do_damp is true
    • remove_wrap: removes the wrap-around when zooming out.
    • new_size: size of the result array. If not provided the same as the input size will be used.

    Examples

    julia> using TestImages, NDTools, View5D
+
+julia> a = Float32.(testimage("resolution"));
+
+julia> b = resample_czt(a, (0.99,1.3)); # just zooming and shrinking
+
+julia> c = resample_czt(a, (x->0.8+x^2/3,x->1.0+0.2*x)); # a more complicated distortion
+
+julia> d = resample_czt(a, (x->1.0,x->1.0), shear=(x->50*x^2,0.0)); # a more complicated distortion
+
+julia> @ve a,b,c,d # visualize distortions
    source
    FourierTools.upsample2 — Function
    upsample2(mat::AbstractArray{T, N}; dims=1:N, fix_center=false, keep_singleton=false)

    Upsamples by a factor of two in all dimensions. The code is optimized for speed by using subpixelshifts rather than Fourier resizing. By default the first pixel maintains its position. However, this leads to a shift of the center (size[d]÷2+1) in the resulting array for uneven array sizes. fix_center=true can be used to remedy this and the result array center position will agree to the source array center position. keep_singleton=true will not upsample dimensions of size one. Note that upsample2 is based on Fourier-shifting and you may have to deal with wrap-around problems.

    julia> upsample2(collect(collect(1.0:9.0)'))
+2×18 Matrix{Float64}:
+ 1.0  0.24123  2.0  3.24123  3.0  2.93582  4.0  5.0  5.0  5.0  6.0  7.06418  7.0  6.75877  8.0  9.75877  9.0  5.0
+ 1.0  0.24123  2.0  3.24123  3.0  2.93582  4.0  5.0  5.0  5.0  6.0  7.06418  7.0  6.75877  8.0  9.75877  9.0  5.0
+
+julia> upsample2(collect(collect(1.0:9.0)'); fix_center=true, keep_singleton=true)
+1×18 Matrix{Float64}:
+ 5.0  1.0  0.24123  2.0  3.24123  3.0  2.93582  4.0  5.0  5.0  5.0  6.0  7.06418  7.0  6.75877  8.0  9.75877  9.0
    source
    FourierTools.upsample2_abs2 — Function
    upsample2_abs2(mat::AbstractArray{T, N}; dims=1:N)

    Upsamples by a factor of two and applies the abs2 operation. The code is optimized for speed.

    source
    FourierTools.barrel_pin — Function
    barrel_pin(arr, rel=0.5)

    emulates a barrel (rel>0) or a pincushion (rel<0) distortion. The distortions are calculated using resample_czt() with separable quadratic zooms.

    See also: resample_czt()

    Examples

    julia> using TestImages, NDTools, View5D
+
+julia> a = Float32.(testimage("resolution"))
+
+julia> b = barrel_pin(a,0.5) # strong barrel distortion
+
+julia> c = barrel_pin(a,-0.5) # strong pin-cushion distortion
+
+julia> @ve a,b,c # visualize distortions
    source
    diff --git a/v0.4.5/rotate/index.html b/v0.4.5/rotate/index.html new file mode 100644 index 0000000..ca7ff8b --- /dev/null +++ b/v0.4.5/rotate/index.html @@ -0,0 +1,11 @@ + +Image Rotation with FFTs · FourierTools.jl
    Edit on GitHub

    Rotation with FFT

    Via shear it is possible to rotate an image. Since we also implemented a shear algorithm, rotation can be implemented as well. For details look at this webpage.

    Examples

    For full interactivity, have a look at this Pluto notebook.

    using Revise, FourierTools, Plots, TestImages, PlutoUI, ImageShow
+
+begin
+    img = Float32.(testimage("fabio_512_gray"))
+    z = zeros(Float32, (768, 768))
+    FourierTools.center_set!(z, img)
+end
+
+
+Gray.(FourierTools.rotate(z, 26))

    Function references

    FourierTools.rotate — Function
    rotate(arr, θ, rotation_plane=(1,2), adapt_size=true, keep_new_size=false)

    Rotate an arr in the plane rotation_plane with an angle θ in rad around the center pixel. Note that, in contrast to ImageTransformations.imrotate, the rotation is done around the Fourier-center pixel (size()÷2+1) and not the geometric mid point.

    Arguments:

    • arr: the array to rotate
    • Θ: the angle (in rad) to rotate by
    • rotation_plane: two dimensions selecting the 2D plane in which a multidimensional dataset is rotated
    • adapt_size: if true (default), the three shears, which make up the rotation, will be allowed to enlarge the size of the array. This is slower but avoids wrap-around artefacts If false, the in-place version of rotate is used with all its problems. Only recommended for very small angles!
    • keep_new_size: if true, the enlarged sizes (only for adapt_size=true) will also be returned. Otherwise the resulting data will be cut down to the original size
    • pad_value: specifies the value that areas outside the visible range (in the source) should be assigend to. A smart choice can reduce edge artefacts.

    rotate! is also available.

    source
    FourierTools.rotate! — Function
    rotate!(arr, θ, rotation_plane=(1,2))

    In-place rotate an arr in the plane spanned by the two dimensions in the tuple rotation_plane with an angle θ in rad around the center pixel. Note that, in contrast to ImageTransformations.imrotate, the rotation is done around the Fourier-center pixel (size()÷2+1) and not the geometric mid point. Note also that due to the operation being performed by successive cyclic shear operations in-place, pixels near the corner will be experiencing a massive wrap-around problem. Use the out-of-place version rotate to avoid this. Note also that this version generates very bad results with the angle approaching π. To fix this, use the out-of-place version of rotate.

    Arguments:

    • arr: the array to rotate
    • Θ: the angle (in rad) to rotate by
    • rotation_plane: two dimensions selecting the 2D plane in which a multidimensional dataset is rotated
    source
    diff --git a/v0.4.5/search/index.html b/v0.4.5/search/index.html new file mode 100644 index 0000000..dd22fc2 --- /dev/null +++ b/v0.4.5/search/index.html @@ -0,0 +1,2 @@ + +Search · FourierTools.jl diff --git a/v0.4.5/search_index.js b/v0.4.5/search_index.js new file mode 100644 index 0000000..37b92bf --- /dev/null +++ b/v0.4.5/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"fractional/#Fractional-Fourier-Transforms","page":"Fractional Fourier Transform","title":"Fractional Fourier Transforms","text":"","category":"section"},{"location":"fractional/","page":"Fractional Fourier Transform","title":"Fractional Fourier Transform","text":"Implementation of the Fast Fractional Fourier Transform based on Javier García, David Mas, and Rainer G. Dorsch, \"Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,\" Appl. Opt. 35, 7013-7018 (1996)","category":"page"},{"location":"fractional/","page":"Fractional Fourier Transform","title":"Fractional Fourier Transform","text":"FourierTools.frfft","category":"page"},{"location":"fractional/#FourierTools.frfft","page":"Fractional Fourier Transform","title":"FourierTools.frfft","text":"frfft(arr, p; shift=false, method=:garcia)\n\nCalculates the fractional Fast Fourier transform (FRFFT) of the order p of arr. No dims argument is supported yet.\n\nIf shift=false the FRFFT is calculated around the first entry. If shift=true the FRFFT is calculated aroound the center.\n\nMethods\n\nSeveral implementation exists. The following are implemented: \n\nmethod=:garcia: A convolutional approach based on 2 FFTs. See García, J., Mas, D., & Dorsch, R. G. (1996). Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm. Applied Optics, 35(35), 7013. doi:10.1364/ao.35.007013 \n\n\n\n\n\n","category":"function"},{"location":"rotate/#Rotation-with-FFT","page":"Image Rotation with FFTs","title":"Rotation with FFT","text":"","category":"section"},{"location":"rotate/","page":"Image Rotation with FFTs","title":"Image Rotation with FFTs","text":"Via shear it is possible to rotate an image. Since we also implemented a shear algorithm, rotation can be implemented as well. For details look at this webpage.","category":"page"},{"location":"rotate/#Examples","page":"Image Rotation with FFTs","title":"Examples","text":"","category":"section"},{"location":"rotate/","page":"Image Rotation with FFTs","title":"Image Rotation with FFTs","text":"For full interactivity, have a look at this Pluto notebook.","category":"page"},{"location":"rotate/","page":"Image Rotation with FFTs","title":"Image Rotation with FFTs","text":"using Revise, FourierTools, Plots, TestImages, PlutoUI, ImageShow\n\nbegin\n img = Float32.(testimage(\"fabio_512_gray\"))\n z = zeros(Float32, (768, 768))\n FourierTools.center_set!(z, img)\nend\n\n\nGray.(FourierTools.rotate(z, 26))","category":"page"},{"location":"rotate/","page":"Image Rotation with FFTs","title":"Image Rotation with FFTs","text":"(Image: )","category":"page"},{"location":"rotate/#Function-references","page":"Image Rotation with FFTs","title":"Function references","text":"","category":"section"},{"location":"rotate/","page":"Image Rotation with FFTs","title":"Image Rotation with FFTs","text":" FourierTools.rotate\n FourierTools.rotate!","category":"page"},{"location":"rotate/#FourierTools.rotate","page":"Image Rotation with FFTs","title":"FourierTools.rotate","text":"rotate(arr, θ, rotation_plane=(1,2), adapt_size=true, keep_new_size=false)\n\nRotate an arr in the plane rotation_plane with an angle θ in rad around the center pixel. Note that, in contrast to ImageTransformations.imrotate, the rotation is done around the Fourier-center pixel (size()÷2+1) and not the geometric mid point.\n\nArguments:\n\narr: the array to rotate\nΘ: the angle (in rad) to rotate by\nrotation_plane: two dimensions selecting the 2D plane in which a multidimensional dataset is rotated \nadapt_size: if true (default), the three shears, which make up the rotation, will be allowed to enlarge the size of the array. This is slower but avoids wrap-around artefacts If false, the in-place version of rotate is used with all its problems. Only recommended for very small angles!\nkeep_new_size: if true, the enlarged sizes (only for adapt_size=true) will also be returned. Otherwise the resulting data will be cut down to the original size\npad_value: specifies the value that areas outside the visible range (in the source) should be assigend to. A smart choice can reduce edge artefacts.\n\nrotate! is also available.\n\n\n\n\n\n","category":"function"},{"location":"rotate/#FourierTools.rotate!","page":"Image Rotation with FFTs","title":"FourierTools.rotate!","text":"rotate!(arr, θ, rotation_plane=(1,2))\n\nIn-place rotate an arr in the plane spanned by the two dimensions in the tuple rotation_plane with an angle θ in rad around the center pixel. Note that, in contrast to ImageTransformations.imrotate, the rotation is done around the Fourier-center pixel (size()÷2+1) and not the geometric mid point. Note also that due to the operation being performed by successive cyclic shear operations in-place, pixels near the corner will be experiencing a massive wrap-around problem. Use the out-of-place version rotate to avoid this. Note also that this version generates very bad results with the angle approaching π. To fix this, use the out-of-place version of rotate.\n\nArguments:\n\narr: the array to rotate\nΘ: the angle (in rad) to rotate by\nrotation_plane: two dimensions selecting the 2D plane in which a multidimensional dataset is rotated \n\n\n\n\n\n","category":"function"},{"location":"shear/#Rotation-with-FFT","page":"Image Shearing with FFTs","title":"Rotation with FFT","text":"","category":"section"},{"location":"shear/","page":"Image Shearing with FFTs","title":"Image Shearing with FFTs","text":"Via shear it is possible to rotate an image. Shearing is basically a shift operation but with different shift distance in each row.","category":"page"},{"location":"shear/#Examples","page":"Image Shearing with FFTs","title":"Examples","text":"","category":"section"},{"location":"shear/","page":"Image Shearing with FFTs","title":"Image Shearing with FFTs","text":"For full interactivity, have a look at this Pluto notebook.","category":"page"},{"location":"shear/","page":"Image Shearing with FFTs","title":"Image Shearing with FFTs","text":"using Revise, FourierTools, Plots, TestImages, PlutoUI, ImageShow\n\nbegin\n img = Float32.(testimage(\"fabio_512_gray\"))\n z = zeros(Float32, (768, 768))\n FourierTools.center_set!(z, img)\nend\n\nGray.(FourierTools.shear(z, -305))","category":"page"},{"location":"shear/","page":"Image Shearing with FFTs","title":"Image Shearing with FFTs","text":"(Image: )","category":"page"},{"location":"shear/#Function-references","page":"Image Shearing with FFTs","title":"Function references","text":"","category":"section"},{"location":"shear/","page":"Image Shearing with FFTs","title":"Image Shearing with FFTs","text":" FourierTools.shear\n FourierTools.shear!","category":"page"},{"location":"shear/#FourierTools.shear","page":"Image Shearing with FFTs","title":"FourierTools.shear","text":"shear(arr, Δ, shear_dir_dim=1, shear_dim=2; fix_nyquist=false, adapt_size=false::Bool, pad_value=zero(eltype(arr)))\n\nShears an array by the amount of Δ pixels via an FFT approach. Δ is the relative shift between the top and bottom row shifted with respect to each other. shear_dir_dim decides the direction of the shear and shear_dim is the second dimension where the shear happens. There is also shear! available.\n\n#Arguments\n\narr: array to shear \nshear_dir_dim: dimension of the shift during shear\nshear_dim: dimension along which to progress and apply variing shears along shear_dir_dim\nfix_nyquist: apply a fix to the highest frequency during the Fourier-space application of the exponential factor\nadapt_size: if true, pad the data prior to the shear. The result array will be larger\npad_value: the value to pad with (only applies if adapt_size=true)\nassign_wrap=assign_wrap: replaces wrap-around areas by pad_value (only of adapt_size is false)\n\nFor complex arrays we use fft, for real array we use rfft.\n\n\n\n\n\n","category":"function"},{"location":"shear/#FourierTools.shear!","page":"Image Shearing with FFTs","title":"FourierTools.shear!","text":"shear!(arr, Δ, shear_dir_dim=1, shear_dim=2; fix_nyquist=false, assign_wrap=false, pad_value=zero(eltype(arr)))\n\nFor more details see shear.\n\nExtra Arguments\n\nassign_wrap: if true wrap-around areas are replaced by pad_value pad_value: the value to replace wrap-around areas with\n\nFor complex arrays we can completely avoid large memory allocations. For real arrays, we need at least allocate on array in the fourier space.\n\n\n\n\n\n","category":"function"},{"location":"resampling/#Resampling","page":"Resampling (sinc Interpolation)","title":"Resampling","text":"","category":"section"},{"location":"resampling/","page":"Resampling (sinc Interpolation)","title":"Resampling (sinc Interpolation)","text":"To sinc interpolate a signal, it is possible to zero pad a signal in Fourier space and to do an inverse Fourier transform effectively evaluating the Fourier series at more samples. If the signal was initially band-limited, sinc interpolation leads to smoother, perfectly interpolated signals","category":"page"},{"location":"resampling/#Sinc-interpolation","page":"Resampling (sinc Interpolation)","title":"Sinc interpolation","text":"","category":"section"},{"location":"resampling/","page":"Resampling (sinc Interpolation)","title":"Resampling (sinc Interpolation)","text":"Below you can find a simple example for up sampling using resample. Furthermore, there is an image interpolation Pluto.jl notebook in the examples folder. We can see that the interpolated signal matches the higher sampled signal well.","category":"page"},{"location":"resampling/","page":"Resampling (sinc Interpolation)","title":"Resampling (sinc Interpolation)","text":" begin\n\tN_low = 128\n\tx_min = 0.0\n\tx_max = 16π\n\t\n\txs_low = range(x_min, x_max, length=N_low+1)[1:N_low]\n\txs_high = range(x_min, x_max, length=5000)[1:end-1]\n\tf(x) = sin(0.5*x) + cos(x) + cos(2 * x) + sin(0.25*x)\n\tarr_low = f.(xs_low)\n\tarr_high = f.(xs_high)\nend\n\nbegin\n\tN = 1000\n\txs_interp = range(x_min, x_max, length=N+1)[1:N]\n\tarr_interp = resample(arr_low, N)\nend\n\nbegin\n\tscatter(xs_low, arr_low, legend=:bottomleft, markersize=2, label=\"Low sampling\")\n\tplot!(xs_interp, arr_interp, label=\"FFT based sinc interpolation\", linestyle=:dash)\n\tplot!(xs_high, arr_high, linestyle=:dashdotdot, label=\"High sampling\")\nend","category":"page"},{"location":"resampling/","page":"Resampling (sinc Interpolation)","title":"Resampling (sinc Interpolation)","text":"(Image: )","category":"page"},{"location":"resampling/#Downsampling","page":"Resampling (sinc Interpolation)","title":"Downsampling","text":"","category":"section"},{"location":"resampling/","page":"Resampling (sinc Interpolation)","title":"Resampling (sinc Interpolation)","text":"32 samples in the downsampled signal should be sufficient for Nyquist sampling. And as we can see, the downsampled signal still matches the original one.","category":"page"},{"location":"resampling/","page":"Resampling (sinc Interpolation)","title":"Resampling (sinc Interpolation)","text":"begin\n\tN_ds = 32\n\txs_ds = range(x_min, x_max, length=N_ds+1)[1:N_ds]\n\tarr_ds = resample(arr_high, N_ds)\nend\n\nbegin\n\tscatter(xs_low, arr_low, legend=:bottomleft, markersize=2, label=\"Low sampling\")\n\tplot!(xs_interp, arr_interp, label=\"FFT based sinc interpolation\", linestyle=:dash)\n\tplot!(xs_ds, arr_ds, label=\"resampled array\", linestyle=:dot)\t\nend","category":"page"},{"location":"resampling/","page":"Resampling (sinc Interpolation)","title":"Resampling (sinc Interpolation)","text":"(Image: )","category":"page"},{"location":"resampling/#Image-Upsampling","page":"Resampling (sinc Interpolation)","title":"Image Upsampling","text":"","category":"section"},{"location":"resampling/","page":"Resampling (sinc Interpolation)","title":"Resampling (sinc Interpolation)","text":"Having a Nyquist sampled image, it is possible to perform a sinc interpolation and creating visually much nicer images. However, the information content does not change between both images. The full Pluto notebook is here. The right image is the upsampled version of the left one.","category":"page"},{"location":"resampling/","page":"Resampling (sinc Interpolation)","title":"Resampling (sinc Interpolation)","text":"(Image: ) (Image: )","category":"page"},{"location":"resampling/","page":"Resampling (sinc Interpolation)","title":"Resampling (sinc Interpolation)","text":"There are fast versions (upsample2) for upsampling by a factor of 2.","category":"page"},{"location":"resampling/#Function-References","page":"Resampling (sinc Interpolation)","title":"Function References","text":"","category":"section"},{"location":"resampling/","page":"Resampling (sinc Interpolation)","title":"Resampling (sinc Interpolation)","text":"FourierTools.resample\nFourierTools.resample_by_FFT\nFourierTools.resample_by_RFFT\nFourierTools.resample_nfft\nFourierTools.resample_czt\nFourierTools.upsample2\nFourierTools.upsample2_abs2\nFourierTools.barrel_pin","category":"page"},{"location":"resampling/#FourierTools.resample","page":"Resampling (sinc Interpolation)","title":"FourierTools.resample","text":"resample(arr, new_size [, normalize=true])\n\nCalculates the sinc interpolation of an arr on a new array size new_size. It is a re-evaluation of the Fourier series at new grid points. new_size can be arbitrary. Each dimension is then independently either up or downsampled.\n\nThis method is based on FFTs and therefore implicitly assumes periodic boundaries and a finite frequency support.\n\nnormalize=true by default multiplies by an appropriate factor so that the array size is included in the scaling. This results in an array having roughly the same mean intensity.\n\nBasic Principle\n\nIf size(new_size)[i] > size(arr)[i], we apply zero padding in Fourier space.\n\nIf size(new_size)[i] < size(arr)[i], we cut out a centered part of the Fourier spectrum.\n\nWe apply some tricks at the boundary to increase accuracy of highest frequencies. \n\nFor real arrays we use rfft based operations, for complex one we use fft based ones.\n\nExamples\n\nsinc interpolation of 2 datapoints result in an approximation of cosine.\n\njulia> resample([2.0, 0.0], (6,))\n6-element Vector{Float64}:\n 2.0\n 1.5\n 0.5\n 0.0\n 0.5\n 1.5\n\njulia> resample([2.0, 0.0], (6,)) ≈ 1 .+ cos.(2π .* (0:5)./6)\ntrue\n\n\n\n\n\n","category":"function"},{"location":"resampling/#FourierTools.resample_by_FFT","page":"Resampling (sinc Interpolation)","title":"FourierTools.resample_by_FFT","text":"resample_by_FFT(mat, new_size)\n\nDoes a resampling based on fft. This function is called by resampling.\n\n\n\n\n\n","category":"function"},{"location":"resampling/#FourierTools.resample_by_RFFT","page":"Resampling (sinc Interpolation)","title":"FourierTools.resample_by_RFFT","text":"resample_by_RFFT(mat, new_size)\n\nDoes a resampling based on rfft. This function is called by resampling.\n\n\n\n\n\n","category":"function"},{"location":"resampling/#FourierTools.resample_nfft","page":"Resampling (sinc Interpolation)","title":"FourierTools.resample_nfft","text":"resample_nfft(img, new_pos, dst_size=nothing; pixel_coords=false, is_local_shift=false, is_src_coords=true, reltol=1e-9)\n\nresamples an ND-array to a set of new positions new_pos measured in either in pixels (pixel_coords=true) or relative (Fourier-) image coordinates (pixel_coords=false). new_pos can be \n\nan array of Tuples specifying the zoom along each direction\nan N+1 dimensional array (for N-dimensional input data img) of destination positions, the last dimension enumerating the respective destination coordinate dimension.\na function accepting a coordinate Tuple and yielding a destination position Tuple.\n\nresample_nfft can perform a large range of possible resamplings. Note that the default setting is is_src_coords=true which means that the source coordinates of each destination position have to be specified. This has the advantage that the result has usually less artefacts, but the positions may be more less convenient to specify.\n\nArguments\n\nimg: the image to apply resampling to\nnew_pos`: specifies the resampling. See description above.\ndst_size: this argument optionally defines the output size. If you require a different result size for new_pos being a function or with is_src_coords=true, state it here. By default (dst_size=nothing) the destination size will be inferred form the argument new_pos or assumed to be size(img).\nis_local_shift: specifies, whether the resampling coordinates refer to a relative shift or absoluter coordinates\nis_in_pixels: specifies whether the coordinates (or relative distances) are given in pixel pitch units (is_in_pixels=true) or in units relative to the array sizes (Fourier convention) \nis_src_coords: specifies, whether the resampling positions refer to sampling at source (is_src_coords=true) or destination coordinates \nreltol: will be used as an argument to the nfft function spedifying the relative precision to calculate to\n\nSee also: resample, resample_czt\n\nExamples\n\njulia> using FourierTools, TestImages, NDTools, View5D, IndexFunArrays\n\njulia> a = Float32.(testimage(\"resolution\"));\n\njulia> b = resample_nfft(a, t -> (2.5f0 *sign(t[1])*t[1]^2, t[2]*(0.5f0+t[1]))); # a complicated deformation\n\njulia> sz = size(a);\n\n# stacking only the displacement along the last dimension:\njulia> new_pos = cat(xx(sz,scale=ScaFT), zeros(sz), dims=3);\n\njulia> c = resample_nfft(a, new_pos, is_local_shift=true); # stretch along x using an array\n\njulia> new_pos = cat(.-xx(sz,scale=ScaFT)./2, zeros(sz), dims=3);\n\njulia> c2 = resample_nfft(a, new_pos, is_local_shift=true, is_src_coords=false); # stretch along x using an array\n\n# Notice the difference in brightness between c and c2\njulia> @ve a b c c2 # visualize distortion and x-shrinks. \n\n# Lets try a 2D rotation:\n# define a rotation operation\njulia> rot_alpha(a, t) = (cosd(a)*t[1] + sind(a)*t[2], -sind(a)*t[1]+cosd(a)*t[2])\n\n# positions as an array of tuples\njulia> new_pos = rot_alpha.(10.0, idx(a, scale=ScaFT))\n\n# lets do the resampling, this time by specifying the destination coordinates:\njulia> d = resample_nfft(a, new_pos, is_src_coords=false);\n\n#display the result\njulia> @ve a d\n\n#how about a spiral deformation?\njulia> new_pos = rot_alpha.(rr(a), idx(a, scale=ScaFT))\n\njulia> e = resample_nfft(a, new_pos);\n\njulia> f = resample_nfft(a, new_pos, is_src_coords=false);\n\n# observe the artefacts generated by undersampling in the destination grid\njulia> @ve a e f\n\n\n\n\n\n","category":"function"},{"location":"resampling/#FourierTools.resample_czt","page":"Resampling (sinc Interpolation)","title":"FourierTools.resample_czt","text":"resample_czt(arr, rel_zoom; shear=nothing, shear_dim=nothing, fix_nyquist=false, new_size = size(arr), \n do_damp=false, rel_pad=0.2, remove_wrap=true)\n\nresamples the image with fixed factors or a list of separable functions using the chirp z transform algorithm. The data is first padded by a relative amount rel_pad which is needed to avoid wrap-around problems. As opposed to resample(), this routine allows for arbitrary non-integer zoom factors. It is reasonably fast but only allows a stretch (via rel_zoom) and a shift (via shear in pixels) per line or column.\n\nNote that each entry of the tuple in rel_zoom or shear describes the zoom or shear to apply to all other dimensions individually per entry along this dimension number. \n\nArguments:\n\narr: array to resample\nrel_zoom: factors to zoom as a tuple or a tuple of functions defining the zooms\nshear: a tuple of shears or a tuple of shear functions defining the shears\nshear_dim: which dimension to shear\nfix_nyquist: defines whether to apply fix_nyquist when using the applyshiftstrength! function.\ndo_damp: applies a padding and damping outside the region to zoom, to avoid artefacts\nrel_pad: amount of padding to apply, if do_damp is true\nremove_wrap: removes the wrap-around when zooming out.\nnew_size: size of the result array. If not provided the same as the input size will be used.\n\nExamples\n\njulia> using TestImages, NDTools, View5D\n\njulia> a = Float32.(testimage(\"resolution\"));\n\njulia> b = resample_czt(a, (0.99,1.3)); # just zooming and shrinking\n\njulia> c = resample_czt(a, (x->0.8+x^2/3,x->1.0+0.2*x)); # a more complicated distortion\n\njulia> d = resample_czt(a, (x->1.0,x->1.0), shear=(x->50*x^2,0.0)); # a more complicated distortion\n\njulia> @ve a,b,c,d # visualize distortions\n\n\n\n\n\n","category":"function"},{"location":"resampling/#FourierTools.upsample2","page":"Resampling (sinc Interpolation)","title":"FourierTools.upsample2","text":"upsample2(mat::AbstractArray{T, N}; dims=1:N, fix_center=false, keep_singleton=false)\n\nUpsamples by a factor of two in all dimensions. The code is optimized for speed by using subpixelshifts rather than Fourier resizing. By default the first pixel maintains its position. However, this leads to a shift of the center (size[d]÷2+1) in the resulting array for uneven array sizes. fix_center=true can be used to remedy this and the result array center position will agree to the source array center position. keep_singleton=true will not upsample dimensions of size one. Note that upsample2 is based on Fourier-shifting and you may have to deal with wrap-around problems.\n\njulia> upsample2(collect(collect(1.0:9.0)'))\n2×18 Matrix{Float64}:\n 1.0 0.24123 2.0 3.24123 3.0 2.93582 4.0 5.0 5.0 5.0 6.0 7.06418 7.0 6.75877 8.0 9.75877 9.0 5.0\n 1.0 0.24123 2.0 3.24123 3.0 2.93582 4.0 5.0 5.0 5.0 6.0 7.06418 7.0 6.75877 8.0 9.75877 9.0 5.0\n\njulia> upsample2(collect(collect(1.0:9.0)'); fix_center=true, keep_singleton=true)\n1×18 Matrix{Float64}:\n 5.0 1.0 0.24123 2.0 3.24123 3.0 2.93582 4.0 5.0 5.0 5.0 6.0 7.06418 7.0 6.75877 8.0 9.75877 9.0\n\n\n\n\n\n","category":"function"},{"location":"resampling/#FourierTools.upsample2_abs2","page":"Resampling (sinc Interpolation)","title":"FourierTools.upsample2_abs2","text":"upsample2_abs2(mat::AbstractArray{T, N}; dims=1:N)\n\nUpsamples by a factor of two and applies the abs2 operation. The code is optimized for speed.\n\n\n\n\n\n","category":"function"},{"location":"resampling/#FourierTools.barrel_pin","page":"Resampling (sinc Interpolation)","title":"FourierTools.barrel_pin","text":"barrel_pin(arr, rel=0.5)\n\nemulates a barrel (rel>0) or a pincushion (rel<0) distortion. The distortions are calculated using resample_czt() with separable quadratic zooms.\n\nSee also: resample_czt()\n\nExamples\n\njulia> using TestImages, NDTools, View5D\n\njulia> a = Float32.(testimage(\"resolution\"))\n\njulia> b = barrel_pin(a,0.5) # strong barrel distortion\n\njulia> c = barrel_pin(a,-0.5) # strong pin-cushion distortion\n\njulia> @ve a,b,c # visualize distortions\n\n\n\n\n\n","category":"function"},{"location":"czt/#CZTs","page":"CZT","title":"CZTs","text":"","category":"section"},{"location":"czt/","page":"CZT","title":"CZT","text":"Chirp Z Transformations: Allows Fourier-transformation and at the same time zooming into the result, which is why it is also called the Zoomed-FFT algorithm. The algorithm is loosely based on a publication [Rabiner, Schafer, Rader, The Chirp z-Transform Algorithm, IEEE Trans AU 17(1969) p. 86]. It needs three FFTs to perform its work but one can be precalculated by using plan_czt. Variable zooms, transform dimensions, array center positions as well as output sizes are supported along wiht a low-level interface by specifingy a and w. ","category":"page"},{"location":"czt/","page":"CZT","title":"CZT","text":"FourierTools.czt\nFourierTools.plan_czt\nFourierTools.iczt\nFourierTools.czt_1d\nFourierTools.plan_czt_1d\nFourierTools.CZTPlan_1D\nFourierTools.CZTPlan_ND","category":"page"},{"location":"czt/#FourierTools.czt","page":"CZT","title":"FourierTools.czt","text":"czt(xin, scale, dims=1:ndims(xin), dsize=size(xin,d); a=nothing, w=nothing, damp=ones(ndims(xin)), \n src_center=size(xin,d)÷2+1, dst_center=dsize÷2+1, remove_wrap=false, fft_flags=FFTW.ESTIMATE)\n\nChirp z transform of the ND array xin The tuple scale defines the zoom factors in the Fourier domain. Each has to be bigger than one.\n\nSee also: iczt, czt_1d\n\nThe code is based on Rabiner, Schafer & Rader 1969, IEEE Trans. on Audio and Electroacoustics, 17,86-92\n\nArguments:\n\nxin: array to transform\nscale: a tuple of factors (one for each dimension) to zoom into during the czt. Note that a factor of nothing (or 1.0) needs to be provided, if a dimension is not transformed.\ndims: a tuple of dimensions over which to apply the czt.\ndsize: a tuple specifying the destination size\na: defines the starting phase of the result CZT. This relates to the where the center of the destination array should be. The default is nothing which means it is calculated from the src_center argument.\nw: defines the consecutive phases of the result array, i.e. the zoom. It is (default nothing) usually automatically calculated from the scaled and the damp argument. You only need to state it, if you want to use the low-level interface (e.g. for the Laplace transform).\ndamp: a multiplicative factor to apply as a damping coefficient to w.\nsrc_center: position of the nominal central (zero-position) pixel in the source array. By default the Fourier-center size(src).÷2 .+1 is used.\ndst_center: the center (zero-position) of the destination array. By default the Fourier-center size(dst).÷2 .+1 is used.\nremove_wrap: if true, the positions that represent a wrap-around will be set to zero\n\nExample:\n\njulia> using IndexFunArrays\n\njulia> sz = (10,10);\n\njulia> xin = disc(sz,4)\n10×10 Matrix{Float64}:\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0\n 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0\n 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0\n 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0\n 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0\n 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0\n 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0\n\njulia> xft = czt(xin,(1.2,1.3));\n\njulia> zoomed = real.(ift(xft))\n10×10 Matrix{Float64}:\n -0.0197423 0.0233008 -0.0449251 0.00295724 0.205593 -0.166546 0.205593 0.00295724 -0.0449251 0.0233008\n 0.0239759 -0.028264 0.0541186 -0.0116475 -0.261294 0.312719 -0.261294 -0.0116475 0.0541186 -0.028264\n -0.0569 0.0666104 -0.122277 0.140354 0.78259 1.34381 0.78259 0.140354 -0.122277 0.0666104\n 0.00540611 -0.0117886 0.0837357 1.30651 1.8283 1.50127 1.8283 1.30651 0.0837357 -0.0117886\n 0.11892 -0.147731 0.368046 1.76537 1.33218 1.66119 1.33218 1.76537 0.368046 -0.147731\n -0.00389861 0.0145979 1.21842 1.52989 1.67375 1.543 1.67375 1.52989 1.21842 0.0145979\n 0.11892 -0.147731 0.368046 1.76537 1.33218 1.66119 1.33218 1.76537 0.368046 -0.147731\n 0.00540611 -0.0117886 0.0837357 1.30651 1.8283 1.50127 1.8283 1.30651 0.0837357 -0.0117886\n -0.0569 0.0666104 -0.122277 0.140354 0.78259 1.34381 0.78259 0.140354 -0.122277 0.0666104\n 0.0239759 -0.028264 0.0541186 -0.0116475 -0.261294 0.312719 -0.261294 -0.0116475 0.0541186 -0.028264\n\n\n\n\n\n","category":"function"},{"location":"czt/#FourierTools.plan_czt","page":"CZT","title":"FourierTools.plan_czt","text":"plan_czt(xin, scale, dims, dsize=size(xin); a=nothing, w=nothing, damp=ones(ndims(xin)), \n src_center=size(xin).÷2 .+1, dst_center=dsize.÷2 .+1, remove_wrap=false, fft_flags=FFTW.ESTIMATE)\n\ncreates a plan for an N-dimensional chirp z-transformation (CZT). The generated plan is then applied via muliplication. For details about the arguments, see czt().\n\n\n\n\n\n","category":"function"},{"location":"czt/#FourierTools.iczt","page":"CZT","title":"FourierTools.iczt","text":"iczt(xin ,scale, dims=1:length(size(xin)), dsize=size(xin,d); a=nothing, w=nothing, damp=1.0, \n src_center=size(xin,d)÷2+1, dst_center=dsize÷2+1, remove_wrap=false, fft_flags=FFTW.ESTIMATE)\n\nInverse chirp z transform of the ND array xin The tuple scale defines the zoom factors in the Fourier domain. Each has to be bigger than one. The code is based on Rabiner, Schafer & Rader 1969, IEEE Trans. on Audio and Electroacoustics, 17,86-92\n\nArguments:\n\nxin: array to transform\nscaled: factor to zoom into during the 1-dimensional czt. \nd: single dimension to transform (as a tuple)\ndsize: size of the destination array\na: defines the starting phase of the result CZT. This relates to the where the center of the destination array should be. The default is nothing which means it is calculated from the src_center argument.\nw: defines the consecutive phases of the result array, i.e. the zoom. It is (default nothing) usually automatically calculated from the scaled and the damp argument. You only need to state it, if you want to use the low-level interface (e.g. for the Laplace transform).\ndamp: a multiplicative factor to apply as a damping coefficient to w.\nsrc_center: position of the nominal central (zero-position) pixel in the source array. By default the Fourier-center size(src).÷2 .+1 is used.\ndst_center: the center (zero-position) of the destination array. By default the Fourier-center size(dst).÷2 .+1 is used.\nremove_wrap: if true, the positions that represent a wrap-around will be set to zero\npad_value: the value to pad wrapped data with. \n\nSee also: czt, czt_1d\n\nExample\n\n\njulia> using IndexFunArrays\n\njulia> sz = (10,10);\n\njulia> xin = disc(sz,4)\n10×10 Matrix{Float64}:\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0\n 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0\n 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0\n 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0\n 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0\n 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0\n 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0\n\njulia> xft = ft(xin);\n\njulia> iczt(xft,(1.2,1.3))\n10×10 Matrix{ComplexF64}:\n 0.00648614+0.0213779im 0.0165456+0.0357733im 0.0389356+0.0482465im -0.235491-0.156509im … 0.178484-0.0730099im -0.245418-5.88331e-5im 0.0471654-0.0814548im 0.0141525+0.0734892im\n -0.104602-0.160481im -0.163859-0.125535im 0.495205+0.135059im 0.660335+0.00736904im 0.764045-0.0497007im 0.67753+0.263814im 0.48095-0.0775406im -0.159713-0.0637132im\n 0.139304+0.111348im 0.454973+0.106869im 0.970263-0.0249785im 1.25999-0.166495im 1.07328-0.0481437im 1.24013-0.14664im 0.986722-0.0414382im 0.450186+0.111656im\n -0.035645-0.0311352im 1.03899-0.0589268im 1.1463-0.0940003im 0.790545+0.283668im 0.994255+0.134865im 0.80774-0.0124851im 1.13205+0.151519im 1.04314-0.130321im\n 0.292575+0.0853233im 0.929883+0.0687029im 1.06514-0.0649952im 0.989483-0.019913im 1.02311+0.018235im 0.979555-0.136654im 1.07337+0.0317868im 0.92749+0.0405597im\n 1.12254-0.0464723im 1.03467-0.0239316im 0.92709+0.0822984im 1.0521-0.0992709im … 0.983655-0.0663123im 1.0521+0.0992709im 0.92709-0.0822984im 1.03467+0.0239316im\n 0.287928-0.0306724im 0.92749-0.0405597im 1.07337-0.0317868im 0.979555+0.136654im 1.01648+0.0597475im 0.989483+0.019913im 1.06514+0.0649952im 0.929883-0.0687029im\n -0.0275957+0.169775im 1.04314+0.130321im 1.13205-0.151519im 0.80774+0.0124851im 1.00574+0.0629632im 0.790545-0.283668im 1.1463+0.0940003im 1.03899+0.0589268im\n 0.130009-0.120643im 0.450186-0.111656im 0.986722+0.0414382im 1.24013+0.14664im 1.06002+0.0348813im 1.25999+0.166495im 0.970263+0.0249785im 0.454973-0.106869im\n -0.0965531+0.0404296im -0.159713+0.0637132im 0.48095+0.0775406im 0.67753-0.263814im 0.77553-0.121603im 0.660335-0.00736904im 0.495205-0.135059im -0.163859+0.125535im\n\n\n\n\n\n","category":"function"},{"location":"czt/#FourierTools.czt_1d","page":"CZT","title":"FourierTools.czt_1d","text":"czt_1d(xin , scaled , d; remove_wrap=false, pad_value=zero(eltype(xin)))\n\nChirp z transform along a single direction d of an ND array xin. Note that the result type is defined by eltype(xin) and not by scales.\n\nThe code is based on Rabiner, Schafer & Rader 1969, IEEE Trans. on Audio and Electroacoustics, 17,86-92\n\nArguments:\n\nxin: array to transform\nscaled: factor to zoom into during the 1-dimensional czt. \nd: single dimension to transform (as a tuple)\ndsize: size of the destination array\na: defines the starting phase of the result CZT. This relates to the where the center of the destination array should be. The default is nothing which means it is calculated from the src_center argument.\nw: defines the consecutive phases of the result array, i.e. the zoom. It is (default nothing) usually automatically calculated from the scaled and the damp argument. You only need to state it, if you want to use the low-level interface (e.g. for the Laplace transform).\ndamp: a multiplicative factor to apply as a damping coefficient to w.\nsrc_center: position of the nominal central (zero-position) pixel in the source array. By default the F ourier-center size(src).÷2 .+1 is used.\ndst_center: the center (zero-position) of the destination array. By default the Fourier-center size(dst).÷2 .+1 is used.\nextra_phase: a phase ramp to apply to the final result relating to the srccenter. By default nothing which calculates this phase according to the `srccenter`.\nglobal_phase: the initial phase of the destitation array. By default nothing which calculates this phase according to the centers.\nremove_wrap: if true, the positions that represent a wrap-around will be set to zero\npad_value: the value to pad wrapped data with. \n\n\n\n\n\nczt_1d(xin , plan::CZTPlan_1D)\n\nChirp z transform along a single direction d of an ND array xin. Note that the result type is defined by eltype(xin) and not by scales. The plan can also be applied via multiplication with xin.\n\nThe code is based on Rabiner, Schafer & Rader 1969, IEEE Trans. on Audio and Electroacoustics, 17,86-92\n\nArguments\n\n`plan`: A plan created via plan_czt_1d()\n\n\n\n\n\n","category":"function"},{"location":"czt/#FourierTools.plan_czt_1d","page":"CZT","title":"FourierTools.plan_czt_1d","text":"plan_czt_1d(xin, scaled, d, dsize=size(xin,d); a=nothing, w=nothing, damp=1.0, src_center=(size(xin,d)+1)/2, \n dst_center=dsize÷2+1, remove_wrap=false, fft_flags=FFTW.ESTIMATE)\n\ncreates a plan for an one-dimensional chirp z-transformation (CZT). The generated plan is then applied via muliplication. For details about the arguments, see czt_1d().\n\n\n\n\n\n","category":"function"},{"location":"czt/#FourierTools.CZTPlan_1D","page":"CZT","title":"FourierTools.CZTPlan_1D","text":"CZTPlan_1D{CT<:Complex, D<:Integer, AT<:AbstractArray{CT, D}, PT<:Number, PFFT<:AbstractFFTs.Plan, PIFFT<:AbstractFFTs.ScaledPlan}\n\ntype used for the onedimensional plan of the chirp Z transformation (CZT). containing\n\nMembers:\n\n`d`: dimension (only one!) to transform with this plan\n`pad_value`: the value to pad wrapped data with (zero is already handled by the `wd` term, if wanted). \n`pad_ranges` :: tuple of two ranges of invalid positions, which can be replaced by pad values\n`aw`: factor to multiply input with\n`fft_fv`: fourier-transform (FFTW) of the convolutio kernel\n`wd`: factor to multiply the result of the convolution by\n`fftw_plan!`: plan for the forward FFTW of the convolution kernel\n`ifftw_plan!`: plan for the inverse FFTW of the convolution kernel\n\n\n\n\n\n","category":"type"},{"location":"czt/#FourierTools.CZTPlan_ND","page":"CZT","title":"FourierTools.CZTPlan_ND","text":"CZTPlan_ND{CT, D} # <: AbstractArray{T,D}\n\ntype used for the onedimensional plan of the chirp Z transformation (CZT). containing\n\nMembers:\n\n`plans`: vector of CZTPlan_1D for each of the directions of the ND array to transform\n\n\n\n\n\n","category":"type"},{"location":"shifting/#Shifting","page":"Shifting with FFTs","title":"Shifting","text":"","category":"section"},{"location":"shifting/","page":"Shifting with FFTs","title":"Shifting with FFTs","text":"Using the Fourier shift property one can implement shifting of arrays not only over pixel but also sub-pixel amount.","category":"page"},{"location":"shifting/#Examples","page":"Shifting with FFTs","title":"Examples","text":"","category":"section"},{"location":"shifting/","page":"Shifting with FFTs","title":"Shifting with FFTs","text":"For full interactivity, have a look at this Pluto notebook.","category":"page"},{"location":"shifting/","page":"Shifting with FFTs","title":"Shifting with FFTs","text":"begin\n f(x) = cos(4π * x / 30)\n x1 = 1:30\n x2 = x1 .+ 3\nend\n\nbegin\n y1 = f.(x1)\n y2 = f.(x2)\n offset = 2.01\n y3 = shift(y2, tuple(offset))\nend\n\nbegin\n plot(y1, label=\"Original signal\")\n plot!(y2, label=\"Shifted signal\")\n plot!(y3, label=\"Fourier shifted with $offset\")\nend","category":"page"},{"location":"shifting/","page":"Shifting with FFTs","title":"Shifting with FFTs","text":"(Image: )","category":"page"},{"location":"shifting/#Function-references","page":"Shifting with FFTs","title":"Function references","text":"","category":"section"},{"location":"shifting/","page":"Shifting with FFTs","title":"Shifting with FFTs","text":" FourierTools.shift\n FourierTools.shift!","category":"page"},{"location":"shifting/#FourierTools.shift","page":"Shifting with FFTs","title":"FourierTools.shift","text":"shift(arr, shifts)\n\nReturning a shifted array. See shift! for more details\n\n\n\n\n\n","category":"function"},{"location":"shifting/#FourierTools.shift!","page":"Shifting with FFTs","title":"FourierTools.shift!","text":"shift!(arr, shifts; kwargs...)\n\nShifts an array in-place. For real arrays it is based on rfft. For complex arrays based on fft. shifts can be non-integer, for integer shifts one should prefer circshift or ShiftedArrays.circshift because a FFT-based methods introduces numerical errors.\n\nkwargs...\n\nfix_nyquist_frequency=false: Fourier shifting of even-sized arrays is not revertible. However, if you did shift(x, δ) you can it revert by shift(x, δ, fix_nyquist_frequency=true). This only works if δ is the same.\ntake_real=true: For even-sized arrays we take by default the real part of the exponential phase at the Nyquist frequency. This satisfies the property of real valuedness and the aliasing of the Nyquist term.\n\nMemory Usage\n\nNote that for complex arrays we can avoid any large memory allocations because of fft!. For rfft there does not exist a usable implementation yet, so for real arrays there might be a temporary larger memory usage.\n\nExamples\n\njulia> x = [1.0 2.0 3.0; 4.0 5.0 6.0]\n2×3 Matrix{Float64}:\n 1.0 2.0 3.0\n 4.0 5.0 6.0\n\njulia> shift!(x, (1, 2))\n2×3 Matrix{Float64}:\n 5.0 6.0 4.0\n 2.0 3.0 1.0\n\njulia> x = [0, 1.0, 0.0, 1.0]\n4-element Vector{Float64}:\n 0.0\n 1.0\n 0.0\n 1.0\n\njulia> shift!(x, 0.5)\n4-element Vector{Float64}:\n 0.49999999999999994\n 0.5\n 0.49999999999999994\n 0.5\n\n\n\n\n\n","category":"function"},{"location":"utils/#Some-Utility-Functions","page":"Utility Functions","title":"Some Utility Functions","text":"","category":"section"},{"location":"utils/","page":"Utility Functions","title":"Utility Functions","text":"FourierTools.δ\nFourierTools.select_region\nFourierTools.center_set!\nFourierTools.get_indices_around_center\nFourierTools.center_extract\nFourierTools.odd_view\nFourierTools.fourier_reverse!","category":"page"},{"location":"utils/#FourierTools.δ","page":"Utility Functions","title":"FourierTools.δ","text":"δ([T,] sz, pos=FourierTools.fft_center.(sz))\n\nReturn an array which has 1 at pos in the array of size sz.\n\nExamples\n\njulia> δ((3, 3))\n3×3 Matrix{Int64}:\n 0 0 0\n 0 1 0\n 0 0 0\n\njulia> δ(Float32, (4, 3))\n4×3 Matrix{Float32}:\n 0.0 0.0 0.0\n 0.0 0.0 0.0\n 0.0 1.0 0.0\n 0.0 0.0 0.0\n\njulia> δ(Float32, (3, 3), (1,1))\n3×3 Matrix{Float32}:\n 1.0 0.0 0.0\n 0.0 0.0 0.0\n 0.0 0.0 0.0\n\n\n\n\n\n","category":"function"},{"location":"utils/#FourierTools.select_region","page":"Utility Functions","title":"FourierTools.select_region","text":"select_region(mat; new_size)\n\nperforms the necessary Fourier-space operations of resampling in the space of ft (meaning the already circshifted version of fft).\n\nnew_size. The size of the array view after the operation finished. \n\ncenter. Specifies the center of the new view in coordinates of the old view. By default an alignment of the Fourier-centers is assumed.\n\nExamples\n\njulia> using FFTW, FourierTools\n\njulia> select_region(ones(3,3),new_size=(7,7),center=(1,3))\n7×7 PaddedView(0.0, OffsetArray(::Matrix{Float64}, 4:6, 2:4), (Base.OneTo(7), Base.OneTo(7))) with eltype Float64:\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 1.0 1.0 1.0 0.0 0.0 0.0\n 0.0 1.0 1.0 1.0 0.0 0.0 0.0\n 0.0 1.0 1.0 1.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n\n\n\n\n\n","category":"function"},{"location":"utils/#FourierTools.center_set!","page":"Utility Functions","title":"FourierTools.center_set!","text":"center_set!(arr_large, arr_small)\n\nPuts the arr_small central into arr_large. The convention, where the center is, is the same as the definition as for FFT based centered. Function works both for even and uneven arrays.\n\nExamples\n\njulia> FourierTools.center_set!([1, 1, 1, 1, 1, 1], [5, 5, 5])\n6-element Array{Int64,1}:\n 1\n 1\n 5\n 5\n 5\n 1\n\n\n\n\n\n","category":"function"},{"location":"utils/#FourierTools.get_indices_around_center","page":"Utility Functions","title":"FourierTools.get_indices_around_center","text":"get_indices_around_center(i_in, i_out)\n\nA function which provides two output indices i1 and i2 where i2 - i1 = i_out The indices are chosen in a way that the set i1:i2 cuts the interval 1:i_in in a way that the center frequency stays at the center position. Works for both odd and even indices\n\n\n\n\n\n","category":"function"},{"location":"utils/#FourierTools.center_extract","page":"Utility Functions","title":"FourierTools.center_extract","text":"center_extract(arr, new_size_array)\n\nExtracts a center of an array. new_size_array must be list of sizes indicating the output size of each dimension. Centered means that a center frequency stays at the center position. Works for even and uneven. If length(new_size_array) < length(ndims(arr)) the remaining dimensions are untouched and copied.\n\nExamples\n\njulia> FourierTools.center_extract([1 2; 3 4], [1])\n1×2 view(::Matrix{Int64}, 2:2, 1:2) with eltype Int64:\n 3 4\n\njulia> FourierTools.center_extract([1 2; 3 4], [1, 1])\n1×1 view(::Matrix{Int64}, 2:2, 2:2) with eltype Int64:\n 4\n\njulia> FourierTools.center_extract([1 2 3; 3 4 5; 6 7 8], [2 2])\n2×2 view(::Matrix{Int64}, 1:2, 1:2) with eltype Int64:\n 1 2\n 3 4\n\n\n\n\n\n","category":"function"},{"location":"utils/#FourierTools.odd_view","page":"Utility Functions","title":"FourierTools.odd_view","text":"odd_view(arr)\n\ncreates a view of arr that for each even dimension excludes the starting index yielding a view of the array with only odd dimensions. This is useful for operations in Fourier-space which should leave the first index unaltered such as reverse! Note that an array reversal can also be achieved by using two ffts instead of one fft and one ifft.\n\nExamples\n\njulia> odd_view([1 2 3; 4 5 6])\n1×3 view(::Matrix{Int64}, 2:2, 1:3) with eltype Int64:\n 4 5 6\n\n\n\n\n\n","category":"function"},{"location":"utils/#FourierTools.fourier_reverse!","page":"Utility Functions","title":"FourierTools.fourier_reverse!","text":"fourier_reverse!(arr; dims=1:ndims(arr))\n\nreverses the dimensions of the input array arr in place. This effectively mirrors these array. Note that for even-sized dimensions the first index is excluded from the reverse operation along this dimensions. \n\nExample\n\njulia> a = [1 2 3;4 5 6;7 8 9;10 11 12]\n4×3 Matrix{Int64}:\n 1 2 3\n 4 5 6\n 7 8 9\n 10 11 12\n\njulia> fourier_reverse!(a);\n\njulia> a\n4×3 Matrix{Int64}:\n 3 2 1\n 12 11 10\n 9 8 7\n 6 5 4\n\n\n\n\n\n","category":"function"},{"location":"nfft/#NFFT-helpers","page":"NFFT","title":"NFFT helpers","text":"","category":"section"},{"location":"nfft/","page":"NFFT","title":"NFFT","text":"Based on NFFT.jl we provide some convenient helper functions:","category":"page"},{"location":"nfft/","page":"NFFT","title":"NFFT","text":"FourierTools.nfft_nd\nFourierTools.plan_nfft_nd","category":"page"},{"location":"nfft/#FourierTools.nfft_nd","page":"NFFT","title":"FourierTools.nfft_nd","text":"nfft_nd(src, dst_coords, dst_size=nothing; is_in_pixels=false, is_local_shift=false)\n\nperforms an n-dimensional non-uniform FFT on grids with a regular topology. In comparison to the nfft() routine, which this computed is based on, this version does not require any reshape operations. See plan_nfft_nd for details on the arguments and usage examples. Note that the input can be Real valued and will be automatically converted to Complex.\n\n# A Zoomed transform in 3D\njulia> nfft_nd(rand(10,12,12), (t)-> (0.8*t[1], 0.7*t[2], 0.6*t[3]))\n\n\n\n\n\n","category":"function"},{"location":"nfft/#FourierTools.plan_nfft_nd","page":"NFFT","title":"FourierTools.plan_nfft_nd","text":"plan_nfft_nd(src, dst_coords; is_in_pixels=false, is_local_shift=false, pad_value=nothing, reltol=1e-9)\n\nPlans an n-dimensional non-uniform FFT on grids with a regular topology. In comparison to the nfft() routine, which this computed is based on, this version does not require any reshape operations.\n\nArguments\n\nsrc: source array\ndst_coords: array of destination coordinates. This can be either an Array of Tuple or an Array with the last dimension ofsizelength(size(dst_coords))-1` referring to the destination coordinates where the FFT needs to be computed. Alternatively also a function mapping a tuple (of source index positions) to a tuple (of destination index positions). In the recommended mode, the indices are normalized to to Fouier frequency range (roughly speaking from -0.5:0.5).\nis_in_pixels: A Boolean flag indicating whether dstcoords refers to coordinates in pixels or (default) in relative frequency positions. If `isin_pixels=true` is selected, destination coordinates (1-based indexing) as typical for array indexing is assumed and internally converted to relative positions.\nis_local_shift: A Boolean controlling wether dst_coords refers to the destination coordinates or the relative distance from standard grid coordinates (size determind from dst_coordinates). \npad_value: if supplied, values outside the valid pixel range (roughly -0.5:0.5) are replaced by this complex-valued pad value.\nis_adjoint: if true this plan is based on the adjoint rather than the ordinary plan\nreltol: The numerical precision to which the results are computed. This is passed to the nfft routine. Worse precision is faster.\n\n# Lets try a 2D rotation:\njulia> using TestImages, NDTools, View5D, IndexFunArrays, FourierTools\n\njulia> img = Float64.(testimage(\"resolution\"));\n\n# define a rotation operation\njulia> rot_alpha(a, t) = (cosd(a)*t[1] - sind(a)*t[2], sind(a)*t[1]+cosd(a)*t[2])\n\njulia> new_pos = rot_alpha.(10.0, idx(img, scale=ScaFT))\n\njulia> f = ift(img)\n\njulia> p = plan_nfft_nd(f, new_pos; is_local_shift=false, is_in_pixels=false)\n\njulia> g = real.(p * f)\n\n#display the result\njulia> @ve img, g\n\n\n\n\n\nplan_nfft_nd(src::AbstractArray{T,D}, dst_fkt::Function, dst_size=size(src); is_in_pixels=false, is_adjoint=false, kwargs...)\n\nPlans an n-dimensional non-uniform FFT on grids with a regular topology defined via the function dst_fkt.\n\nArguments\n\nsrc: source array\ndst_fkt: a function mapping a tuple (of source index positions) to a tuple (of destination index positions). In the recommended mode, the indices are normalized to to Fouier frequency range (roughly speaking from -0.5:0.5). If the named argument is_in_pixels is provided, the function is expected to act on one-index based pixel coordinates. This option is particularly interesting in combination with the argument is_loca_shift.\ndst_size: this argument is only used for functions. If you require a different result size for dst_coords being a function, state it here.By defaul (dst_size=nothing) the destination size will be inferred form the argument new_pos or assumed to be size(src).\nis_in_pixels: A Boolean flag indicating whether dstcoords refers to coordinates in pixels or (default) in relative frequency positions. If `isin_pixels=true` is selected, destination coordinates (1-based indexing) as typical for array indexing is assumed and internally converted to relative positions.\nis_adjoint: if true this plan is based on the adjoint rather than the ordinary plan\n\nFor other arguments and examples see the array-version of plan_nfft above.\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FFT-helpers","page":"FFT Helpers","title":"FFT helpers","text":"","category":"section"},{"location":"helpers/","page":"FFT Helpers","title":"FFT Helpers","text":"ffts\nffts!\niffts\nrffts\nirffts\nFourierTools.fftshift_view\nFourierTools.ifftshift_view\nFourierTools.rfftshift_view\nFourierTools.irfftshift_view\nft\nift\nrft\nirft","category":"page"},{"location":"helpers/#FourierTools.ffts","page":"FFT Helpers","title":"FourierTools.ffts","text":"ffts(A [, dims])\n\nResult is semantically equivalent to fftshift(fft(A, dims), dims) However, the shift is done with ShiftedArrays and therefore doesn't allocate memory.\n\nSee also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts, \n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ffts!","page":"FFT Helpers","title":"FourierTools.ffts!","text":"ffts!(A [, dims])\n\nResult is semantically equivalent to fftshift(fft!(A, dims), dims). A is in-place modified. However, the shift is done with ShiftedArrays and therefore doesn't allocate memory.\n\nSee also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts, \n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.iffts","page":"FFT Helpers","title":"FourierTools.iffts","text":"iffts(A [, dims])\n\nResult is semantically equivalent to ifft(ifftshift(A, dims), dims). A is in-place modified. However, the shift is done with ShiftedArrays and therefore doesn't allocate memory.\n\nSee also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts, \n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.rffts","page":"FFT Helpers","title":"FourierTools.rffts","text":"rffts(A [, dims])\n\nCalculates a rfft(A, dims) and then shift the frequencies to the center. dims[1] is not shifted, because there is no negative and positive frequency. The shift is done with ShiftedArrays and therefore doesn't allocate memory.\n\nSee also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts, \n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.irffts","page":"FFT Helpers","title":"FourierTools.irffts","text":"irffts(A, d, [, dims])\n\nCalculates a irfft(A, d, dims) and then shift the frequencies back to the corner. dims[1] is not shifted, because there is no negative and positive frequency. The shift is done with ShiftedArrays and therefore doesn't allocate memory.\n\nSee also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts, \n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.fftshift_view","page":"FFT Helpers","title":"FourierTools.fftshift_view","text":"fftshift_view(A [, dims])\n\nResult is semantically equivalent to fftshift(A, dims) but returns a view instead. \n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ifftshift_view","page":"FFT Helpers","title":"FourierTools.ifftshift_view","text":"ifftshift_view(A [, dims])\n\nResult is semantically equivalent to fftshift(A, dims) but returns a view instead. \n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.rfftshift_view","page":"FFT Helpers","title":"FourierTools.rfftshift_view","text":"rfftshift_view(A, dims)\n\nShifts the frequencies to the center expect for dims[1] because there os no negative and positive frequency.\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.irfftshift_view","page":"FFT Helpers","title":"FourierTools.irfftshift_view","text":"irfftshift_view(A, dims)\n\nShifts the frequencies back to the corner except for dims[1] because there os no negative and positive frequency.\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ft","page":"FFT Helpers","title":"FourierTools.ft","text":"ft(A [, dims])\n\nDigital Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(fft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with both coordinate systems in real and Fourier-space being centered at position CtrFT == size÷2+1\n\nThe following identities are true:\n\njulia> sz = (5,5)\n(5, 5)\n\njulia> ft(ones(sz)) ≈ prod(sz) .* δ(sz)\ntrue\n\njulia> ft(δ(sz)) ≈ ones(sz)\ntrue\n\nSee also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts, \n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ift","page":"FFT Helpers","title":"FourierTools.ift","text":"ift(A [, dims])\n\nDigital inverse Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(ifft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with both coordinate systems in real and Fourier-space being centered at position CtrFT == size÷2+1\n\nThe following identities are true:\n\njulia> sz = (5,6,7)\n(5, 6, 7)\n\njulia> ift(ones(sz)) ≈ δ(sz)\ntrue\n\njulia> ift(δ(sz)) ≈ ones(sz) ./ prod(sz)\ntrue\n\nSee also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts, \n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.rft","page":"FFT Helpers","title":"FourierTools.rft","text":"rft(A [, dims])\n\nDigital real-valued Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(rfft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with the coordinate systems in real space centered at CtrFT == size÷2+1 and in (half) Fourier-space being centered at CtrRFT == setindex(size÷2 +1,1,1).\n\nThe following identities are true:\n\njulia> sz = (6,6)\n(6, 6)\n\njulia> rft(δ(sz)) ≈ ones(rft_size(sz))\ntrue\n\nSee also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts, \n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.irft","page":"FFT Helpers","title":"FourierTools.irft","text":"irft(A, d, [, dims])\n\nDigital real-valued inverse Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(irfft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with the coordinate systems in real space centered at CtrFT == size÷2+1 and in (half) Fourier-space being centered at CtrRFT == setindex(size÷2 +1,1,1). Note that the size d of the first transform direction [1] is a required argument.\n\nThe following identities are true:\n\njulia> sz = (6,6)\n(6, 6)\n\njulia> irft(ones(rft_size(sz)),sz[1]) ≈ δ(sz)\ntrue\n\nSee also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts, \n\n\n\n\n\n","category":"function"},{"location":"helpers/#FFT-Utils","page":"FFT Helpers","title":"FFT Utils","text":"","category":"section"},{"location":"helpers/","page":"FFT Helpers","title":"FFT Helpers","text":"FourierTools.fftpos\nFourierTools.fft_center\nFourierTools.rft_size\nFourierTools.rfft_size\nFourierTools.ft_center_diff\nFourierTools.rft_center_diff\nFourierTools.center_pos","category":"page"},{"location":"helpers/#FourierTools.fftpos","page":"FFT Helpers","title":"FourierTools.fftpos","text":"fftpos(L, N, around=CenterFirst::Center)\n\nConstruct a range from -L/2 to L/2 around around\n\nHowever, we ensure that those positions are in a way which they are useful for FFT operations. This means, that depending on the center a small offset is subtracted.\n\nSee NDTools.Center for all center options. You need to load using NDTools to access all center options.\n\nExamples\n\njulia> collect(fftpos(1,4))\n4-element Vector{Float64}:\n 0.0\n 0.2916666666666667\n 0.5833333333333334\n 0.875\n\njulia> collect(fftpos(1,5))\n5-element Vector{Float64}:\n 0.0\n 0.225\n 0.45\n 0.675\n 0.9\n\njulia> using NDTools\n\njulia> collect(fftpos(1,4, CenterFirst))\n4-element Vector{Float64}:\n 0.0\n 0.2916666666666667\n 0.5833333333333334\n 0.875\n\njulia> collect(fftpos(1,4, CenterFT))\n4-element Vector{Float64}:\n -0.5833333333333333\n -0.29166666666666663\n 3.70074341541719e-17\n 0.2916666666666667\n\njulia> collect(fftpos(1,4, CenterMiddle))\n4-element Vector{Float64}:\n -0.4375\n -0.14583333333333334\n 0.14583333333333334\n 0.4375\n\n\n\n\n\nfftpos(l, N, around)\n\nAnother fftpos method where the range is constructed around around. around is here a number indicating the index position around the range is constructed\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.fft_center","page":"FFT Helpers","title":"FourierTools.fft_center","text":"fft_center(x)\n\nReturns the center of a size in Fourier sense and Julia 1-based indices.\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.rft_size","page":"FFT Helpers","title":"FourierTools.rft_size","text":"rft_size(sz::NTuple{Int})\n\nReturns the size of an rft or rfft performed on the data x, without performing the rfft. sz: corresponding real space size to obtain the rft size for\n\n\n\n\n\nrft_size(arr)\n\nReturns the size of an rft or rfft performed on the data x, without performing the rfft.\n\narr: array to optain the corresponding rft size for\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.rfft_size","page":"FFT Helpers","title":"FourierTools.rfft_size","text":"rfft_size(size, dims)\n\nReturns the size rfft would return if applied to a real array. size is the input size to rfft and dims the dimensions the rfft transforms over. Actually we only would need first(dims).\n\njulia> using FFTW\n\njulia> rfft((ones((4,3,2))), (2,3)) |> size\n(4, 2, 2)\n\njulia> FourierTools.rfft_size((4,3,2), (2, 3))\n(4, 2, 2)\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ft_center_diff","page":"FFT Helpers","title":"FourierTools.ft_center_diff","text":"ft_center_diff(s [, dims])\n\nCalculates how much each dimension must be shifted that the center frequency is at the Fourier center. This if for a normal fft\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.rft_center_diff","page":"FFT Helpers","title":"FourierTools.rft_center_diff","text":"rft_center_diff(s [, dims])\n\nCalculates how much each dimension must be shifted that the center frequency is at the Fourier center. This is for rfft. The dims[1] must be therefore not shifted!\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.center_pos","page":"FFT Helpers","title":"FourierTools.center_pos","text":"center_pos(x)\n\nCalculate the position of the center frequency. Size of the array is x\n\nExamples\n\njulia> FourierTools.center_pos(3)\n2\njulia> FourierTools.center_pos(4)\n3\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FFT-Helpers-2D","page":"FFT Helpers","title":"FFT Helpers 2D","text":"","category":"section"},{"location":"helpers/","page":"FFT Helpers","title":"FFT Helpers","text":"FourierTools.fftshift2d\nFourierTools.ifftshift2d\nFourierTools.fftshift2d_view\nFourierTools.ifftshift2d_view\nfft2d\nifft2d\nrfft2d\nirfft2d\nffts2d\nffts2d!\niffts2d\nrffts2d\nirffts2d\nft2d\nift2d\nrft2d\nirft2d","category":"page"},{"location":"helpers/#FourierTools.fftshift2d","page":"FFT Helpers","title":"FourierTools.fftshift2d","text":"fftshift2d(mat::AbstractArray{T, N}) where {T, N}\n\nShort-hand for fftshift(mat, (1,2)). See fftshift for details.\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ifftshift2d","page":"FFT Helpers","title":"FourierTools.ifftshift2d","text":"ifftshift2d(mat::AbstractArray{T, N}) where {T, N}\n\nShort-hand for ifftshift(mat, (1,2)). See ifftshift for details.\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.fftshift2d_view","page":"FFT Helpers","title":"FourierTools.fftshift2d_view","text":"fftshift2d_view(mat::AbstractArray{T, N}) where {T, N}\n\nShort-hand for fftshift_view(mat, (1,2)). See fftshift_view for details.\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ifftshift2d_view","page":"FFT Helpers","title":"FourierTools.ifftshift2d_view","text":"ifftshift2d_view(mat::AbstractArray{T, N}) where {T, N}\n\nShort-hand for ifftshift_view(mat, (1,2)) performing only a 2D inverse ft. See ifft for details.\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.fft2d","page":"FFT Helpers","title":"FourierTools.fft2d","text":"fft2d(mat::AbstractArray{T, N}) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ifft2d","page":"FFT Helpers","title":"FourierTools.ifft2d","text":"ifft2d(mat::AbstractArray{T, N}) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.rfft2d","page":"FFT Helpers","title":"FourierTools.rfft2d","text":"rfft2d(mat::AbstractArray{T, N}) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.irfft2d","page":"FFT Helpers","title":"FourierTools.irfft2d","text":"irfft2d(mat::AbstractArray{T, N}, d) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ffts2d","page":"FFT Helpers","title":"FourierTools.ffts2d","text":"ft2d(mat::AbstractArray{T, N}) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ffts2d!","page":"FFT Helpers","title":"FourierTools.ffts2d!","text":"fft2ds!(mat::AbstractArray{T, N}) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.iffts2d","page":"FFT Helpers","title":"FourierTools.iffts2d","text":"iffts2d(mat::AbstractArray{T, N}) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.rffts2d","page":"FFT Helpers","title":"FourierTools.rffts2d","text":"rffts2d(mat::AbstractArray{T, N}) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.irffts2d","page":"FFT Helpers","title":"FourierTools.irffts2d","text":"riffts2d(mat::AbstractArray{T, N}, d) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ft2d","page":"FFT Helpers","title":"FourierTools.ft2d","text":"ft2d(mat::AbstractArray{T, N}) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.ift2d","page":"FFT Helpers","title":"FourierTools.ift2d","text":"ift2d(mat::AbstractArray{T, N}) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.rft2d","page":"FFT Helpers","title":"FourierTools.rft2d","text":"rft2d(mat::AbstractArray{T, N}) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"helpers/#FourierTools.irft2d","page":"FFT Helpers","title":"FourierTools.irft2d","text":"irft2d(mat::AbstractArray{T, N}, d) where {T, N}\n\nOnly over dims=(1,2).\n\n\n\n\n\n","category":"function"},{"location":"convolutions/#Function-References","page":"FFT Based Convolutions and Cross-Correlation","title":"Function References","text":"","category":"section"},{"location":"convolutions/","page":"FFT Based Convolutions and Cross-Correlation","title":"FFT Based Convolutions and Cross-Correlation","text":"conv\nccorr\nconv_psf\nplan_conv\nplan_conv_psf\nplan_conv_buffer\nplan_conv_psf_buffer","category":"page"},{"location":"convolutions/#FourierTools.conv","page":"FFT Based Convolutions and Cross-Correlation","title":"FourierTools.conv","text":"conv(u, v[, dims])\n\nConvolve u with v over dims dimensions with an FFT based method. Note, that this method introduces wrap-around artifacts without proper padding/windowing.\n\nArguments\n\nu is an array in real space.\nv is the array to be convolved in real space as well.\nPer default ntuple(+, min(N, M))) means that we perform the convolution over all dimensions of that array which has less dimensions. If dims is an array with integers, we perform convolution only over these dimensions. Eg. dims=[1,3] would perform the convolution over the first and third dimension. Second dimension is not convolved.\n\nIf u and v are both a real valued array we use rfft and hence the output is real as well. If either u or v is complex we use fft and output is hence complex.\n\nExamples\n\n1D with FFT over all dimensions. We choose v to be a delta peak. Therefore convolution should act as identity.\n\njulia> u = [1 2 3 4 5]\n1×5 Array{Int64,2}:\n 1 2 3 4 5\njulia> v = [0 0 1 0 0]\n1×5 Array{Int64,2}:\n 0 0 1 0 0\n\njulia> conv(u, v)\n1×5 Matrix{Float64}:\n 4.0 5.0 1.0 2.0 3.0\n\n2D with FFT with different dims arguments.\n\njulia> u = 1im .* [1 2 3; 4 5 6]\n2×3 Matrix{Complex{Int64}}:\n 0+1im 0+2im 0+3im\n 0+4im 0+5im 0+6im\n\njulia> v = [1im 0 0; 1im 0 0]\n2×3 Matrix{Complex{Int64}}:\n 0+1im 0+0im 0+0im\n 0+1im 0+0im 0+0im\n\njulia> conv(u, v)\n2×3 Matrix{ComplexF64}:\n -5.0+0.0im -7.0+0.0im -9.0+0.0im\n -5.0+0.0im -7.0+0.0im -9.0+0.0im\n\n\n\n\n\n","category":"function"},{"location":"convolutions/#FourierTools.ccorr","page":"FFT Based Convolutions and Cross-Correlation","title":"FourierTools.ccorr","text":"ccorr(u, v[, dims]; centered=false)\n\nCalculates the cross-correlation between u and v along dims. centered=true moves the output of the cross-correlation to the Fourier center.\n\nIf u and v are both a real valued array we use rfft and hence the output is real as well. If either u or v is complex we use fft and output is hence complex.\n\nPer default the correlation is performed along min(ndims(u), ndims(v)).\n\njulia> ccorr([1,1,0,0], [1,1,0,0], centered=true)\n4-element Vector{Float64}:\n 0.0\n 1.0\n 2.0\n 1.0\n\njulia> ccorr([1,1,0,0], [1,1,0,0])\n4-element Vector{Float64}:\n 2.0\n 1.0\n 0.0\n 1.0\n\njulia> ccorr([1im,0,0,0], [0,1im,0,0])\n4-element Vector{ComplexF64}:\n 0.0 + 0.0im\n 0.0 + 0.0im\n 0.0 + 0.0im\n 1.0 + 0.0im\n\njulia> ccorr([1im,0,0,0], [0,1im,0,0], centered=true)\n4-element Vector{ComplexF64}:\n 0.0 + 0.0im\n 1.0 + 0.0im\n 0.0 + 0.0im\n 0.0 + 0.0im\n\n\n\n\n\n","category":"function"},{"location":"convolutions/#FourierTools.conv_psf","page":"FFT Based Convolutions and Cross-Correlation","title":"FourierTools.conv_psf","text":"conv_psf(u, psf[, dims])\n\nconv_psf is a shorthand for conv(u,ifftshift(psf)). For examples see conv.\n\n\n\n\n\n","category":"function"},{"location":"convolutions/#FourierTools.plan_conv","page":"FFT Based Convolutions and Cross-Correlation","title":"FourierTools.plan_conv","text":"plan_conv(u, v [, dims]; kwargs...)\n\nPre-plan an optimized convolution for arrays shaped like u and v (based on pre-plan FFT) along the given dimenions dims. dims = 1:ndims(u) per default. The 0 frequency of u must be located at the first entry.\n\nWe return two arguments: The first one is v_ft (obtained by fft(v) or rfft(v)). The second return is the convolution function pconv. pconv itself has two arguments. pconv(u, v_ft=v_ft) where u is the object and v_ft the v_ft. This function achieves faster convolution than conv(u, u). Depending whether u is real or complex we do ffts or rffts Additionally, it is possible to provide flags=FFTW.MEASURE as kwargs to change the planning of the FFT.\n\nExamples\n\njulia> u = [1 2 3 4 5]\n1×5 Matrix{Int64}:\n 1 2 3 4 5\n\njulia> v = [1 0 0 0 0]\n1×5 Matrix{Int64}:\n 1 0 0 0 0\n\njulia> v_ft, pconv = plan_conv(u, v);\n\njulia> pconv(u, v_ft)\n1×5 Matrix{Float64}:\n 1.0 2.0 3.0 4.0 5.0\n\njulia> pconv(u)\n1×5 Matrix{Float64}:\n 1.0 2.0 3.0 4.0 5.0\n\n\n\n\n\n","category":"function"},{"location":"convolutions/#FourierTools.plan_conv_psf","page":"FFT Based Convolutions and Cross-Correlation","title":"FourierTools.plan_conv_psf","text":"plan_conv_psf(u, psf [, dims]; kwargs...) where {T, N}\n\nplan_conv_psf is a shorthand for plan_conv(u, ifftshift(psf)). For examples see plan_conv.\n\n\n\n\n\n","category":"function"},{"location":"convolutions/#FourierTools.plan_conv_buffer","page":"FFT Based Convolutions and Cross-Correlation","title":"FourierTools.plan_conv_buffer","text":"plan_conv_buffer(u, v [, dims]; kwargs...)\n\nSimilar to plan_conv but instead uses buffers to prevent memory allocations. Not AD friendly!\n\n\n\n\n\n","category":"function"},{"location":"convolutions/#FourierTools.plan_conv_psf_buffer","page":"FFT Based Convolutions and Cross-Correlation","title":"FourierTools.plan_conv_psf_buffer","text":"plan_conv_psf_buffer(u, psf [, dims]; kwargs...) where {T, N}\n\nplan_conv_psf_buffer is a shorthand for plan_conv_buffer(u, ifftshift(psf)). For examples see plan_conv.\n\n\n\n\n\n","category":"function"},{"location":"#FourierTools.jl","page":"FourierTools.jl","title":"FourierTools.jl","text":"","category":"section"},{"location":"","page":"FourierTools.jl","title":"FourierTools.jl","text":"This package contains various functions that are useful for working with and in Fourier space.","category":"page"},{"location":"","page":"FourierTools.jl","title":"FourierTools.jl","text":"Install FourierTools.jl via the package manager. Currently with","category":"page"},{"location":"","page":"FourierTools.jl","title":"FourierTools.jl","text":"julia> add FourierTools ","category":"page"},{"location":"","page":"FourierTools.jl","title":"FourierTools.jl","text":"Check out the sub pages for different applications.","category":"page"}] +} diff --git a/v0.4.5/shear/index.html b/v0.4.5/shear/index.html new file mode 100644 index 0000000..f3cbad7 --- /dev/null +++ b/v0.4.5/shear/index.html @@ -0,0 +1,10 @@ + +Image Shearing with FFTs · FourierTools.jl
    Edit on GitHub

    Rotation with FFT

    Via shear it is possible to rotate an image. Shearing is basically a shift operation but with different shift distance in each row.

    Examples

    For full interactivity, have a look at this Pluto notebook.

    using Revise, FourierTools, Plots, TestImages, PlutoUI, ImageShow
+
+begin
+    img = Float32.(testimage("fabio_512_gray"))
+    z = zeros(Float32, (768, 768))
+    FourierTools.center_set!(z, img)
+end
+
+Gray.(FourierTools.shear(z, -305))

    Function references

    FourierTools.shear — Function
    shear(arr, Δ, shear_dir_dim=1, shear_dim=2; fix_nyquist=false, adapt_size=false::Bool, pad_value=zero(eltype(arr)))

    Shears an array by the amount of Δ pixels via an FFT approach. Δ is the relative shift between the top and bottom row shifted with respect to each other. shear_dir_dim decides the direction of the shear and shear_dim is the second dimension where the shear happens. There is also shear! available.

    #Arguments

    • arr: array to shear
    • shear_dir_dim: dimension of the shift during shear
    • shear_dim: dimension along which to progress and apply variing shears along shear_dir_dim
    • fix_nyquist: apply a fix to the highest frequency during the Fourier-space application of the exponential factor
    • adapt_size: if true, pad the data prior to the shear. The result array will be larger
    • pad_value: the value to pad with (only applies if adapt_size=true)
    • assign_wrap=assign_wrap: replaces wrap-around areas by pad_value (only of adapt_size is false)

    For complex arrays we use fft, for real array we use rfft.

    source
    FourierTools.shear! — Function
    shear!(arr, Δ, shear_dir_dim=1, shear_dim=2; fix_nyquist=false, assign_wrap=false, pad_value=zero(eltype(arr)))

    For more details see shear.

    Extra Arguments

    assign_wrap: if true wrap-around areas are replaced by pad_value pad_value: the value to replace wrap-around areas with

    For complex arrays we can completely avoid large memory allocations. For real arrays, we need at least allocate on array in the fourier space.

    source
    diff --git a/v0.4.5/shifting/index.html b/v0.4.5/shifting/index.html new file mode 100644 index 0000000..410835e --- /dev/null +++ b/v0.4.5/shifting/index.html @@ -0,0 +1,41 @@ + +Shifting with FFTs · FourierTools.jl
    Edit on GitHub

    Shifting

    Using the Fourier shift property one can implement shifting of arrays not only over pixel but also sub-pixel amount.

    Examples

    For full interactivity, have a look at this Pluto notebook.

    begin
+    f(x) = cos(4π * x / 30)
+    x1 = 1:30
+    x2 = x1 .+ 3
+end
+
+begin
+    y1 = f.(x1)
+    y2 = f.(x2)
+    offset = 2.01
+    y3 = shift(y2, tuple(offset))
+end
+
+begin
+    plot(y1, label="Original signal")
+    plot!(y2, label="Shifted signal")
+    plot!(y3, label="Fourier shifted with $offset")
+end

    Function references

    FourierTools.shift! — Function
    shift!(arr, shifts; kwargs...)

    Shifts an array in-place. For real arrays it is based on rfft. For complex arrays based on fft. shifts can be non-integer, for integer shifts one should prefer circshift or ShiftedArrays.circshift because a FFT-based methods introduces numerical errors.

    kwargs...

    • fix_nyquist_frequency=false: Fourier shifting of even-sized arrays is not revertible. However, if you did shift(x, δ) you can it revert by shift(x, δ, fix_nyquist_frequency=true). This only works if δ is the same.
    • take_real=true: For even-sized arrays we take by default the real part of the exponential phase at the Nyquist frequency. This satisfies the property of real valuedness and the aliasing of the Nyquist term.

    Memory Usage

    Note that for complex arrays we can avoid any large memory allocations because of fft!. For rfft there does not exist a usable implementation yet, so for real arrays there might be a temporary larger memory usage.

    Examples

    julia> x = [1.0 2.0 3.0; 4.0 5.0 6.0]
+2×3 Matrix{Float64}:
+ 1.0  2.0  3.0
+ 4.0  5.0  6.0
+
+julia> shift!(x, (1, 2))
+2×3 Matrix{Float64}:
+ 5.0  6.0  4.0
+ 2.0  3.0  1.0
+
+julia> x = [0, 1.0, 0.0, 1.0]
+4-element Vector{Float64}:
+ 0.0
+ 1.0
+ 0.0
+ 1.0
+
+julia> shift!(x, 0.5)
+4-element Vector{Float64}:
+ 0.49999999999999994
+ 0.5
+ 0.49999999999999994
+ 0.5
    source
    diff --git a/v0.4.5/siteinfo.js b/v0.4.5/siteinfo.js new file mode 100644 index 0000000..66cc2ae --- /dev/null +++ b/v0.4.5/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.4.5"; diff --git a/v0.4.5/utils/index.html b/v0.4.5/utils/index.html new file mode 100644 index 0000000..a32e5f9 --- /dev/null +++ b/v0.4.5/utils/index.html @@ -0,0 +1,63 @@ + +Utility Functions · FourierTools.jl
    Edit on GitHub

    Some Utility Functions

    FourierTools.δ — Function
    δ([T,] sz, pos=FourierTools.fft_center.(sz))

    Return an array which has 1 at pos in the array of size sz.

    Examples

    julia> δ((3, 3))
+3×3 Matrix{Int64}:
+ 0  0  0
+ 0  1  0
+ 0  0  0
+
+julia> δ(Float32, (4, 3))
+4×3 Matrix{Float32}:
+ 0.0  0.0  0.0
+ 0.0  0.0  0.0
+ 0.0  1.0  0.0
+ 0.0  0.0  0.0
+
+julia> δ(Float32, (3, 3), (1,1))
+3×3 Matrix{Float32}:
+ 1.0  0.0  0.0
+ 0.0  0.0  0.0
+ 0.0  0.0  0.0
    source
    FourierTools.select_region — Function
    select_region(mat; new_size)

    performs the necessary Fourier-space operations of resampling in the space of ft (meaning the already circshifted version of fft).

    new_size. The size of the array view after the operation finished.

    center. Specifies the center of the new view in coordinates of the old view. By default an alignment of the Fourier-centers is assumed.

    Examples

    julia> using FFTW, FourierTools
+
+julia> select_region(ones(3,3),new_size=(7,7),center=(1,3))
+7×7 PaddedView(0.0, OffsetArray(::Matrix{Float64}, 4:6, 2:4), (Base.OneTo(7), Base.OneTo(7))) with eltype Float64:
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0  1.0  1.0  1.0  0.0  0.0  0.0
+ 0.0  1.0  1.0  1.0  0.0  0.0  0.0
+ 0.0  1.0  1.0  1.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0  0.0  0.0
    source
    FourierTools.center_set! — Function
    center_set!(arr_large, arr_small)

    Puts the arr_small central into arr_large. The convention, where the center is, is the same as the definition as for FFT based centered. Function works both for even and uneven arrays.

    Examples

    julia> FourierTools.center_set!([1, 1, 1, 1, 1, 1], [5, 5, 5])
+6-element Array{Int64,1}:
+ 1
+ 1
+ 5
+ 5
+ 5
+ 1
    source
    FourierTools.get_indices_around_center — Function
    get_indices_around_center(i_in, i_out)

    A function which provides two output indices i1 and i2 where i2 - i1 = i_out The indices are chosen in a way that the set i1:i2 cuts the interval 1:i_in in a way that the center frequency stays at the center position. Works for both odd and even indices

    source
    FourierTools.center_extract — Function
    center_extract(arr, new_size_array)

    Extracts a center of an array. new_size_array must be list of sizes indicating the output size of each dimension. Centered means that a center frequency stays at the center position. Works for even and uneven. If length(new_size_array) < length(ndims(arr)) the remaining dimensions are untouched and copied.

    Examples

    julia> FourierTools.center_extract([1 2; 3 4], [1])
+1×2 view(::Matrix{Int64}, 2:2, 1:2) with eltype Int64:
+ 3  4
+
+julia> FourierTools.center_extract([1 2; 3 4], [1, 1])
+1×1 view(::Matrix{Int64}, 2:2, 2:2) with eltype Int64:
+ 4
+
+julia> FourierTools.center_extract([1 2 3; 3 4 5; 6 7 8], [2 2])
+2×2 view(::Matrix{Int64}, 1:2, 1:2) with eltype Int64:
+ 1  2
+ 3  4
    source
    FourierTools.odd_view — Function
    odd_view(arr)

    creates a view of arr that for each even dimension excludes the starting index yielding a view of the array with only odd dimensions. This is useful for operations in Fourier-space which should leave the first index unaltered such as reverse! Note that an array reversal can also be achieved by using two ffts instead of one fft and one ifft.

    Examples

    julia> odd_view([1 2 3; 4 5 6])
+1×3 view(::Matrix{Int64}, 2:2, 1:3) with eltype Int64:
+ 4  5  6
    source
    FourierTools.fourier_reverse! — Function
    fourier_reverse!(arr; dims=1:ndims(arr))

    reverses the dimensions of the input array arr in place. This effectively mirrors these array. Note that for even-sized dimensions the first index is excluded from the reverse operation along this dimensions.

    Example

    julia> a = [1 2 3;4 5 6;7 8 9;10 11 12]
+4×3 Matrix{Int64}:
+  1   2   3
+  4   5   6
+  7   8   9
+ 10  11  12
+
+julia> fourier_reverse!(a);
+
+julia> a
+4×3 Matrix{Int64}:
+  3   2   1
+ 12  11  10
+  9   8   7
+  6   5   4
    source
    diff --git a/versions.js b/versions.js index 1efa0de..53cd7ec 100644 --- a/versions.js +++ b/versions.js @@ -6,5 +6,5 @@ var DOC_VERSIONS = [ "v0.1", "dev", ]; -var DOCUMENTER_NEWEST = "v0.4.4"; +var DOCUMENTER_NEWEST = "v0.4.5"; var DOCUMENTER_STABLE = "stable";