VTK-5.0.0/Hybrid/vtkThinPlateSplineTransform.cxx

/*=========================================================================

  Program:   Visualization Toolkit
  Module:    $RCSfile: vtkThinPlateSplineTransform.cxx,v $

  Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen
  All rights reserved.
  See Copyright.txt or http://www.kitware.com/Copyright.htm for details.

     This software is distributed WITHOUT ANY WARRANTY; without even
     the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
     PURPOSE.  See the above copyright notice for more information.

=========================================================================*/
#include "vtkThinPlateSplineTransform.h"

#include "vtkMath.h"
#include "vtkObjectFactory.h"
#include "vtkPoints.h"

vtkCxxRevisionMacro(vtkThinPlateSplineTransform, "$Revision: 1.33 $");
vtkStandardNewMacro(vtkThinPlateSplineTransform);

//------------------------------------------------------------------------
// some dull matrix things

inline double** vtkNewMatrix(int rows, int cols) 
{
  double *matrix = new double[rows*cols];
  double **m = new double *[rows];
  for(int i = 0; i < rows; i++) 
    {
    m[i] = &matrix[i*cols];
    }
  return m;
}

//------------------------------------------------------------------------
inline void vtkDeleteMatrix(double **m) 
{
  delete [] *m;
  delete [] m;
}

//------------------------------------------------------------------------
inline void vtkZeroMatrix(double **m, int rows, int cols) 
{
  for(int i = 0; i < rows; i++) 
    {
    for(int j = 0; j < cols; j++) 
      {
      m[i][j] = 0.0;
      }
    }
}

//------------------------------------------------------------------------
inline void vtkMatrixMultiply(double **a, double **b, double **c,
                              int arows, int acols, int brows, int bcols) 
{
  if(acols != brows) 
    {
    return;     // acols must equal br otherwise we can't proceed
    }
        
  // c must have size arows*bcols (we assume this)

  for(int i = 0; i < arows; i++) 
    {
    for(int j = 0; j < bcols; j++) 
      {
      c[i][j] = 0.0;
      for(int k = 0; k < acols; k++)
        {
        c[i][j] += a[i][k]*b[k][j];
        }
      }
    }
}

//------------------------------------------------------------------------
inline void vtkMatrixTranspose(double **a, double **b, int rows, int cols)
{
  for(int i = 0; i < rows; i++) 
    {
    for(int j = 0; j < cols; j++) 
      {
      double tmp = a[i][j];
      b[i][j] = a[j][i];
      b[j][i] = tmp;
      }
    }
}

//------------------------------------------------------------------------
vtkThinPlateSplineTransform::vtkThinPlateSplineTransform()
{
  this->SourceLandmarks=NULL;
  this->TargetLandmarks=NULL;
  this->Sigma = 1.0;

  // If the InverseFlag is set, then we use an iterative
  // method to invert the transformation.  
  // The InverseTolerance sets the precision to which we want to 
  // calculate the inverse.
  this->InverseTolerance = 0.001;
  this->InverseIterations = 500;

  this->Basis = -1;
  this->SetBasisToR2LogR();

  this->NumberOfPoints = 0;
  this->MatrixW = NULL;
}

//------------------------------------------------------------------------
vtkThinPlateSplineTransform::~vtkThinPlateSplineTransform()
{
  if (this->SourceLandmarks)
    {
    this->SourceLandmarks->Delete();
    }
  if (this->TargetLandmarks)
    {
    this->TargetLandmarks->Delete();
    }
  if (this->MatrixW)
    {
    vtkDeleteMatrix(this->MatrixW);
    this->MatrixW = NULL;
    }
}

//------------------------------------------------------------------------
void vtkThinPlateSplineTransform::SetSourceLandmarks(vtkPoints *source)
{
  if (this->SourceLandmarks == source)
    {
    return;
    }

  if (this->SourceLandmarks)
    {
    this->SourceLandmarks->Delete();
    }

  source->Register(this);
  this->SourceLandmarks = source;

  this->Modified();
}

//------------------------------------------------------------------------
void vtkThinPlateSplineTransform::SetTargetLandmarks(vtkPoints *target)
{
  if (this->TargetLandmarks == target)
    {
    return;
    }

  if (this->TargetLandmarks)
    {
    this->TargetLandmarks->Delete();
    }

  target->Register(this);
  this->TargetLandmarks = target;
  this->Modified();
}

//------------------------------------------------------------------------
unsigned long vtkThinPlateSplineTransform::GetMTime()
{
  unsigned long result = this->vtkWarpTransform::GetMTime();
  unsigned long mtime;

  if (this->SourceLandmarks)
    {
    mtime = this->SourceLandmarks->GetMTime(); 
    if (mtime > result)
      {
      result = mtime;
      }
    }
  if (this->TargetLandmarks)
    {
    mtime = this->TargetLandmarks->GetMTime(); 
    if (mtime > result)
      {
      result = mtime;
      }
    }
  return result;
}

//------------------------------------------------------------------------
void vtkThinPlateSplineTransform::InternalUpdate()
{
  if (this->SourceLandmarks == NULL || this->TargetLandmarks == NULL)
    {
    if (this->MatrixW)
      {
      vtkDeleteMatrix(this->MatrixW);
      }
    this->MatrixW = NULL;
    this->NumberOfPoints = 0;
    return;
    }

  if (this->SourceLandmarks->GetNumberOfPoints() !=
      this->TargetLandmarks->GetNumberOfPoints())
    {
    vtkErrorMacro("Update: Source and Target Landmarks contain a different number of points");
    return;
    }

  const vtkIdType N = this->SourceLandmarks->GetNumberOfPoints();
  const int D = 3; // dimensions

  // the output weights matrix
  double **W = vtkNewMatrix(N+D+1,D); 
  double **A = &W[N+1];  // the linear rotation + scale matrix 
  double *C = W[N];      // the linear translation 

  if (N >= 3)
    {
    // Notation and inspiration from:
    // Fred L. Bookstein (1997) "Shape and the Information in Medical Images: 
    // A Decade of the Morphometric Synthesis" Computer Vision and Image 
    // Understanding 66(2):97-118
    // and online work published by Tim Cootes (http://www.wiau.man.ac.uk/~bim)
        
    // the input matrices
    double **L = vtkNewMatrix(N+D+1,N+D+1);
    double **X = vtkNewMatrix(N+D+1,D);

    // build L
    // will leave the bottom-right corner with zeros
    vtkZeroMatrix(L,N+D+1,N+D+1);

    int q,c;
    double p[3],p2[3];
    double dx,dy,dz;
    double r;
    double (*phi)(double) = this->BasisFunction;
    
    for(q = 0; q < N; q++)
      {
      this->SourceLandmarks->GetPoint(q,p);
      // fill in the top-right and bottom-left corners of L (Q)
      L[N][q] = L[q][N] = 1.0;
      L[N+1][q] = L[q][N+1] = p[0];
      L[N+2][q] = L[q][N+2] = p[1];
      L[N+3][q] = L[q][N+3] = p[2];
      // fill in the top-left corner of L (K), using symmetry
      for(c = 0; c < q; c++)
        {
        this->SourceLandmarks->GetPoint(c,p2);
        dx = p[0]-p2[0]; dy = p[1]-p2[1]; dz = p[2]-p2[2];
        r = sqrt(dx*dx + dy*dy + dz*dz);
        L[q][c] = L[c][q] = phi(r/this->Sigma);
        }
      }
    
    // build X
    vtkZeroMatrix(X,N+D+1,D);
    for (q = 0; q < N; q++)
      {
      this->TargetLandmarks->GetPoint(q,p);
      X[q][0] = p[0];
      X[q][1] = p[1];
      X[q][2] = p[2];
      }
    
    // solve for W, where W = Inverse(L)*X; 

    // this is done via eigenvector decomposition so
    // that we can avoid singular values
    // W = V*Inverse(w)*U*X  
    
    double *values = new double[N+D+1];
    double **V = vtkNewMatrix(N+D+1,N+D+1);
    double **w = vtkNewMatrix(N+D+1,N+D+1);
    double **U = L;  // reuse the space
    vtkMath::JacobiN(L,N+D+1,values,V);
    vtkMatrixTranspose(V,U,N+D+1,N+D+1);

    vtkIdType i, j;
    double maxValue = 0.0; // maximum eigenvalue
    for (i = 0; i < N+D+1; i++)
      {
      double tmp = fabs(values[i]);
      if (tmp > maxValue)
        {
        maxValue = tmp;
        }
      } 

    for (i = 0; i < N+D+1; i++)
      {
      for (j = 0; j < N+D+1; j++)
        {
        w[i][j] = 0.0;
        }
      // here's the trick: don't invert the singular values
      if (fabs(values[i]/maxValue) > 1e-16)
        {
        w[i][i] = 1.0/values[i];
        }
      }
    delete [] values;
    
    vtkMatrixMultiply(U,X,W,N+D+1,N+D+1,N+D+1,D);
    vtkMatrixMultiply(w,W,X,N+D+1,N+D+1,N+D+1,D);
    vtkMatrixMultiply(V,X,W,N+D+1,N+D+1,N+D+1,D);
    
    vtkDeleteMatrix(V);
    vtkDeleteMatrix(w);
    vtkDeleteMatrix(U);
    vtkDeleteMatrix(X);

    // now the linear portion of the warp must be checked
    // (this is a very poor check for now)
    if (fabs(vtkMath::Determinant3x3((double (*)[3]) *A)) < 1e-16)
      {
      for (i = 0; i < 3; i++)
        {
        if (sqrt(A[0][i]*A[0][i] + A[1][i]*A[1][i] + A[2][i]*A[2][i])
            < 1e-16)
          {
          A[0][i] = A[1][i] = A[2][i] = A[i][0] = A[i][1] = A[i][2] = 0;
          A[i][i] = 1.0;
          }
        }
      }
    }
  // special cases, I added these to ensure that this class doesn't 
  // misbehave if the user supplied fewer than 3 landmarks
  else // (N < 3)
    {
    vtkIdType i,j;
    // set nonlinear portion of transformation to zero
    for (i = 0; i < N; i++)
      {
      for (j = 0; j < D; j++)
        {
        W[i][j] = 0;
        }
      }

    if (N == 2)
      { // two landmarks, construct a similarity transformation
      double s0[3],t0[3],s1[3],t1[3];
      this->SourceLandmarks->GetPoint(0,s0);
      this->TargetLandmarks->GetPoint(0,t0);
      this->SourceLandmarks->GetPoint(1,s1);
      this->TargetLandmarks->GetPoint(1,t1);

      double ds[3],dt[3],as[3],at[3];
      double rs = 0, rt = 0;
      for (i = 0; i < 3; i++)
        {
        as[i] = (s0[i] + s1[i])/2;  // average of endpoints
        ds[i] = s1[i] - s0[i];      // vector between points
        rs += ds[i]*ds[i];
        at[i] = (t0[i] + t1[i])/2;
        dt[i] = t1[i] - t0[i];
        rt += dt[i]*dt[i];
        }

      // normalize the two vectors
      rs = sqrt(rs);
      ds[0] /= rs; ds[1] /= rs; ds[2] /= rs; 
      rt = sqrt(rt);
      dt[0] /= rt; dt[1] /= rt; dt[2] /= rt; 

      double w,x,y,z;
      // take dot & cross product
      w = ds[0]*dt[0] + ds[1]*dt[1] + ds[2]*dt[2];
      x = ds[1]*dt[2] - ds[2]*dt[1];
      y = ds[2]*dt[0] - ds[0]*dt[2];
      z = ds[0]*dt[1] - ds[1]*dt[0];

      double r = sqrt(x*x + y*y + z*z);
      double theta = atan2(r,w);

      // construct quaternion
      w = cos(theta/2);
      if (r != 0)
        {
        r = sin(theta/2)/r;
        x = x*r;
        y = y*r;
        z = z*r;
        }
      else // rotation by 180 degrees
        {
        // rotate around a vector perpendicular to ds
        vtkMath::Perpendiculars(ds,dt,0,0);
        r = sin(theta/2);
        x = dt[0]*r;
        y = dt[1]*r;
        z = dt[2]*r;
        }
      
      // now r is scale factor for matrix
      r = rt/rs;

      // build a rotation + scale matrix
      A[0][0] = (w*w + x*x - y*y - z*z)*r;
      A[0][1] = (x*y + w*z)*2*r;
      A[0][2] = (x*z - w*y)*2*r;

      A[1][0] = (x*y - w*z)*2*r;
      A[1][1] = (w*w - x*x + y*y - z*z)*r;
      A[1][2] = (y*z + w*x)*2*r;

      A[2][0] = (x*z + w*y)*2*r;
      A[2][1] = (y*z - w*x)*2*r;
      A[2][2] = (w*w - x*x - y*y + z*z)*r;

      // include the translation
      C[0] = at[0] - as[0]*A[0][0] - as[1]*A[1][0] - as[2]*A[2][0];
      C[1] = at[1] - as[0]*A[0][1] - as[1]*A[1][1] - as[2]*A[2][1];
      C[2] = at[2] - as[0]*A[0][2] - as[1]*A[1][2] - as[2]*A[2][2];
      }
    else if (N == 1) // one landmark, translation only
      {
      double p[3],p2[3];
      this->SourceLandmarks->GetPoint(0,p);
      this->TargetLandmarks->GetPoint(0,p2);
      
      for (i = 0; i < D; i++)
        {
        for (j = 0; j < D; j++)
          {
          A[i][j] = 0;
          }
        A[i][i] = 1;
        C[i] = p2[i] - p[i];
        }
      }

    else // if no landmarks, set to identity
      {
      for (i = 0; i < D; i++)
        {
        for (j = 0; j < D; j++)
          {
          A[i][j] = 0;
          }
        A[i][i] = 1;
        C[i] = 0;
        }
      }
    }

  // left in for debug purposes
  /*
  cerr << "W =\n";
  for (int i = 0; i < N+1+D; i++)
    {
    cerr << W[i][0] << ' ' << W[i][1] << ' ' << W[i][2] << '\n';
    }
  cerr << "\n";
  */

  if (this->MatrixW)
    {
    vtkDeleteMatrix(this->MatrixW);
    }
  this->MatrixW = W;
  this->NumberOfPoints = N;
}

//------------------------------------------------------------------------
// The matrix W was created by Update.  Not much has to be done to
// apply the transform:  do an affine transformation, then do
// perturbations based on the landmarks.
template<class T>
inline void vtkThinPlateSplineForwardTransformPoint(vtkThinPlateSplineTransform *self,
                                                    double **W, int N,
                                                    double (*phi)(double),
                                                    const T point[3], T output[3])
{
  if (N == 0)
    {
    output[0] = point[0];
    output[1] = point[1];
    output[2] = point[2];
    return;
    }

  double *C = W[N]; 
  double **A = &W[N+1];

  double dx,dy,dz;
  double p[3];
  double U,r;
  double invSigma = 1.0/self->GetSigma();

  double x = 0, y = 0, z = 0; 

  vtkPoints *sourceLandmarks = self->GetSourceLandmarks();

  // do the nonlinear stuff
  for(vtkIdType i = 0; i < N; i++)
    {
    sourceLandmarks->GetPoint(i,p);
    dx = point[0]-p[0]; dy = point[1]-p[1]; dz = point[2]-p[2];
    r = sqrt(dx*dx + dy*dy + dz*dz);
    U = phi(r*invSigma);
    x += U*W[i][0];
    y += U*W[i][1];
    z += U*W[i][2];
    }

  // finish off with the affine transformation
  x += C[0] + point[0]*A[0][0] + point[1]*A[1][0] + point[2]*A[2][0];
  y += C[1] + point[0]*A[0][1] + point[1]*A[1][1] + point[2]*A[2][1];
  z += C[2] + point[0]*A[0][2] + point[1]*A[1][2] + point[2]*A[2][2];

  output[0] = x;
  output[1] = y;
  output[2] = z;
}

void vtkThinPlateSplineTransform::ForwardTransformPoint(const double point[3], 
                                                        double output[3])
{
  vtkThinPlateSplineForwardTransformPoint(this, this->MatrixW, 
                                          this->NumberOfPoints, 
                                          this->BasisFunction,
                                          point, output);
}

void vtkThinPlateSplineTransform::ForwardTransformPoint(const float point[3], 
                                                        float output[3])
{
  vtkThinPlateSplineForwardTransformPoint(this, this->MatrixW, 
                                          this->NumberOfPoints,
                                          this->BasisFunction,
                                          point, output);
}

//----------------------------------------------------------------------------
// calculate the thin plate spline as well as the jacobian
template<class T>
inline void vtkThinPlateSplineForwardTransformDerivative(
  vtkThinPlateSplineTransform *self,
  double **W, int N,
  double (*phi)(double, double&),
  const T point[3], T output[3],
  T derivative[3][3])
{
  if (N == 0)
    {
    for (int i = 0; i < 3; i++)
      {
      output[i] = point[i];
      derivative[i][0] = derivative[i][1] = derivative[i][2] = 0.0;
      derivative[i][i] = 1.0;
      }
    return;
    }

  double *C = W[N]; 
  double **A = &W[N+1];

  double dx,dy,dz;
  double p[3];
  double r, U, f, Ux, Uy, Uz;
  double x = 0, y = 0, z = 0; 
  double invSigma = 1.0/self->GetSigma();

  derivative[0][0] = derivative[0][1] = derivative[0][2] = 0;
  derivative[1][0] = derivative[1][1] = derivative[1][2] = 0;
  derivative[2][0] = derivative[2][1] = derivative[2][2] = 0;

  vtkPoints *sourceLandmarks = self->GetSourceLandmarks();

  // do the nonlinear stuff
  for(vtkIdType i = 0; i < N; i++)
    {
    sourceLandmarks->GetPoint(i,p);
    dx = point[0]-p[0]; dy = point[1]-p[1]; dz = point[2]-p[2];
    r = sqrt(dx*dx + dy*dy + dz*dz);

    // get both U and its derivative and do the sigma-mangling
    U = 0;
    f = 0;
    if (r != 0)
      {
      U = phi(r*invSigma,f);
      f *= invSigma/r;
      }

    Ux = f*dx;
    Uy = f*dy;
    Uz = f*dz;

    x += U*W[i][0];
    y += U*W[i][1];
    z += U*W[i][2];

    derivative[0][0] += Ux*W[i][0];
    derivative[0][1] += Uy*W[i][0];
    derivative[0][2] += Uz*W[i][0];
    derivative[1][0] += Ux*W[i][1];
    derivative[1][1] += Uy*W[i][1];
    derivative[1][2] += Uz*W[i][1];
    derivative[2][0] += Ux*W[i][2];
    derivative[2][1] += Uy*W[i][2];
    derivative[2][2] += Uz*W[i][2];
    }

  // finish with the affine transformation
  x += C[0] + point[0]*A[0][0] + point[1]*A[1][0] + point[2]*A[2][0];
  y += C[1] + point[0]*A[0][1] + point[1]*A[1][1] + point[2]*A[2][1];
  z += C[2] + point[0]*A[0][2] + point[1]*A[1][2] + point[2]*A[2][2];

  output[0] = x;
  output[1] = y;
  output[2] = z;

  derivative[0][0] += A[0][0];
  derivative[0][1] += A[1][0];
  derivative[0][2] += A[2][0];
  derivative[1][0] += A[0][1];
  derivative[1][1] += A[1][1];
  derivative[1][2] += A[2][1];
  derivative[2][0] += A[0][2];
  derivative[2][1] += A[1][2];
  derivative[2][2] += A[2][2];
}  

void vtkThinPlateSplineTransform::ForwardTransformDerivative(
                                                  const double point[3],
                                                  double output[3],
                                                  double derivative[3][3])
{
  vtkThinPlateSplineForwardTransformDerivative(this, this->MatrixW, 
                                               this->NumberOfPoints,
                                               this->BasisDerivative,
                                               point, output, derivative);
}

void vtkThinPlateSplineTransform::ForwardTransformDerivative(
                                                  const float point[3],
                                                  float output[3],
                                                  float derivative[3][3])
{
  vtkThinPlateSplineForwardTransformDerivative(this, this->MatrixW, 
                                               this->NumberOfPoints,
                                               this->BasisDerivative,
                                               point, output, derivative);
}

//------------------------------------------------------------------------
void vtkThinPlateSplineTransform::PrintSelf(ostream& os, vtkIndent indent)
{
  this->Superclass::PrintSelf(os,indent);
  
  os << indent << "Sigma: " << this->Sigma << "\n";
  os << indent << "Basis: " << this->GetBasisAsString() << "\n";
  os << indent << "Source Landmarks: " << this->SourceLandmarks << "\n";
  if (this->SourceLandmarks)
    {
    this->SourceLandmarks->PrintSelf(os,indent.GetNextIndent());
    }
  os << indent << "Target Landmarks: " << this->TargetLandmarks << "\n";
  if (this->TargetLandmarks)
    {
    this->TargetLandmarks->PrintSelf(os,indent.GetNextIndent());
    }
}

//----------------------------------------------------------------------------
vtkAbstractTransform *vtkThinPlateSplineTransform::MakeTransform()
{
  return vtkThinPlateSplineTransform::New(); 
}

//----------------------------------------------------------------------------
void vtkThinPlateSplineTransform::InternalDeepCopy(
                                      vtkAbstractTransform *transform)
{
  vtkThinPlateSplineTransform *t = (vtkThinPlateSplineTransform *)transform;

  this->SetInverseTolerance(t->InverseTolerance);
  this->SetInverseIterations(t->InverseIterations);
  this->SetSigma(t->Sigma);
  this->SetBasis(t->GetBasis());
  this->SetSourceLandmarks(t->SourceLandmarks);
  this->SetTargetLandmarks(t->TargetLandmarks);

  if (this->InverseFlag != t->InverseFlag)
    {
    this->InverseFlag = t->InverseFlag;
    this->Modified();
    }
}

//------------------------------------------------------------------------
// a very basic radial basis function
double vtkRBFr(double r)
{
  return r;
}

// calculate both phi(r) its derivative wrt r
double vtkRBFDRr(double r, double &dUdr)
{
  dUdr = 1;
  return r;
}

//------------------------------------------------------------------------
// the standard 2D thin plate spline basis function
double vtkRBFr2logr(double r)
{
  if (r)
    {
    return r*r*log(r);
    }
  else
    {
    return 0;
    }
}

// calculate both phi(r) its derivative wrt r
double vtkRBFDRr2logr(double r, double &dUdr)
{
  if (r)
    {
    double tmp = log(r);
    dUdr = r*(1+2*tmp);
    return r*r*tmp;
    }
  else
    {
    dUdr = 0;
    return 0;
    }
}

//------------------------------------------------------------------------
void vtkThinPlateSplineTransform::SetBasis(int basis)
{
  if (basis == this->Basis)
    {
    return;
    }

  switch (basis)
    {
    case VTK_RBF_CUSTOM:
      break;
    case VTK_RBF_R:
      this->BasisFunction = &vtkRBFr;
      this->BasisDerivative = &vtkRBFDRr;
      break;
    case VTK_RBF_R2LOGR:
      this->BasisFunction = &vtkRBFr2logr;
      this->BasisDerivative = &vtkRBFDRr2logr;
      break;
    default:
      vtkErrorMacro(<< "SetBasisFunction: Unrecognized basis function");
      break;
    }

  this->Basis = basis;
  this->Modified();
}  

//------------------------------------------------------------------------
const char *vtkThinPlateSplineTransform::GetBasisAsString()
{
  switch (this->Basis)
    {
    case VTK_RBF_CUSTOM:
      return "Custom";
    case VTK_RBF_R:
      return "R";
    case VTK_RBF_R2LOGR:
      return "R2LogR";
     }
  return "Unknown";
}