ITK  5.4.0
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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.

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, (

"Evaluation of 2D/3D bias correction with 1+1ES-optimization" Martin Styner, Prof. Dr. G. Gerig (IKT, BIWI, ETH Zuerich), TR-197 (

Definition at line 77 of file itkMultivariateLegendrePolynomial.h.

+ Collaboration diagram for itk::MultivariateLegendrePolynomial:


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)

Member Typedef Documentation

◆ 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

Definition at line 82 of file itkMultivariateLegendrePolynomial.h.

◆ IndexType

Definition at line 95 of file itkMultivariateLegendrePolynomial.h.

◆ LongArrayType

Definition at line 84 of file itkMultivariateLegendrePolynomial.h.

◆ ParametersType

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<unsigned long>

Definition at line 83 of file itkMultivariateLegendrePolynomial.h.

Constructor & Destructor Documentation

◆ MultivariateLegendrePolynomial()

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


◆ ~MultivariateLegendrePolynomial()

virtual itk::MultivariateLegendrePolynomial::~MultivariateLegendrePolynomial ( )


Member Function Documentation

◆ CalculateXCoef()

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

◆ CalculateYCoef()

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

◆ Evaluate()

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

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

Gets the degree (the degree of Legendre polynomials).

Definition at line 112 of file itkMultivariateLegendrePolynomial.h.

◆ GetDimension()

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

Gets the dimension.

Definition at line 105 of file itkMultivariateLegendrePolynomial.h.

◆ GetDomainSize()

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

Gets each dimension's size.

Definition at line 131 of file itkMultivariateLegendrePolynomial.h.

◆ GetNumberOfCoefficients() [1/2]

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

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

See also

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 

◆ Print()

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

◆ PrintSelf()

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

◆ SetCoefficients() [1/2]

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

Sets the Legendre polynomials' parameters.

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)

Member Data Documentation

◆ m_CachedXCoef

CoefficientArrayType itk::MultivariateLegendrePolynomial::m_CachedXCoef {}

Definition at line 317 of file itkMultivariateLegendrePolynomial.h.

◆ m_CachedYCoef

CoefficientArrayType itk::MultivariateLegendrePolynomial::m_CachedYCoef {}

Definition at line 318 of file itkMultivariateLegendrePolynomial.h.

◆ m_CachedZCoef

CoefficientArrayType itk::MultivariateLegendrePolynomial::m_CachedZCoef {}

Definition at line 319 of file itkMultivariateLegendrePolynomial.h.

◆ m_CoefficientArray

CoefficientArrayType itk::MultivariateLegendrePolynomial::m_CoefficientArray {}

Definition at line 316 of file itkMultivariateLegendrePolynomial.h.

◆ m_Degree

unsigned int itk::MultivariateLegendrePolynomial::m_Degree {}

Definition at line 313 of file itkMultivariateLegendrePolynomial.h.

◆ m_Dimension

unsigned int itk::MultivariateLegendrePolynomial::m_Dimension {}

Definition at line 312 of file itkMultivariateLegendrePolynomial.h.

◆ m_DomainSize

DomainSizeType itk::MultivariateLegendrePolynomial::m_DomainSize {}

Definition at line 311 of file itkMultivariateLegendrePolynomial.h.

◆ m_NormFactor

DoubleArrayType itk::MultivariateLegendrePolynomial::m_NormFactor {}

Definition at line 321 of file itkMultivariateLegendrePolynomial.h.

◆ m_NumberOfCoefficients

unsigned int itk::MultivariateLegendrePolynomial::m_NumberOfCoefficients {}

Definition at line 314 of file itkMultivariateLegendrePolynomial.h.

◆ m_PrevY

IndexValueType itk::MultivariateLegendrePolynomial::m_PrevY {}

Definition at line 322 of file itkMultivariateLegendrePolynomial.h.

◆ m_PrevZ

IndexValueType itk::MultivariateLegendrePolynomial::m_PrevZ {}

Definition at line 323 of file itkMultivariateLegendrePolynomial.h.

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