itkNumericInverse.h

/*
 For more information, please see: http://software.sci.utah.edu
 
 The MIT License
 
 Copyright (c) 2016 Scientific Computing and Imaging Institute,
 University of Utah.
 
 
 Permission is hereby granted, free of charge, to any person obtaining a
 copy of this software and associated documentation files (the "Software"),
 to deal in the Software without restriction, including without limitation
 the rights to use, copy, modify, merge, publish, distribute, sublicense,
 and/or sell copies of the Software, and to permit persons to whom the
 Software is furnished to do so, subject to the following conditions:
 
 The above copyright notice and this permission notice shall be included
 in all copies or substantial portions of the Software.
 
 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
 OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
 THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
 FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
 DEALINGS IN THE SOFTWARE.
*/

// File         : itkNumericTransform.h
// Author       : Pavel A. Koshevoy
// Created      : 2005/06/03 10:16
// Copyright    : (C) 2004-2008 University of Utah
// Description  : A numerical inverse transform class, based on the
//                Newton-Raphson method for nonlinear systems of equations.

#ifndef __itkNumericInverse_h
#define __itkNumericInverse_h

// system includes:
#include <iostream>
#include <vector>

// ITK includes:
#include <itkTransform.h>
#include <itkMacro.h>

// VXL includes:
#include <vnl/algo/vnl_svd.h>

namespace help
{

//----------------------------------------------------------------
// nonlinear_system_evaluator_t
// 
template <class ScalarType>
class nonlinear_system_evaluator_t
{
public:
  virtual ~nonlinear_system_evaluator_t() {}
  
  // evaluate the nonlinear system of equations
  // and its Jacobian (dF/dx) at a given point:
  virtual void
  operator() (const std::vector<ScalarType> & x,
              std::vector<ScalarType> & F,
              std::vector<std::vector<ScalarType> > & J) const = 0;
};

//----------------------------------------------------------------
// NewtonRaphson
// 
template <class ScalarType>
bool
NewtonRaphson(const nonlinear_system_evaluator_t<ScalarType> & usrfun,
              std::vector<ScalarType> & x, // estimated root point
              const unsigned int & ntrial, // number of iterations
              const ScalarType & tolx, // convergence tolerance in x
              const ScalarType & tolf) // convergence tolerance in F
{
  // shortcut:
  const unsigned int n = x.size();
  
  std::vector<ScalarType> F(n);
  std::vector<std::vector<ScalarType> > J(n);
  for (unsigned int i = 0; i < n; i++) J[i].resize(n);
  
  vnl_matrix<ScalarType> A(n, n);
  vnl_vector<ScalarType> b(n);
  
  for (unsigned int k = 0; k < ntrial; k++)
  {
    // evaluate the function at the current position:
    usrfun(x, F, J);
    
    // check for function convergence:
    ScalarType errf = ScalarType(0);
    for (unsigned int i = 0; i < n; i++)
    {
      errf += fabs(F[i]);
    }
    if (errf <= tolf) break;
    
    // setup the left hand side of the linear system:
    for (unsigned int i = 0; i < n; i++)
    {
      for (unsigned int j = 0; j < n; j++)
      {
        A[i][j] = J[i][j];
      }
    }
    
    // setup the right hand side of the linear system:
    for (unsigned int i = 0; i < n; i++)
    {
      b[i] = -F[i];
    }
    
    vnl_svd<double> svd(A);
    vnl_vector<double> dx = svd.solve(b);
    
    // check for root convergence:
    ScalarType errx = ScalarType(0);
    for (unsigned int i = 0; i < n; i++)
    {
      errx += fabs(dx[i]);
      x[i] += dx[i];
    }
    if (errx <= tolx) break;
  }
  
  return true;
}

} // namespace help

namespace itk
{


//----------------------------------------------------------------
// NumericInverse
//
template <class TTransform>
class NumericInverse :
  public help::nonlinear_system_evaluator_t<typename TTransform::ScalarType>
{
public:
  typedef TTransform TransformType;
  typedef typename TTransform::ScalarType ScalarType;
  
  NumericInverse(const TransformType & transform):
    transform_(transform)
  {}
  
  // virtual:
  virtual
  void operator() (const std::vector<ScalarType> & x,
                   std::vector<ScalarType> & F,
                   std::vector<std::vector<ScalarType> > & J) const
  {
    transform_.eval(x, F, J);
    
    const unsigned int n = x.size();
    for (unsigned int i = 0; i < n; i++)
    {
      F[i] -= y_[i];
    }
  }
  
  // If y = Transform(x), then x = BackTransform(y).
  // given y, find x:
  bool transform(const std::vector<ScalarType> & y,
     std::vector<ScalarType> & x,
     bool x_is_initialized = false) const
  {
    y_ = y;
    if (!x_is_initialized) x  = y;
    return NewtonRaphson(*this, x, 50, 1e-12, 1e-12);
  }
  
private:
  // the transform whose inverse we are trying to evaluate:
  const TransformType & transform_;
  
  // the point for which we are tryying to find the inverse mapping:
  mutable std::vector<ScalarType> y_;
};

} // namespace itk

#endif // __itkNumericInverse_h