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gmmlib.c
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gmmlib.c
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/*
* Copyright (c) 2020 Wellcome Centre for Human Neuroimaging
* John Ashburner, Mikael Brudfors & Yael Balbastre
* $Id: gmmlib.c 8065 2021-02-15 12:42:04Z john $
*/
#include "spm_mex.h"
#include<math.h>
#include<stdio.h>
#include<stdlib.h>
#define EXP(x) fastexp(x)
typedef struct
{
double *s2;
double *s1;
double *s0;
} SStype;
typedef struct
{
mwSize P;
double *mu;
double *b;
double *W;
double *nu;
double *gam;
double *conN;
double *conT;
} GMMtype;
typedef struct
{
mwSize Po;
mwSize Pm;
unsigned char *observed;
double *Wt;
double *L_mm;
double *m_m;
double *m_o;
unsigned char *obs;
} MissInfType;
static const double pi = 3.1415926535897931;
#define MaxChan ((mwSize)50) /* largest integer valued float is 2^52 */
#define Undefined ((mwSize)0xFFFFFFFFFFFFF)
/* A (hopefully) faster approximation to exp.
* Note that some precision is lost for values
* of x further from integers. */
static double fastexp(double x)
{
double r, rr;
mwSignedIndex i;
static double lkp_mem[256], *exp_lkp = lkp_mem+128;
/* exp(i+r) = exp(i)*exp(r), where:
* exp(i) (where i is an integer) is from the lookup table;
* exp(r) (residual) is from a generalised continued fraction
* https://en.wikipedia.org/wiki/Exponential_function#Continued_fractions_for_ex
*
* Should not encounter values more extreme than -128 or 127,
* particularly as the upper limit of x will be 0 and values
* of x below log(eps)=-36.04 should be numerically equivalent.*/
i = (mwSignedIndex)rint(x);
if (i<-128) i = -128;
if (i> 127) i = 127;
if (exp_lkp[i]==0.0) exp_lkp[i] = exp((double)i);
r = x - (double)i;
rr = r*r;
/* return exp_lkp[i] * (1.0+2.0*r/(2.0-r+rr/(6.0+rr/(10.0+rr/14.0))));
* return exp_lkp[i] * (1.0+2.0*r/(2.0-r+rr/(6.0+rr/(10.0)))); */
return exp_lkp[i] * (1.0+2.0*r/(2.0-r+rr/6.0));
}
static mwSize is_observed(mwSize code, mwSize i)
{
return (code>>i) & (mwSize)1;
}
static mwSize num_observed(mwSize code, mwSize P)
{
mwSize i, Po;
for(i=0, Po=0; i<P; i++)
Po += is_observed(code,i);
return Po;
}
/* Allocate memory for a data structure, and assign pointers from
* the structure to the appropriate parts of s0_ptr, s1_ptr and s2_ptr.
* P - number of image channels
* K - number of Gaussians
* s0_ptr, s1_ptr & s2_ptr - memory allocated for storing sufficient
* statistics.
* Returns the allocated data structure, with pointers assigned.
*/
static /*@null@*/ SStype *suffstat_pointers(mwSize P, mwSize K, double *s0_ptr, double *s1_ptr, double *s2_ptr)
{
SStype /*@NULL@*/ *suffstat;
suffstat = (SStype *)calloc((size_t)1<<(size_t)P,sizeof(SStype));
if (suffstat != NULL)
{
mwSize code,o0=0,o1=0,o2=0;
for(code=0; code<((mwSize)1<<P); code++)
{
mwSize Po;
Po = num_observed(code,P);
suffstat[code].s0 = &(s0_ptr[o0]);
suffstat[code].s1 = &(s1_ptr[o1]);
suffstat[code].s2 = &(s2_ptr[o2]);
o0 += K;
o1 += K*Po;
o2 += K*Po*Po;
}
}
return suffstat;
}
/* Read vector of mean and variance.
* N1 - number of voxels each volume of mf and mv
* P - number of volumes in mf and mv
* mf - E[f], size N1 x P
* vf - diag(Var[f]), size N1 x P
* x - extracted means, length P
* v - extracted variances, length P
* Returns a code indicating which volumes have a finite value (ie data not missing)
*/
static mwSize get_vox(mwSize N1, mwSize P, float mf[], float vf[], /*@out@*/ double x[], /*@out@*/ double v[])
{
mwSize j, j1, o, code;
for(j=0, j1=0, o=0, code=0; j<P; j++, o+=N1)
{
double tmp = (double)mf[o];
if (isfinite(tmp)!=0)
{
x[j1] = tmp;
v[j1] = vf[o];
code |= (mwSize)1<<j;
j1++;
}
}
return code;
}
/* log-sum-exp function: log(sum(exp(q)))
* K - length of q
*
static double lse(mwSize K, double q[])
{
mwSize k;
double mx, s;
for(k=1, mx=q[0]; k<K; k++)
{
if (q[k]>mx) mx = q[k];
}
for(k=0, s=0.0; k<K; k++)
s += EXP(q[k]-mx);
return log(s)+mx;
}
*/
/* softmax function: exp(q)/sum(exp(q))
* K - length of q or p
* q - input data
* p - output data
* Input and output could be the same
* Returns lse(q)
*/
static double softmax1(mwSize K, double q[], /*@out@*/ double p[])
{
mwSize k;
double mx, s;
for(k=1, mx=q[0]; k<K; k++)
if (q[k]>mx) mx = q[k];
for(k=0, s=0.0; k<K; k++)
s += (p[k] = EXP(q[k]-mx));
for(k=0; k<K; k++)
p[k] /= s;
return log(s) + mx;
}
/* softmax function: exp(q)/(sum(exp(q)) + 1)
* K - length of q or p
* q - input data
* p - output data
* Input and output could be the same
* Returns lse([q 0])
*/
static double /*@unused@*/ softmax(mwSize K, double q[], /*@out@*/ double p[])
{
mwSize k;
double mx, s;
for(k=0, mx=0; k<K; k++)
if (q[k]>mx) mx = q[k];
for(k=0, s=EXP(-mx); k<K; k++)
s += (p[k] = EXP(q[k]-mx));
for(k=0; k<K; k++)
p[k] /= s;
return log(s) + mx;
}
/* \Del^2 = (\mu-x)^T W (\mu-x) + trace(W diag(v))
* P - dimensions of mu, W, x & v
* mu - mean, size P x 1
* W - covariance, size P x P
* x - expectation of data, size P x 1
* v - variance of data, size P x 1 (ie diagonal of covariance)
*/
static double del2(mwSize P, double mu[], double W[], double x[], double v[])
{
mwSize j,i;
double d=0.0, r, *wj;
for(j=0,wj=W; j<P; j++, wj+=P)
{
r = x[j]-mu[j];
d += wj[j]*(r*r+v[j]);
for(i=j+1; i<P; i++)
d += 2.0*r*wj[i]*(x[i]-mu[i]);
}
return d;
}
/* psi / digamma function
* From http://web.science.mq.edu.au/~mjohnson/code/digamma.c
* by Mark Johnson, Professor of Language Sciences (CORE),
* Department of Computing, Faculty of Science,
* Macquarie University, Sydney, Australia.
*/
static double psi(double z)
{
double f = 0, r, r2, r4;
/* psi(z) = psi(z+1) - 1/z */
for (f=0; z<7.0; z++) f -= 1.0/z;
z -= 1.0/2.0;
r = 1.0/z;
r2 = r*r;
r4 = r2*r2;
f += log(z)+(1.0/24.0)*r2-(7.0/960.0)*r4+(31.0/8064.0)*r4*r2-(127.0/30720.0)*r4*r4;
return f;
}
/* Add log-likelihoods from Normal distributions, accounting for uncertainty of
* the parameters
*
* K - number of Gaussians.
* gmm - data structure for Gaussian distributions with different
* combinations of missing data.
* code - Indicates which data are missing.
* x - E[x]
* v - Var[x]
* p - on input: a vector of logs of prior probabilities
* on output: logs of likelihoods are added to the log priors
*/
static void Nloglikelihoods(mwSize K, GMMtype gmm[], mwSize code, double x[], double v[], double p[])
{
mwSize P, k;
double *mu, *b, *W, *nu, *gam, *con;
P = gmm[code].P;
mu = gmm[code].mu;
b = gmm[code].b;
W = gmm[code].W;
nu = gmm[code].nu;
gam = gmm[code].gam;
con = gmm[code].conN;
for(k=0; k<K; k++, W+=P*P, mu+=P)
p[k] += con[k] - 0.5*nu[k]*del2(P, mu, W, x, v);
}
/* Add log-likelihoods from Student's T distributions, accounting
* for uncertainty of the parameters
*
* K - number of Gaussians.
* gmm - data structure for Gaussian distributions with different
* combinations of missing data.
* code - Indicates which data are missing.
* x - E[x]
* v - Var[x]
* p - on input: a vector of logs of prior probabilities
* on output: logs of likelihoods are added to the log priors
*/
static void Tloglikelihoods(mwSize K, GMMtype gmm[], mwSize code, double x[], double v[], double p[])
{
mwSize P, k;
double *mu, *b, *W, *nu, *con;
/*
Compute other log-likelihoods from a mixture of Student's t distributions.
See Eqns. 10.78-10.82 & B.68-B.72 in Bishop's PRML book.
In practice, it only improves probabilities by a tiny amount.
ln St(x|mu,Lam,tau)
gammaln((tau + P)/2.0) - gammaln(tau/2.0) + ld1/2 - (P/2.0)*log(tau*pi) - ((tau+P)/2)*log(1 + del2/tau)
where:
Lam = W*(nu+1-P)*beta/(1+beta)
tau = nu+1-P
del2 = (x-mu)'*Lam*(x-mu)
ld1 = log(det(Lam)) = log(det(W)) + P*log((nu+1-P)*beta/(1+beta))
This gives:
gammaln((nu+1)/2.0) - gammaln((nu+1-P)/2.0) + (ld + P*log((nu+1-P)*beta/(1+beta)))/2 - ...
(P/2.0)*log((nu+1-P)*pi) - ...
((nu+1)/2)*log(1 + beta/(beta+1)*((x-mu)'*W*(x-mu) + sum(diag(W).*vf)))
*/
P = gmm[code].P;
mu = gmm[code].mu;
b = gmm[code].b;
W = gmm[code].W;
nu = gmm[code].nu;
con = gmm[code].conT;
for(k=0; k<K; k++, W+=P*P, mu+=P)
p[k] += con[k] - 0.5*(nu[k]+1.0)*log(1.0 + b[k]/(b[k]+1.0)*del2(P, mu, W, x, v));
}
/*
* lnP = E[\ln A]
* E[\ln {\bf a}_k] = \psi({\bf a}_k) - \psi(\sum_i a_{ik})
*
*/
static void Dloglikelihoods(mwSize i, mwSize K, double *lnP, double *p)
{
mwSize k;
for(k=0; k<K; k++)
p[k] += lnP[i+256*k];
}
/* Construct a vector of log tissue priors for a voxel
*
* N1 - Number of voxels in each 3D volume.
* lp - Pointer to first voxel in the volumes.
* K - Number of Gaussian distributions
* lkp - Lookup table indicating which Gaussian is
* associated with which tissue class.
* p - Output vector of log tissue priors.
*/
static int logpriors(mwSize N1, float *lp, mwSize K, mwSize *lkp, /*@out@*/ double *p)
{
mwSize k;
/* double l; */
for(k=0; k<K; k++)
{
double lpk;
lpk = (double)lp[N1*lkp[k]];
if (isfinite(lpk)==0)
return 0;
p[k] = lpk;
}
/*
l = lse(K,p);
for(k=0; k<K; k++)
p[k] -= l;
*/
return 1;
}
/* Cholesky decomposition
* n - dimension of matrix a
* a - an n \times n matrix
* p - an n \times 1 vector
*
* A triangle of the input matrix is partially overwritten
* by the output. Diagonal elements are stored in p.
*/
static void choldc(mwSize n, double a[], /*@out@*/ double p[])
{
mwSize i, j;
mwSignedIndex k;
double sm, sm0;
sm0 = 1e-40;
for(i=0; i<n; i++) sm0 = sm0 + a[i*n+i];
sm0 *= 1e-7;
sm0 *= sm0;
for(i=0; i<n; i++)
{
for(j=i; j<n; j++)
{
sm = a[i*n+j];
for(k=(mwSignedIndex)i-1; k>=0; k--)
sm -= a[i*n+k] * a[j*n+k];
if(i==j)
{
if(sm <= sm0) sm = sm0;
p[i] = sqrt(sm);
}
else
a[j*n+i] = sm / p[i];
}
}
}
/* Solve a least squares problem with the results from a
* Cholesky decomposition
*
* n - Dimension of matrix and data.
* a & p - Cholesky decomposed matrix.
* b - Vector of input data.
* x - Vector or outputs.
*/
static void cholls(mwSize n, const double a[], const double p[],
const double b[], /*@out@*/ double x[])
{
mwSignedIndex i, k;
double sm;
for(i=0; i<(mwSignedIndex)n; i++)
{
sm = b[i];
for(k=i-1; k>=0; k--)
sm -= a[i*n+k]*x[k];
x[i] = sm/p[i];
}
for(i=(mwSignedIndex)n-1; i>=0; i--)
{
sm = x[i];
for(k=i+1; k<(mwSignedIndex)n; k++)
sm -= a[k*n+i]*x[k];
x[i] = sm/p[i];
}
}
/* n! */
static mwSize factorial(mwSize n)
{
static mwSize products[21];
if (products[0]==0)
{
mwSize i;
products[0] = 1;
for(i=1; i<21; i++)
products[i] = products[i-1]*i;
}
return products[n];
}
/* Compute space required for storing sufficient statistics.
*
* P - Number of image volumes.
* K - Number of tissue classes.
* *m0, *m1 & *m2 - Space needed for the zeroeth,
* first and second moments.
*/
void space_needed(mwSize P, mwSize K, mwSize *m0, mwSize *m1, mwSize *m2)
{
mwSize m;
for(m=0, *m0=0, *m1=0, *m2=0; m<=P; m++)
{
mwSize nel;
nel = K*factorial(P)/(factorial(m)*factorial(P - m));
*m0 += nel;
*m1 += nel*m;
*m2 += nel*m*m;
}
}
/* Allocate memory for a data structure for representing
* GMMs with missing data
*
* P - Number of images/channels.
* K - Number of Gaussians
*/
static /*@null@*/ GMMtype *allocate_gmm(mwSize P, mwSize K)
{
mwSize o, code, i, n0=0,n1=0,n2=0;
double *buf;
unsigned char *bytes;
GMMtype /*@NULL@*/ *gmm;
space_needed(P, K, &n0, &n1, &n2);
o = ((mwSize)1<<P)*sizeof(GMMtype);
bytes = calloc((size_t)o+(size_t)(n0*(mwSize)5+n1+n2)*sizeof(double),1);
gmm = (GMMtype *)bytes;
if (gmm!=NULL)
{
buf = (double *)(bytes + o);
o = 0;
for(code=0; code<((mwSize)1<<P); code++)
{
mwSize nel = 0;
for(i=0; i<code; i++) nel += is_observed(code,i);
gmm[code].P = nel;
gmm[code].mu = buf+o; o += K*nel;
gmm[code].b = buf+o; o += K;
gmm[code].W = buf+o; o += K*nel*nel;
gmm[code].nu = buf+o; o += K;
gmm[code].gam = buf+o; o += K;
gmm[code].conN = buf+o; o += K;
gmm[code].conT = buf+o; o += K;
}
}
return gmm;
}
/* Invert a matrix
*
* P - Matrix dimensions
* W - Matrix (input, P \times P)
* S - Matrix inverse (output, P \times P)
* T - Scratch space (P*(P+1))
*/
static double invert(mwSize P, double *W /* P*P */, double *S /* P*P */, double *T /* P*(P+1) */)
{
mwSize i, j, PP=P*P;
double ld = 0.0, *p;
for(i=0; i<PP; i++) T[i] = W[i];
p = T+PP;
choldc(P,T,p);
for(j=0; j<P; j++)
{
ld += log(p[j]);
/* Column of identity matrix */
for(i=0; i<P; i++) S[i+j*P]=0.0;
S[j+j*P] = 1.0;
cholls(P, T, p, S+j*P, S+j*P);
}
return -2.0*ld;
}
static /*@null@*/ MissInfType *allocate_missinf(mwSize P, mwSize K)
{
mwSize o, code, m, mem;
unsigned char *bytes;
MissInfType /*@NULL@*/ *missinf;
for(m=0, mem=0; m<=P; m++)
{
mwSize nel;
nel = factorial(P)/(factorial(m)*factorial(P - m));
mem += nel*K*(P-m)*m*sizeof(double); /* L_mo */
mem += nel*K*P*sizeof(double);
mem += nel*K*m*m*sizeof(double);
mem += nel*((P+sizeof(double)-1)/sizeof(double))*sizeof(double);
}
o = ((mwSize)1<<P)*sizeof(MissInfType);
bytes = calloc((size_t)(o + mem), (size_t)1);
missinf = (MissInfType *)bytes;
if (missinf!=NULL)
{
double *buf;
buf = (double *)(bytes + o);
o = 0;
for(code=0; code<((mwSize)1<<P); code++)
{
mwSize Po = num_observed(code,P);
mwSize Pm = P-Po;
missinf[code].Po = Po;
missinf[code].Pm = Pm;
missinf[code].Wt = buf+o; o += K*Po*Pm;
missinf[code].L_mm = buf+o; o += K*Pm*Pm;
missinf[code].m_m = buf+o; o += K*Pm;
missinf[code].m_o = buf+o; o += K*Po;
missinf[code].obs = (unsigned char *)(buf+o);
o += (P+sizeof(double)-1)/sizeof(double);
}
}
return missinf;
}
static void dispmat(double *f, mwSize d1, mwSize d2, mwSize d3)
{
mwSize i, j, k;
for(k=0; k<d3; k++)
{
for(i=0; i<d1; i++)
{
for(j=0; j<d2; j++)
printf(" %g", f[i+d1*(d2*k+j)]);
printf("\n");
}
printf("--\n");
}
}
static MissInfType /*@NULL@*/ *prepare_missinf(mwSize P, mwSize K, double *L /* P*P */, double *m)
{
MissInfType *missinf;
mwSize code;
missinf = allocate_missinf(P, K);
if (missinf==NULL) return missinf;
for(code=0; code<(mwSize)1<<P; code++)
{
mwSize i, j, im, io, jm, jo, k;
mwSize Po, Pm;
double p[64], *Lmm, *Lmo, *mm, *mo;
unsigned char *obs;
Po = missinf[code].Po;
Pm = missinf[code].Pm;
Lmm = missinf[code].L_mm; /* partially decomposed on output */
mm = missinf[code].m_m;
mo = missinf[code].m_o;
Lmo = missinf[code].Wt;
obs = missinf[code].obs;
for(j=0, jm=0, jo=0; j<P; j++)
{
if (is_observed(code,j)==0)
{/* missing */
obs[j] = (unsigned char)0;
jm ++;
}
else
{/* observed */
obs[j] = (unsigned char)1;
jo ++;
}
}
for(k=0; k<K; k++)
{
for(i=0, im=0, io=0; i<P; i++)
{
if (is_observed(code,i)==0)
{
for(j=0, jo=0,jm=0; j<P; j++)
{
if (is_observed(code,j)==0)
Lmm[im+Pm*(jm++)] = L[i+P*(j+P*k)];
else
Lmo[im+Pm*(jo++)] = L[i+P*(j+P*k)];
}
mm[im++] = m[i+P*k];
}
else
mo[io++] = m[i+P*k];
}
/* Wt = Lmm\Lmo */
choldc(Pm,Lmm,p);
for(jo=0; jo<Po; jo++)
cholls(Pm,Lmm, p, Lmo+Pm*jo, Lmo+Pm*jo);
/* Update pointers to k'th vector/matrix */
mm += Pm;
mo += Po;
Lmm += Pm*Pm;
Lmo += Pm*Po;
}
}
/* For debugging
for(code=0; code<(mwSize)1<<P; code++)
{
mwSize Po, Pm;
Po = missinf[code].Po;
Pm = missinf[code].Pm;
printf("\ncode=%d Po=%d Pm=%d\n", code, Po, Pm);
printf("\nLmm\n"); dispmat(missinf[code].L_mm,Pm,Pm,K);
printf("\nmm\n"); dispmat(missinf[code].m_m,1,Pm,K);
printf("\nmo\n"); dispmat(missinf[code].m_o,1,Po,K);
printf("\nLmo\n"); dispmat(missinf[code].Wt,Pm,Po,K);
}
*/
return missinf;
}
/* Construct a data structure for storing a variational Gaussian
* mixture model for handling missing data.
*
* P - Dimension
* K - Number of Gaussians
* mu,b,W,nu - Variational Bayesian GMM parameters
* mu - P \times K
* b - 1 \times K
* W - P \times P \times K
* nu - 1 \times K
* gam - Mixing proportions (1 \times K).
*
* The function returns the data structure.
*/
static /*@null@*/ GMMtype *sub_gmm(mwSize P, mwSize K, double *mu, double *b, double *W, double *nu, double *gam)
{
const double log2pi = log(2*pi), log2 = log(2.0);
double *S, *Si;
GMMtype *gmm;
mwSize k, code, PP = P*P;
if ((gmm = allocate_gmm(P,K)) == NULL) return gmm;
if ((S = (double *)calloc((size_t)(P*((mwSize)3*P+(mwSize)1)),sizeof(double))) == NULL)
{
(void)free((void *)gmm);
return NULL;
}
Si = S + PP;
for(k=0; k<K; k++)
{
double lgam = log(gam[k]);
(void)invert(P,W+PP*k,S,S+PP);
for(code=0; code<(mwSize)1<<P; code++)
{
mwSize j,j1, Po;
double ld, ld1, eld;
Po = gmm[code].P;
gmm[code].nu[k] = nu[k] - (double)(P-Po);
gmm[code].b[k] = b[k];
gmm[code].gam[k] = lgam;
for(j=0, j1=0; j<P; j++)
{
if ((((mwSize)1<<j) & code) != 0)
{
mwSize i, i1;
gmm[code].mu[j1+Po*k] = mu[j+P*k];
for(i=0, i1=0; i<P; i++)
{
if ((((mwSize)1<<i) & code) != 0)
{
Si[i1+Po*j1] = S[i+P*j];
i1++;
}
}
j1++;
}
}
ld = invert(Po,Si,gmm[code].W+k*Po*Po,Si+Po*Po);
/* Constant term for VB mixture of Gaussians
E[ln N(x | m, L^{-1})] w.r.t. Gaussian-Wishart */
for(j=0,eld=0.0; j<Po; j++) eld += psi((gmm[code].nu[k]-(double)j)*0.5);
eld += Po*log2 + ld;
gmm[code].conN[k] = 0.5*(eld - (double)Po*(log2pi+1.0/gmm[code].b[k])) + lgam;
/* Constant term for VB mixture of T distributions */
ld1 = ld + (double)Po*log((gmm[code].nu[k]+1.0-(double)Po)*b[k]/(b[k]+1.0));
gmm[code].conT[k] = lgamma(0.5*(gmm[code].nu[k]+1.0)) - lgamma(0.5*(gmm[code].nu[k]+1.0-(double)Po)) +
0.5*ld1 - 0.5*Po*log((gmm[code].nu[k]+1.0-(double)Po)*pi) + lgam;
}
}
(void)free((void *)S);
return gmm;
}
/* Compute sufficient statistics in a way that handles missing data
*
* nf - Vector of dimensions (n_x, n_y, n_z, P).
* mf - E[f], dimensions nf.
* vf - Var[f], dimensions nf.
* gmm - Gaussian mixture model data structure.
* nm - Dimensions of log tissue priors (4 elements).
* skip - Sampling density of tissue priors (in x, y and z).
* lkp - Lookup table relating Gaussians to tissue classes.
* lp - Log tissue priors
* suffstat - Data stucture to hold resulting sufficient statistics.
*/
static double suffstats_missing(mwSize nf[], float mf[], float vf[], unsigned char label[],
mwSize K, GMMtype gmm[], double lnP[],
mwSize nm[], mwSize skip[], mwSize lkp[], float lp[],
SStype suffstat[], double H[])
{
mwSize K1, i0,i1,i2, n2,n1,n0, P, Nf, Nm, code;
double ll = 0.0, mx[MaxChan], vx[MaxChan], p[128];
P = nf[3];
Nf = nf[0]*nf[1]*nf[2];
K1 = nm[3];
Nm = nm[0]*nm[1]*nm[2];
n2 = nm[2]/skip[2]; if (n2>nf[2]) n2 = nf[2];
n1 = nm[1]/skip[1]; if (n1>nf[1]) n1 = nf[1];
n0 = nm[0]/skip[0]; if (n0>nf[0]) n0 = nf[0];
for(i2=0; i2<n2; i2++)
{
for(i1=0; i1<n1; i1++)
{
mwSize off_f, off_m;
off_f = nf[0]*(i1 + nf[1]*i2);
off_m = nm[0]*(i1*skip[1] + nm[1]*i2*skip[2]);
for(i0=0; i0<n0; i0++)
{
mwSize i, im;
i = i0 + off_f;
im = i0*skip[0] + off_m;
code = get_vox(Nf,P,mf+i,vf+i,mx,vx);
if (code>0 && logpriors(Nm, lp+im, K, lkp, p)!=0)
{
mwSize j, j1, k, Po;
double *s0, *s1, *s2;
Nloglikelihoods(K, gmm, code, mx, vx, p);
if (label!=NULL)
{
mwSize labi = (mwSize)(label[im]);
Dloglikelihoods(labi, K, lnP, p);
ll += softmax1(K,p,p);
for(k=0; k<K; k++)
H[labi + 256*k] += p[k];
}
else
ll += softmax1(K,p,p);
Po = gmm[code].P;
s0 = suffstat[code].s0;
s1 = suffstat[code].s1;
s2 = suffstat[code].s2;
for(k=0; k<K; k++, s2+=Po*Po, s1+=Po, s0++)
{
double pk = p[k];
*s0 += pk;
for(j=0; j<Po; j++)
{
double mxj = mx[j];
double px = pk*mxj;
s1[j] += px;
s2[j+Po*j] += pk*(mxj*mxj+vx[j]);
for(j1=j+1; j1<Po; j1++)
s2[j1+Po*j] += px*mx[j1];
}
}
}
}
}
}
/* Add in upper triangle second order sufficiant statistics */
for(code=1; code<((mwSize)1<<P); code++)
{
mwSize j, j1, k, Po;
double *s2;
Po = gmm[code].P;
s2 = suffstat[code].s2;
for(k=0; k<K; k++, s2+=Po*Po)
{
for(j=0; j<Po; j++)
{
for(j1=j+1; j1<Po; j1++)
s2[j+Po*j1] = s2[j1+Po*j];
}
}
}
return ll;
}
/* Constructs a gmm structure from mu, b, W, nu, and gam, as well as
* a structure containing pointers to the sufficient statistics.
* It then calls suffstats_missing before freeing up the structures.
*/
double call_suffstats_missing(mwSize nf[], float mf[], float vf[], unsigned char label[],
mwSize K, double mu[], double b[], double W[], double nu[], double gam[], double lnP[],
mwSize nm[], mwSize skip[], mwSize lkp[], float lp[],
double s0_ptr[], double s1_ptr[], double s2_ptr[], double H[])
{
mwSize P = nf[3];
GMMtype *gmm;
SStype *suffstat;
double ll=0.0;
if (P>=MaxChan || K>=128) return NAN;
if ((gmm = sub_gmm(P, K, mu, b, W, nu, gam))==NULL) return NAN;
if ((suffstat = suffstat_pointers(P, K, s0_ptr, s1_ptr, s2_ptr)) == NULL)
{
(void)free((void *)gmm);
return NAN;
}
ll = suffstats_missing(nf, mf, vf, label, K, gmm, lnP, nm, skip, lkp, lp, suffstat, H);
(void)free((void *)gmm);
(void)free((void *)suffstat);
return ll;
}
/* Compute responsibilities in a way that handles missing data.
* Responsibilities used for fitting the GMM are constructed
* from a VB GMM, whereas those not used ar constructed from
* a VB mixture of Student's T distributions.
*
* nf - Vector of dimensions (n_x, n_y, n_z, P).
* mf - E[f], dimensions nf.
* vf - Var[f], dimensions nf.
* gmm - Gaussian mixture model data structure.
* nm - Dimensions of log tissue priors (4 elements).
* skip - Sampling density for GMM vs TMM (in x, y and z).
* lkp - Lookup table relating Gaussians to tissue classes.
* lp - Log tissue priors
* r - Responsibilities (n_x, n_y, n_z, max(lkp)).
*/
static double responsibilities(mwSize nf[], mwSize skip[], float mf[], float vf[], unsigned char label[],
mwSize K, GMMtype *gmm, double lnP[],
mwSize K1, mwSize lkp[], float lp[],
float r[])
{
mwSize P, N1, i0,i1,i2;
double ll = 0.0, mx[MaxChan], vx[MaxChan], p[128];
P = nf[3];
N1 = nf[0]*nf[1]*nf[2];
for(i2=0; i2<nf[2]; i2++)
{
for(i1=0; i1<nf[1]; i1++)
{
mwSize off_f;
off_f = nf[0]*(i1 + nf[1]*i2);
for(i0=0; i0<nf[0]; i0++)
{
mwSize i, code, k, k1;
i = i0+off_f;
code = get_vox(N1,P,mf+i,vf+i,mx,vx);
if (logpriors(N1, lp+i, K, lkp, p)!=0)
{
if (code!=0)
{
if (label!=NULL) Dloglikelihoods((mwSize)(label[i]), K, lnP, p);
if ((i2%skip[2])==0 && ((i1%skip[1])==0) & ((i0%skip[0])==0))
Nloglikelihoods(K, gmm, code, mx, vx, p);
else
Tloglikelihoods(K, gmm, code, mx, vx, p);
ll += softmax1(K,p,p);
for(k=0; k<K; k++)
{
k1 = lkp[k];
if (k1<K1-1)
r[i+k1*N1] += p[k];
}
}
else
{
for(k1=0; k1<K1-1; k1++)
r[i+k1*N1] = NAN;
/*
(void)softmax(K1,p,p);
for(k1=0; k1<K1-1; k1++)
r[i+k1*N1] += p[k1];
*/
}
}
else
for(k1=0; k1<K1-1; k1++)
r[i+k1*N1] = NAN;
}
}
}
return ll;
}
/* Constructs a gmm structure from mu, b, W, nu, and gam, using this
* to call responsibilities.
*/
double call_responsibilities(mwSize nf[], mwSize skip[], float mf[], float vf[], unsigned char label[],
mwSize K, double mu[], double b[], double W[], double nu[], double gam[], double lnP[],
mwSize K1, mwSize lkp[], float lp[],
float r[])
{
mwSize P = nf[3];
GMMtype *gmm;
double ll;
if (P>=MaxChan || K>=128) return NAN;
if ((gmm = sub_gmm(P, K, mu, b, W, nu, gam))==NULL) return NAN;
ll = responsibilities(nf, skip, mf, vf, label, K, gmm, lnP, K1, lkp, lp, r);
(void)free((void *)gmm);
return ll;
}
/* Gradient and Hessian for INU updates
*
* The computations (two channels only) can be checked with
% Some MATLAB Symbolic Toolbox working...
syms w_11 w_12 w_22 mu_1 mu_2 x_1 x_2 b_1 b_2 mx_1 mx_2 real
syms vx_1 vx_2 positive
W = [w_11 w_12; w_12 w_22]; % Precision of Gaussian
mu = [mu_1; mu_2]; % Mean of Gaussian
x = [x_1; x_2];
mx = [mx_1; mx_2]; % E[x]
B = diag([b_1; 0]); % INU as a funciton of b_1
% Objective function for a single Gaussian. Extending to more is trivial.
E0 = (x-expm(-B)*mu)'*(expm(B)'*W*expm(B))*(x-expm(-B)*mu)/2 - log(det(expm(B)'*W*expm(B)))/2;
% The above objective function is equivalent to:
E = (expm(B)*x-mu)'*W*(expm(B)*x-mu)/2 - log(det(expm(B)'*W*expm(B)))/2;