 ITK  5.4.0 Insight Toolkit
itk::MultivariateLegendrePolynomial Class Reference

#include <itkMultivariateLegendrePolynomial.h>

## Detailed Description

2D and 3D multivariate Legendre Polynomial

In 2D,

$f(x_{vector}, parameter_{vector}) = \sum_i^l \left( \sum_j^{l-i} \left( parameter_ {ij} * P_i(x) *P_j(y)) \right) \right)$

where P_i() denoting a Legendre polynomial of degree i and l it the degree of the polynomial

In 3D,

$f(x_{vector}, parameter_{vector}) = \sum_i^l \left( \sum_j^{l-i} \left( \sum_k^{l-i-j} \left( parameter_{ijk} * P_i(x) * P_j(y) * P_k(z) \right) \right) \right)$

The size of the parameter vector for 2D is $$\frac{(l+1)\cdot(1+2)}{2}$$, and for 3D is $$\frac{(l+1)*(l+2)*(l+3)}{3!}$$

To get the size of the parameter vector, users can use one of the two GetNumberOfCoefficients() member functions

To get function result, users can use the operator() or its SimpleForwardIterator's Get() method.

This is a part of the bias correction methods and implementations that was initially developed and implemented by Martin Styner, Univ. of North Carolina at Chapel Hill, and his colleagues.

Note
For more details. refer to the following articles. "Parametric estimate of intensity inhomogeneities applied to MRI" Martin Styner, G. Gerig, Christian Brechbuehler, Gabor Szekely, IEEE TRANSACTIONS ON MEDICAL IMAGING; 19(3), pp. 153-165, 2000, (https://www.cs.unc.edu/~styner/docs/tmi99.pdf)

"Evaluation of 2D/3D bias correction with 1+1ES-optimization" Martin Styner, Prof. Dr. G. Gerig (IKT, BIWI, ETH Zuerich), TR-197 (https://www.cs.unc.edu/~styner/docs/StynerTR97.pdf)

Definition at line 77 of file itkMultivariateLegendrePolynomial.h. Collaboration diagram for itk::MultivariateLegendrePolynomial:

## Classes

class  CoefficientVectorSizeMismatch

class  SimpleForwardIterator

## Public Types

using CoefficientArrayType = DoubleArrayType

using DomainSizeType = ULongArrayType

using DoubleArrayType = std::vector< double >

using IndexType = LongArrayType

using LongArrayType = std::vector< long >

using ParametersType = Array< double >

using Self = MultivariateLegendrePolynomial

using ULongArrayType = std::vector< unsigned long >

## Public Member Functions

const CoefficientArrayTypeGetCoefficients () const

unsigned int GetDegree () const

unsigned int GetDimension () const

const DomainSizeTypeGetDomainSize () const

unsigned int GetNumberOfCoefficients () const

unsigned int GetNumberOfCoefficients (unsigned int dimension, unsigned int degree)

MultivariateLegendrePolynomial (unsigned int dimension, unsigned int degree, const DomainSizeType &domainSize)

void SetCoefficients (const CoefficientArrayType &coefficients)

void SetCoefficients (const ParametersType &coefficients)

virtual ~MultivariateLegendrePolynomial ()

double Evaluate (IndexType &index)

DomainSizeType m_DomainSize {}

unsigned int m_Dimension {}

unsigned int m_Degree {}

unsigned int m_NumberOfCoefficients {}

CoefficientArrayType m_CoefficientArray {}

CoefficientArrayType m_CachedXCoef {}

CoefficientArrayType m_CachedYCoef {}

CoefficientArrayType m_CachedZCoef {}

DoubleArrayType m_NormFactor {}

IndexValueType m_PrevY {}

IndexValueType m_PrevZ {}

void Print (std::ostream &os) const

void PrintSelf (std::ostream &os, Indent indent) const

double LegendreSum (const double x, int n, const CoefficientArrayType &coef, int offset=0)

void CalculateXCoef (double norm_y, const CoefficientArrayType &coef)

void CalculateYCoef (double norm_z, const CoefficientArrayType &coef)

## ◆ CoefficientArrayType

Internal coefficient storage type.

Definition at line 87 of file itkMultivariateLegendrePolynomial.h.

## ◆ DomainSizeType

The size of the domain.

Definition at line 94 of file itkMultivariateLegendrePolynomial.h.

## ◆ DoubleArrayType

 using itk::MultivariateLegendrePolynomial::DoubleArrayType = std::vector

Definition at line 82 of file itkMultivariateLegendrePolynomial.h.

## ◆ IndexType

Definition at line 95 of file itkMultivariateLegendrePolynomial.h.

## ◆ LongArrayType

 using itk::MultivariateLegendrePolynomial::LongArrayType = std::vector

Definition at line 84 of file itkMultivariateLegendrePolynomial.h.

## ◆ ParametersType

 using itk::MultivariateLegendrePolynomial::ParametersType = Array

Same as CoefficientArray This type definition will be used by EnergyFunction object.

Definition at line 91 of file itkMultivariateLegendrePolynomial.h.

## ◆ Self

Definition at line 80 of file itkMultivariateLegendrePolynomial.h.

## ◆ ULongArrayType

 using itk::MultivariateLegendrePolynomial::ULongArrayType = std::vector

Definition at line 83 of file itkMultivariateLegendrePolynomial.h.

## ◆ MultivariateLegendrePolynomial()

 itk::MultivariateLegendrePolynomial::MultivariateLegendrePolynomial ( unsigned int dimension, unsigned int degree, const DomainSizeType & domainSize )

Constructor.

## ◆ ~MultivariateLegendrePolynomial()

 virtual itk::MultivariateLegendrePolynomial::~MultivariateLegendrePolynomial ( )
virtual

Destructor.

## ◆ CalculateXCoef()

 void itk::MultivariateLegendrePolynomial::CalculateXCoef ( double norm_y, const CoefficientArrayType & coef )
protected

## ◆ CalculateYCoef()

 void itk::MultivariateLegendrePolynomial::CalculateYCoef ( double norm_z, const CoefficientArrayType & coef )
protected

## ◆ Evaluate()

 double itk::MultivariateLegendrePolynomial::Evaluate ( IndexType & index )
inline

In the case which the bias field is 2D, it returns bias value at the point which is specified by the index

Definition at line 170 of file itkMultivariateLegendrePolynomial.h.

## ◆ GetCoefficients()

 const CoefficientArrayType& itk::MultivariateLegendrePolynomial::GetCoefficients ( ) const

Gets Legendre polynomials' coefficients.

## ◆ GetDegree()

 unsigned int itk::MultivariateLegendrePolynomial::GetDegree ( ) const
inline

Gets the degree (the degree of Legendre polynomials).

Definition at line 112 of file itkMultivariateLegendrePolynomial.h.

## ◆ GetDimension()

 unsigned int itk::MultivariateLegendrePolynomial::GetDimension ( ) const
inline

Gets the dimension.

Definition at line 105 of file itkMultivariateLegendrePolynomial.h.

## ◆ GetDomainSize()

 const DomainSizeType& itk::MultivariateLegendrePolynomial::GetDomainSize ( ) const
inline

Gets each dimension's size.

Definition at line 131 of file itkMultivariateLegendrePolynomial.h.

## ◆ GetNumberOfCoefficients() [1/2]

 unsigned int itk::MultivariateLegendrePolynomial::GetNumberOfCoefficients ( ) const
inline

Returns the number of coefficients of the polynomial This number is computed from the degree of the polynomial the SetCoefficients() method expects an array of this size, an exception is thrown otherwise

SetCoefficients

Definition at line 124 of file itkMultivariateLegendrePolynomial.h.

## ◆ GetNumberOfCoefficients() [2/2]

 unsigned int itk::MultivariateLegendrePolynomial::GetNumberOfCoefficients ( unsigned int dimension, unsigned int degree )

Gets the number of coefficients.

## ◆ LegendreSum()

 double itk::MultivariateLegendrePolynomial::LegendreSum ( const double x, int n, const CoefficientArrayType & coef, int offset = 0 )
protected

## ◆ Print()

 void itk::MultivariateLegendrePolynomial::Print ( std::ostream & os ) const

## ◆ PrintSelf()

 void itk::MultivariateLegendrePolynomial::PrintSelf ( std::ostream & os, Indent indent ) const
protected

## ◆ SetCoefficients() [1/2]

 void itk::MultivariateLegendrePolynomial::SetCoefficients ( const CoefficientArrayType & coefficients )

Sets the Legendre polynomials' parameters.

Warning
The number of coefficients provided should match the number returned by GetNumberOfCoefficients() otherwise an exception is thrown.

## ◆ SetCoefficients() [2/2]

 void itk::MultivariateLegendrePolynomial::SetCoefficients ( const ParametersType & coefficients )

## ◆ m_CachedXCoef

 CoefficientArrayType itk::MultivariateLegendrePolynomial::m_CachedXCoef {}
private

Definition at line 317 of file itkMultivariateLegendrePolynomial.h.

## ◆ m_CachedYCoef

 CoefficientArrayType itk::MultivariateLegendrePolynomial::m_CachedYCoef {}
private

Definition at line 318 of file itkMultivariateLegendrePolynomial.h.

## ◆ m_CachedZCoef

 CoefficientArrayType itk::MultivariateLegendrePolynomial::m_CachedZCoef {}
private

Definition at line 319 of file itkMultivariateLegendrePolynomial.h.

## ◆ m_CoefficientArray

 CoefficientArrayType itk::MultivariateLegendrePolynomial::m_CoefficientArray {}
private

Definition at line 316 of file itkMultivariateLegendrePolynomial.h.

## ◆ m_Degree

 unsigned int itk::MultivariateLegendrePolynomial::m_Degree {}
private

Definition at line 313 of file itkMultivariateLegendrePolynomial.h.

## ◆ m_Dimension

 unsigned int itk::MultivariateLegendrePolynomial::m_Dimension {}
private

Definition at line 312 of file itkMultivariateLegendrePolynomial.h.

## ◆ m_DomainSize

 DomainSizeType itk::MultivariateLegendrePolynomial::m_DomainSize {}
private

Definition at line 311 of file itkMultivariateLegendrePolynomial.h.

## ◆ m_NormFactor

 DoubleArrayType itk::MultivariateLegendrePolynomial::m_NormFactor {}
private

Definition at line 321 of file itkMultivariateLegendrePolynomial.h.

## ◆ m_NumberOfCoefficients

 unsigned int itk::MultivariateLegendrePolynomial::m_NumberOfCoefficients {}
private

Definition at line 314 of file itkMultivariateLegendrePolynomial.h.

## ◆ m_PrevY

 IndexValueType itk::MultivariateLegendrePolynomial::m_PrevY {}
private

Definition at line 322 of file itkMultivariateLegendrePolynomial.h.

## ◆ m_PrevZ

 IndexValueType itk::MultivariateLegendrePolynomial::m_PrevZ {}
private

Definition at line 323 of file itkMultivariateLegendrePolynomial.h.

The documentation for this class was generated from the following file: