bool FitGaussian(const vector<Vector3>& pts,Vector3& mean,Matrix3& R,Vector3& axes)
{
mean = GetMean(pts);
Matrix A(pts.size(),3);
for(size_t i=0;i<pts.size();i++)
(pts[i]-mean).get(A(i,0),A(i,1),A(i,2));
SVDecomposition<Real> svd;
if(!svd.set(A)) {
return false;
}
svd.sortSVs();
axes.set(svd.W(0),svd.W(1),svd.W(2));
for(int i=0;i<3;i++)
for(int j=0;j<3;j++)
R(i,j) = svd.V(i,j);
return true;
}
Real RotationFit(const vector<Vector3>& a,const vector<Vector3>& b,Matrix3& R)
{
Assert(a.size() == b.size());
assert(a.size() >= 3);
Matrix3 C;
C.setZero();
for(size_t k=0;k<a.size();k++) {
for(int i=0;i<3;i++)
for(int j=0;j<3;j++)
C(i,j) += a[k][j]*b[k][i];
}
//let A=[a1 ... an]^t, B=[b1 ... bn]^t
//solve for min sum of squares of E=ARt-B
//let C=AtB
//solution is given by CCt = RtCtCR
//Solve C^tR = R^tC with SVD CC^t = R^tC^tCR
//CtRX = RtCX
//C = RCtR
//Ct = RtCRt
//=> CCt = RCtCRt
//solve SVD of C and Ct (giving eigenvectors of CCt and CtC
//C = UWVt => Ct=VWUt
//=> UWUt = RVWVtRt
//=> U=RV => R=UVt
Matrix mC(3,3),mCtC(3,3);
Copy(C,mC);
SVDecomposition<Real> svd;
if(!svd.set(mC)) {
cerr<<"RotationFit: Couldn't set svd of covariance matrix"<<endl;
R.setIdentity();
return Inf;
}
Matrix mR;
mR.mulTransposeB(svd.U,svd.V);
Copy(mR,R);
if(R.determinant() < 0) { //it's a mirror
svd.sortSVs();
if(!FuzzyZero(svd.W(2),(Real)1e-2)) {
cerr<<"RotationFit: Uhh... what do we do? SVD of rotation doesn't have a zero singular value"<<endl;
/*
cerr<<svd.W<<endl;
cerr<<"Press any key to continue"<<endl;
getchar();
*/
}
//negate the last column of V
Vector vi;
svd.V.getColRef(2,vi);
vi.inplaceNegative();
mR.mulTransposeB(svd.V,svd.U);
Copy(mR,R);
Assert(R.determinant() > 0);
}
Real sum=0;
for(size_t k=0;k<a.size();k++)
sum += b[k].distanceSquared(R*a[k]);
return sum;
}
