-
Notifications
You must be signed in to change notification settings - Fork 1
/
Section1.tex
756 lines (641 loc) · 22 KB
/
Section1.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
\documentclass{beamer}
\usepackage[latin1]{inputenc}
\usepackage[3D]{movie15}
%\usetheme{Antibes}
\usetheme{Warsaw}
%\usetheme{Marburg}
%\usetheme[secheader]{Boadilla}
%\usetheme{default}
%\usetheme{Dresden}
%\usetheme{Madrid}
\usecolortheme{seahorse}
%\usecolortheme{crane}
%\usecolortheme{albatross}
%\usecolortheme{whale}
%\usecolortheme{beaver}
\usefonttheme{structuresmallcapsserif}
\title[Anatomical Features]{Multivariate models of inter-subject anatomical variability}
\author{John Ashburner}
\institute[[email protected]]{Wellcome Trust Centre for Neuroimaging,\\
UCL Institute of Neurology,\\
12 Queen Square,\\
London WC1N 3BG,\\
UK.}
\date{}
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\titlepage
\end{frame}
\section{Introduction}
\subsection{Prediction}
\begin{frame}
%\frametitle{There is more than just classification}
\begin{center}
{\huge ``\emph{The only relevant test of the validity of a hypothesis is comparison of prediction with experience}.''\par}
\vspace{1cm}
Milton Friedman
\end{center}
\end{frame}
\begin{frame}
\frametitle{Choosing models/hypotheses/theories}
\begin{columns}[c]
\column{.3\textwidth}
\includegraphics[width=\textwidth]{mackay}
\vspace{.5cm}
{\small
MacKay, DJC. ``Bayesian interpolation.'' Neural computation 4, no. 3 (1992): 415-447.\par}
\column{.7\textwidth}
\includegraphics[width=\textwidth]{mackay1992}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Evidence-based Science}
...also just known as ``science''.
\vspace{0.5cm}
\begin{itemize}
\item{Researchers claim to find differences between groups. Do those findings actually discriminate?}
\item{How can we most accurately diagnose a disorder from image data?}
\item{Pharma wants biomarkers. How do we most effectively identify them?}
\item{There are lots of potential imaging biomarkers. Which are most (cost) effective?}
\end{itemize}
Pattern recognition provides a framework to compare data (or preprocessing strategy) to determine the most accurate approach.
\end{frame}
%\begin{frame}
%\frametitle{Inter-subject Variability}
%Why focus on anatomy?
%\begin{itemize}
%\item{Many medical applications involve understanding differences among individuals/populations.}
%\item{In image data, most of the differences we can see are anatomical in nature.}
%\item{Understanding growth and development requires us to look at growth and development (anatomy).}
%\end{itemize}
%\end{frame}
%\begin{frame}
%\frametitle{Biological variability is multivariate}
%\begin{center}
%\includegraphics[height=0.7\textwidth]{faces}
%\end{center}
%\end{frame}
\subsection{Binary Classification}
\begin{frame}
\frametitle{A generative classification approach}
\begin{center}
\includegraphics[height=.8\textheight]{fld}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Discriminative classification approaches}
\begin{center}
\includegraphics[height=.8\textheight]{linear_discrimination}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Bayesian classification}
\begin{center}
\includegraphics[height=.8\textheight]{logistic_regr}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Why Bayesian?}
\begin{itemize}
\item To deal with different priors.
\begin{itemize}
\item Consider a method with 90\% sensitivity and specificity.
\item Consider using this to screen for a disease afflicting 1\% of the population.
\item On average, out of 100 people there would be 10 wrongly assigned to the disease group.
\item A positive diagnosis suggests only about a 10\% chance of having the disease.
{\small
\begin{eqnarray*}
P(\text{Disease} | \text{Pred+}) & = \frac{P(\text{Pred+} | \text{Disease}) P(\text{Disease})}{P(\text{Pred+} | \text{Disease}) P(\text{Disease}) + P(\text{Pred+} | \text{Healthy}) P(\text{Healthy})}\cr
& = \frac{\text{Sensitivity} \times P(\text{Disease})}{\text{Sensitivity} \times P(\text{Disease}) + (1-Specificity) \times P(\text{Healthy})}
\end{eqnarray*}
\par
}
\end{itemize}
\item Better decision-making by accounting for utility functions.
\end{itemize}
\end{frame}
%\subsection{Kernel methods}
%\begin{frame}
%\frametitle{There is more than just classification}
%\begin{columns}[c]
%\column{.4\textwidth}
%\begin{itemize}
%\item Canonical Correlation Analysis
%\item Multi-Way, Multi-View Learning
%\item Other existing methods
%\item Methods not yet invented
%\end{itemize}
%\column{.6\textwidth}
%\includegraphics[width=\textwidth]{huopaniemi}
%{\itshape\tiny Huopaniemi, Ilkka, Tommi Suvitaival, Janne Nikkil\"{a}, Matej Ore\v{s}i\v{c}, and Samuel Kaski. ``Multi-way, multi-view learning.'' arXiv preprint arXiv:0912.3211 (2009).\par}
%\end{columns}
%\end{frame}
%\begin{frame}
%\frametitle{Kernel Matrices}
%Linear kernel matrices may computed from the raw features.
%\begin{eqnarray*}
%{\bf C} = {\bf V}{\bf V}^T
%\end{eqnarray*}
%A simple spatial feature selection may be considered as the following, where ${\bf A}$ is a diagonal matrix of ones and zeros:
%\begin{eqnarray*}
%{\bf C} = {\bf V}{\bf A}{\bf V}^T
%\end{eqnarray*}
%However, ${\bf A}$ may be more complicated, for example encoding spatial smoothing, high-pass filtering or any number of other things.
%\end{frame}
%\begin{frame}
%\frametitle{Inner Products}
%This gives us an alternative way of measuring distances between vectors in a linear way, where ${\bf A}$ is symmetric and positive definite.
%\begin{eqnarray*}
%d({\bf v}_1,{\bf v}_2) = \sqrt{({\bf v}_1 - {\bf v}_2)^T {\bf A} ({\bf v}_1 - {\bf v}_2)}
%\end{eqnarray*}
%Usually, the operation ${\bf A}{\bf v}^T$ is performed as a convolution. For example, when dealing with 2D data, we may convolve with the Laplacian operator.
%\begin{eqnarray*}
%{\nabla}^2 {\bf v} = {\bf v} \ast \begin{pmatrix} 0 & -1 & 0\cr -1 & 4 & -1\cr 0 & -1 & 0\end{pmatrix}
%\end{eqnarray*}
%Note that the actual form of ${\bf A}$ can vary, so we need to figure out what metric tensor is optimal.
%\end{frame}
\subsection{Curse of dimensionality}
\begin{frame}
\frametitle{Curse of dimensionality}
\begin{center}
{\Huge Large $p$, small $n$.\par}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Nearest-neighbour classification}
\begin{columns}[c]
\column{0.7\textwidth}
\includegraphics[width=\textwidth]{voronoi}
\column{0.3\textwidth}
\begin{itemize}
\item Not nice smooth separations.
\item Lots of sharp corners.
\item May be improved with \emph{K-nearest neighbours}.
\end{itemize}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Rule-based approaches}
\begin{columns}[c]
\column{0.7\textwidth}
\includegraphics[width=\textwidth]{rule_based}
\column{0.3\textwidth}
\begin{itemize}
\item Not nice smooth separations.
\item Lots of sharp corners.
\end{itemize}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Corners matter in high-dimensions}
\begin{columns}[c]
\column{0.4\textwidth}
\includegraphics[width=\textwidth]{circle}
\column{0.6\textwidth}
\includegraphics[width=\textwidth]{sphere}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Corners matter in high-dimensions}
\begin{columns}[c]
\column{0.2\textwidth}
\includegraphics[width=.8\textwidth]{circle}
\includegraphics[width=\textwidth]{sphere}
\column{0.8\textwidth}
\includegraphics[width=\textwidth]{corners}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Dimensionality $\ne$ number of voxels}
\begin{itemize}
\item Little evidence to suggest that most voxel-based feature selection methods help.
\begin{itemize}
\item Little or no increase in predictive accuracy.
\item Commonly perceived as being more ``interpretable''.
\end{itemize}
\item Prior knowledge derived from independent data is the most reliable way to improve accuracy.
\begin{itemize}
\item e.g. search the literature for clues about which regions to weight more heavily.
\end{itemize}
\end{itemize}
\vspace{1cm}
{\tiny Cuingnet, R\'emi, Emilie Gerardin, J\'er\^ome Tessieras, Guillaume Auzias, St\'ephane Leh\'ericy, Marie-Odile Habert, Marie Chupin, Habib Benali, and Olivier Colliot. ``Automatic classification of patients with Alzheimer's disease from structural MRI: a comparison of ten methods using the ADNI database.'' Neuroimage 56, no. 2 (2011): 766-781.\par}
{\tiny Chu, Carlton, Ai-Ling Hsu, Kun-Hsien Chou, Peter Bandettini, and ChingPo Lin. ``Does feature selection improve classification accuracy? Impact of sample size and feature selection on classification using anatomical magnetic resonance images.'' Neuroimage 60, no. 1 (2012): 59-70.\par}
{\tiny See winning strategies in \url{http://www.ebc.pitt.edu/PBAIC.html}\par}
\end{frame}
\begin{frame}
\frametitle{Linear versus Nonlinear methods}
\begin{columns}[c]
\column{0.7\textwidth}
\begin{itemize}
\item Linear methods are more interpretable.
\item Nonlinear methods usually increase dimensionality.
\item Better to preprocess to obtain features that behave more linearly.
\end{itemize}
\includegraphics[width=\textwidth]{bmi}
\column{0.3\textwidth}
\includegraphics[width=\textwidth]{log_transformed}
\end{columns}
\end{frame}
\section{Geometric Variability}
\subsection{Manifolds}
\begin{frame}
\frametitle{Transformed images fall on manifolds}
Rotating an image leads to points on a 1D manifold.\par
\includegraphics[width=\textwidth]{manifold_rotation}
Rigid-body motion leads to a 6-dimensional manifold (not shown).
\end{frame}
\begin{frame}
\frametitle{Local linearisation through smoothing}
\begin{columns}[c]
\column{0.7\textwidth}
\includegraphics[width=\textwidth]{manifold_unsmoothed}\par
\includegraphics[width=\textwidth]{manifold_smoothed}
\column{0.3\textwidth}
Spatial smoothing can make the manifolds more linear with respect to small misregistrations.
Some information is inevitably lost.
\end{columns}
\end{frame}
\subsection{Principal Components}
\begin{frame}
\frametitle{One mode of geometric variability}
\begin{columns}[c]
\column{0.5\textwidth}
Simulated images\par
\includegraphics[height=0.9\textwidth]{circles}
\column{0.5\textwidth}
Principal components\par
\includegraphics[height=0.9\textwidth]{circles_pca}
\end{columns}
A suitable model would reduce these data to a single dimension.
\end{frame}
\begin{frame}
\frametitle{Two modes of geometric variability}
\begin{columns}[c]
\column{0.5\textwidth}
Simulated images\par
\includegraphics[height=0.9\textwidth]{things}
\column{0.5\textwidth}
Principal components\par
\includegraphics[height=0.9\textwidth]{things_pca}
\end{columns}
A suitable model would reduce these data to two dimensions.
\end{frame}
\section{Similarity Measures}
\subsection{Distances}
\begin{frame}
\frametitle{Similarity Measures}
\begin{itemize}
\item Many methods are based on similarity measures.
\item A common similarity measure is the dot product.
\begin{eqnarray*}
\text{Similarity: } k({\bf x},{\bf y}) = \sum_k x_k y_k
\end{eqnarray*}
\item Nonlinear methods are often based on distances.
\begin{eqnarray*}
\text{Distance: } & d({\bf x},{\bf y}) = & \sqrt{\sum_k (x_k-y_k)^2}\cr
\text{Similarity: } & k({\bf x},{\bf y}) = & exp(-\lambda d({\bf x},{\bf y})^2)
\end{eqnarray*}
\item How do we best measure distances between brain images?
\end{itemize}
\end{frame}
%\begin{frame}
%\frametitle{No Free Ducklings}
%{\bf No Free Lunch theorem} says that learning is impossible without prior knowledge (\url{http://en.wikipedia.org/wiki/No\_free\_lunch\_in\_search\_and\_optimization}).
%{\bf Ugly Duckling theorem} says that things are all equivalently similar to each other without prior knowledge (\url{http://en.wikipedia.org/wiki/Ugly\_duckling\_theorem}).
%\vspace{1cm}
%What prior knowledge do we have about the variability among people that can be measured using MRI?
%How do we use this knowledge?
%\end{frame}
\begin{frame}
\frametitle{Image Registration}
\begin{columns}[c]
\column{0.7\textwidth}
\begin{itemize}
\item Image registration measures distances between images.
\item Often involves minimising the sum of two terms:
\begin{itemize}
\item Distance between the image intensities.
\item Distance of the deformation from zero.
\end{itemize}
\item The sum of these terms gives the distance.
\end{itemize}
\column{0.3\textwidth}
\includegraphics[width=\textwidth]{shoot2d}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Different ways of measuring distances}
\begin{columns}[c]
\column{.3\textwidth}
\includegraphics[width=\textwidth]{mumford}
\column{.7\textwidth}
\includegraphics[width=\textwidth]{mumford_fig}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Different ways of measuring distances}
\begin{columns}[c]
\column{.2\textwidth}
\begin{center}
Two simulated images\par
\includegraphics[width=\textwidth]{figure2Di}
\end{center}
\column{.8\textwidth}
\includegraphics[width=\textwidth]{figure2Dii}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Metrics}
Distances need to satisfy the properties of a \emph{metric}:
\begin{enumerate}
\item $d({\bf x}, {\bf y}) \ge 0$ (non-negativity)
\item $d({\bf x}, {\bf y}) = 0$ if and only if ${\bf x} = {\bf y}$ (identity of indiscernibles)
\item $d({\bf x}, {\bf y}) = d({\bf y}, {\bf x})$ (symmetry)
\item $d({\bf x}, {\bf z}) \le d({\bf x}, {\bf y}) + d({\bf y}, {\bf z})$ (triangle inequality).
\end{enumerate}
Satisfying (3) requires inverse-consistent image registration.
Satisfying (4) requires a specific class of image registration models.
\end{frame}
\begin{frame}
\frametitle{Non-Euclidean geometry}
\begin{columns}[c]
\column{0.5\textwidth}
\begin{itemize}
\item Distances are not always measured along a straight line.
\item ``\emph{Shapes are the ultimate non-linear sort of thing}''
\end{itemize}
\column{0.5\textwidth}
\includegraphics[width=\textwidth]{Globe}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{Linear approximations to nonlinear problems}
%\begin{columns}[c]
%\column{0.2\textwidth}
%Dealing with non-Euclidean geometry.
%Linear approximation around the average.
%\begin{center}
%\column{0.8\textwidth}
\begin{center}
\includegraphics[width=1.2\textwidth]{spheres}
\end{center}
%\end{columns}
\end{frame}
%\begin{frame}
%\frametitle{D'Arcy Thompson's Generative Model}
%\begin{quote}
%``...diverse and dissimilar fishes can be referred as a whole to identical functions of very different co-ordinate systems...''
%\end{quote}
%\begin{center}
%\includegraphics[height=0.4\textheight]{OGAF}
%\includegraphics[height=0.4\textheight]{fish}
%\end{center}
%We can compute relative shapes using image registration.
%\end{frame}
\subsection{Examples of features}
\begin{frame}
\frametitle{Example Images}
Some example (non-brain) images.
\begin{center}
\includegraphics[width=0.9\textwidth]{original}
\end{center}
\end{frame}
%\begin{frame}
%\frametitle{Samples from a Linear Generative Model}
%Did the images come from a model like this?
%\begin{center}
%\includegraphics[width=0.9\textwidth]{simulated_lin}
%\end{center}
%\end{frame}
%\begin{frame}
%\frametitle{Samples from a Geometric Generative Model}
%Or one like this?
%\begin{center}
%\includegraphics[width=0.9\textwidth]{simulated}
%\end{center}
%\end{frame}
\begin{frame}
\frametitle{Registered Images}
We could register the images to their average shape...
\begin{center}
\includegraphics[width=0.9\textwidth]{warped}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Deformations}
...and study the deformations...
\begin{center}
\includegraphics[width=0.9\textwidth]{deformations}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Jacobian Determinants}
...or the relative volumes...
\begin{center}
\includegraphics[width=0.9\textwidth]{jacobians}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Scalar Momentum}
... or ``scalar momentum'' (Singh et al, MICCAI 2010).
\begin{center}
\includegraphics[width=0.9\textwidth]{alpha}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Reconstructed Images}
Reconstructions from template and scalar momenta.
\begin{center}
\includegraphics[width=0.9\textwidth]{reconstructed}
\end{center}
\end{frame}
%\begin{frame}
%\frametitle{Evolution}
%\begin{center}
%\includegraphics[width=.8\textwidth]{evolution}\par
%\begin{tiny}
%Singh, Fletcher, Preston, Ha, King, Marron, Wiener \& Joshi (2010). \emph{Multivariate Statistical Analysis of Deformation Momenta Relating Anatomical Shape to Neuropsychological Measures}. T. Jiang et al. (Eds.): MICCAI 2010, Part III, LNCS 6363, pp. 529--537, 2010.\par
%\end{tiny}
%\end{center}
%\end{frame}
\section{Real data}
\subsection{Data}
\begin{frame}
\frametitle{Real data}
\begin{columns}[c]
\column{0.5\textwidth}
Used 550 T1w brain MRI from IXI ({\bf I}nformation e{\bf X}traction from {\bf I}mages) dataset.
\url{http://www.brain-development.org/}
Data from three different hospitals in London:
\begin{itemize}
\item{Hammersmith Hospital using a Philips 3T system}
\item{Guy's Hospital using a Philips 1.5T system}
\item{Institute of Psychiatry using a GE 1.5T system}
\end{itemize}
\column{0.5\textwidth}
\includegraphics[width=0.9\textwidth]{orig_ixi}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Grey and White Matter}
\begin{columns}[c]
\column{0.26\textwidth}
Segmented into GM and WM.
Approximately aligned via rigid-body.
\column{0.75\textwidth}
\includegraphics[width=0.5\textwidth]{gm_ixi}
\includegraphics[width=0.5\textwidth]{wm_ixi}
\end{columns}
\begin{tiny}
Ashburner, J \& Friston, KJ. \emph{Unified segmentation}. NeuroImage 26(3):839--851 (2005).
\end{tiny}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Diffeomorphic Alignment}
All GM and WM were diffeomorphically aligned to their common average-shaped template.
\begin{center}
\includegraphics[width=.8\textwidth]{template}
\end{center}
\begin{tiny}
Ashburner, J \& Friston, KJ. \emph{Diffeomorphic registration using geodesic shooting and Gauss-Newton optimisation}. NeuroImage 55(3):954--967 (2011).
Ashburner, J \& Friston, KJ. \emph{Computing average shaped tissue probability templates}. NeuroImage 45(2):333--341 (2009).
\end{tiny}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Features}
\begin{frame}
\frametitle{Volumetric Features}
\begin{columns}[c]
\column{0.33\textwidth}
A number of features were used for pattern recognition.
Firstly, two features relating to relative volumes.
Initial velocity divergence is similar to logarithms of Jacobian determinants.
\column{0.33\textwidth}
Jacobian Determinants
\includegraphics[width=1\textwidth]{jac_ixi}
\column{0.33\textwidth}
Initial Velocity Divergence
\includegraphics[width=1\textwidth]{div_ixi}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Grey Matter Features}
\begin{columns}[c]
\column{0.33\textwidth}
Rigidly Registered GM
\includegraphics[width=1\textwidth]{gm_ixi}
\column{0.33\textwidth}
Nonlinearly Registered GM
\includegraphics[width=1\textwidth]{wc1_ixi}
\column{0.33\textwidth}
Registered and Jacobian Scaled GM
\includegraphics[width=1\textwidth]{mwc1_ixi}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{``Scalar Momentum'' Features}
\begin{columns}[c]
\column{0.33\textwidth}
``Scalar momentum'' actually has two components because GM was matched with GM and WM was matched with WM.
\column{0.33\textwidth}
First Momentum Component
\includegraphics[width=1\textwidth]{resids1_ixi}
\column{0.33\textwidth}
Second Momentum Component
\includegraphics[width=1\textwidth]{resids2_ixi}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Results}
\begin{frame}
\frametitle{Age Regression}
Linear Gaussian Process Regression to predict subject ages.
\begin{columns}[c]
\column{0.5\textwidth}
\includegraphics[width=1\textwidth]{age_loglikelihood}
\column{0.5\textwidth}
\includegraphics[width=1\textwidth]{age_rms}
\end{columns}
\begin{tiny}
Rasmussen, CE \& Williams, CKI. \emph{Gaussian processes for machine learning}. Springer (2006).
\end{tiny}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Sex Classification}
Linear Gaussian Process Classification (EP) to predict sexes.
\begin{columns}[c]
\column{0.5\textwidth}
\includegraphics[width=1\textwidth]{sex_loglikelihood}
\column{0.5\textwidth}
\includegraphics[width=1\textwidth]{sex_auc_GP}
\end{columns}
\begin{tiny}
Rasmussen, CE \& Williams, CKI. \emph{Gaussian processes for machine learning}. Springer (2006).
\end{tiny}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}
%\frametitle{Sex Classification}
%Linear SVM versus Gaussian Process Classification (EP).
%\begin{columns}[c]
%\column{0.5\textwidth}
%\includegraphics[width=1\textwidth]{sex_SVM_v_GP_acc}
%\column{0.5\textwidth}
%\includegraphics[width=1\textwidth]{sex_SVM_v_GP_AUC}
%\end{columns}
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Predictive Accuracies}
\begin{columns}[c]
\column{0.5\textwidth}
Age
\includegraphics[width=1\textwidth]{age_predictions}
\column{0.5\textwidth}
Sex
\includegraphics[width=1\textwidth]{sex_roc}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Conclusions}
\begin{itemize}
\item{Scalar momentum (with about 10mm smoothing) appears to be a useful feature set.}
\item{Jacobian-scaled warped GM is surprisingly poor.}
%\item{SVC slightly more accurate than GP (but we knew that already).}
\item{Amount of spatial smoothing makes a big difference.}
\item{Further dependencies on the details of the registration still need exploring.}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}
%\frametitle{Additional references}
%\begin{itemize}
%\item{Ashburner, J \& Kl\"oppel, K. \emph{Multivariate models of inter-subject anatomical variability}. NeuroImage 56(2):422--439 (2011).}
%\item{Ashburner, J \& Friston, KJ. \emph{Diffeomorphic registration using geodesic shooting and Gauss-Newton optimisation}. NeuroImage 55(3):954--967 (2011).}
%\end{itemize}
%\end{frame}
\begin{frame}
\begin{center}
\includegraphics[width=.8\textwidth]{hyper_male}
\end{center}
\end{frame}
\begin{frame}
\begin{center}
\includegraphics[width=.8\textwidth]{avgT1}
\end{center}
\end{frame}
\begin{frame}
\begin{center}
\includegraphics[width=.8\textwidth]{hyper_female}
\end{center}
\end{frame}
\end{document}