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Stamatios Sotiropoulos authoredStamatios Sotiropoulos authored
diffmodels.cc 98.77 KiB
/* Diffusion model fitting
Timothy Behrens, Saad Jbabdi, Stam Sotiropoulos - FMRIB Image Analysis Group
Copyright (C) 2005 University of Oxford */
/* CCOPYRIGHT */
#include "Bingham_Watson_approx.h"
#include "diffmodels.h"
////////////////////////////////////////////////
// DIFFUSION TENSOR MODEL
////////////////////////////////////////////////
void DTI::linfit(){
ColumnVector logS(npts);
ColumnVector Dvec(7);
for (int i=1;i<=npts; i++){
if(Y(i)>0)
logS(i)=log(Y(i));
else
logS(i)=0;
}
Dvec = -iAmat*logS;
if(Dvec(7)>-23)
m_s0=exp(-Dvec(7));
else
m_s0=Y.MaximumAbsoluteValue();
for (int i=1;i<=Y.Nrows();i++){
if(m_s0<Y.Sum()/Y.Nrows()){ m_s0=Y.MaximumAbsoluteValue(); }
logS(i)=(Y(i)/m_s0)>0.01 ? log(Y(i)):log(0.01*m_s0);
}
Dvec = -iAmat*logS;
m_sse = (Amat*Dvec+logS).SumSquare();
m_s0=exp(-Dvec(7));
if(m_s0<Y.Sum()/Y.Nrows()){ m_s0=Y.Sum()/Y.Nrows(); }
vec2tens(Dvec);
calc_tensor_parameters();
m_covar.ReSize(7);
float dof=logS.Nrows()-7;
float sig2=m_sse/dof;
m_covar << sig2*(Amat.t()*Amat).i();
}
ColumnVector DTI::calc_md_grad(const ColumnVector& _tens)const{
ColumnVector g(6);
g = 0;
g(1) = 1/3.0;
g(4) = 1/3.0;
g(6) = 1/3.0;
return g;
}
// this will only work if the determinant is strictly positive
ReturnMatrix DTI::calc_fa_grad(const ColumnVector& _intens)const{
ColumnVector gradv(6),ik(6),k(6);
float m = (_intens(1)+_intens(4)+_intens(6))/3.0;
SymmetricMatrix K(3),iK(3),M(3);
// rescale input matrix
vec2tens(_intens,M);
//M /=m;
m = M.Trace()/3.0;
K = M - m*IdentityMatrix(3);
tens2vec(K,k);
iK << K.i();
tens2vec(iK,ik);
float p = K.SumSquare()/6.0;
float q = K.Determinant()/2.0;
float h = std::sqrt(p*p*p-q*q)/q;
float phi = std::atan(h)/3.0;
if(q<0)phi+=M_PI;
float _l1 = m + 2.0*std::sqrt(p)*std::cos(phi);
float _l2 = m - std::sqrt(p)*(std::cos(phi)+std::sqrt(3.0)*std::sin(phi));
float _l3 = m - std::sqrt(p)*(std::cos(phi)-std::sqrt(3.0)*std::sin(phi));
float t = 6.0/9.0*(_l1*_l1+_l2*_l2+_l3*_l3 - _l1*_l2-_l1*_l3-_l2*_l3);
float b = _l1*_l1+_l2*_l2+_l3*_l3;
float _fa = std::sqrt(3.0/2.0)*std::sqrt(t/b);
float dfadl1 = 3.0/4.0/_fa * ( 6.0/9.0*(2.0*_l1-_l2-_l3)/b - t/b/b*2.0*_l1 );
float dfadl2 = 3.0/4.0/_fa * ( 6.0/9.0*(2.0*_l2-_l1-_l3)/b - t/b/b*2.0*_l2 );
float dfadl3 = 3.0/4.0/_fa * ( 6.0/9.0*(2.0*_l3-_l1-_l2)/b - t/b/b*2.0*_l3 );
// determine dkdx
ColumnVector dkdx(6);
dkdx << 2.0/3.0 << 1.0 << 1.0 << 2.0/3.0 << 1.0 << 2.0/3.0;
for(int i=1;i<=6;i++){
float dL1dx=0,dL2dx=0,dL3dx=0;
if(i==1||i==4||i==6){
dL1dx=1.0/3.0;dL2dx=1.0/3.0;dL3dx=1.0/3.0;
}
//
float p_p = k(i)/3.0 * dkdx(i);
float q_p = q*ik(i) * dkdx(i);
float h_p = (3.0*p*p*p_p - 2.0*q_p*q*(1+h*h))/2.0/h/q/q;
float phi_p = h_p/(1+h*h)/3.0;
dL1dx += p_p/std::sqrt(p)*std::cos(phi) - 2.0*std::sqrt(p)*phi_p*std::sin(phi);
dL2dx -= std::sqrt(p)*(.5*p_p*(m-_l2)+phi_p*(std::sin(phi)-std::sqrt(3.0)*std::cos(phi)));
dL3dx -= std::sqrt(p)*(.5*p_p*(m-_l3)+phi_p*(std::sin(phi)+std::sqrt(3.0)*std::cos(phi)));
//
gradv(i) = dfadl1*dL1dx + dfadl2*dL2dx + dfadl3*dL3dx;
}
gradv.Release();
return gradv;
}
float DTI::calc_fa_var()const{
ColumnVector grd;
ColumnVector vtens;
tens2vec(m_tens,vtens);
grd = calc_fa_grad(vtens);
ColumnVector g(7);
g.SubMatrix(1,6,1,1) = grd;
g(7) = 0;
return((g.t()*m_covar*g).AsScalar());
}
void DTI::rot2angles(const Matrix& rot,float& th1,float& th2,float& th3)const{
if(rot(3,1)!=1 && rot(3,1)!=-1){
th2 = -asin(rot(3,1));
float c=std::cos(th2);
th1 = atan2(rot(3,2)/c,rot(3,3)/c);
th3 = atan2(rot(2,1)/c,rot(1,1)/c);
}
else{ th1 = atan2(rot(1,2),rot(1,3));
th2 = -sign(rot(3,1))*M_PI/2;
th3 = 0;
}
}
void DTI::angles2rot(const float& th1,const float& th2,const float& th3,Matrix& rot)const{
float c1=std::cos(th1),s1=std::sin(th1);
float c2=std::cos(th2),s2=std::sin(th2);
float c3=std::cos(th3),s3=std::sin(th3);
rot(1,1) = c2*c3; rot(1,2) = s1*s2*c3-c1*s3; rot(3,1) = c1*s2*c3+s1*s3;
rot(2,1) = c2*s3; rot(2,2) = s1*s2*s3+c1*c3; rot(3,2) = c1*s2*s3-s1*c3;
rot(3,1) = -s2; rot(3,2) = s1*c2; rot(3,3) = c1*c2;
}
// nonlinear tensor fitting
void DTI::nonlinfit(){
// initialise with linear fitting
linfit();
print();
// set starting parameters
// params = s0, log(l1),log(l2), log(l3), th1, th2, th3
ColumnVector start(nparams);
start(1) = m_s0;
// eigenvalues
start(2) = m_l1>0?std::log(m_l1):std::log(1e-5);
start(3) = m_l2>0?std::log(m_l2):std::log(1e-5);
start(4) = m_l3>0?std::log(m_l3):std::log(1e-5);
// angles
float th1,th2,th3;
Matrix rot(3,3);
rot.Row(1) = m_v1.t();
rot.Row(2) = m_v2.t();
rot.Row(3) = m_v3.t();
rot2angles(rot,th1,th2,th3);
start(5) = th1;
start(6) = th2;
start(7) = th3;
// do the fit
NonlinParam lmpar(start.Nrows(),NL_LM);
lmpar.SetGaussNewtonType(LM_L);
lmpar.SetStartingEstimate(start);
NonlinOut status;
status = nonlin(lmpar,(*this));
ColumnVector final_par(nparams);
final_par = lmpar.Par();
// finalise parameters
m_s0 = final_par(1);
m_l1 = exp(final_par(2));
m_l2 = exp(final_par(3));
m_l3 = exp(final_par(4));
angles2rot(final_par(5),final_par(6),final_par(7),rot);
m_v1 = rot.Row(1).t();
m_v2 = rot.Row(2).t();
m_v3 = rot.Row(3).t();
sort();
m_tens << m_l1*m_v1*m_v1.t() + m_l2*m_v2*m_v2.t() + m_l3*m_v3*m_v3.t();
calc_tensor_parameters();
print();
//exit(1);
}
void DTI::sort(){
vector< pair<float,int> > ls(3);
vector<ColumnVector> vs(3);
ls[0].first=m_l1;
ls[0].second=0;
ls[1].first=m_l2;
ls[1].second=1;
ls[2].first=m_l3;
ls[2].second=2;
vs[0]=m_v1;vs[1]=m_v2;vs[2]=m_v3;
std::sort(ls.begin(),ls.end());
m_l1 = ls[2].first;
m_v1 = vs[ ls[2].second ];
m_l2 = ls[1].first;
m_v2 = vs[ ls[1].second ];
m_l3 = ls[0].first;
m_v3 = vs[ ls[0].second ];
}
void DTI::calc_tensor_parameters(){
Matrix Vd;
DiagonalMatrix Dd(3);
// mean, eigenvalues and eigenvectors
EigenValues(m_tens,Dd,Vd);
m_md = Dd.Sum()/Dd.Nrows();
m_l1 = Dd(3,3);
m_l2 = Dd(2,2);
m_l3 = Dd(1,1);
m_v1 = Vd.Column(3);
m_v2 = Vd.Column(2);
m_v3 = Vd.Column(1);
// mode
float e1=m_l1-m_md, e2=m_l2-m_md, e3=m_l3-m_md;
float n = (e1 + e2 - 2*e3)*(2*e1 - e2 - e3)*(e1 - 2*e2 + e3);
float d = (e1*e1 + e2*e2 + e3*e3 - e1*e2 - e2*e3 - e1*e3);
d = sqrt(bigger(0, d));
d = 2*d*d*d;
m_mo = smaller(bigger(d ? n/d : 0.0, -1),1);
// fa
float numer=1.5*((m_l1-m_md)*(m_l1-m_md)+(m_l2-m_md)*(m_l2-m_md)+(m_l3-m_md)*(m_l3-m_md));
float denom=(m_l1*m_l1+m_l2*m_l2+m_l3*m_l3);
if(denom>0) m_fa=numer/denom;
else m_fa=0;
if(m_fa>0){m_fa=sqrt(m_fa);}
else{m_fa=0;}
}
// now the nonlinear fitting routines
double DTI::cf(const NEWMAT::ColumnVector& p)const{
//cout << "CF" << endl;
//OUT(p.t());
double cfv = 0.0;
double err = 0.0;
////////////////////////////////////
ColumnVector ls(3);
Matrix rot(3,3);
angles2rot(p(5),p(6),p(7),rot);
for(int k=2;k<=4;k++){
ls(k-1) = exp(p(k));
}
////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){ err = p(1)*anisoterm(i,ls,rot) - Y(i);
cfv += err*err;
}
//OUT(cfv);
//cout<<"--------"<<endl;
return(cfv);
}
NEWMAT::ReturnMatrix DTI::grad(const NEWMAT::ColumnVector& p)const{
NEWMAT::ColumnVector gradv(p.Nrows());
cout<<"grad"<<endl;
OUT(p.t());
gradv = 0.0;
////////////////////////////////////
ColumnVector ls(3);
Matrix rot(3,3);
Matrix rot1(3,3),rot2(3,3),rot3(3,3);
angles2rot(p(5),p(6),p(7),rot);
angles2rot(p(5)+M_PI/2.0,p(6),p(7),rot1);
angles2rot(p(5),p(6)+M_PI/2.0,p(7),rot2);
angles2rot(p(5),p(6),p(7)+M_PI/2.0,rot3);
for(int k=2;k<=4;k++){
ls(k-1) = exp(p(k));
}
////////////////////////////////////
Matrix J(npts,nparams);
ColumnVector x(3);
ColumnVector diff(npts);
float sig;
for(int i=1;i<=Y.Nrows();i++){
sig = p(1)*anisoterm(i,ls,rot);
J(i,1) = sig/p(1);
x = rotproduct(bvecs.Column(i),rot);
J(i,2) = -bvals(1,i)*x(1)*sig*ls(1);
J(i,3) = -bvals(1,i)*x(2)*sig*ls(2);
J(i,4) = -bvals(1,i)*x(3)*sig*ls(3);
x = rotproduct(bvecs.Column(i),rot1,rot);
J(i,5) = -2.0*bvals(1,i)*(ls(1)*x(1)+ls(2)*x(2)+ls(3)*x(3))*sig;
x = rotproduct(bvecs.Column(i),rot2,rot);
J(i,6) = -2.0*bvals(1,i)*(ls(1)*x(1)+ls(2)*x(2)+ls(3)*x(3))*sig;
x = rotproduct(bvecs.Column(i),rot3,rot);
J(i,7) = -2.0*bvals(1,i)*(ls(1)*x(1)+ls(2)*x(2)+ls(3)*x(3))*sig;
diff(i) = sig - Y(i);
}
OUT(diff.t());
OUT(J.t());
gradv = 2.0*J.t()*diff;
OUT(gradv.t());
cout<<"------"<<endl;
gradv.Release();
return gradv;
}
//this uses Gauss-Newton approximation
boost::shared_ptr<BFMatrix> DTI::hess(const NEWMAT::ColumnVector& p,boost::shared_ptr<BFMatrix> iptr)const{
boost::shared_ptr<BFMatrix> hessm;
if (iptr && iptr->Nrows()==(unsigned int)p.Nrows() && iptr->Ncols()==(unsigned int)p.Nrows()) hessm = iptr;
else hessm = boost::shared_ptr<BFMatrix>(new FullBFMatrix(p.Nrows(),p.Nrows()));
cout<<"hess"<<endl;
OUT(p.t());
////////////////////////////////////
ColumnVector ls(3);
Matrix rot(3,3);
Matrix rot1(3,3),rot2(3,3),rot3(3,3);
angles2rot(p(5),p(6),p(7),rot);
angles2rot(p(5)+M_PI/2,p(6),p(7),rot1);
angles2rot(p(5),p(6)+M_PI/2,p(7),rot2);
angles2rot(p(5),p(6),p(7)+M_PI/2,rot3);
for(int k=2;k<=4;k++){
ls(k-1) = exp(p(k));
}
////////////////////////////////////
Matrix J(npts,nparams);
ColumnVector x(3);
float sig;
for(int i=1;i<=Y.Nrows();i++){
sig = p(1)*anisoterm(i,ls,rot);
J(i,1) = sig/p(1);
x = rotproduct(bvecs.Column(i),rot);
J(i,2) = -bvals(1,i)*x(1)*sig*ls(1);
J(i,3) = -bvals(1,i)*x(2)*sig*ls(2);
J(i,4) = -bvals(1,i)*x(3)*sig*ls(3);
x = rotproduct(bvecs.Column(i),rot1,rot);
J(i,5) = -2.0*bvals(1,i)*(ls(1)*x(1)+ls(2)*x(2)+ls(3)*x(3))*sig;
x = rotproduct(bvecs.Column(i),rot2,rot);
J(i,6) = -2.0*bvals(1,i)*(ls(1)*x(1)+ls(2)*x(2)+ls(3)*x(3))*sig;
x = rotproduct(bvecs.Column(i),rot3,rot);
J(i,7) = -2.0*bvals(1,i)*(ls(1)*x(1)+ls(2)*x(2)+ls(3)*x(3))*sig;
}
for (int i=1; i<=p.Nrows(); i++){
for (int j=i; j<=p.Nrows(); j++){
sig = 0.0;
for(int k=1;k<=J.Nrows();k++)
sig += 2.0*(J(k,i)*J(k,j));
hessm->Set(i,j,sig);
}
}
for (int j=1; j<=p.Nrows(); j++) {
for (int i=j+1; i<=p.Nrows(); i++) {
hessm->Set(i,j,hessm->Peek(j,i));
}
}
hessm->Print();
cout<<"------"<<endl;
return(hessm);
}
ColumnVector DTI::rotproduct(const ColumnVector& x,const Matrix& R)const{
ColumnVector ret(3);
for(int i=1;i<=3;i++)
ret(i) = x(1)*x(1)*R(1,i)*R(1,i)+x(2)*x(2)*R(2,i)*R(2,i)+x(3)*x(3)*R(3,i)*R(3,i)
+2.0*( x(1)*R(1,i)*(x(2)*R(2,i)+x(3)*R(3,i)) +x(2)*x(3)*R(2,i)*R(3,i) );
return ret;
}
ColumnVector DTI::rotproduct(const ColumnVector& x,const Matrix& R1,const Matrix& R2)const{
ColumnVector ret(3);
for(int i=1;i<=3;i++)
ret(i) = x(1)*x(1)*R1(1,i)*R2(1,i)+x(2)*x(2)*R1(2,i)*R2(2,i)+x(3)*x(3)*R1(3,i)*R2(3,i)
+( x(1)*R1(1,i)*(x(2)*R2(2,i)+x(3)*R2(3,i)) +x(2)*x(3)*R1(2,i)*R2(3,i) )
+( x(1)*R2(1,i)*(x(2)*R1(2,i)+x(3)*R1(3,i)) +x(2)*x(3)*R2(2,i)*R1(3,i) );
return ret;
}
float DTI::anisoterm(const int& pt,const ColumnVector& ls,const Matrix& rot)const{
ColumnVector x(3);
x = rotproduct(bvecs.Column(pt),rot);
return exp(-bvals(1,pt)*(ls(1)*x(1)+ls(2)*x(2)+ls(3)*x(3)));
}
/////////////////////////////////////////////////////////////////////////
// PARTIAL VOLUME MODEL - SINGLE SHELL
// Constrained Optimization for the diffusivity, fractions and their sum<1
//////////////////////////////////////////////////////////////////////////
void PVM_single_c::fit(){
// initialise with a tensor
DTI dti(Y,bvecs,bvals);
dti.linfit();
// set starting parameters for nonlinear fitting
float _th,_ph;
cart2sph(dti.get_v1(),_th,_ph);
ColumnVector start(nparams);
//Initialize the non-linear fitter. Use the DTI estimates for most parameters, apart from the volume fractions
start(1) = dti.get_s0();
//start(2) = d2lambda(dti.get_md()>0?dti.get_md()*2:0.001); // empirically found that d~2*MD
start(2) = d2lambda(dti.get_l1()>0?dti.get_l1():0.002); // empirically found that d~L1
start(4) = _th;
start(5) = _ph;
for(int ii=2,i=6;ii<=nfib;ii++,i+=3){
cart2sph(dti.get_v(ii),_th,_ph);
start(i+1) = _th;
start(i+2) = _ph;
}
// do a better job for initializing the volume fractions
fit_pvf(start);
// do the fit
NonlinParam lmpar(start.Nrows(),NL_LM);
lmpar.SetGaussNewtonType(LM_L);
lmpar.SetStartingEstimate(start);
NonlinOut status;
status = nonlin(lmpar,(*this));
ColumnVector final_par(nparams);
final_par = lmpar.Par();
if (m_eval_BIC){
double RSS=cf(final_par); //get the sum of squared residuals
m_BIC=npts*log(RSS/npts)+log(npts)*nparams; //evaluate BIC
}
// finalise parameters
m_s0 = final_par(1);
m_d = lambda2d(final_par(2));
for(int k=1;k<=nfib;k++){
int kk = 3 + 3*(k-1);
m_f(k) = beta2f(final_par(kk))*partial_fsum(m_f,k-1); m_th(k) = final_par(kk+1);
m_ph(k) = final_par(kk+2);
}
if (m_return_fanning)
Fanning_angles_from_Hessian();
if (m_include_f0)
m_f0=beta2f(final_par(nparams))*partial_fsum(m_f,nfib);
sort();
}
void PVM_single_c::sort(){
vector< pair<float,int> > fvals(nfib);
ColumnVector ftmp(nfib),thtmp(nfib),phtmp(nfib),fantmp;
vector<ColumnVector> Hess_vec_tmp; vector<Matrix> Hess;
ftmp=m_f;thtmp=m_th;phtmp=m_ph;
if (m_return_fanning){
fantmp=m_fanning_angles;
Hess_vec_tmp=m_invprHes_e1;
Hess=m_Hessian;
}
for(int i=1;i<=nfib;i++){
pair<float,int> p(m_f(i),i);
fvals[i-1] = p;
}
std::sort(fvals.begin(),fvals.end());
for(int i=1,ii=nfib-1;ii>=0;i++,ii--){
m_f(i) = ftmp(fvals[ii].second);
m_th(i) = thtmp(fvals[ii].second);
m_ph(i) = phtmp(fvals[ii].second);
if (m_return_fanning){
m_fanning_angles(i)=fantmp(fvals[ii].second);
m_invprHes_e1[i-1]=Hess_vec_tmp[fvals[ii].second-1];
m_Hessian[i-1]=Hess[fvals[ii].second-1];
}
}
}
//Returns 1-Sum(f_j), 1<=j<=ii. (ii<=nfib)
//Used for transforming beta to f and vice versa
float PVM_single_c::partial_fsum(ColumnVector& fs, int ii) const{
float fsum=1.0;
for(int j=1;j<=ii;j++)
fsum-=fs(j);
return fsum;
}
//If the sum of the fractions is >1, then zero as many fractions
//as necessary, so that the sum becomes smaller than 1.
void PVM_single_c::fix_fsum(ColumnVector& fs)const{
float sumf=0;
for(int i=1;i<=nfib;i++){
sumf+=fs(i);
if(sumf>=1){
for(int j=i;j<=nfib;j++)
fs(j)=FSMALL; //make the fraction almost zero
break;
}
}
}
//Find the volume fractions given all the other model
//parameters using Linear Least Squaresvoid PVM_single_c::fit_pvf(ColumnVector& x)const{
ColumnVector fs(nfib);
ColumnVector Y_I(npts);
Matrix M(npts,nfib),dir(3,nfib);
float s0=x(1),d=lambda2d(x(2)), f0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
dir(1,k) = sin(x(kk+1))*cos(x(kk+2));
dir(2,k) = sin(x(kk+1))*sin(x(kk+2));
dir(3,k) = cos(x(kk+1));
}
////////////////////////////////////
for(int i=1;i<=npts;i++){
float Iso_term=isoterm(i,d);
Y_I(i) = Y(i)-s0*Iso_term;
for(int k=1;k<=nfib;k++){
M(i,k)=s0*(anisoterm(i,d,dir.Column(k))-Iso_term);
}
//if (m_include_f0)
//M(i,f_num)=s0*(1-Iso_term);
}
fs = pinv(M)*Y_I;
if (m_include_f0){
f0=FSMALL; fs(1)-=f0; //Initialize f0 with a very small value
}
for(int k=1;k<=nfib;k++)
fs(k)=fabs(fs(k)); //make sure that the initial values for the fractions are positive
fix_fsum(fs);
for(int k=1;k<=nfib;k++){
float tmpr=fs(k)/partial_fsum(fs,k-1);
if (tmpr>1.0) tmpr=1; //This can be due to numerical errors
x(3+3*(k-1))=f2beta(tmpr);
}
if (m_include_f0){
float tmpr=f0/partial_fsum(fs,nfib);
if (tmpr>1.0) tmpr=1; //This can be due to numerical errors
x(nparams)=f2beta(tmpr);
}
}
//Print the final estimates (after having them transformed)
void PVM_single_c::print()const{
cout << "PVM (Single) FIT RESULTS " << endl;
cout << "S0 :" << m_s0 << endl;
cout << "D :" << m_d << endl;
for(int i=1;i<=nfib;i++){
cout << "F" << i << " :" << m_f(i) << endl;
ColumnVector x(3);
x << sin(m_th(i))*cos(m_ph(i)) << sin(m_th(i))*sin(m_ph(i)) << cos(m_th(i));
if(x(3)<0)x=-x;
float _th,_ph;cart2sph(x,_th,_ph);
cout << "TH" << i << " :" << _th << endl;
cout << "PH" << i << " :" << _ph << endl;
cout << "DIR" << i << " : " << x(1) << " " << x(2) << " " << x(3) << endl;
}
if (m_include_f0)
cout << "F0 :" << m_f0 << endl;
if (m_eval_BIC)
cout<< "BIC :"<<m_BIC<<endl;
}
//Print the estimates using a vector with the untransformed parameter valuesvoid PVM_single_c::print(const ColumnVector& p)const{
ColumnVector f(nfib);
cout << "PARAMETER VALUES " << endl;
cout << "S0 :" << p(1) << endl;
cout << "D :" << lambda2d(p(2)) << endl;
for(int i=3,ii=1;ii<=nfib;i+=3,ii++){
f(ii) = beta2f(p(i))*partial_fsum(f,ii-1);
cout << "F" << ii << " :" << f(ii) << endl;
cout << "TH" << ii << " :" << p(i+1)*180.0/M_PI << " deg" << endl;
cout << "PH" << ii << " :" << p(i+2)*180.0/M_PI << " deg" << endl;
}
if (m_include_f0)
cout << "F0 :" << beta2f(p(nparams))*partial_fsum(f,nfib);
}
ReturnMatrix PVM_single_c::get_prediction()const{
ColumnVector pred(npts);
ColumnVector p(nparams);
ColumnVector fs(nfib);
fs=m_f;
p(1) = m_s0;
p(2) = d2lambda(m_d);
for(int i=3,ii=1;ii<=nfib;i+=3,ii++){
float tmpr=m_f(ii)/partial_fsum(fs,ii-1);
if (tmpr>1.0) tmpr=1; //This can be due to numerical errors
p(i) = f2beta(tmpr);
p(i+1) = m_th(ii);
p(i+2) = m_ph(ii);
}
if (m_include_f0){
float tmpr=m_f0/partial_fsum(fs,nfib);
if (tmpr>1.0) tmpr=1; //This can be due to numerical errors
p(nparams)=f2beta(tmpr);
}
pred = forwardModel(p);
pred.Release();
return pred;
}
NEWMAT::ReturnMatrix PVM_single_c::forwardModel(const NEWMAT::ColumnVector& p)const{
ColumnVector pred(npts);
pred = 0;
float val;
float _d = lambda2d(p(2));
////////////////////////////////////
ColumnVector fs(nfib);
Matrix x(nfib,3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1));
}
////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
val = 0.0;
for(int k=1;k<=nfib;k++){
val += fs(k)*anisoterm(i,_d,x.Row(k).t());
}
if (m_include_f0){ float temp_f0=beta2f(p(nparams))*partial_fsum(fs,nfib);
pred(i) = p(1)*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_d)+val);
}
else
pred(i) = p(1)*((1-sumf)*isoterm(i,_d)+val);
}
pred.Release();
return pred;
}
//Cost Function, sum of squared residuals
//assume that parameter values p are untransformed (e.g. need to transform them to get d, f's)
double PVM_single_c::cf(const NEWMAT::ColumnVector& p)const{
//cout<<"CF"<<endl;
//OUT(p.t());
double cfv = 0.0;
double err;
float _d = lambda2d(p(2));
////////////////////////////////////
ColumnVector fs(nfib);
Matrix x(nfib,3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1));
}
////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
err = 0.0;
for(int k=1;k<=nfib;k++){
err += fs(k)*anisoterm(i,_d,x.Row(k).t());
}
if (m_include_f0){
float temp_f0=beta2f(p(nparams))*partial_fsum(fs,nfib);
err = (p(1)*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_d)+err) - Y(i));
}
else
err = (p(1)*((1-sumf)*isoterm(i,_d)+err) - Y(i));
cfv += err*err;
}
return(cfv);
}
NEWMAT::ReturnMatrix PVM_single_c::grad(const NEWMAT::ColumnVector& p)const{
//cout<<"GRAD"<<endl;
//OUT(p.t());
NEWMAT::ColumnVector gradv(p.Nrows());
gradv = 0.0;
float _d = lambda2d(p(2));
////////////////////////////////////
ColumnVector fs(nfib);
ColumnVector bs(nfib);
Matrix x(nfib,3);ColumnVector xx(3); ColumnVector yy(3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
bs(k)=p(kk);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1)); }
////////////////////////////////////
Matrix f_deriv;
//Compute the derivatives with respect to betas, i.e the transformed volume fraction variables
f_deriv=fractions_deriv(nfib, fs, bs);
Matrix J(npts,nparams); //Get the Jacobian of the model equation. The derivatives of the cost function
//for parameter j will then be: Grad_j=Sum(2*(F(x_i)-Y_i)J(i,j)), Sum across data points i
ColumnVector diff(npts);
float sig, Iso_term;
ColumnVector Aniso_terms(nfib);
for(int i=1;i<=Y.Nrows();i++){
Iso_term=isoterm(i,_d); //Precompute some terms for this datapoint
for(int k=1;k<=nfib;k++){
xx = x.Row(k).t();
Aniso_terms(k)=anisoterm(i,_d,xx);
}
sig = 0;
J.Row(i)=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
xx = x.Row(k).t();
sig += fs(k)*Aniso_terms(k); //Total signal
// other stuff for derivatives
// lambda (i.e. d)
J(i,2) += p(1)*fs(k)*anisoterm_lambda(i,p(2),xx);
// beta (i.e. f)
J(i,kk)=0;
for (int j=1; j<=nfib; j++){
if (f_deriv(j,k)!=0)
J(i,kk) += p(1)*(Aniso_terms(j)-Iso_term)*f_deriv(j,k);
}
// th
J(i,kk+1) = p(1)*fs(k)*anisoterm_th(i,_d,xx,p(kk+1),p(kk+2));
// ph
J(i,kk+2) = p(1)*fs(k)*anisoterm_ph(i,_d,xx,p(kk+1),p(kk+2));
}
if (m_include_f0){
float temp_f0=beta2f(p(nparams))*partial_fsum(fs,nfib);
//derivative with respect to f0
J(i,nparams)= p(1)*(1-Iso_term)*sin(2*p(nparams))*partial_fsum(fs,nfib);
sig=p(1)*(temp_f0+(1-sumf-temp_f0)*Iso_term+sig);
J(i,2) += p(1)*(1-sumf-temp_f0)*isoterm_lambda(i,p(2));
}
else{
sig = p(1)*((1-sumf)*Iso_term+sig);
J(i,2) += p(1)*(1-sumf)*isoterm_lambda(i,p(2)); //lambda
}
diff(i) = sig - Y(i);
J(i,1) = sig/p(1); //S0
}
gradv = 2*J.t()*diff;
gradv.Release();
return gradv;
}
//this uses Gauss-Newton approximation, i.e Hij ~ Sum(Gj*Gk), with G the derivative of the cost function with respect to parameter j
//and the Sum over all data points.
boost::shared_ptr<BFMatrix> PVM_single_c::hess(const NEWMAT::ColumnVector& p,boost::shared_ptr<BFMatrix> iptr)const{
//cout<<"HESS"<<endl;
//OUT(p.t());
boost::shared_ptr<BFMatrix> hessm;
if (iptr && iptr->Nrows()==(unsigned int)p.Nrows() && iptr->Ncols()==(unsigned int)p.Nrows()) hessm = iptr;
else hessm = boost::shared_ptr<BFMatrix>(new FullBFMatrix(p.Nrows(),p.Nrows()));
float _d = lambda2d(p(2)); ////////////////////////////////////
ColumnVector fs(nfib);
ColumnVector bs(nfib);
Matrix x(nfib,3);ColumnVector xx(3); ColumnVector yy(3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
bs(k)=p(kk);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1));
}
////////////////////////////////////
Matrix f_deriv;
f_deriv=fractions_deriv(nfib, fs, bs);
Matrix J(npts,nparams);
float sig, Iso_term;
ColumnVector Aniso_terms(nfib);
for(int i=1;i<=Y.Nrows();i++){
Iso_term=isoterm(i,_d); //Precompute some terms for this datapoint
for(int k=1;k<=nfib;k++){
xx = x.Row(k).t();
Aniso_terms(k)=anisoterm(i,_d,xx);
}
sig = 0;
J.Row(i)=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
xx = x.Row(k).t();
sig += fs(k)*Aniso_terms(k); //Total signal
// other stuff for derivatives
// lambda (i.e. d)
J(i,2) += p(1)*fs(k)*anisoterm_lambda(i,p(2),xx);
// beta (i.e. f)
J(i,kk)=0;
for (int j=1; j<=nfib; j++){
if (f_deriv(j,k)!=0)
J(i,kk) += p(1)*(Aniso_terms(j)-Iso_term)*f_deriv(j,k);
}
// th
J(i,kk+1) = p(1)*fs(k)*anisoterm_th(i,_d,xx,p(kk+1),p(kk+2));
// ph
J(i,kk+2) = p(1)*fs(k)*anisoterm_ph(i,_d,xx,p(kk+1),p(kk+2));
}
if (m_include_f0){
float temp_f0=beta2f(p(nparams))*partial_fsum(fs,nfib);
//derivative with respect to f0
J(i,nparams)= p(1)*(1-Iso_term)*sin(2*p(nparams))*partial_fsum(fs,nfib);
sig=p(1)*(temp_f0+(1-sumf-temp_f0)*Iso_term+sig);
J(i,2) += p(1)*(1-sumf-temp_f0)*isoterm_lambda(i,p(2));
}
else{
sig = p(1)*((1-sumf)*Iso_term+sig);
J(i,2) += p(1)*(1-sumf)*isoterm_lambda(i,p(2)); //lambda
}
J(i,1) = sig/p(1); //S0
}
for (int i=1; i<=p.Nrows(); i++){
for (int j=i; j<=p.Nrows(); j++){
sig = 0.0;
for(int k=1;k<=J.Nrows();k++)
sig += 2*(J(k,i)*J(k,j));
hessm->Set(i,j,sig);
} }
for (int j=1; j<=p.Nrows(); j++) {
for (int i=j+1; i<=p.Nrows(); i++) {
hessm->Set(i,j,hessm->Peek(j,i));
}
}
return(hessm);
}
//Once the model is fitted: For each fibre, compute a 3x3 Hessian of the Cost function at the cartesian (x,y,z) coordinates of the orientation,
//evaluated at the estimated parameters. Return the second eigenvector of the (inverse) Hessian - No need to invert though!
void PVM_single_c::eval_Hessian_at_peaks(){
vector<ColumnVector> V;
ColumnVector temp_vec(3);
float fiso=1,dot, Saniso,dSaniso_dx,dSaniso_dy,dSaniso_dz;
ColumnVector S(npts), F(npts);
Matrix dS_dx(npts,nfib), dS_dy(npts,nfib), dS_dz(npts,nfib);
Matrix dS_dxdx(npts,nfib), dS_dydy(npts,nfib), dS_dzdz(npts,nfib), dS_dxdy(npts,nfib),dS_dxdz(npts,nfib),dS_dydz(npts,nfib);
for (int n=1; n<=nfib; n++){
temp_vec<< cos(m_ph(n))*sin(m_th(n)) << sin(m_ph(n))*sin(m_th(n)) << cos(m_th(n));
V.push_back(temp_vec);
fiso-=m_f(n);
}
for (int i=1; i<=npts; i++){
float bd=bvals(1,i)*m_d;
S(i)=fiso*exp(-bd);
for (int n=1; n<=nfib; n++){
dot=V[n-1](1)*bvecs(1,i)+V[n-1](2)*bvecs(2,i)+V[n-1](3)*bvecs(3,i);
Saniso=exp(-bd*dot*dot);
S(i)=S(i)+m_f(n)*Saniso;
float dot1=-2*bd*dot*Saniso; float dot2=-2*bd*dot; float dot3=-2*bd*Saniso; float fS0=m_s0*m_f(n);
//First Derivatives of the signal wrt x,y,z
dSaniso_dx=dot1*bvecs(1,i); dSaniso_dy=dot1*bvecs(2,i); dSaniso_dz=dot1*bvecs(3,i);
dS_dx(i,n)=fS0*dSaniso_dx; dS_dy(i,n)=fS0*dSaniso_dy; dS_dz(i,n)=fS0*dSaniso_dz;
//Second Derivatives of the signal wrt x,y,z
dS_dxdx(i,n)=fS0*bvecs(1,i)*(dSaniso_dx*dot2+dot3*bvecs(1,i)); dS_dydy(i,n)=fS0*bvecs(2,i)*(dSaniso_dy*dot2+dot3*bvecs(2,i));
dS_dzdz(i,n)=fS0*bvecs(3,i)*(dSaniso_dz*dot2+dot3*bvecs(3,i)); dS_dxdy(i,n)=fS0*bvecs(1,i)*(dSaniso_dy*dot2+dot3*bvecs(2,i));
dS_dxdz(i,n)=fS0*bvecs(1,i)*(dSaniso_dz*dot2+dot3*bvecs(3,i)); dS_dydz(i,n)=fS0*bvecs(2,i)*(dSaniso_dz*dot2+dot3*bvecs(3,i));
}
F(i)=Y(i)-m_s0*S(i);
}
for (int n=1; n<=nfib; n++){ //For each fibre, compute the Hessian matrix of the cost function at (x,y,z)
SymmetricMatrix H(3); H=0;
for (int i=1; i<=npts; i++){
H(1,1)-=dS_dx(i,n)*dS_dx(i,n)-F(i)*dS_dxdx(i,n);
H(2,2)-=dS_dy(i,n)*dS_dy(i,n)-F(i)*dS_dydy(i,n);
H(3,3)-=dS_dz(i,n)*dS_dz(i,n)-F(i)*dS_dzdz(i,n);
H(1,2)-=dS_dx(i,n)*dS_dy(i,n)-F(i)*dS_dxdy(i,n);
H(1,3)-=dS_dx(i,n)*dS_dz(i,n)-F(i)*dS_dxdz(i,n);
H(2,3)-=dS_dy(i,n)*dS_dz(i,n)-F(i)*dS_dydz(i,n);
}
H=-2*H;
m_Hessian.push_back(H); //store the Hessian
}
}
//For each fibre, get the projection of the inverse Hessian to the fanning plane.
//Its first eigenvector will be utilized to get a fanning angle in [0,pi).
void PVM_single_c::Fanning_angles_from_Hessian(){
Matrix Rth(3,3), Rph(3,3), R(3,3), H, A(3,3), P(3,3), E;
DiagonalMatrix L; SymmetricMatrix Q(3);
ColumnVector e1(3),vfib(3),v2(3),v3(3);
eval_Hessian_at_peaks(); //Compute the Hessian for each fibre orientation
//Then project the inverse Hessian to the fanning plane (perpendicular to the orientation) and obtain its first eigenvector
for (int n=1; n<=nfib; n++){ //For each fitted fibre
P << 0 << 0 << 0
<< 0 << 1 << 0
<< 0 << 0 << 1;
H=m_Hessian[n-1];
float sinth=sin(m_th(n)), costh=cos(m_th(n));
float sinph=sin(m_ph(n)), cosph=cos(m_ph(n));
vfib<< sinth*cosph << sinth*sinph << costh; //Corresponding fibre orientation
//Define two vectors that are orthogonal to vfib
if (vfib(1)==0 && vfib(2)==0) //then, we have a [0 0 1] orientation
v2<< 1 << 0 << 0;
else
v2 << vfib(2) << -vfib(1) << 0; //define v2, so that vfib*v2=0;
float mag=sqrt(v2(1)*v2(1)+v2(2)*v2(2)+v2(3)*v2(3));
v2=v2/mag;
v3 << vfib(2)*v2(3)-vfib(3)*v2(2) << vfib(3)*v2(1)-vfib(1)*v2(3) << vfib(1)*v2(2)-vfib(2)*v2(1); //Now get the cross product
//Define a Projection Matrix to the plane perpendicular to the fibre orientation vfib
A.Row(1)<<vfib.t(); A.Row(2)<<v2.t(); A.Row(3)<<v3.t();
P= P*A;
try{ //If the Hessian is invertible (might not be in some background voxels)
Q << P*H.i()*P.t(); //Project the inverse Hessian to this plane
EigenValues(Q,L,E); //Eigendecompose the projected inverse Hessian
e1 = E.Column(3); //Projected Hessian 1st eigenvector
}
catch(...){ //Otherwise give a random value and proceed
e1<<1<<0<<0;
}
e1 << A.t()*e1;
m_invprHes_e1.push_back(e1);
//float dot=DotProduct(e1,vfib); ColumnVector p;
//if (dot<0) {e1=-e1; dot=-dot; }
//p=e1-dot*vfib; //Projection of e2 on a plane perpendicular to vfib
//p=p/sqrt(p(1)*p(1)+p(2)*p(2)+p(3)*p(3));
//e1=p;
Rth<<costh << 0 << -sinth
<<0 << 1 << 0
<<sinth << 0 << costh;
Rph<<cosph << sinph <<0
<<-sinph<< cosph <<0
<<0 << 0 <<1;
R=Rth*Rph; //Rotation Matrix for vfib to become parallel to z
e1<< R*e1; //Rotate the fanning plane effectively to xy plane
m_fanning_angles(n)=atan2(-e1(1),e1(2)); //this gives [-pi,pi] range
if (m_fanning_angles(n)<0) m_fanning_angles(n)+=M_PI; //transform it to [0,pi]
//cout<<m_fanning_angles(n)<<endl<<endl;
}
}
////////////////////////////////////////////////
// PARTIAL VOLUME MODEL - SINGLE SHELL (OLD)
////////////////////////////////////////////////
void PVM_single::fit(){
// initialise with a tensor
DTI dti(Y,bvecs,bvals);
dti.linfit();
//dti.print();
// set starting parameters for nonlinear fitting float _th,_ph;
cart2sph(dti.get_v1(),_th,_ph);
ColumnVector start(nparams);
start(1) = dti.get_s0();
//start(2) = dti.get_md()>0?dti.get_md()*2:0.001; // empirically found that d~2*MD
start(2) = dti.get_l1()>0?dti.get_l1():0.002; // empirically found that d~L1
start(3) = dti.get_fa()<1?f2x(dti.get_fa()):f2x(0.95); // first pvf = FA
start(4) = _th;
start(5) = _ph;
float sumf=x2f(start(3));
float tmpsumf=sumf;
for(int ii=2,i=6;ii<=nfib;ii++,i+=3){
float denom=2;
do{
start(i) = f2x(x2f(start(i-3))/denom);
denom *= 2;
tmpsumf = sumf + x2f(start(i));
}while(tmpsumf>=1);
sumf += x2f(start(i));
cart2sph(dti.get_v(ii),_th,_ph);
start(i+1) = _th;
start(i+2) = _ph;
}
if (m_include_f0)
start(nparams)=f2x(FSMALL);
// do the fit
NonlinParam lmpar(start.Nrows(),NL_LM);
lmpar.SetGaussNewtonType(LM_L);
lmpar.SetStartingEstimate(start);
NonlinOut status;
status = nonlin(lmpar,(*this));
ColumnVector final_par(nparams);
final_par = lmpar.Par();
// finalise parameters
m_s0 = final_par(1);
m_d = std::abs(final_par(2));
for(int k=1;k<=nfib;k++){
int kk = 3 + 3*(k-1);
m_f(k) = x2f(final_par(kk));
m_th(k) = final_par(kk+1);
m_ph(k) = final_par(kk+2);
}
if (m_include_f0)
m_f0=x2f(final_par(nparams));
sort();
fix_fsum();
}
void PVM_single::sort(){
vector< pair<float,int> > fvals(nfib);
ColumnVector ftmp(nfib),thtmp(nfib),phtmp(nfib);
ftmp=m_f;thtmp=m_th;phtmp=m_ph;
for(int i=1;i<=nfib;i++){
pair<float,int> p(m_f(i),i);
fvals[i-1] = p;
}
std::sort(fvals.begin(),fvals.end());
for(int i=1,ii=nfib-1;ii>=0;i++,ii--){
m_f(i) = ftmp(fvals[ii].second);
m_th(i) = thtmp(fvals[ii].second);
m_ph(i) = phtmp(fvals[ii].second);
}
}
void PVM_single::fix_fsum(){ float sumf=0;
if (m_include_f0)
sumf=m_f0;
for(int i=1;i<=nfib;i++){
sumf+=m_f(i);
if(sumf>=1){for(int j=i;j<=nfib;j++)m_f(j)=FSMALL; break;}
}
}
ReturnMatrix PVM_single::get_prediction()const{
ColumnVector pred(npts);
ColumnVector p(nparams);
p(1) = m_s0;
p(2) = m_d;
for(int i=3,ii=1;ii<=nfib;i+=3,ii++){
p(i) = f2x(m_f(ii));
p(i+1) = m_th(ii);
p(i+2) = m_ph(ii);
}
if (m_include_f0)
p(nparams)=f2x(m_f0);
pred = forwardModel(p);
pred.Release();
return pred;
}
NEWMAT::ReturnMatrix PVM_single::forwardModel(const NEWMAT::ColumnVector& p)const{
//cout<<"FORWARD"<<endl;
//OUT(p.t());
ColumnVector pred(npts);
pred = 0;
float val;
float _d = std::abs(p(2));
////////////////////////////////////
ColumnVector fs(nfib);
Matrix x(nfib,3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
fs(k) = x2f(p(kk));
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1));
}
////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
val = 0.0;
for(int k=1;k<=nfib;k++){
val += fs(k)*anisoterm(i,_d,x.Row(k).t());
}
if (m_include_f0){
float temp_f0=x2f(p(nparams));
pred(i) = p(1)*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_d)+val);
}
else
pred(i) = p(1)*((1-sumf)*isoterm(i,_d)+val);
}
pred.Release();
//cout<<"----"<<endl;
return pred;
}
double PVM_single::cf(const NEWMAT::ColumnVector& p)const{
//cout<<"CF"<<endl;
//OUT(p.t());
double cfv = 0.0; double err;
float _d = std::abs(p(2));
////////////////////////////////////
ColumnVector fs(nfib);
Matrix x(nfib,3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
fs(k) = x2f(p(kk));
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1));
}
////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
err = 0.0;
for(int k=1;k<=nfib;k++){
err += fs(k)*anisoterm(i,_d,x.Row(k).t());
}
if (m_include_f0){
float temp_f0=x2f(p(nparams));
err = (p(1)*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_d)+err) - Y(i));
}
else
err = (p(1)*((1-sumf)*isoterm(i,_d)+err) - Y(i));
cfv += err*err;
}
//OUT(cfv);
//cout<<"----"<<endl;
return(cfv);
}
NEWMAT::ReturnMatrix PVM_single::grad(const NEWMAT::ColumnVector& p)const{
//cout<<"GRAD"<<endl;
//OUT(p.t());
NEWMAT::ColumnVector gradv(p.Nrows());
gradv = 0.0;
float _d = std::abs(p(2));
////////////////////////////////////
ColumnVector fs(nfib);
Matrix x(nfib,3);ColumnVector xx(3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
fs(k) = x2f(p(kk));
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1));
}
////////////////////////////////////
Matrix J(npts,nparams);
ColumnVector diff(npts);
float sig;
for(int i=1;i<=Y.Nrows();i++){
sig = 0;
J.Row(i)=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
xx = x.Row(k).t();
sig += fs(k)*anisoterm(i,_d,xx);
// other stuff for derivatives
// d
J(i,2) += (p(2)>0?1.0:-1.0)*p(1)*fs(k)*anisoterm_d(i,_d,xx);
// f
J(i,kk) = p(1)*(anisoterm(i,_d,xx)-isoterm(i,_d)) * two_pi*sign(p(kk))*1/(1+p(kk)*p(kk));
// th
J(i,kk+1) = p(1)*fs(k)*anisoterm_th(i,_d,xx,p(kk+1),p(kk+2)); // ph
J(i,kk+2) = p(1)*fs(k)*anisoterm_ph(i,_d,xx,p(kk+1),p(kk+2));
}
if (m_include_f0){
float temp_f0=x2f(p(nparams));
//derivative with respect to f0
J(i,nparams)= p(1)*(1-isoterm(i,_d)) * two_pi*sign(p(nparams))*1/(1+p(nparams)*p(nparams));
sig=p(1)*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_d)+sig);
J(i,2) += (p(2)>0?1.0:-1.0)*p(1)*(1-sumf-temp_f0)*isoterm_d(i,_d);
}
else{
sig = p(1)*((1-sumf)*isoterm(i,_d)+sig);
J(i,2) += (p(2)>0?1.0:-1.0)*p(1)*(1-sumf)*isoterm_d(i,_d);
}
diff(i) = sig - Y(i);
J(i,1) = sig/p(1);
}
gradv = 2*J.t()*diff;
gradv.Release();
return gradv;
}
//this uses Gauss-Newton approximation
boost::shared_ptr<BFMatrix> PVM_single::hess(const NEWMAT::ColumnVector& p,boost::shared_ptr<BFMatrix> iptr)const{
//cout<<"HESS"<<endl;
//OUT(p.t());
boost::shared_ptr<BFMatrix> hessm;
if (iptr && iptr->Nrows()==(unsigned int)p.Nrows() && iptr->Ncols()==(unsigned int)p.Nrows()) hessm = iptr;
else hessm = boost::shared_ptr<BFMatrix>(new FullBFMatrix(p.Nrows(),p.Nrows()));
float _d = std::abs(p(2));
////////////////////////////////////
ColumnVector fs(nfib);
Matrix x(nfib,3);ColumnVector xx(3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
fs(k) = x2f(p(kk));
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1));
}
////////////////////////////////////
Matrix J(npts,nparams);
float sig;
for(int i=1;i<=Y.Nrows();i++){
sig = 0;
J.Row(i)=0;
for(int k=1;k<=nfib;k++){
int kk = 3+3*(k-1);
xx = x.Row(k).t();
sig += fs(k)*anisoterm(i,_d,xx);
// other stuff for derivatives
// d
J(i,2) += (p(2)>0?1.0:-1.0)*p(1)*fs(k)*anisoterm_d(i,_d,xx);
// f
J(i,kk) = p(1)*(anisoterm(i,_d,xx)-isoterm(i,_d)) * two_pi*sign(p(kk))*1/(1+p(kk)*p(kk));
// th
J(i,kk+1) = p(1)*fs(k)*anisoterm_th(i,_d,xx,p(kk+1),p(kk+2));
// ph
J(i,kk+2) = p(1)*fs(k)*anisoterm_ph(i,_d,xx,p(kk+1),p(kk+2));
}
if (m_include_f0){
float temp_f0=x2f(p(nparams));
//derivative with respect to f0
J(i,nparams)= p(1)*(1-isoterm(i,_d)) * two_pi*sign(p(nparams))*1/(1+p(nparams)*p(nparams)); sig=p(1)*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_d)+sig);
J(i,2) += (p(2)>0?1.0:-1.0)*p(1)*(1-sumf-temp_f0)*isoterm_d(i,_d);
}
else{
sig = p(1)*((1-sumf)*isoterm(i,_d)+sig);
J(i,2) += (p(2)>0?1.0:-1.0)*p(1)*(1-sumf)*isoterm_d(i,_d);
}
J(i,1) = sig/p(1);
}
for (int i=1; i<=p.Nrows(); i++){
for (int j=i; j<=p.Nrows(); j++){
sig = 0.0;
for(int k=1;k<=J.Nrows();k++)
sig += 2*(J(k,i)*J(k,j));
hessm->Set(i,j,sig);
}
}
for (int j=1; j<=p.Nrows(); j++) {
for (int i=j+1; i<=p.Nrows(); i++) {
hessm->Set(i,j,hessm->Peek(j,i));
}
}
//hessm->Print();
//cout<<"----"<<endl;
return(hessm);
}
////////////////////////////////////////////////
// PARTIAL VOLUME MODEL - MULTIPLE SHELLS
////////////////////////////////////////////////
void PVM_multi::fit(){
// initialise with simple pvm
PVM_single_c pvm1(Y,bvecs,bvals,nfib,false,m_include_f0);
pvm1.fit();
//cout<<"Init with single"<<endl;
//pvm1.print();
float _a,_b;
_a = 1.0; // start with d=d_std
_b = pvm1.get_d();
ColumnVector start(nparams);
start(1) = pvm1.get_s0();
start(2) = _a;
start(3) = _b;
for(int i=1,ii=4;i<=nfib;i++,ii+=3){
start(ii) = f2x(pvm1.get_f(i));
start(ii+1) = pvm1.get_th(i);
start(ii+2) = pvm1.get_ph(i);
}
if (m_include_f0)
start(nparams)=f2x(pvm1.get_f0());
// do the fit
NonlinParam lmpar(start.Nrows(),NL_LM);
lmpar.SetGaussNewtonType(LM_L);
lmpar.SetStartingEstimate(start);
NonlinOut status;
status = nonlin(lmpar,(*this));
ColumnVector final_par(nparams);
final_par = lmpar.Par();
// finalise parameters
m_s0 = final_par(1);
m_d = std::abs(final_par(2)*final_par(3));
m_d_std = std::sqrt(std::abs(final_par(2)*final_par(3)*final_par(3)));
for(int i=4,k=1;k<=nfib;i+=3,k++){
m_f(k) = x2f(final_par(i));
m_th(k) = final_par(i+1);
m_ph(k) = final_par(i+2);
}
if (m_include_f0)
m_f0=x2f(final_par(nparams));
sort();
fix_fsum();
}
void PVM_multi::sort(){
vector< pair<float,int> > fvals(nfib);
ColumnVector ftmp(nfib),thtmp(nfib),phtmp(nfib);
ftmp=m_f;thtmp=m_th;phtmp=m_ph;
for(int i=1;i<=nfib;i++){
pair<float,int> p(m_f(i),i);
fvals[i-1] = p;
}
std::sort(fvals.begin(),fvals.end());
for(int i=1,ii=nfib-1;ii>=0;i++,ii--){
m_f(i) = ftmp(fvals[ii].second);
m_th(i) = thtmp(fvals[ii].second);
m_ph(i) = phtmp(fvals[ii].second);
}
}
void PVM_multi::fix_fsum(){
float sumf=0;
if (m_include_f0)
sumf=m_f0;
for(int i=1;i<=nfib;i++){
if (m_f(i)==0) m_f(i)=FSMALL;
sumf+=m_f(i);
if(sumf>=1){for(int j=i;j<=nfib;j++)m_f(j)=FSMALL;break;}
}
}
ReturnMatrix PVM_multi::get_prediction()const{
ColumnVector pred(npts);
ColumnVector p(nparams);
p(1) = m_s0;
p(2) = m_d*m_d/m_d_std/m_d_std;
p(3) = m_d_std*m_d_std/m_d; // =1/beta
for(int k=1;k<=nfib;k++){
int kk = 4+3*(k-1);
p(kk) = f2x(m_f(k));
p(kk+1) = m_th(k);
p(kk+2) = m_ph(k);
}
if (m_include_f0)
p(nparams)=f2x(m_f0);
pred = forwardModel(p);
pred.Release();
return pred;
}
NEWMAT::ReturnMatrix PVM_multi::forwardModel(const NEWMAT::ColumnVector& p)const{
ColumnVector pred(npts);
pred = 0;
float val;
float _a = std::abs(p(2));
float _b = std::abs(p(3)); ////////////////////////////////////
ColumnVector fs(nfib);
Matrix x(nfib,3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 4+3*(k-1);
fs(k) = x2f(p(kk));
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1));
}
////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
val = 0.0;
for(int k=1;k<=nfib;k++){
val += fs(k)*anisoterm(i,_a,_b,x.Row(k).t());
}
if (m_include_f0){
float temp_f0=x2f(p(nparams));
pred(i) = std::abs(p(1))*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_a,_b)+val);
}
else
pred(i) = std::abs(p(1))*((1-sumf)*isoterm(i,_a,_b)+val);
}
pred.Release();
return pred;
}
double PVM_multi::cf(const NEWMAT::ColumnVector& p)const{
//cout<<"CF"<<endl;
//OUT(p.t());
double cfv = 0.0;
double err;
float _a = std::abs(p(2));
float _b = std::abs(p(3));
////////////////////////////////////
ColumnVector fs(nfib);
Matrix x(nfib,3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 4+3*(k-1);
fs(k) = x2f(p(kk));
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1));
}
////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
err = 0.0;
for(int k=1;k<=nfib;k++){
err += fs(k)*anisoterm(i,_a,_b,x.Row(k).t());
}
if (m_include_f0){
float temp_f0=x2f(p(nparams));
err = (std::abs(p(1))*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_a,_b)+err) - Y(i));
}
else
err = (std::abs(p(1))*((1-sumf)*isoterm(i,_a,_b)+err) - Y(i));
cfv += err*err;
}
//OUT(cfv);
//cout<<"----"<<endl;
return(cfv);
}
NEWMAT::ReturnMatrix PVM_multi::grad(const NEWMAT::ColumnVector& p)const{ //cout<<"GRAD"<<endl;
//OUT(p.t());
NEWMAT::ColumnVector gradv(p.Nrows());
gradv = 0.0;
float _a = std::abs(p(2));
float _b = std::abs(p(3));
////////////////////////////////////
ColumnVector fs(nfib);
Matrix x(nfib,3);ColumnVector xx(3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 4+3*(k-1);
fs(k) = x2f(p(kk));
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1));
}
////////////////////////////////////
Matrix J(npts,nparams);
ColumnVector diff(npts);
float sig;
for(int i=1;i<=Y.Nrows();i++){
sig = 0;
J.Row(i)=0;
for(int k=1;k<=nfib;k++){
int kk = 4+3*(k-1);
xx = x.Row(k).t();
sig += fs(k)*anisoterm(i,_a,_b,xx);
// other stuff for derivatives
// alpha
J(i,2) += (p(2)>0?1.0:-1.0)*std::abs(p(1))*fs(k)*anisoterm_a(i,_a,_b,xx);
// beta
J(i,3) += (p(3)>0?1.0:-1.0)*std::abs(p(1))*fs(k)*anisoterm_b(i,_a,_b,xx);
// f
J(i,kk) = std::abs(p(1))*(anisoterm(i,_a,_b,xx)-isoterm(i,_a,_b)) * two_pi*sign(p(kk))*1/(1+p(kk)*p(kk));
// th
J(i,kk+1) = std::abs(p(1))*fs(k)*anisoterm_th(i,_a,_b,xx,p(kk+1),p(kk+2));
// ph
J(i,kk+2) = std::abs(p(1))*fs(k)*anisoterm_ph(i,_a,_b,xx,p(kk+1),p(kk+2));
}
if (m_include_f0){
float temp_f0=x2f(p(nparams));
//derivative with respect to f0
J(i,nparams)= std::abs(p(1))*(1-isoterm(i,_a,_b))*two_pi*sign(p(nparams))*1/(1+p(nparams)*p(nparams));
sig=std::abs(p(1))*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_a,_b)+sig);
J(i,2) += (p(2)>0?1.0:-1.0)*std::abs(p(1))*(1-sumf-temp_f0)*isoterm_a(i,_a,_b);
J(i,3) += (p(3)>0?1.0:-1.0)*std::abs(p(1))*(1-sumf-temp_f0)*isoterm_b(i,_a,_b);
}
else{
sig = std::abs(p(1))*((1-sumf)*isoterm(i,_a,_b)+sig);
J(i,2) += (p(2)>0?1.0:-1.0)*std::abs(p(1))*(1-sumf)*isoterm_a(i,_a,_b);
J(i,3) += (p(3)>0?1.0:-1.0)*std::abs(p(1))*(1-sumf)*isoterm_b(i,_a,_b);
}
diff(i) = sig - Y(i);
J(i,1) = (p(1)>0?1.0:-1.0)*sig/p(1);
}
gradv = 2*J.t()*diff;
//OUT(gradv.t());
//cout<<"----"<<endl;
gradv.Release();
return gradv;
}
//this uses Gauss-Newton approximation
boost::shared_ptr<BFMatrix> PVM_multi::hess(const NEWMAT::ColumnVector& p,boost::shared_ptr<BFMatrix> iptr)const{
//cout<<"HESS"<<endl; //OUT(p.t());
boost::shared_ptr<BFMatrix> hessm;
if (iptr && iptr->Nrows()==(unsigned int)p.Nrows() && iptr->Ncols()==(unsigned int)p.Nrows()) hessm = iptr;
else hessm = boost::shared_ptr<BFMatrix>(new FullBFMatrix(p.Nrows(),p.Nrows()));
float _a = std::abs(p(2));
float _b = std::abs(p(3));
////////////////////////////////////
ColumnVector fs(nfib);
Matrix x(nfib,3);ColumnVector xx(3);
float sumf=0;
for(int k=1;k<=nfib;k++){
int kk = 4+3*(k-1);
fs(k) = x2f(p(kk));
sumf += fs(k);
x(k,1) = sin(p(kk+1))*cos(p(kk+2));
x(k,2) = sin(p(kk+1))*sin(p(kk+2));
x(k,3) = cos(p(kk+1));
}
////////////////////////////////////
Matrix J(npts,nparams);
float sig;
for(int i=1;i<=Y.Nrows();i++){
sig = 0;
J.Row(i)=0;
for(int k=1;k<=nfib;k++){
int kk = 4+3*(k-1);
xx = x.Row(k).t();
sig += fs(k)*anisoterm(i,_a,_b,xx);
// other stuff for derivatives
// change of variable
float cov = two_pi*sign(p(kk))*1/(1+p(kk)*p(kk));
// alpha
J(i,2) += (p(2)>0?1.0:-1.0)*std::abs(p(1))*fs(k)*anisoterm_a(i,_a,_b,xx);
// beta
J(i,3) += (p(3)>0?1.0:-1.0)*std::abs(p(1))*fs(k)*anisoterm_b(i,_a,_b,xx);
// f
J(i,kk) = std::abs(p(1))*(anisoterm(i,_a,_b,xx)-isoterm(i,_a,_b)) * cov;
// th
J(i,kk+1) = std::abs(p(1))*fs(k)*anisoterm_th(i,_a,_b,xx,p(kk+1),p(kk+2));
// ph
J(i,kk+2) = std::abs(p(1))*fs(k)*anisoterm_ph(i,_a,_b,xx,p(kk+1),p(kk+2));
}
if (m_include_f0){
float temp_f0=x2f(p(nparams));
//derivative with respect to f0
J(i,nparams)= std::abs(p(1))*(1-isoterm(i,_a,_b))*two_pi*sign(p(nparams))*1/(1+p(nparams)*p(nparams));
sig=std::abs(p(1))*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_a,_b)+sig);
J(i,2) += (p(2)>0?1.0:-1.0)*std::abs(p(1))*(1-sumf-temp_f0)*isoterm_a(i,_a,_b);
J(i,3) += (p(3)>0?1.0:-1.0)*std::abs(p(1))*(1-sumf-temp_f0)*isoterm_b(i,_a,_b);
}
else{
sig = std::abs(p(1))*((1-sumf)*isoterm(i,_a,_b)+sig);
J(i,2) += (p(2)>0?1.0:-1.0)*std::abs(p(1))*(1-sumf)*isoterm_a(i,_a,_b);
J(i,3) += (p(3)>0?1.0:-1.0)*std::abs(p(1))*(1-sumf)*isoterm_b(i,_a,_b);
}
J(i,1) = (p(1)>0?1.0:-1.0)*sig/p(1);
}
for (int i=1; i<=p.Nrows(); i++){
for (int j=i; j<=p.Nrows(); j++){
sig = 0.0;
for(int k=1;k<=J.Nrows();k++)
sig += 2*(J(k,i)*J(k,j));
hessm->Set(i,j,sig);
}
}
for (int j=1; j<=p.Nrows(); j++) {
for (int i=j+1; i<=p.Nrows(); i++) {
hessm->Set(i,j,hessm->Peek(j,i)); }
}
//hessm->Print();
//cout<<"----"<<endl;
return(hessm);
}
///////////////////////////////////////////////////////////////////////////
// FANNING MODEL - BALL & BINGHAMS
// Constrained Optimization for the diffusivity, fractions and their sum<1,
// and the Bingham eigenvalues
//////////////////////////////////////////////////////////////////////////
void PVM_Ball_Binghams::fit(){
// Fit the ball & stick first to initialize some of the parameters
PVM_single_c pvmbs(Y,bvecs,bvals,nfib,false,m_include_f0,true); //Return a fanning angle estimate as well
pvmbs.fit();
// pvmbs.print();
ColumnVector k1_init, w_init;
ColumnVector final_par(nparams);
double minRSS=1e20;
if (!m_gridsearch){ //Initialize the fanning eigenvalues using a grid or a set of intermediate values
k1_init.ReSize(1); k1_init<<20;
w_init.ReSize(1); w_init<<5.0;
}
else{
k1_init.ReSize(6); k1_init<< 10 << 20 << 50 << 100 << 500 << 1000;
w_init.ReSize(6); w_init<< 1 << 5 << 10 << 20 << 50 << 90;
}
for (int n1=1; n1<=k1_init.Nrows(); n1++)
for (int n2=1; n2<=w_init.Nrows(); n2++){
ColumnVector start(nparams);
ColumnVector fs(nfib); fs=0;
//Initialize the non-linear fitter. Transform all initial values to the unconstrained parameter space
start(1) = pvmbs.get_s0();
start(2) = d2lambda(pvmbs.get_d());
for(int n=1,i=3; n<=nfib; n++,i+=nparams_per_fibre){
fs(n)=pvmbs.get_f(n);
float tmpr=fs(n)/partial_fsum(fs,n-1);
if (tmpr>1) tmpr=1; //This can be true due to numerical errors
start(i) = f2beta(tmpr);
start(i+1) = pvmbs.get_th(n);
start(i+2) = pvmbs.get_ph(n);
start(i+3) = pvmbs.get_fanning_angle(n);
start(i+4) = k12l1(k1_init(n1));
start(i+5) = w2gam(w_init(n2));
}
if (m_include_f0){
float tmpr=pvmbs.get_f0()/partial_fsum(fs,nfib);
if (tmpr>1) tmpr=1; //This can be true due to numerical errors
start(nparams)=f2beta(tmpr);
}
// do the fit
NonlinParam lmpar(start.Nrows(),NL_LM);
lmpar.SetGaussNewtonType(LM_LM);
lmpar.SetStartingEstimate(start);
//lmpar.LogCF(true);
NonlinOut status;
status = nonlin(lmpar,(*this));
ColumnVector tmp_par(nparams);
tmp_par = lmpar.Par();
//cout<<"Number of Iterations: "<<lmpar.NIter()<<endl;
//vector<double> Cf=lmpar.CFHistory();
//for (int n=0; n<(int)Cf.size(); n++)
// cout<<Cf[n]<<" ";
//cout<<endl;
double RSS=cf(tmp_par); //get the sum of squared residuals
if (RSS<=minRSS){
final_par=tmp_par;
minRSS=RSS;
}
}
if (m_eval_BIC){
m_BIC=npts*log(minRSS/npts)+log(npts)*nparams; //evaluate BIC
//cout<<"RSS="<<RSS<<". BIC="<<m_BIC<<endl;
}
// finalise parameters
m_s0 = final_par(1);
m_d = lambda2d(final_par(2));
for(int n=1; n<=nfib; n++){
int kk=3+nparams_per_fibre*(n-1);
m_f(n) = beta2f(final_par(kk))*partial_fsum(m_f,n-1);
m_th(n) = final_par(kk+1);
m_ph(n) = final_par(kk+2);
m_psi(n)= final_par(kk+3);
m_k1(n) = l12k1(final_par(kk+4));
m_k2(n) = m_k1(n)/gam2w(final_par(kk+5));
}
if (m_include_f0)
m_f0=beta2f(final_par(nparams))*partial_fsum(m_f,nfib);
sort();
}
void PVM_Ball_Binghams::sort(){
vector< pair<float,int> > fvals(nfib);
ColumnVector ftmp(nfib),thtmp(nfib),phtmp(nfib),psitmp(nfib),k1tmp(nfib),k2tmp(nfib);
ftmp=m_f;thtmp=m_th;phtmp=m_ph; psitmp=m_psi; k1tmp=m_k1; k2tmp=m_k2;
for(int i=1;i<=nfib;i++){
pair<float,int> p(m_f(i),i);
fvals[i-1] = p;
}
std::sort(fvals.begin(),fvals.end());
for(int i=1,ii=nfib-1;ii>=0;i++,ii--){
m_f(i) = ftmp(fvals[ii].second);
m_th(i) = thtmp(fvals[ii].second);
m_ph(i) = phtmp(fvals[ii].second);
m_psi(i)= psitmp(fvals[ii].second);
m_k1(i)= k1tmp(fvals[ii].second);
m_k2(i)= k2tmp(fvals[ii].second);
}
}
//Returns 1-Sum(f_j), 1<=j<=ii. (ii<=nfib)
//Used for transforming beta to f and vice versa
float PVM_Ball_Binghams::partial_fsum(ColumnVector& fs, int ii) const{
float fsum=1.0;
for(int j=1;j<=ii;j++)
fsum-=fs(j);
if (fsum==0) //Very rare cases
fsum=tiny;
return fsum;
}
//Print the final estimates (after having them transformed)
void PVM_Ball_Binghams::print()const{
cout << endl<<"Ball & Bingham FIT RESULTS " << endl;
cout << "S0 :" << m_s0 << endl;
cout << "D :" << m_d << endl;
for(int i=1;i<=nfib;i++){
cout << "F" << i << " :" << m_f(i) << endl;
ColumnVector x(3),fan_vec;
x << sin(m_th(i))*cos(m_ph(i)) << sin(m_th(i))*sin(m_ph(i)) << cos(m_th(i));
fan_vec=get_fanning_vector(i);
float _th,_ph,_psi;cart2sph(x,_th,_ph); _psi=m_psi(i);
if(x(3)<0) {x=-x; _psi=M_PI-m_psi(i); }
cout << "TH" << i << " : " << _th*180.0/M_PI << " deg" << endl;
cout << "PH" << i << " : " << _ph*180.0/M_PI << " deg" << endl;
cout << "PSI" << i << " : " <<_psi<<endl;
cout << "DIR" << i << " : " << x(1) << " " << x(2) << " " << x(3) << endl;
cout << "FAN_DIR" << i << " : " << fan_vec(1) << " " << fan_vec(2) << " " << fan_vec(3) << endl;
cout << "K1_" << i << " : " <<m_k1(i)<<endl;
cout << "K2_" << i << " : " <<m_k2(i)<<endl;
}
if (m_include_f0)
cout << "F0 :" << m_f0 << endl;
if (m_eval_BIC)
cout<< "BIC :"<<m_BIC<<endl;
}
//Print the estimates using a vector that contains the transformed parameter values
//i.e. need to untransform them to get d,f's etc
void PVM_Ball_Binghams::print(const ColumnVector& p)const{
ColumnVector f(nfib);
cout << "PARAMETER VALUES " << endl;
cout << "S0 :" << p(1) << endl;
cout << "D :" << lambda2d(p(2)) << endl;
for(int i=3,ii=1;ii<=nfib;i+=3,ii++){
f(ii) = beta2f(p(i))*partial_fsum(f,ii-1);
float k1=l12k1(p(i+4));
cout << "F" << ii << " :" << f(ii) << endl;
cout << "TH" << ii << " :" << p(i+1)*180.0/M_PI << " deg" << endl;
cout << "PH" << ii << " :" << p(i+2)*180.0/M_PI << " deg" << endl;
cout << "PSI" << ii << " :"<< p(i+3)*180.0/M_PI << " deg" << endl;
cout << "k1_" << ii << " :"<< k1 << endl;
cout << "k2_" << ii << " :"<< k1/gam2w(p(i+4))<< endl;
}
if (m_include_f0)
cout << "F0 :" << beta2f(p(nparams))*partial_fsum(f,nfib);
}
//Applies the forward model and gets the model predicted signal using the estimated parameter values (true,non-transformed space)
ReturnMatrix PVM_Ball_Binghams::get_prediction()const{
ColumnVector pred(npts);
ColumnVector p(nparams);
ColumnVector fs(nfib);
fs=m_f;
p(1) = m_s0; //Transform parameters to the space where they are uncostrained
p(2) = d2lambda(m_d);
for(int i=3,ii=1;ii<=nfib;i+=nparams_per_fibre,ii++){
float tmpr=m_f(ii)/partial_fsum(fs,ii-1);
if (tmpr>1.0) tmpr=1; //This can be due to numerical errors
p(i) = f2beta(tmpr);
p(i+1) = m_th(ii);
p(i+2) = m_ph(ii); p(i+3) = m_psi(ii);
p(i+4) = k12l1(m_k1(ii));
p(i+5) = w2gam(m_k1(ii)/m_k2(ii));
}
if (m_include_f0){
float tmpr=m_f0/partial_fsum(fs,nfib);
if (tmpr>1.0) tmpr=1; //This can be due to numerical errors
p(nparams)=f2beta(tmpr);
}
pred = forwardModel(p);
pred.Release();
return pred;
}
//Applies the forward model and gets a model predicted signal using the parameter values in p (transformed parameter space)
NEWMAT::ReturnMatrix PVM_Ball_Binghams::forwardModel(const NEWMAT::ColumnVector& p)const{
ColumnVector pred(npts);
pred = 0;
float val;
float _d = lambda2d(p(2));
////////////////////////////////////
ColumnVector fs(nfib); ColumnVector temp_vec(3), denom(3);
Matrix Rpsi(3,3), Rth(3,3), Rph(3,3), R(3,3);
vector<Matrix> B; vector<ColumnVector> approx_denomB;
DiagonalMatrix L(3); SymmetricMatrix Q(3);
L=0; Rpsi=0; Rth=0; Rph=0; Rth(2,2)=1; Rph(3,3)=1; Rpsi(3,3)=1;
float sumf=0; fs=0;
for(int k=1;k<=nfib;k++){
int kk = 3+nparams_per_fibre*(k-1);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
sumf += fs(k);
float cosph, sinph,cospsi,sinpsi,costh,sinth,k1,k2;
costh=cos(p(kk+1)); sinth=sin(p(kk+1)); cosph=cos(p(kk+2)); sinph=sin(p(kk+2));
cospsi=cos(p(kk+3)); sinpsi=sin(p(kk+3));
k1=l12k1(p(kk+4)); k2=k1/gam2w(p(kk+5));
L(1)=-k1; L(2)=-k2; denom<<L(1)<<L(2)<<0;
Rth(1,1)=costh; Rth(1,3)=-sinth; Rth(3,1)=sinth; Rth(3,3)=costh;
Rph(1,1)=cosph; Rph(1,2)=sinph; Rph(2,1)=-sinph; Rph(2,2)=cosph;
Rpsi(1,1)=cospsi; Rpsi(1,2)=sinpsi; Rpsi(2,1)=-sinpsi; Rpsi(2,2)=cospsi;
R=Rpsi*Rth*Rph;
R<<R.t()*L*R;
B.push_back(R);
temp_vec=approx_denominatorB(denom);
approx_denomB.push_back(temp_vec);
}
////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
val = 0.0;
for(int k=1;k<=nfib;k++){
Q<<B[k-1]-_d*bvecs_dyadic[i-1];
EigenValues(Q,L); temp_vec<<L(1)<<L(2)<<L(3);
val += fs(k)*hyp_SratioB_knowndenom(temp_vec,approx_denomB[k-1]);
}
if (m_include_f0){
float temp_f0=beta2f(p(nparams))*partial_fsum(fs,nfib);
pred(i) = p(1)*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_d)+val);
}
else
pred(i) = p(1)*((1-sumf)*isoterm(i,_d)+val);
}
pred.Release();
return pred;
}
//Instead of returning the model predicted signal for each direction
//returns the individual signal contributions i.e. isotropic, anisotropic1, anisotropic2,etc. //Weighting with the fractions, scaling with S0 and summing those gives the signal.
//A Matrix npts x (nfib+1) is returned
NEWMAT::ReturnMatrix PVM_Ball_Binghams::forwardModel_compartments(const NEWMAT::ColumnVector& p) const{
Matrix Sig(npts,nfib+1);
float _d = lambda2d(p(2));
////////////////////////////////////
ColumnVector fs(nfib); ColumnVector temp_vec(3), denom(3);
Matrix Rpsi(3,3), Rth(3,3), Rph(3,3), R(3,3);
vector<Matrix> B; vector<ColumnVector> approx_denomB;
DiagonalMatrix L(3); SymmetricMatrix Q(3);
L=0; Rpsi=0; Rth=0; Rph=0; Rth(2,2)=1; Rph(3,3)=1; Rpsi(3,3)=1;
float sumf=0; fs=0;
for(int k=1;k<=nfib;k++){
int kk = 3+nparams_per_fibre*(k-1);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
sumf += fs(k);
float cosph, sinph,cospsi,sinpsi,costh,sinth,k1,k2;
costh=cos(p(kk+1)); sinth=sin(p(kk+1)); cosph=cos(p(kk+2)); sinph=sin(p(kk+2));
cospsi=cos(p(kk+3)); sinpsi=sin(p(kk+3));
k1=l12k1(p(kk+4)); k2=k1/gam2w(p(kk+5));
L(1)=-k1; L(2)=-k2; denom<<L(1)<<L(2)<<0;
Rth(1,1)=costh; Rth(1,3)=-sinth; Rth(3,1)=sinth; Rth(3,3)=costh;
Rph(1,1)=cosph; Rph(1,2)=sinph; Rph(2,1)=-sinph; Rph(2,2)=cosph;
Rpsi(1,1)=cospsi; Rpsi(1,2)=sinpsi; Rpsi(2,1)=-sinpsi; Rpsi(2,2)=cospsi;
R=Rpsi*Rth*Rph;
R<<R.t()*L*R;
B.push_back(R);
temp_vec=approx_denominatorB(denom);
approx_denomB.push_back(temp_vec);
}
////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
Sig(i,1) = isoterm(i,_d);
for(int k=1;k<=nfib;k++){
Q<<B[k-1]-_d*bvecs_dyadic[i-1];
EigenValues(Q,L); temp_vec<<L(1)<<L(2)<<L(3);
Sig(i,k+1)= hyp_SratioB_knowndenom(temp_vec,approx_denomB[k-1]);
}
}
Sig.Release();
return Sig;
}
//Builds up the model predicted signal for each direction by using precomputed individual compartment signals, stored in Matrix Sig.
//Weights them with the fractions, scales with S0 and sums to get the signal.
NEWMAT::ReturnMatrix PVM_Ball_Binghams::pred_from_compartments(const NEWMAT::ColumnVector& p, const NEWMAT::Matrix& Sig) const{
ColumnVector pred(npts);
float val;
ColumnVector fs(nfib);
float sumf=0; fs=0;
for(int k=1;k<=nfib;k++){
int kk = 3+nparams_per_fibre*(k-1);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
sumf += fs(k);
}
for(int i=1;i<=Y.Nrows();i++){
val = 0.0;
for(int k=1;k<=nfib;k++)
val += fs(k)*Sig(i,k+1);
if (m_include_f0){
float temp_f0=beta2f(p(nparams))*partial_fsum(fs,nfib);
pred(i) = p(1)*(temp_f0+(1-sumf-temp_f0)*Sig(i,1)+val); }
else
pred(i) = p(1)*((1-sumf)*Sig(i,1)+val);
}
pred.Release();
return pred;
}
//Builds up the model predicted signal for each direction by using precomputed individual compartment signals, stored in Matrix Sig.
//Weights them with the fractions, scales with S0 and sums to get the signal.
//The signal of the fibre compartment with index fib is recalculated.
NEWMAT::ReturnMatrix PVM_Ball_Binghams::pred_from_compartments(const NEWMAT::ColumnVector& p, const NEWMAT::Matrix& Sig,const int& fib) const{
ColumnVector pred(npts); Matrix newSig;
float val;
float _d = lambda2d(p(2));
////////////////////////////////////
ColumnVector fs(nfib); ColumnVector temp_vec(3), denom(3);
Matrix Rpsi(3,3), Rth(3,3), Rph(3,3), R(3,3);
DiagonalMatrix L(3); SymmetricMatrix Q(3);
L=0; Rpsi=0; Rth=0; Rph=0; Rth(2,2)=1; Rph(3,3)=1; Rpsi(3,3)=1;
float sumf=0; fs=0;
for(int k=1;k<=nfib;k++){
int kk = 3+nparams_per_fibre*(k-1);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
sumf += fs(k);
}
///////////////////////////////////////
int kk = 3+nparams_per_fibre*(fib-1); float cosph, sinph,cospsi,sinpsi,costh,sinth,k1,k2;
costh=cos(p(kk+1)); sinth=sin(p(kk+1)); cosph=cos(p(kk+2)); sinph=sin(p(kk+2));
cospsi=cos(p(kk+3)); sinpsi=sin(p(kk+3));
k1=l12k1(p(kk+4)); k2=k1/gam2w(p(kk+5));
L(1)=-k1; L(2)=-k2; denom<<L(1)<<L(2)<<0;
Rth(1,1)=costh; Rth(1,3)=-sinth; Rth(3,1)=sinth; Rth(3,3)=costh;
Rph(1,1)=cosph; Rph(1,2)=sinph; Rph(2,1)=-sinph; Rph(2,2)=cosph;
Rpsi(1,1)=cospsi; Rpsi(1,2)=sinpsi; Rpsi(2,1)=-sinpsi; Rpsi(2,2)=cospsi;
R=Rpsi*Rth*Rph;
R<<R.t()*L*R;
temp_vec=approx_denominatorB(denom);
denom=temp_vec;
newSig=Sig; //Get the new Signal for compartment fib
for(int i=1;i<=Y.Nrows();i++){
Q<<R-_d*bvecs_dyadic[i-1];
EigenValues(Q,L); temp_vec<<L(1)<<L(2)<<L(3);
newSig(i,fib+1)=hyp_SratioB_knowndenom(temp_vec,denom);
}
///////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
val = 0.0;
for(int k=1;k<=nfib;k++)
val += fs(k)*newSig(i,k+1);
if (m_include_f0){
float temp_f0=beta2f(p(nparams))*partial_fsum(fs,nfib);
pred(i) = p(1)*(temp_f0+(1-sumf-temp_f0)*newSig(i,1)+val);
}
else
pred(i) = p(1)*((1-sumf)*newSig(i,1)+val);
}
pred.Release();
return pred;
}
//Cost Function, sum of squared residuals
//assume that parameter values p are transformed (e.g. need to untransform them to get d, f's,etc)
double PVM_Ball_Binghams::cf(const NEWMAT::ColumnVector& p)const{
double cfv = 0.0;
double err;
ColumnVector S;
S=forwardModel(p); //Model predictions
for(int i=1;i<=npts;i++){
err=S(i)-Y(i); //Residual
cfv+=err*err; //Sum of squared residuals
}
//cout<<"CF="<<cfv<<endl; OUT(p.t());
return(cfv);
}
/*
//Slower implementation
NEWMAT::ReturnMatrix PVM_Ball_Binghams::grad(const ColumnVector& p)const
{
ColumnVector gradv(nparams);//This is the gradient of the cost function
Matrix J(npts,nparams); //This is the Jacobian matrix of the model equation
//The derivative of the cost function w.r.t. parameter j
//will then be: Grad_j=Sum(2*(F(x_i)-Y_i)*J(i,j)), with Sum across data points i
ColumnVector diff(npts); //Residuals
ColumnVector p_plus_h, S_trial,S;
//Compute the Jacobian first using finite differences for each element
double step;
ColumnVector typical_scale(nparams);
typical_scale=1;
for (int k=1; k<=nfib; k++){
int kk = 3+nparams_per_fibre*(k-1);
typical_scale(kk+4)=100;
typical_scale(kk+5)=1;
}
S=forwardModel(p);
diff=S-Y;
for (int i=1; i<=npts; i++)
J(i,1)=S(i)/p(1); //derivatives with respect to S0 are analytic: S_i/S0
for (int n=2; n<=nparams; n++) {
p_plus_h = p;
step = SQRTtiny*nonzerosign(p(n))*max(fabs(p(n))*typical_scale(n),1.0);
step = nonzerosign(tiny)*min(max(fabs(step),1.0e-8),0.1); //check that 1e-8<step<0.1
p_plus_h(n)=p(n)+step;
S_trial=forwardModel(p_plus_h);
for (int i=1; i<=npts; i++)
J(i,n)=(S_trial(i)-S(i))/step;
}
gradv =2*J.t()*diff;
gradv.Release();
return(gradv);
}
//Slower implementation
//this uses Gauss-Newton approximation
boost::shared_ptr<BFMatrix> PVM_Ball_Binghams::hess(const NEWMAT::ColumnVector& p,boost::shared_ptr<BFMatrix> iptr)const{
boost::shared_ptr<BFMatrix> hessm;
if (iptr && iptr->Nrows()==(unsigned int)p.Nrows() && iptr->Ncols()==(unsigned int)p.Nrows()) hessm = iptr;
else hessm = boost::shared_ptr<BFMatrix>(new FullBFMatrix(p.Nrows(),p.Nrows()));
Matrix J(npts,nparams); //This is the Jacobian matrix of the model equation ColumnVector p_plus_h, S_trial,S;
//Compute the Jacobian first using finite differences for each element
double step,sig;
ColumnVector typical_scale(nparams);
typical_scale=1;
for (int k=1; k<=nfib; k++){
int kk = 3+nparams_per_fibre*(k-1);
typical_scale(kk+4)=100;
typical_scale(kk+5)=1;
}
S=forwardModel(p);
for (int i=1; i<=npts; i++)
J(i,1)=S(i)/p(1); //derivatives with respect to S0 are analytic: S_i/S0
for (int n=2; n<=nparams; n++) {
p_plus_h = p;
step = SQRTtiny*nonzerosign(p(n))*max(fabs(p(n))*typical_scale(n),1.0);
step = nonzerosign(tiny)*min(max(fabs(step),1.0e-8),0.1); //check that 1e-8<step<0.1
p_plus_h(n)=p(n)+step;
S_trial=forwardModel(p_plus_h);
for (int i=1; i<=npts; i++)
J(i,n)=(S_trial(i)-S(i))/step;
}
for (int i=1; i<=p.Nrows(); i++){
for (int j=i; j<=p.Nrows(); j++){
sig = 0.0;
for(int k=1;k<=J.Nrows();k++)
sig += 2*(J(k,i)*J(k,j));
hessm->Set(i,j,sig);
}
}
for (int j=1; j<=p.Nrows(); j++) {
for (int i=j+1; i<=p.Nrows(); i++) {
hessm->Set(i,j,hessm->Peek(j,i));
}
}
return(hessm);
}
*/
NEWMAT::ReturnMatrix PVM_Ball_Binghams::grad(const ColumnVector& p)const
{
ColumnVector gradv(nparams);//This is the gradient of the cost function
Matrix J(npts,nparams); //This is the Jacobian matrix of the model equation
//The derivative of the cost function w.r.t. parameter j
//will then be: Grad_j=Sum(2*(F(x_i)-Y_i)*J(i,j)), with Sum across data points i
ColumnVector diff(npts); //Residuals
ColumnVector p_plus_h, S_trial,S;
Matrix Sig;
ColumnVector fs(nfib), bs(nfib);
//Compute the Jacobian first using finite differences for each element. Derivatives are analytic for S0 and the volume fractions
double step;
ColumnVector typical_scale(nparams);
typical_scale=1;
for (int k=1; k<=nfib; k++){
int kk = 3+nparams_per_fibre*(k-1);
bs(k)=p(kk);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
typical_scale(kk+4)=100;
typical_scale(kk+5)=1;
}
//Compute the derivatives with respect to betas, i.e the transformed volume fraction variables Matrix f_deriv;
f_deriv=fractions_deriv(nfib, fs, bs);
Sig=forwardModel_compartments(p);
S=pred_from_compartments(p, Sig);
diff=S-Y;
for (int i=1; i<=npts; i++)
J(i,1)=S(i)/p(1); //derivatives with respect to S0 are analytic: S_i/S0
for (int n=2; n<=nparams; n++) {
p_plus_h = p;
step = SQRTtiny*nonzerosign(p(n))*max(fabs(p(n))*typical_scale(n),1.0);
step = nonzerosign(tiny)*min(max(fabs(step),1.0e-8),0.1); //check that 1e-8<step<0.1
p_plus_h(n)=p(n)+step;
if (n<3 || (n==nparams && m_include_f0)){ //for d all compartments will change. Also for f0, if included. Use for those params numerical differentiation
S_trial=forwardModel(p_plus_h);
for (int i=1; i<=npts; i++)
J(i,n)=(S_trial(i)-S(i))/step;
}
else{ //for the other params, update only the signal of the relevant fibre compartment
int fib_indx=(int)ceil((float)(n-2)/(float)nparams_per_fibre); //index indicating in which fibre compartment parameter n belongs to
if (n==3+nparams_per_fibre*(fib_indx-1)){ //Then we have a volume fraction, derivative is analytic.
for (int i=1; i<=npts; i++){
J(i,n)=0;
for (int j=1; j<=nfib; j++){
if (f_deriv(j,fib_indx)!=0)
J(i,n) += p(1)*(Sig(i,j+1)-Sig(i,1))*f_deriv(j,fib_indx);
}
}
}
else{ //for all other params, use numerical differentiation
S_trial=pred_from_compartments(p_plus_h, Sig,fib_indx);
for (int i=1; i<=npts; i++){
J(i,n)=(S_trial(i)-S(i))/step;
//if (J(i,n)==0) cout<<"Zero gradient!!"<<endl;//J(i,n)=1e-8; //stabilize LM in case differentiation has failed due to step size
}
}
}
}
gradv =2*J.t()*diff;
gradv.Release();
return(gradv);
}
//this uses Gauss-Newton approximation
boost::shared_ptr<BFMatrix> PVM_Ball_Binghams::hess(const NEWMAT::ColumnVector& p,boost::shared_ptr<BFMatrix> iptr)const{
boost::shared_ptr<BFMatrix> hessm;
if (iptr && iptr->Nrows()==(unsigned int)p.Nrows() && iptr->Ncols()==(unsigned int)p.Nrows()) hessm = iptr;
else hessm = boost::shared_ptr<BFMatrix>(new FullBFMatrix(p.Nrows(),p.Nrows()));
Matrix J(npts,nparams); //This is the Jacobian matrix of the model equation
ColumnVector p_plus_h, S_trial,S, fs(nfib), bs(nfib);
Matrix Sig;
//Compute the Jacobian first using finite differences for each element. Derivatives are analytic for S0 and the volume fractions
double step,sigt;
ColumnVector typical_scale(nparams);
typical_scale=1;
for (int k=1; k<=nfib; k++){
int kk = 3+nparams_per_fibre*(k-1);
bs(k)=p(kk);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
typical_scale(kk+4)=100;
typical_scale(kk+5)=1;
}
//Compute the derivatives with respect to betas, i.e the transformed volume fraction variables Matrix f_deriv;
f_deriv=fractions_deriv(nfib, fs, bs);
Sig=forwardModel_compartments(p);
S=pred_from_compartments(p, Sig);
for (int i=1; i<=npts; i++)
J(i,1)=S(i)/p(1); //derivatives with respect to S0 are analytic: S_i/S0
for (int n=2; n<=nparams; n++) {
p_plus_h = p;
step = SQRTtiny*nonzerosign(p(n))*max(fabs(p(n))*typical_scale(n),1.0);
step = nonzerosign(tiny)*min(max(fabs(step),1.0e-8),0.1); //check that 1e-8<step<0.1
p_plus_h(n)=p(n)+step;
if (n<3 || (n==nparams && m_include_f0)){ //for d all compartments will change. Also for f0, if included. Use for those params numerical differentiation
S_trial=forwardModel(p_plus_h);
for (int i=1; i<=npts; i++)
J(i,n)=(S_trial(i)-S(i))/step;
}
else{ //for the other params, update only the signal of the relevant fibre compartment
int fib_indx=(int)ceil((float)(n-2)/(float)nparams_per_fibre); //index indicating in which fibre compartment parameter n belongs to
if (n==3+nparams_per_fibre*(fib_indx-1)){ //Then we have a volume fraction, derivative is analytic.
for (int i=1; i<=npts; i++){
J(i,n)=0;
for (int j=1; j<=nfib; j++){
if (f_deriv(j,fib_indx)!=0)
J(i,n) += p(1)*(Sig(i,j+1)-Sig(i,1))*f_deriv(j,fib_indx);
}
}
}
else{ //for all other params, use numerical differentiation
S_trial=pred_from_compartments(p_plus_h, Sig,fib_indx);
for (int i=1; i<=npts; i++){
J(i,n)=(S_trial(i)-S(i))/step;
//if (J(i,n)==0) J(i,n)=1e-8; //stabilize LM in case differentiation has failed due to step size
}
}
}
}
for (int i=1; i<=p.Nrows(); i++){
for (int j=i; j<=p.Nrows(); j++){
sigt = 0.0;
for(int k=1;k<=J.Nrows();k++)
sigt += 2*(J(k,i)*J(k,j));
hessm->Set(i,j,sigt);
}
}
for (int j=1; j<=p.Nrows(); j++) {
for (int i=j+1; i<=p.Nrows(); i++) {
hessm->Set(i,j,hessm->Peek(j,i));
}
}
return(hessm);
}
float PVM_Ball_Binghams::isoterm(const int& pt,const float& _d)const{
return(std::exp(-bvals(1,pt)*_d));
}
NEWMAT::ReturnMatrix PVM_Ball_Binghams::fractions_deriv(const int& nfib, const ColumnVector& fs, const ColumnVector& bs) const{
NEWMAT::Matrix Deriv(nfib,nfib);
float fsum;
Deriv=0;
for (int j=1; j<=nfib; j++)
for (int k=1; k<=nfib; k++){ if (j==k){
fsum=1;
for (int n=1; n<=j-1; n++)
fsum-=fs(n);
Deriv(j,k)=sin(2*bs(k))*fsum;
}
else if (j>k){
fsum=0;
for (int n=1; n<=j-1; n++)
fsum+=Deriv(n,k);
Deriv(j,k)=-pow(sin(bs(j)),2.0)*fsum;
}
}
Deriv.Release();
return Deriv;
}
//Returns a vector that indicates the fanning orientation
NEWMAT::ReturnMatrix PVM_Ball_Binghams:: get_fanning_vector(const int& i) const{
ColumnVector fan_vec(3);
float t_th=m_th(i); float t_ph=m_ph(i); float t_psi=m_psi(i);
float costh=cos(t_th); float sinth=sin(t_th);
float cosph=cos(t_ph); float sinph=sin(t_ph);
float cospsi=cos(t_psi); float sinpsi=sin(t_psi);
/*
Matrix Rpsi(3,3), Rth(3,3), Rph(3,3), R(3,3);
Rpsi=0; Rth=0; Rph=0; Rth(2,2)=1; Rph(3,3)=1; Rpsi(3,3)=1;
Rth(1,1)=costh; Rth(1,3)=-sinth; Rth(3,1)=sinth; Rth(3,3)=costh;
Rph(1,1)=cosph; Rph(1,2)=sinph; Rph(2,1)=-sinph; Rph(2,2)=cosph;
Rpsi(1,1)=cospsi; Rpsi(1,2)=sinpsi; Rpsi(2,1)=-sinpsi; Rpsi(2,2)=cospsi;
R=Rpsi*Rth*Rph; */
//fan_vec(1)=R(2,1); fan_vec(2)=R(2,2); fan_vec(3)=R(2,3);
fan_vec(1)=-sinpsi*costh*cosph-cospsi*sinph; fan_vec(2)=-sinpsi*costh*sinph+cospsi*cosph; fan_vec(3)=sinpsi*sinth;
fan_vec.Release();
return fan_vec;
}
///////////////////////////////////////////////////////////////////////////
// FANNING MODEL - BALL & WATSONS
// Constrained Optimization for the diffusivity, fractions and their sum<1,
// and the Bingham eigenvalues
//////////////////////////////////////////////////////////////////////////
void PVM_Ball_Watsons::fit(){
// Fit the ball & stick first to initialize some of the parameters
PVM_single_c pvmbs(Y,bvecs,bvals,nfib,false,m_include_f0);
pvmbs.fit();
// pvmbs.print();
ColumnVector k_init;
ColumnVector final_par(nparams);
double minRSS=1e20;
if (!m_gridsearch){
k_init.ReSize(1); k_init<<20;
}
else{
k_init.ReSize(6); k_init<< 10 << 20 << 50 << 100 << 500 << 1000;
}
for (int n1=1; n1<=k_init.Nrows(); n1++){
ColumnVector start(nparams);
ColumnVector fs(nfib); fs=0; //Initialize the non-linear fitter. Transform all initial values to the uncostrained parameter space
start(1) = pvmbs.get_s0();
start(2) = d2lambda(pvmbs.get_d());
for(int n=1,i=3; n<=nfib; n++,i+=nparams_per_fibre){
fs(n)=pvmbs.get_f(n);
float tmpr=fs(n)/partial_fsum(fs,n-1);
if (tmpr>1) tmpr=1; //This can be true due to numerical errors
start(i) = f2beta(tmpr);
start(i+1) = pvmbs.get_th(n);
start(i+2) = pvmbs.get_ph(n);
start(i+3) = k12l1(k_init(n1));
}
if (m_include_f0){
float tmpr=pvmbs.get_f0()/partial_fsum(fs,nfib);
if (tmpr>1) tmpr=1; //This can be true due to numerical errors
start(nparams)=f2beta(tmpr);
}
// do the fit
NonlinParam lmpar(start.Nrows(),NL_LM);
lmpar.SetGaussNewtonType(LM_LM);
lmpar.SetStartingEstimate(start);
//lmpar.LogCF(true);
NonlinOut status;
status = nonlin(lmpar,(*this));
ColumnVector tmp_par(nparams);
tmp_par = lmpar.Par();
/*cout<<"Number of Iterations: "<<lmpar.NIter()<<endl;
vector<double> Cf=lmpar.CFHistory();
for (int n=0; n<(int)Cf.size(); n++)
cout<<Cf[n]<<" ";
cout<<endl;
*/
double RSS=cf(tmp_par); //get the sum of squared residuals
if (RSS<=minRSS){
final_par=tmp_par;
minRSS=RSS;
}
}
if (m_eval_BIC){
m_BIC=npts*log(minRSS/npts)+log(npts)*nparams; //evaluate BIC
}
// finalise parameters
m_s0 = final_par(1);
m_d = lambda2d(final_par(2));
for(int n=1; n<=nfib; n++){
int kk=3+nparams_per_fibre*(n-1);
m_f(n) = beta2f(final_par(kk))*partial_fsum(m_f,n-1);
m_th(n) = final_par(kk+1);
m_ph(n) = final_par(kk+2);
m_k(n) = l12k1(final_par(kk+3));
}
if (m_include_f0)
m_f0=beta2f(final_par(nparams))*partial_fsum(m_f,nfib);
sort();
}
void PVM_Ball_Watsons::sort(){
vector< pair<float,int> > fvals(nfib);
ColumnVector ftmp(nfib),thtmp(nfib),phtmp(nfib),ktmp(nfib); ftmp=m_f;thtmp=m_th;phtmp=m_ph; ktmp=m_k;
for(int i=1;i<=nfib;i++){
pair<float,int> p(m_f(i),i);
fvals[i-1] = p;
}
std::sort(fvals.begin(),fvals.end());
for(int i=1,ii=nfib-1;ii>=0;i++,ii--){
m_f(i) = ftmp(fvals[ii].second);
m_th(i) = thtmp(fvals[ii].second);
m_ph(i) = phtmp(fvals[ii].second);
m_k(i)= ktmp(fvals[ii].second);
}
}
//Returns 1-Sum(f_j), 1<=j<=ii. (ii<=nfib)
//Used for transforming beta to f and vice versa
float PVM_Ball_Watsons::partial_fsum(ColumnVector& fs, int ii) const{
float fsum=1.0;
for(int j=1;j<=ii;j++)
fsum-=fs(j);
if (fsum==0) //Very rare cases
fsum=tiny;
return fsum;
}
//Print the final estimates (after having them transformed)
void PVM_Ball_Watsons::print()const{
cout << endl<<"Ball & Watson FIT RESULTS " << endl;
cout << "S0 :" << m_s0 << endl;
cout << "D :" << m_d << endl;
for(int i=1;i<=nfib;i++){
cout << "F" << i << " :" << m_f(i) << endl;
ColumnVector x(3);
x << sin(m_th(i))*cos(m_ph(i)) << sin(m_th(i))*sin(m_ph(i)) << cos(m_th(i));
float _th,_ph;cart2sph(x,_th,_ph);
if(x(3)<0) x=-x;
cout << "TH" << i << " : " << _th*180.0/M_PI << " deg" << endl;
cout << "PH" << i << " : " << _ph*180.0/M_PI << " deg" << endl;
cout << "DIR" << i << " : " << x(1) << " " << x(2) << " " << x(3) << endl;
cout << "K_" << i << " : " <<m_k(i)<<endl;
}
if (m_include_f0)
cout << "F0 :" << m_f0 << endl;
if (m_eval_BIC)
cout<< "BIC :"<<m_BIC<<endl;
}
//Print the estimates using a vector that contains the transformed parameter values
//i.e. need to untransform them to get d,f's etc
void PVM_Ball_Watsons::print(const ColumnVector& p)const{
ColumnVector f(nfib);
cout << "PARAMETER VALUES " << endl;
cout << "S0 :" << p(1) << endl;
cout << "D :" << lambda2d(p(2)) << endl;
for(int i=3,ii=1;ii<=nfib;i+=3,ii++){
f(ii) = beta2f(p(i))*partial_fsum(f,ii-1);
float _k=l12k1(p(i+3));
cout << "F" << ii << " :" << f(ii) << endl;
cout << "TH" << ii << " :" << p(i+1)*180.0/M_PI << " deg" << endl;
cout << "PH" << ii << " :" << p(i+2)*180.0/M_PI << " deg" << endl;
cout << "K_" << ii << " :"<< _k << endl;
}
if (m_include_f0) cout << "F0 :" << beta2f(p(nparams))*partial_fsum(f,nfib);
}
//Applies the forward model and gets the model predicted signal using the estimated parameter values (true,non-transformed space)
ReturnMatrix PVM_Ball_Watsons::get_prediction()const{
ColumnVector pred(npts);
ColumnVector p(nparams);
ColumnVector fs(nfib);
fs=m_f;
p(1) = m_s0; //Transform parameters to the space where they are uncostrained
p(2) = d2lambda(m_d);
for(int i=3,ii=1;ii<=nfib;i+=nparams_per_fibre,ii++){
float tmpr=m_f(ii)/partial_fsum(fs,ii-1);
if (tmpr>1.0) tmpr=1; //This can be due to numerical errors
p(i) = f2beta(tmpr);
p(i+1) = m_th(ii);
p(i+2) = m_ph(ii);
p(i+3) = k12l1(m_k(ii));
}
if (m_include_f0){
float tmpr=m_f0/partial_fsum(fs,nfib);
if (tmpr>1.0) tmpr=1; //This can be due to numerical errors
p(nparams)=f2beta(tmpr);
}
pred = forwardModel(p);
pred.Release();
return pred;
}
//Applies the forward model and gets a model predicted signal using the parameter values in p (transformed parameter space)
NEWMAT::ReturnMatrix PVM_Ball_Watsons::forwardModel(const NEWMAT::ColumnVector& p)const{
ColumnVector pred(npts);
pred = 0;
float val;
float _d = lambda2d(p(2));
////////////////////////////////////
ColumnVector fs(nfib), ks(nfib); ColumnVector temp_vec(3), denom(3);
Matrix v(nfib,3); vector<ColumnVector> approx_denomW; Matrix A(3,3);
float sumf=0; fs=0;
for(int n=1;n<=nfib;n++){
int nn = 3+nparams_per_fibre*(n-1);
float cosph, sinph,costh,sinth;
fs(n) = beta2f(p(nn))*partial_fsum(fs,n-1);
sumf += fs(n);
costh=cos(p(nn+1)); sinth=sin(p(nn+1)); cosph=cos(p(nn+2)); sinph=sin(p(nn+2));
v(n,1) = cosph*sinth; v(n,2) = sinph*sinth; v(n,3) = costh;
ks(n)=l12k1(p(nn+3));
temp_vec=approx_denominatorW(ks(n));
approx_denomW.push_back(temp_vec);
}
////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
val = 0.0;
float bd=bvals(1,i)*_d;
for(int n=1;n<=nfib;n++){
A=(v.Row(n)).t()*(bvecs.Column(i)).t();
float Q=-2*bd*ks(n)*(pow(A(1,1)+A(2,2)+A(3,3),2.0) - pow(A(2,1)-A(1,2),2.0) - pow(A(1,3)-A(3,1),2.0)-pow(A(2,3)-A(3,2),2.0));
Q=sqrt(ks(n)*ks(n)+bd*bd+Q);
temp_vec<<0.5*(ks(n)-bd+Q)<<0.5*(ks(n)-bd-Q)<<0;
val+= fs(n)*hyp_SratioW_knowndenom(temp_vec,approx_denomW[n-1]);
}
if (m_include_f0){ float temp_f0=beta2f(p(nparams))*partial_fsum(fs,nfib);
pred(i) = p(1)*(temp_f0+(1-sumf-temp_f0)*isoterm(i,_d)+val);
}
else
pred(i) = p(1)*((1-sumf)*isoterm(i,_d)+val);
}
pred.Release();
return pred;
}
//Instead of returning the model predicted signal for each direction
//returns the individual signal contributions i.e. isotropic, anisotropic1, anisotropic2,etc.
//Weighting with the fractions, scaling with S0 and summing those gives the signal.
//A Matrix npts x (nfib+1) is returned
NEWMAT::ReturnMatrix PVM_Ball_Watsons::forwardModel_compartments(const NEWMAT::ColumnVector& p) const{
Matrix Sig(npts,nfib+1);
float _d = lambda2d(p(2));
////////////////////////////////////
ColumnVector ks(nfib); ColumnVector temp_vec(3), denom(3);
Matrix v(nfib,3); vector<ColumnVector> approx_denomW; Matrix A(3,3);
for(int n=1;n<=nfib;n++){
int nn = 3+nparams_per_fibre*(n-1);
float cosph, sinph,costh,sinth;
costh=cos(p(nn+1)); sinth=sin(p(nn+1)); cosph=cos(p(nn+2)); sinph=sin(p(nn+2));
v(n,1) = cosph*sinth; v(n,2) = sinph*sinth; v(n,3) = costh;
ks(n)=l12k1(p(nn+3));
temp_vec=approx_denominatorW(ks(n));
approx_denomW.push_back(temp_vec);
}
////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
Sig(i,1) = isoterm(i,_d);
float bd=bvals(1,i)*_d;
for(int n=1;n<=nfib;n++){
A=(v.Row(n)).t()*(bvecs.Column(i)).t();
float Q=-2*bd*ks(n)*(pow(A(1,1)+A(2,2)+A(3,3),2.0) - pow(A(2,1)-A(1,2),2.0) - pow(A(1,3)-A(3,1),2.0)-pow(A(2,3)-A(3,2),2.0));
Q=sqrt(ks(n)*ks(n)+bd*bd+Q);
temp_vec<<0.5*(ks(n)-bd+Q)<<0.5*(ks(n)-bd-Q)<<0;
Sig(i,n+1)= hyp_SratioW_knowndenom(temp_vec,approx_denomW[n-1]);
}
}
Sig.Release();
return Sig;
}
//Builds up the model predicted signal for each direction by using precomputed individual compartment signals, stored in Matrix Sig.
//Weights them with the fractions, scales with S0 and sums to get the signal.
NEWMAT::ReturnMatrix PVM_Ball_Watsons::pred_from_compartments(const NEWMAT::ColumnVector& p, const NEWMAT::Matrix& Sig) const{
ColumnVector pred(npts);
float val;
ColumnVector fs(nfib);
float sumf=0; fs=0;
for(int n=1;n<=nfib;n++){
int nn = 3+nparams_per_fibre*(n-1);
fs(n) = beta2f(p(nn))*partial_fsum(fs,n-1);
sumf += fs(n);
}
for(int i=1;i<=Y.Nrows();i++){
val = 0.0;
for(int n=1;n<=nfib;n++)
val += fs(n)*Sig(i,n+1);
if (m_include_f0){
float temp_f0=beta2f(p(nparams))*partial_fsum(fs,nfib);
pred(i) = p(1)*(temp_f0+(1-sumf-temp_f0)*Sig(i,1)+val);
}
else
pred(i) = p(1)*((1-sumf)*Sig(i,1)+val);
}
pred.Release();
return pred;
}
//Builds up the model predicted signal for each direction by using precomputed individual compartment signals, stored in Matrix Sig.
//Weights them with the fractions, scales with S0 and sums to get the signal.
//The signal of the fibre compartment with index fib is recalculated.
NEWMAT::ReturnMatrix PVM_Ball_Watsons::pred_from_compartments(const NEWMAT::ColumnVector& p, const NEWMAT::Matrix& Sig,const int& fib) const{
ColumnVector pred(npts); Matrix newSig;
float val;
float _d = lambda2d(p(2));
////////////////////////////////////
ColumnVector fs(nfib); ColumnVector temp_vec(3), denom(3), v(3); Matrix A(3,3);
float sumf=0; fs=0;
for(int n=1;n<=nfib;n++){
int nn = 3+nparams_per_fibre*(n-1);
fs(n) = beta2f(p(nn))*partial_fsum(fs,n-1);
sumf += fs(n);
}
///////////////////////////////////////
int nn = 3+nparams_per_fibre*(fib-1); float cosph, sinph,costh,sinth,_k;
costh=cos(p(nn+1)); sinth=sin(p(nn+1)); cosph=cos(p(nn+2)); sinph=sin(p(nn+2));
v(1) = cosph*sinth; v(2) = sinph*sinth; v(3) = costh;
_k=l12k1(p(nn+3));
temp_vec=approx_denominatorW(_k);
denom=temp_vec;
newSig=Sig; //Get the new Signal for compartment fib
for(int i=1;i<=Y.Nrows();i++){
float bd=bvals(1,i)*_d;
A=v*(bvecs.Column(i)).t();
float Q=-2*bd*_k*(pow(A(1,1)+A(2,2)+A(3,3),2.0) - pow(A(2,1)-A(1,2),2.0) - pow(A(1,3)-A(3,1),2.0)-pow(A(2,3)-A(3,2),2.0));
Q=sqrt(_k*_k+bd*bd+Q);
temp_vec<<0.5*(_k-bd+Q)<<0.5*(_k-bd-Q)<<0;
newSig(i,fib+1)= hyp_SratioW_knowndenom(temp_vec,denom);
}
///////////////////////////////////////
for(int i=1;i<=Y.Nrows();i++){
val = 0.0;
for(int n=1;n<=nfib;n++)
val += fs(n)*newSig(i,n+1);
if (m_include_f0){
float temp_f0=beta2f(p(nparams))*partial_fsum(fs,nfib);
pred(i) = p(1)*(temp_f0+(1-sumf-temp_f0)*newSig(i,1)+val);
}
else
pred(i) = p(1)*((1-sumf)*newSig(i,1)+val);
}
pred.Release();
return pred;
}
//Cost Function, sum of squared residuals
//assume that parameter values p are transformed (e.g. need to untransform them to get d, f's,etc)
double PVM_Ball_Watsons::cf(const NEWMAT::ColumnVector& p)const{
double cfv = 0.0;
double err;
ColumnVector S;
S=forwardModel(p); //Model predictions
for(int i=1;i<=npts;i++){
err=S(i)-Y(i); //Residual
cfv+=err*err; //Sum of squared residuals
}
//cout<<"CF="<<cfv<<endl; OUT(p.t());
return(cfv);
}
NEWMAT::ReturnMatrix PVM_Ball_Watsons::grad(const ColumnVector& p)const
{
ColumnVector gradv(nparams);//This is the gradient of the cost function
Matrix J(npts,nparams); //This is the Jacobian matrix of the model equation
//The derivative of the cost function w.r.t. parameter j
//will then be: Grad_j=Sum(2*(F(x_i)-Y_i)*J(i,j)), with Sum across data points i
ColumnVector diff(npts); //Residuals
ColumnVector p_plus_h, S_trial,S;
Matrix Sig;
ColumnVector fs(nfib), bs(nfib);
//Compute the Jacobian first using finite differences for each element. Derivatives are analytic for S0 and the volume fractions
double step;
ColumnVector typical_scale(nparams);
typical_scale=1;
for (int k=1; k<=nfib; k++){
int kk = 3+nparams_per_fibre*(k-1);
bs(k)=p(kk);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
typical_scale(kk+3)=100;
}
//Compute the derivatives with respect to betas, i.e the transformed volume fraction variables
Matrix f_deriv;
f_deriv=fractions_deriv(nfib, fs, bs);
Sig=forwardModel_compartments(p);
S=pred_from_compartments(p, Sig);
diff=S-Y;
for (int i=1; i<=npts; i++)
J(i,1)=S(i)/p(1); //derivatives with respect to S0 are analytic: S_i/S0
for (int n=2; n<=nparams; n++) {
p_plus_h = p;
step = SQRTtiny*nonzerosign(p(n))*max(fabs(p(n))*typical_scale(n),1.0);
step = nonzerosign(tiny)*min(max(fabs(step),1.0e-8),0.1); //check that 1e-8<step<0.1
p_plus_h(n)=p(n)+step;
if (n<3 || (n==nparams && m_include_f0)){ //for d all compartments will change. Also for f0, if included. Use for those params numerical differentiation
S_trial=forwardModel(p_plus_h);
for (int i=1; i<=npts; i++)
J(i,n)=(S_trial(i)-S(i))/step;
}
else{ //for the other params, update only the signal of the relevant fibre compartment
int fib_indx=(int)ceil((float)(n-2)/(float)nparams_per_fibre); //index indicating in which fibre compartment parameter n belongs to
if (n==3+nparams_per_fibre*(fib_indx-1)){ //Then we have a volume fraction, derivative is analytic.
for (int i=1; i<=npts; i++){
J(i,n)=0;
for (int j=1; j<=nfib; j++){
if (f_deriv(j,fib_indx)!=0)
J(i,n) += p(1)*(Sig(i,j+1)-Sig(i,1))*f_deriv(j,fib_indx);
}
} }
else{ //for all other params, use numerical differentiation
S_trial=pred_from_compartments(p_plus_h, Sig,fib_indx);
for (int i=1; i<=npts; i++)
J(i,n)=(S_trial(i)-S(i))/step;
}
}
}
gradv =2*J.t()*diff;
gradv.Release();
return(gradv);
}
//this uses Gauss-Newton approximation
boost::shared_ptr<BFMatrix> PVM_Ball_Watsons::hess(const NEWMAT::ColumnVector& p,boost::shared_ptr<BFMatrix> iptr)const{
boost::shared_ptr<BFMatrix> hessm;
if (iptr && iptr->Nrows()==(unsigned int)p.Nrows() && iptr->Ncols()==(unsigned int)p.Nrows()) hessm = iptr;
else hessm = boost::shared_ptr<BFMatrix>(new FullBFMatrix(p.Nrows(),p.Nrows()));
Matrix J(npts,nparams); //This is the Jacobian matrix of the model equation
ColumnVector p_plus_h, S_trial,S, fs(nfib), bs(nfib);
Matrix Sig;
//Compute the Jacobian first using finite differences for each element. Derivatives are analytic for S0 and the volume fractions
double step,sigt;
ColumnVector typical_scale(nparams);
typical_scale=1;
for (int k=1; k<=nfib; k++){
int kk = 3+nparams_per_fibre*(k-1);
bs(k)=p(kk);
fs(k) = beta2f(p(kk))*partial_fsum(fs,k-1);
typical_scale(kk+3)=100;
}
//Compute the derivatives with respect to betas, i.e the transformed volume fraction variables
Matrix f_deriv;
f_deriv=fractions_deriv(nfib, fs, bs);
Sig=forwardModel_compartments(p);
S=pred_from_compartments(p, Sig);
for (int i=1; i<=npts; i++)
J(i,1)=S(i)/p(1); //derivatives with respect to S0 are analytic: S_i/S0
for (int n=2; n<=nparams; n++) {
p_plus_h = p;
step = SQRTtiny*nonzerosign(p(n))*max(fabs(p(n))*typical_scale(n),1.0);
step = nonzerosign(tiny)*min(max(fabs(step),1.0e-8),0.1); //check that 1e-8<step<0.1
p_plus_h(n)=p(n)+step;
if (n<3 || (n==nparams && m_include_f0)){ //for d all compartments will change. Also for f0, if included. Use for those params numerical differentiation
S_trial=forwardModel(p_plus_h);
for (int i=1; i<=npts; i++)
J(i,n)=(S_trial(i)-S(i))/step;
}
else{ //for the other params, update only the signal of the relevant fibre compartment
int fib_indx=(int)ceil((float)(n-2)/(float)nparams_per_fibre); //index indicating in which fibre compartment parameter n belongs to
if (n==3+nparams_per_fibre*(fib_indx-1)){ //Then we have a volume fraction, derivative is analytic.
for (int i=1; i<=npts; i++){
J(i,n)=0;
for (int j=1; j<=nfib; j++){
if (f_deriv(j,fib_indx)!=0)
J(i,n) += p(1)*(Sig(i,j+1)-Sig(i,1))*f_deriv(j,fib_indx);
}
}
}
else{ //for all other params, use numerical differentiation
S_trial=pred_from_compartments(p_plus_h, Sig,fib_indx); for (int i=1; i<=npts; i++)
J(i,n)=(S_trial(i)-S(i))/step;
}
}
}
for (int i=1; i<=p.Nrows(); i++){
for (int j=i; j<=p.Nrows(); j++){
sigt = 0.0;
for(int k=1;k<=J.Nrows();k++)
sigt += 2*(J(k,i)*J(k,j));
hessm->Set(i,j,sigt);
}
}
for (int j=1; j<=p.Nrows(); j++) {
for (int i=j+1; i<=p.Nrows(); i++) {
hessm->Set(i,j,hessm->Peek(j,i));
}
}
return(hessm);
}
float PVM_Ball_Watsons::isoterm(const int& pt,const float& _d)const{
return(std::exp(-bvals(1,pt)*_d));
}
NEWMAT::ReturnMatrix PVM_Ball_Watsons::fractions_deriv(const int& nfib, const ColumnVector& fs, const ColumnVector& bs) const{
NEWMAT::Matrix Deriv(nfib,nfib);
float fsum;
Deriv=0;
for (int j=1; j<=nfib; j++)
for (int k=1; k<=nfib; k++){
if (j==k){
fsum=1;
for (int n=1; n<=j-1; n++)
fsum-=fs(n);
Deriv(j,k)=sin(2*bs(k))*fsum;
}
else if (j>k){
fsum=0;
for (int n=1; n<=j-1; n++)
fsum+=Deriv(n,k);
Deriv(j,k)=-pow(sin(bs(j)),2.0)*fsum;
}
}
Deriv.Release();
return Deriv;
}
///////////////////////////////////////////////////////////////////////////////////////////////
// USEFUL FUNCTIONS TO CALCULATE DERIVATIVES
///////////////////////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////
///////Model 1 (Constrained)
/////////////////////////////////////
// functions
float PVM_single_c::isoterm(const int& pt,const float& _d)const{
return(std::exp(-bvals(1,pt)*_d));
}
float PVM_single_c::anisoterm(const int& pt,const float& _d,const ColumnVector& x)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3); return(std::exp(-bvals(1,pt)*_d*dp*dp));
}
// 1st order derivatives
float PVM_single_c::isoterm_lambda(const int& pt,const float& lambda)const{
return(-2*bvals(1,pt)*lambda*std::exp(-bvals(1,pt)*lambda*lambda));
}
float PVM_single_c::anisoterm_lambda(const int& pt,const float& lambda,const ColumnVector& x)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
return(-2*bvals(1,pt)*lambda*dp*dp*std::exp(-bvals(1,pt)*lambda*lambda*dp*dp));
}
float PVM_single_c::anisoterm_th(const int& pt,const float& _d,const ColumnVector& x,const float& _th,const float& _ph)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
float dp1 = cos(_th)*(bvecs(1,pt)*cos(_ph) + bvecs(2,pt)*sin(_ph)) - bvecs(3,pt)*sin(_th);
return(-2*bvals(1,pt)*_d*dp*dp1*std::exp(-bvals(1,pt)*_d*dp*dp));
}
float PVM_single_c::anisoterm_ph(const int& pt,const float& _d,const ColumnVector& x,const float& _th,const float& _ph)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
float dp1 = sin(_th)*(-bvecs(1,pt)*sin(_ph) + bvecs(2,pt)*cos(_ph));
return(-2*bvals(1,pt)*_d*dp*dp1*std::exp(-bvals(1,pt)*_d*dp*dp));
}
NEWMAT::ReturnMatrix PVM_single_c::fractions_deriv(const int& nfib, const ColumnVector& fs, const ColumnVector& bs) const{
NEWMAT::Matrix Deriv(nfib,nfib);
float fsum;
Deriv=0;
for (int j=1; j<=nfib; j++)
for (int k=1; k<=nfib; k++){
if (j==k){
fsum=1;
for (int n=1; n<=j-1; n++)
fsum-=fs(n);
Deriv(j,k)=sin(2*bs(k))*fsum;
}
else if (j>k){
fsum=0;
for (int n=1; n<=j-1; n++)
fsum+=Deriv(n,k);
Deriv(j,k)=-pow(sin(bs(j)),2.0)*fsum;
}
}
Deriv.Release();
return Deriv;
}
/////////////////////////////////////
////////Model 1 (Old)
/////////////////////////////////////
//functions
float PVM_single::isoterm(const int& pt,const float& _d)const{
return(std::exp(-bvals(1,pt)*_d));
}
float PVM_single::anisoterm(const int& pt,const float& _d,const ColumnVector& x)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
return(std::exp(-bvals(1,pt)*_d*dp*dp));
}
float PVM_single::bvecs_fibre_dp(const int& pt,const float& _th,const float& _ph)const{
float angtmp = cos(_ph-beta(pt))*sinalpha(pt)*sin(_th) + cosalpha(pt)*cos(_th);
return(angtmp*angtmp);
}
float PVM_single::bvecs_fibre_dp(const int& pt,const ColumnVector& x)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
return(dp*dp);
}
// 1st order derivatives
float PVM_single::isoterm_d(const int& pt,const float& _d)const{
return(-bvals(1,pt)*std::exp(-bvals(1,pt)*_d));
}
float PVM_single::anisoterm_d(const int& pt,const float& _d,const ColumnVector& x)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3); return(-bvals(1,pt)*dp*dp*std::exp(-bvals(1,pt)*_d*dp*dp));
}
float PVM_single::anisoterm_th(const int& pt,const float& _d,const ColumnVector& x,const float& _th,const float& _ph)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
float dp1 = cos(_th)*(bvecs(1,pt)*cos(_ph) + bvecs(2,pt)*sin(_ph)) - bvecs(3,pt)*sin(_th);
return(-2*bvals(1,pt)*_d*dp*dp1*std::exp(-bvals(1,pt)*_d*dp*dp));
}
float PVM_single::anisoterm_ph(const int& pt,const float& _d,const ColumnVector& x,const float& _th,const float& _ph)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
float dp1 = sin(_th)*(-bvecs(1,pt)*sin(_ph) + bvecs(2,pt)*cos(_ph));
return(-2*bvals(1,pt)*_d*dp*dp1*std::exp(-bvals(1,pt)*_d*dp*dp));
}
// 2nd order derivatives
float PVM_single::isoterm_dd(const int& pt,const float& _d)const{
return(bvals(1,pt)*bvals(1,pt)*std::exp(-bvals(1,pt)*_d));
}
float PVM_single::anisoterm_dd(const int& pt,const float& _d,const ColumnVector& x)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
dp *= dp;
return(bvals(1,pt)*dp*bvals(1,pt)*dp*std::exp(-bvals(1,pt)*_d*dp));
}
float PVM_single::anisoterm_dth(const int& pt,const float& _d,const ColumnVector& x,const float& _th,const float& _ph)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
float dp1 = cos(_th)*(bvecs(1,pt)*cos(_ph) + bvecs(2,pt)*sin(_ph)) - bvecs(3,pt)*sin(_th);
return( -2*bvals(1,pt)*dp*dp1*(1-bvals(1,pt)*_d*dp*dp)*std::exp(-bvals(1,pt)*_d*dp*dp) );
}
float PVM_single::anisoterm_dph(const int& pt,const float& _d,const ColumnVector& x,const float& _th,const float& _ph)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
float dp1 = sin(_th)*(-bvecs(1,pt)*sin(_ph) + bvecs(2,pt)*cos(_ph));
return( -2*bvals(1,pt)*dp*dp1*(1-bvals(1,pt)*_d*dp*dp)*std::exp(-bvals(1,pt)*_d*dp*dp) );
}
float PVM_single::anisoterm_thth(const int& pt,const float& _d,const ColumnVector& x,const float& _th,const float& _ph)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
return( -2*bvals(1,pt)*_d*std::exp(-bvals(1,pt)*_d*dp*dp)* ( (1-2*bvals(1,pt)*dp*dp) -dp*dp ) );
}
float PVM_single::anisoterm_phph(const int& pt,const float& _d,const ColumnVector& x,const float& _th,const float& _ph)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
float dp1 = (1-cos(2*_th))/2.0;
float dp2 = -bvecs(1,pt)*x(1) - bvecs(2,pt)*x(2);
return( -2*bvals(1,pt)*_d*std::exp(-bvals(1,pt)*_d*dp*dp)* ( (1-2*bvals(1,pt)*dp*dp)*dp1 +dp*dp2 ) );
}
float PVM_single::anisoterm_thph(const int& pt,const float& _d,const ColumnVector& x,const float& _th,const float& _ph)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
float dp2 = cos(_th)*(-bvecs(1,pt)*sin(_ph) + bvecs(2,pt)*cos(_ph));
return( -2*bvals(1,pt)*_d*std::exp(-bvals(1,pt)*_d*dp*dp)* ( dp*dp2 ) );
}
////// NOW FOR MULTISHELL
// functions
float PVM_multi::isoterm(const int& pt,const float& _a,const float& _b)const{
return(std::exp(-_a*std::log(1+bvals(1,pt)*_b)));
}
float PVM_multi::anisoterm(const int& pt,const float& _a,const float& _b,const ColumnVector& x)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
return(std::exp(-_a*std::log(1+bvals(1,pt)*_b*(dp*dp))));
}
// 1st order derivatives
float PVM_multi::isoterm_a(const int& pt,const float& _a,const float& _b)const{
return(-std::log(1+bvals(1,pt)*_b)*std::exp(-_a*std::log(1+bvals(1,pt)*_b)));
}
float PVM_multi::isoterm_b(const int& pt,const float& _a,const float& _b)const{
return(-_a*bvals(1,pt)/(1+bvals(1,pt)*_b)*std::exp(-_a*std::log(1+bvals(1,pt)*_b)));
}
float PVM_multi::anisoterm_a(const int& pt,const float& _a,const float& _b,const ColumnVector& x)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
return(-std::log(1+bvals(1,pt)*(dp*dp)*_b)*std::exp(-_a*std::log(1+bvals(1,pt)*(dp*dp)*_b)));
}
float PVM_multi::anisoterm_b(const int& pt,const float& _a,const float& _b,const ColumnVector& x)const{ float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
return(-_a*bvals(1,pt)*(dp*dp)/(1+bvals(1,pt)*(dp*dp)*_b)*std::exp(-_a*std::log(1+bvals(1,pt)*(dp*dp)*_b)));
}
float PVM_multi::anisoterm_th(const int& pt,const float& _a,const float& _b,const ColumnVector& x,const float& _th,const float& _ph)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
float dp1 = cos(_th)*(bvecs(1,pt)*cos(_ph) + bvecs(2,pt)*sin(_ph)) - bvecs(3,pt)*sin(_th);
return(-_a*_b*bvals(1,pt)/(1+bvals(1,pt)*(dp*dp)*_b)*std::exp(-_a*std::log(1+bvals(1,pt)*(dp*dp)*_b))*2*dp*dp1);
}
float PVM_multi::anisoterm_ph(const int& pt,const float& _a,const float& _b,const ColumnVector& x,const float& _th,const float& _ph)const{
float dp = bvecs(1,pt)*x(1)+bvecs(2,pt)*x(2)+bvecs(3,pt)*x(3);
float dp1 = sin(_th)*(-bvecs(1,pt)*sin(_ph) + bvecs(2,pt)*cos(_ph));
return(-_a*_b*bvals(1,pt)/(1+bvals(1,pt)*(dp*dp)*_b)*std::exp(-_a*std::log(1+bvals(1,pt)*(dp*dp)*_b))*2*dp*dp1);
}