itk::KernelTransform2

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#include <itkKernelTransform2.h>

Inherits from itk::AdvancedTransform< TScalarType, NDimensions, NDimensions >, Transform< TScalarType, 3, 3 >

Public Types

	Name
typedef KernelTransform2	Self
typedef AdvancedTransform< TScalarType, NDimensions, NDimensions >	Superclass
typedef SmartPointer< Self >	Pointer
typedef SmartPointer< const Self >	ConstPointer
typedef Superclass::ScalarType	ScalarType
typedef Superclass::ParametersType	ParametersType
typedef Superclass::NumberOfParametersType	NumberOfParametersType
typedef Superclass::JacobianType	JacobianType
typedef Superclass::InputPointType	InputPointType
typedef Superclass::OutputPointType	OutputPointType
typedef Superclass::InputVectorType	InputVectorType
typedef Superclass::OutputVectorType	OutputVectorType
typedef Superclass::InputCovariantVectorType	InputCovariantVectorType
typedef Superclass::OutputCovariantVectorType	OutputCovariantVectorType
typedef Superclass::InputVnlVectorType	InputVnlVectorType
typedef Superclass::OutputVnlVectorType	OutputVnlVectorType
typedef Superclass ::NonZeroJacobianIndicesType	NonZeroJacobianIndicesType
typedef Superclass::SpatialJacobianType	SpatialJacobianType
typedef Superclass ::JacobianOfSpatialJacobianType	JacobianOfSpatialJacobianType
typedef Superclass::SpatialHessianType	SpatialHessianType
typedef Superclass ::JacobianOfSpatialHessianType	JacobianOfSpatialHessianType
typedef Superclass::InternalMatrixType	InternalMatrixType
typedef DefaultStaticMeshTraits< TScalarType, NDimensions, NDimensions, TScalarType, TScalarType >	PointSetTraitsType
typedef PointSet< InputPointType, NDimensions, PointSetTraitsType >	PointSetType
typedef PointSetType::Pointer	PointSetPointer
typedef PointSetType::PointsContainer	PointsContainer
typedef PointSetType::PointsContainerIterator	PointsIterator
typedef PointSetType::PointsContainerConstIterator	PointsConstIterator
typedef VectorContainer< unsigned long, InputVectorType >	VectorSetType
typedef VectorSetType::Pointer	VectorSetPointer
typedef vnl_matrix_fixed< TScalarType, NDimensions, NDimensions >	IMatrixType
typedef vnl_matrix_fixed< TScalarType, NDimensions, NDimensions >	GMatrixType
typedef vnl_matrix< TScalarType >	LMatrixType
typedef vnl_matrix< TScalarType >	KMatrixType
typedef vnl_matrix< TScalarType >	PMatrixType
typedef vnl_matrix< TScalarType >	YMatrixType
typedef vnl_matrix< TScalarType >	WMatrixType
typedef vnl_matrix< TScalarType >	DMatrixType
typedef vnl_matrix_fixed< TScalarType, NDimensions, NDimensions >	AMatrixType
typedef vnl_vector_fixed< TScalarType, NDimensions >	BMatrixType
typedef vnl_matrix_fixed< TScalarType, 1, NDimensions >	RowMatrixType
typedef vnl_matrix_fixed< TScalarType, NDimensions, 1 >	ColumnMatrixType

Protected Types

	Name
typedef vnl_svd< ScalarType >	SVDDecompositionType
typedef vnl_qr< ScalarType >	QRDecompositionType

Public Functions

	Name
	itkTypeMacro(KernelTransform2 , AdvancedTransform )
	itkNewMacro(Self )
	itkStaticConstMacro(SpaceDimension , unsigned int , NDimensions )
virtual NumberOfParametersType	GetNumberOfParameters(void ) const
	itkGetObjectMacro(SourceLandmarks , PointSetType )
virtual void	SetSourceLandmarks(PointSetType * )
	itkGetObjectMacro(TargetLandmarks , PointSetType )
virtual void	SetTargetLandmarks(PointSetType * )
	itkGetObjectMacro(Displacements , VectorSetType )
void	ComputeWMatrix(void )
void	ComputeLInverse(void )
virtual OutputPointType	TransformPoint(const InputPointType & thisPoint) const
virtual OutputVectorType	TransformVector(const InputVectorType & ) const
virtual OutputVnlVectorType	TransformVector(const InputVnlVectorType & ) const
virtual OutputCovariantVectorType	TransformCovariantVector(const InputCovariantVectorType & ) const
virtual void	GetJacobian(const InputPointType & , JacobianType & , NonZeroJacobianIndicesType & ) const
virtual void	SetIdentity(void )
virtual void	SetParameters(const ParametersType & )
virtual void	SetFixedParameters(const ParametersType & )
virtual void	UpdateParameters(void )
virtual const ParametersType &	GetParameters(void ) const
virtual const ParametersType &	GetFixedParameters(void ) const
virtual void	SetStiffness(double stiffness)
	itkGetMacro(Stiffness , double )
virtual void	SetAlpha(TScalarType itkNotUsedAlpha)
virtual TScalarType	GetAlpha(void ) const
	itkSetMacro(PoissonRatio , TScalarType )
virtual const TScalarType	GetPoissonRatio(void ) const
	itkSetMacro(MatrixInversionMethod , std::string )
	itkGetConstReferenceMacro(MatrixInversionMethod , std::string )
virtual void	GetSpatialJacobian(const InputPointType & ipp, SpatialJacobianType & sj) const
virtual void	GetSpatialHessian(const InputPointType & ipp, SpatialHessianType & sh) const
virtual void	GetJacobianOfSpatialJacobian(const InputPointType & ipp, JacobianOfSpatialJacobianType & jsj, NonZeroJacobianIndicesType & nonZeroJacobianIndices) const
virtual void	GetJacobianOfSpatialJacobian(const InputPointType & ipp, SpatialJacobianType & sj, JacobianOfSpatialJacobianType & jsj, NonZeroJacobianIndicesType & nonZeroJacobianIndices) const
virtual void	GetJacobianOfSpatialHessian(const InputPointType & ipp, JacobianOfSpatialHessianType & jsh, NonZeroJacobianIndicesType & nonZeroJacobianIndices) const
virtual void	GetJacobianOfSpatialHessian(const InputPointType & ipp, SpatialHessianType & sh, JacobianOfSpatialHessianType & jsh, NonZeroJacobianIndicesType & nonZeroJacobianIndices) const

Protected Functions

	Name
	KernelTransform2()
virtual	~KernelTransform2()
void	PrintSelf(std::ostream & os, Indent indent) const
virtual void	ComputeG(const InputVectorType & landmarkVector, GMatrixType & GMatrix) const
virtual void	ComputeReflexiveG(PointsIterator , GMatrixType & GMatrix) const
virtual void	ComputeDeformationContribution(const InputPointType & inputPoint, OutputPointType & result) const
void	ComputeK(void )
void	ComputeL(void )
void	ComputeP(void )
void	ComputeY(void )
void	ComputeD(void )
void	ReorganizeW(void )

Public Attributes

	Name
PointSetPointer	m_SourceLandmarks
PointSetPointer	m_TargetLandmarks

Protected Attributes

	Name
double	m_Stiffness
VectorSetPointer	m_Displacements
LMatrixType	m_LMatrix
LMatrixType	m_LMatrixInverse
KMatrixType	m_KMatrix
PMatrixType	m_PMatrix
YMatrixType	m_YMatrix
WMatrixType	m_WMatrix
DMatrixType	m_DMatrix
AMatrixType	m_AMatrix
BMatrixType	m_BVector
bool	m_WMatrixComputed
bool	m_LMatrixComputed
bool	m_LInverseComputed
bool	m_LMatrixDecompositionComputed
SVDDecompositionType *	m_LMatrixDecompositionSVD
QRDecompositionType *	m_LMatrixDecompositionQR
IMatrixType	m_I
NonZeroJacobianIndicesType	m_NonZeroJacobianIndices
NonZeroJacobianIndicesType	m_NonZeroJacobianIndicesTemp
bool	m_FastComputationPossible

Additional inherited members

Public Types inherited from itk::AdvancedTransform< TScalarType, NDimensions, NDimensions >

	Name
typedef Superclass::ParametersValueType	ParametersValueType
typedef Superclass::DerivativeType	DerivativeType
typedef Superclass::InverseTransformBaseType	InverseTransformBaseType
typedef Superclass::InverseTransformBasePointer	InverseTransformBasePointer
typedef Transform< TScalarType, NInputDimensions, NOutputDimensions >	TransformType
typedef TransformType::Pointer	TransformTypePointer
typedef TransformType::ConstPointer	TransformTypeConstPointer
typedef OutputCovariantVectorType	MovingImageGradientType
typedef MovingImageGradientType::ValueType	MovingImageGradientValueType

Public Functions inherited from itk::AdvancedTransform< TScalarType, NDimensions, NDimensions >

	Name
virtual NumberOfParametersType	GetNumberOfNonZeroJacobianIndices(void ) const
	itkGetConstMacro(HasNonZeroSpatialHessian , bool )
	itkGetConstMacro(HasNonZeroJacobianOfSpatialHessian , bool )
virtual void	EvaluateJacobianWithImageGradientProduct(const InputPointType & ipp, const MovingImageGradientType & movingImageGradient, DerivativeType & imageJacobian, NonZeroJacobianIndicesType & nonZeroJacobianIndices) const
virtual void	ComputeJacobianWithRespectToParameters(const InputPointType & itkNotUsedp, JacobianType & itkNotUsedj) const
virtual void	ComputeJacobianWithRespectToPosition(const InputPointType & itkNotUsedp, JacobianType & itkNotUsedj) const

Protected Functions inherited from itk::AdvancedTransform< TScalarType, NDimensions, NDimensions >

	Name
	AdvancedTransform()
	AdvancedTransform(NumberOfParametersType numberOfParameters)
virtual	~AdvancedTransform()

Protected Attributes inherited from itk::AdvancedTransform< TScalarType, NDimensions, NDimensions >

	Name
bool	m_HasNonZeroSpatialHessian
bool	m_HasNonZeroJacobianOfSpatialHessian

Detailed Description

template <class TScalarType ,
unsigned int NDimensions>
class itk::KernelTransform2;

Intended to be a base class for elastic body spline and thin plate spline. This is implemented in as straightforward a manner as possible from the IEEE TMI paper by Davis, Khotanzad, Flamig, and Harms, Vol. 16, No. 3 June 1997. Notation closely follows their paper, so if you have it in front of you, this code will make a lot more sense.

KernelTransform2: Provides support for defining source and target landmarks Defines a number of data types used in the computations Defines the mathematical framework used to compute all splines, so that subclasses need only provide a kernel specific to that spline

This formulation allows the stiffness of the spline to be adjusted, allowing the spline to vary from interpolating the landmarks to approximating the landmarks. This part of the formulation is based on the short paper by R. Sprengel, K. Rohr, H. Stiehl. "Thin-Plate Spline Approximation for Image Registration". In 18th International Conference of the IEEE Engineering in Medicine and Biology Society. 1996.

This class was modified to support its use in the ITK registration framework by Rupert Brooks, McGill Centre for Intelligent Machines, Montreal, Canada March 2007. See the Insight Journal Paper by Brooks, R., Arbel, T. "Improvements to the itk::KernelTransform and its subclasses."

Modified to include it in elastix:

• style
• make it inherit from AdvancedTransform
• make it threadsafe, like was done in the itk as well.
• Support for matrix inversion by QR decomposition, instead of SVD. QR is much faster. Used in SetParameters() and SetFixedParameters().
• Much faster Jacobian computation for some of the derived kernel transforms.

Public Types Documentation

typedef Self

typedef KernelTransform2 itk::KernelTransform2< TScalarType, NDimensions >::Self;

Standard class typedefs.

typedef Superclass

typedef AdvancedTransform< TScalarType, NDimensions, NDimensions > itk::KernelTransform2< TScalarType, NDimensions >::Superclass;

typedef Pointer

typedef SmartPointer< Self > itk::KernelTransform2< TScalarType, NDimensions >::Pointer;

typedef ConstPointer

typedef SmartPointer< const Self > itk::KernelTransform2< TScalarType, NDimensions >::ConstPointer;

typedef ScalarType

typedef Superclass::ScalarType itk::KernelTransform2< TScalarType, NDimensions >::ScalarType;

Typedefs.

typedef ParametersType

typedef Superclass::ParametersType itk::KernelTransform2< TScalarType, NDimensions >::ParametersType;

typedef NumberOfParametersType

typedef Superclass::NumberOfParametersType itk::KernelTransform2< TScalarType, NDimensions >::NumberOfParametersType;

typedef JacobianType

typedef Superclass::JacobianType itk::KernelTransform2< TScalarType, NDimensions >::JacobianType;

typedef InputPointType

typedef Superclass::InputPointType itk::KernelTransform2< TScalarType, NDimensions >::InputPointType;

typedef OutputPointType

typedef Superclass::OutputPointType itk::KernelTransform2< TScalarType, NDimensions >::OutputPointType;

typedef InputVectorType

typedef Superclass::InputVectorType itk::KernelTransform2< TScalarType, NDimensions >::InputVectorType;

typedef OutputVectorType

typedef Superclass::OutputVectorType itk::KernelTransform2< TScalarType, NDimensions >::OutputVectorType;

typedef InputCovariantVectorType

typedef Superclass::InputCovariantVectorType itk::KernelTransform2< TScalarType, NDimensions >::InputCovariantVectorType;

typedef OutputCovariantVectorType

typedef Superclass::OutputCovariantVectorType itk::KernelTransform2< TScalarType, NDimensions >::OutputCovariantVectorType;

typedef InputVnlVectorType

typedef Superclass::InputVnlVectorType itk::KernelTransform2< TScalarType, NDimensions >::InputVnlVectorType;

typedef OutputVnlVectorType

typedef Superclass::OutputVnlVectorType itk::KernelTransform2< TScalarType, NDimensions >::OutputVnlVectorType;

typedef NonZeroJacobianIndicesType

typedef Superclass ::NonZeroJacobianIndicesType itk::KernelTransform2< TScalarType, NDimensions >::NonZeroJacobianIndicesType;

AdvancedTransform typedefs.

typedef SpatialJacobianType

typedef Superclass::SpatialJacobianType itk::KernelTransform2< TScalarType, NDimensions >::SpatialJacobianType;

typedef JacobianOfSpatialJacobianType

typedef Superclass ::JacobianOfSpatialJacobianType itk::KernelTransform2< TScalarType, NDimensions >::JacobianOfSpatialJacobianType;

typedef SpatialHessianType

typedef Superclass::SpatialHessianType itk::KernelTransform2< TScalarType, NDimensions >::SpatialHessianType;

typedef JacobianOfSpatialHessianType

typedef Superclass ::JacobianOfSpatialHessianType itk::KernelTransform2< TScalarType, NDimensions >::JacobianOfSpatialHessianType;

typedef InternalMatrixType

typedef Superclass::InternalMatrixType itk::KernelTransform2< TScalarType, NDimensions >::InternalMatrixType;

typedef PointSetTraitsType

typedef DefaultStaticMeshTraits< TScalarType, NDimensions, NDimensions, TScalarType, TScalarType > itk::KernelTransform2< TScalarType, NDimensions >::PointSetTraitsType;

PointList typedef. This type is used for maintaining lists of points, specifically, the source and target landmark lists.

typedef PointSetType

typedef PointSet< InputPointType, NDimensions, PointSetTraitsType > itk::KernelTransform2< TScalarType, NDimensions >::PointSetType;

typedef PointSetPointer

typedef PointSetType::Pointer itk::KernelTransform2< TScalarType, NDimensions >::PointSetPointer;

typedef PointsContainer

typedef PointSetType::PointsContainer itk::KernelTransform2< TScalarType, NDimensions >::PointsContainer;

typedef PointsIterator

typedef PointSetType::PointsContainerIterator itk::KernelTransform2< TScalarType, NDimensions >::PointsIterator;

typedef PointsConstIterator

typedef PointSetType::PointsContainerConstIterator itk::KernelTransform2< TScalarType, NDimensions >::PointsConstIterator;

typedef VectorSetType

typedef VectorContainer< unsigned long, InputVectorType > itk::KernelTransform2< TScalarType, NDimensions >::VectorSetType;

VectorSet typedef.

typedef VectorSetPointer

typedef VectorSetType::Pointer itk::KernelTransform2< TScalarType, NDimensions >::VectorSetPointer;

typedef IMatrixType

typedef vnl_matrix_fixed< TScalarType, NDimensions, NDimensions > itk::KernelTransform2< TScalarType, NDimensions >::IMatrixType;

'I' (identity) matrix typedef.

typedef GMatrixType

typedef vnl_matrix_fixed< TScalarType, NDimensions, NDimensions > itk::KernelTransform2< TScalarType, NDimensions >::GMatrixType;

'G' matrix typedef.

typedef LMatrixType

typedef vnl_matrix< TScalarType > itk::KernelTransform2< TScalarType, NDimensions >::LMatrixType;

'L' matrix typedef.

typedef KMatrixType

typedef vnl_matrix< TScalarType > itk::KernelTransform2< TScalarType, NDimensions >::KMatrixType;

'K' matrix typedef.

typedef PMatrixType

typedef vnl_matrix< TScalarType > itk::KernelTransform2< TScalarType, NDimensions >::PMatrixType;

'P' matrix typedef.

typedef YMatrixType

typedef vnl_matrix< TScalarType > itk::KernelTransform2< TScalarType, NDimensions >::YMatrixType;

'Y' matrix typedef.

typedef WMatrixType

typedef vnl_matrix< TScalarType > itk::KernelTransform2< TScalarType, NDimensions >::WMatrixType;

'W' matrix typedef.

typedef DMatrixType

typedef vnl_matrix< TScalarType > itk::KernelTransform2< TScalarType, NDimensions >::DMatrixType;

'D' matrix typedef. Deformation component

typedef AMatrixType

typedef vnl_matrix_fixed< TScalarType, NDimensions, NDimensions > itk::KernelTransform2< TScalarType, NDimensions >::AMatrixType;

'A' matrix typedef. Rotational part of the Affine component

typedef BMatrixType

typedef vnl_vector_fixed< TScalarType, NDimensions > itk::KernelTransform2< TScalarType, NDimensions >::BMatrixType;

'B' matrix typedef. Translational part of the Affine component

typedef RowMatrixType

typedef vnl_matrix_fixed< TScalarType, 1, NDimensions > itk::KernelTransform2< TScalarType, NDimensions >::RowMatrixType;

Row matrix typedef.

typedef ColumnMatrixType

typedef vnl_matrix_fixed< TScalarType, NDimensions, 1 > itk::KernelTransform2< TScalarType, NDimensions >::ColumnMatrixType;

Column matrix typedef.

Protected Types Documentation

typedef SVDDecompositionType

typedef vnl_svd< ScalarType > itk::KernelTransform2< TScalarType, NDimensions >::SVDDecompositionType;

Decompositions, needed for the L matrix. These decompositions are cached for performance reasons during registration. During registration, in every iteration SetParameters() is called, which in turn calls ComputeWMatrix(). The L matrix is not changed however, and therefore it is not needed to redo the decomposition.

typedef QRDecompositionType

typedef vnl_qr< ScalarType > itk::KernelTransform2< TScalarType, NDimensions >::QRDecompositionType;

Public Functions Documentation

function itkTypeMacro

itkTypeMacro(
    KernelTransform2 ,
    AdvancedTransform 
)

Run-time type information (and related methods).

function itkNewMacro

itkNewMacro(
    Self 
)

New macro for creation of through a Smart Pointer.

function itkStaticConstMacro

itkStaticConstMacro(
    SpaceDimension ,
    unsigned int ,
    NDimensions 
)

Dimension of the domain space.

function GetNumberOfParameters

inline virtual NumberOfParametersType GetNumberOfParameters(
    void 
) const

Return the number of parameters that completely define the Transform.

function itkGetObjectMacro

itkGetObjectMacro(
    SourceLandmarks ,
    PointSetType 
)

Get the source landmarks list, which we will denote \( p \).

function SetSourceLandmarks

virtual void SetSourceLandmarks(
    PointSetType * 
)

Set the source landmarks list.

function itkGetObjectMacro

itkGetObjectMacro(
    TargetLandmarks ,
    PointSetType 
)

Get the target landmarks list, which we will denote \( q \).

function SetTargetLandmarks

virtual void SetTargetLandmarks(
    PointSetType * 
)

Set the target landmarks list.

function itkGetObjectMacro

itkGetObjectMacro(
    Displacements ,
    VectorSetType 
)

Get the displacements list, which we will denote \( d \), where \( d_i = q_i - p_i \).

function ComputeWMatrix

void ComputeWMatrix(
    void 
)

Compute W matrix.

function ComputeLInverse

void ComputeLInverse(
    void 
)

Compute L matrix inverse.

function TransformPoint

virtual OutputPointType TransformPoint(
    const InputPointType & thisPoint
) const

Compute the position of point in the new space

function TransformVector

inline virtual OutputVectorType TransformVector(
    const InputVectorType & 
) const

These vector transforms are not implemented for this transform.

function TransformVector

inline virtual OutputVnlVectorType TransformVector(
    const InputVnlVectorType & 
) const

function TransformCovariantVector

inline virtual OutputCovariantVectorType TransformCovariantVector(
    const InputCovariantVectorType & 
) const

function GetJacobian

virtual void GetJacobian(
    const InputPointType & ,
    JacobianType & ,
    NonZeroJacobianIndicesType & 
) const

Compute the Jacobian of the transformation.

function SetIdentity

virtual void SetIdentity(
    void 
)

Set the Transformation Parameters to be an identity transform.

function SetParameters

virtual void SetParameters(
    const ParametersType & 
)

Set the Transformation Parameters and update the internal transformation. The parameters represent the source landmarks. Each landmark point is represented by NDimensions doubles. All the landmarks are concatenated to form one flat Array.

function SetFixedParameters

virtual void SetFixedParameters(
    const ParametersType & 
)

Set Transform Fixed Parameters: To support the transform file writer this function was added to set the target landmarks similar to the SetParameters function setting the source landmarks

function UpdateParameters

virtual void UpdateParameters(
    void 
)

Update the Parameters array from the landmarks coordinates.

function GetParameters

virtual const ParametersType & GetParameters(
    void 
) const

Get the Transformation Parameters - Gets the source landmarks.

function GetFixedParameters

virtual const ParametersType & GetFixedParameters(
    void 
) const

Get Transform Fixed Parameters - Gets the target landmarks.

function SetStiffness

inline virtual void SetStiffness(
    double stiffness
)

Stiffness of the spline. A stiffness of zero results in the standard interpolating spline. A non-zero stiffness allows the spline to approximate rather than interpolate the landmarks. Stiffness values are usually rather small, typically in the range of 0.001 to 0.1. The approximating spline formulation is based on the short paper by R. Sprengel, K. Rohr, H. Stiehl. "Thin-Plate Spline Approximation for Image Registration". In 18th International Conference of the IEEE Engineering in Medicine and Biology Society. 1996.

function itkGetMacro

itkGetMacro(
    Stiffness ,
    double 
)

function SetAlpha

inline virtual void SetAlpha(
    TScalarType  itkNotUsedAlpha
)

This method makes only sense for the ElasticBody splines. Declare here, so that you can always call it if you don't know the type of kernel beforehand. It will be overridden in the ElasticBodySplineKernelTransform and in the ElasticBodyReciprocalSplineKernelTransform.

function GetAlpha

inline virtual TScalarType GetAlpha(
    void 
) const

function itkSetMacro

itkSetMacro(
    PoissonRatio ,
    TScalarType 
)

This method makes only sense for the ElasticBody splines. Declare here, so that you can always call it if you don't know the type of kernel beforehand. It will be overridden in the ElasticBodySplineKernelTransform and in the ElasticBodyReciprocalSplineKernelTransform.

function GetPoissonRatio

inline virtual const TScalarType GetPoissonRatio(
    void 
) const

function itkSetMacro

itkSetMacro(
    MatrixInversionMethod ,
    std::string 
)

Matrix inversion by SVD or QR decomposition.

function itkGetConstReferenceMacro

itkGetConstReferenceMacro(
    MatrixInversionMethod ,
    std::string 
)

function GetSpatialJacobian

inline virtual void GetSpatialJacobian(
    const InputPointType & ipp,
    SpatialJacobianType & sj
) const

Reimplements: itk::AdvancedTransform::GetSpatialJacobian

Must be provided.

function GetSpatialHessian

inline virtual void GetSpatialHessian(
    const InputPointType & ipp,
    SpatialHessianType & sh
) const

Reimplements: itk::AdvancedTransform::GetSpatialHessian

Compute the spatial Hessian of the transformation.

The spatial Hessian is the vector of matrices of partial second order derivatives of the transformation components with respect to the spatial position \(x\), evaluated at a point \(p\).

$$sH=\left[ \begin{array}{cc} \frac{\partial^2 T_{i}}{\partial x_{1} \partial x_{1}}(p) & \frac{\partial^2 T_{i}}{\partial x_{1} \partial x_{2}}(p) \\ \frac{\partial^2 T_{i}}{\partial x_{1} \partial x_{2}}(p) & \frac{\partial^2 T_{i}}{\partial x_{2} \partial x_{2}}(p) \\ \end{array}\right],$$ with i the i-th component of the transformation.

function GetJacobianOfSpatialJacobian

inline virtual void GetJacobianOfSpatialJacobian(
    const InputPointType & ipp,
    JacobianOfSpatialJacobianType & jsj,
    NonZeroJacobianIndicesType & nonZeroJacobianIndices
) const

function GetJacobianOfSpatialJacobian

inline virtual void GetJacobianOfSpatialJacobian(
    const InputPointType & ipp,
    SpatialJacobianType & sj,
    JacobianOfSpatialJacobianType & jsj,
    NonZeroJacobianIndicesType & nonZeroJacobianIndices
) const

function GetJacobianOfSpatialHessian

inline virtual void GetJacobianOfSpatialHessian(
    const InputPointType & ipp,
    JacobianOfSpatialHessianType & jsh,
    NonZeroJacobianIndicesType & nonZeroJacobianIndices
) const

function GetJacobianOfSpatialHessian

inline virtual void GetJacobianOfSpatialHessian(
    const InputPointType & ipp,
    SpatialHessianType & sh,
    JacobianOfSpatialHessianType & jsh,
    NonZeroJacobianIndicesType & nonZeroJacobianIndices
) const

Protected Functions Documentation

function KernelTransform2

KernelTransform2()

function ~KernelTransform2

virtual ~KernelTransform2()

function PrintSelf

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

function ComputeG

virtual void ComputeG(
    const InputVectorType & landmarkVector,
    GMatrixType & GMatrix
) const

Reimplemented by: itk::ThinPlateSplineKernelTransform2::ComputeG

Compute G(x) This is essentially the kernel of the transform. By overriding this method, we can obtain (among others): Elastic body spline Thin plate spline Volume spline.

function ComputeReflexiveG

virtual void ComputeReflexiveG(
    PointsIterator ,
    GMatrixType & GMatrix
) const

Compute a G(x) for a point to itself (i.e. for the block diagonal elements of the matrix K. Parameter indicates for which landmark the reflexive G is to be computed. The default implementation for the reflexive contribution is a diagonal matrix where the diagonal elements are the stiffness of the spline.

function ComputeDeformationContribution

virtual void ComputeDeformationContribution(
    const InputPointType & inputPoint,
    OutputPointType & result
) const

Reimplemented by: itk::ThinPlateSplineKernelTransform2::ComputeDeformationContribution

Compute the contribution of the landmarks weighted by the kernel function to the global deformation of the space.

function ComputeK

void ComputeK(
    void 
)

Compute K matrix.

function ComputeL

void ComputeL(
    void 
)

Compute L matrix.

function ComputeP

void ComputeP(
    void 
)

Compute P matrix.

function ComputeY

void ComputeY(
    void 
)

Compute Y matrix.

function ComputeD

void ComputeD(
    void 
)

Compute displacements \( q_i - p_i \).

function ReorganizeW

void ReorganizeW(
    void 
)

Warning: This method release the memory of the W Matrix.

Reorganize the components of W into D (deformable), A (rotation part of affine) and B (translational part of affine ) components.

Public Attributes Documentation

variable m_SourceLandmarks

PointSetPointer m_SourceLandmarks;

The list of source landmarks, denoted 'p'.

variable m_TargetLandmarks

PointSetPointer m_TargetLandmarks;

The list of target landmarks, denoted 'q'.

Protected Attributes Documentation

variable m_Stiffness

double m_Stiffness;

Stiffness parameter.

variable m_Displacements

VectorSetPointer m_Displacements;

The list of displacements. d[i] = q[i] - p[i];

variable m_LMatrix

LMatrixType m_LMatrix;

The L matrix.

variable m_LMatrixInverse

LMatrixType m_LMatrixInverse;

The inverse of L, which we also cache.

variable m_KMatrix

KMatrixType m_KMatrix;

The K matrix.

variable m_PMatrix

PMatrixType m_PMatrix;

The P matrix.

variable m_YMatrix

YMatrixType m_YMatrix;

The Y matrix.

variable m_WMatrix

WMatrixType m_WMatrix;

The W matrix.

variable m_DMatrix

DMatrixType m_DMatrix;

The Deformation matrix. This is an auxiliary matrix that will hold the Deformation (non-affine) part of the transform. Those are the coefficients that will multiply the Kernel function.

variable m_AMatrix

AMatrixType m_AMatrix;

Rotational/Shearing part of the Affine component of the Transformation.

variable m_BVector

BMatrixType m_BVector;

Translational part of the Affine component of the Transformation.

variable m_WMatrixComputed

bool m_WMatrixComputed;

The G matrix. It used to be mutable because m_GMatrix was made an ivar only to avoid copying the matrix at return time but this is not necessary. SK: we don't need this matrix anymore as a member. Has the W matrix been computed?

variable m_LMatrixComputed

bool m_LMatrixComputed;

Has the L matrix been computed?

variable m_LInverseComputed

bool m_LInverseComputed;

Has the L inverse matrix been computed?

variable m_LMatrixDecompositionComputed

bool m_LMatrixDecompositionComputed;

Has the L matrix decomposition been computed?

variable m_LMatrixDecompositionSVD

SVDDecompositionType * m_LMatrixDecompositionSVD;

variable m_LMatrixDecompositionQR

QRDecompositionType * m_LMatrixDecompositionQR;

variable m_I

IMatrixType m_I;

Identity matrix.

variable m_NonZeroJacobianIndices

NonZeroJacobianIndicesType m_NonZeroJacobianIndices;

Precomputed nonzero Jacobian indices (simply all params)

variable m_NonZeroJacobianIndicesTemp

NonZeroJacobianIndicesType m_NonZeroJacobianIndicesTemp;

for old GetJacobian() method:

variable m_FastComputationPossible

bool m_FastComputationPossible;

The Jacobian can be computed much faster for some of the derived kerbel transforms, most notably the TPS.

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Updated on 2022-07-23 at 17:50:02 -0600