#include <itkWindowedSincInterpolateImageFunction.h>

Inheritance diagram for itk::WindowedSincInterpolateImageFunction< TInputImage, VRadius, TWindowFunction, TBoundaryCondition, TCoordRep >:

Collaboration diagram for itk::WindowedSincInterpolateImageFunction< TInputImage, VRadius, TWindowFunction, TBoundaryCondition, TCoordRep >:

List of all members.

Public Types
typedef SmartPointer< const Self >	ConstPointer
typedef Superclass::ContinuousIndexType	ContinuousIndexType
typedef TInputImage	ImageType
typedef Superclass::IndexType	IndexType
typedef Superclass::IndexValueType	IndexValueType
typedef Superclass::InputImageType	InputImageType
typedef Superclass::OutputType	OutputType
typedef SmartPointer< Self >	Pointer
typedef Superclass::RealType	RealType
typedef WindowedSincInterpolateImageFunction	Self
typedef InterpolateImageFunction < TInputImage, TCoordRep >	Superclass
Public Member Functions
virtual ::itk::LightObject::Pointer	CreateAnother (void) const
virtual OutputType	EvaluateAtContinuousIndex (const ContinuousIndexType &index) const
virtual const char *	GetNameOfClass () const
virtual void	SetInputImage (const ImageType *image)
Static Public Member Functions
static Pointer	New ()
Static Public Attributes
static const unsigned int	ImageDimension = Superclass::ImageDimension
Protected Member Functions
void	PrintSelf (std::ostream &os, Indent indent) const
	WindowedSincInterpolateImageFunction ()
virtual	~WindowedSincInterpolateImageFunction ()
Private Types
typedef ConstNeighborhoodIterator < ImageType, TBoundaryCondition >	IteratorType
Private Member Functions
void	operator= (const Self &)
double	Sinc (double x) const
	WindowedSincInterpolateImageFunction (const Self &)
Private Attributes
unsigned int *	m_OffsetTable
unsigned int	m_OffsetTableSize
unsigned int **	m_WeightOffsetTable
TWindowFunction	m_WindowFunction
Static Private Attributes
static const unsigned int	m_WindowSize

Detailed Description

template<class TInputImage, unsigned int VRadius, class TWindowFunction = Function::HammingWindowFunction< VRadius >, class TBoundaryCondition = ConstantBoundaryCondition< TInputImage >, class TCoordRep = double>
class itk::WindowedSincInterpolateImageFunction< TInputImage, VRadius, TWindowFunction, TBoundaryCondition, TCoordRep >

Use the windowed sinc function to interpolate.

Author:: Paul A. Yushkevich

THEORY

This function is intended to provide an interpolation function that has minimum aliasing artifacts, in contrast to linear interpolation. According to sampling theory, the infinite-support sinc filter, whose Fourier transform is the box filter, is optimal for resampling a function. In practice, the infinite support sinc filter is approximated using a limited support 'windowed' sinc filter.

: This function is based on the following publication:

: Erik H. W. Meijering, Wiro J. Niessen, Josien P. W. Pluim, Max A. Viergever: Quantitative Comparison of Sinc-Approximating Kernels for Medical Image Interpolation. MICCAI 1999, pp. 210-217

: In this work, several 'windows' are estimated. In two dimensions, the interpolation at a position (x,y) is given by the following expression:

: $I(x,y) = \sum_{i = \lfloor x \rfloor + 1 - m}^{\lfloor x \rfloor + m} \sum_{j = \lfloor y \rfloor + 1 - m}^{\lfloor y \rfloor + m} I_{i,j} K(x-i) K(y-j),$

: where m is the 'radius' of the window, (3,4 are reasonable numbers), and K(t) is the kernel function, composed of the sinc function and one of several possible window functions:

: $K(t) = w(t) \textrm{sinc}(t) = w(t) \frac{\sin(\pi t)}{\pi t}$

: Several window functions are provided here in the itk::Function namespace. The conclusions of the referenced paper suggest to use the Welch, Cosine, Kaiser, and Lanczos windows for m = 4,5. These are based on error in rotating medical images w.r.t. the linear interpolation method. In some cases the results achieve a 20-fold improvement in accuracy.

USING THIS FILTER

Use this filter the way you would use any ImageInterpolationFunction, so for instance, you can plug it into the ResampleImageFilter class. In order to initialize the filter you must choose several template parameters.

: The first (TInputImage) is the image type, that's standard.

: The second (VRadius) is the radius of the kernel, i.e., the from the formula above.

: The third (TWindowFunction) is the window function object, which you can choose from about five different functions defined in this header. The default is the Hamming window, which is commonly used but not optimal according to the cited paper.

: The fourth (TBoundaryCondition) is the boundary condition class used to determine the values of pixels that fall off the image boundary. This class has the same meaning here as in the NeighborhoodItetator classes.

: The fifth (TCoordRep) is again standard for interpolating functions, and should be float or double.

CAVEATS

There are a few improvements that an enthusiasting ITK developer could make to this filter. One issue is with the way that the kernel is applied. The computational expense comes from two sources: computing the kernel weights K(t) and multiplying the pixels in the window by the kernel weights. The first is done more or less efficiently in $ 2 m d $ operations (where d is the dimensionality of the image). The second can be done better. Presently, each pixel $ I(i,j,k) $ is multiplied by the weights $ K(x-i), K(y-j), K(z-k) $ and added to the running total. This results in $ d (2m)^d $ multiplication operations. However, by keeping intermediate sums, it would be possible to do the operation in $ O ( (2m)^d ) $ operations. This would require some creative coding. In addition, in the case when one of the coordinates is integer, the computation could be reduced by an order of magnitude.