ITK
5.0.0
Insight Segmentation and Registration Toolkit
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#include <itkBSplineTransform.h>
Deformable transform using a BSpline representation.
This class encapsulates a deformable transform of points from one N-dimensional space to another N-dimensional space. The deformation field is modelled using B-splines. A deformation is defined on a sparse regular grid of control points and is varied by defining a deformation of each control point. The deformation at any point is obtained by using a B-spline interpolation kernel.
The deformation field grid is defined by a user specified transform domain (origin, physical dimensions, direction) and B-spline mesh size where the mesh size is the number of polynomial patches comprising the finite domain of support. The relationship between the mesh size ( number of polynomical pieces) and the number of control points in any given dimension is
mesh size = number of control points - spline order
Each grid/control point has associated with it N deformation coefficients , representing the N directional components of the deformation. Deformation outside the grid plus support region for the BSpline interpolation is assumed to be zero.
The parameters for this transform is N x N-D grid of spline coefficients. The user specifies the parameters as one flat array: each N-D grid is represented by an array in the same way an N-D image is represented in the buffer; the N arrays are then concatentated together on form a single array.
The following illustrates the typical usage of this class:
An alternative way to set the B-spline coefficients is via array of images. The fixed parameters of the transform are taken directly from the first image. It is assumed that the subsequent images are the same buffered region. The following illustrates the API:
The class is templated coordinate representation type (float or double), the space dimension and the spline order.
Definition at line 107 of file itkBSplineTransform.h.
Public Types | |
using | CoefficientImageArray = typename Superclass::CoefficientImageArray |
using | ConstPointer = SmartPointer< const Self > |
using | ContinuousIndexType = typename Superclass::ContinuousIndexType |
using | DirectionType = typename Superclass::DirectionType |
using | FixedParametersType = typename Superclass::FixedParametersType |
using | FixedParametersValueType = typename Superclass::FixedParametersValueType |
using | ImagePointer = typename Superclass::ImagePointer |
using | ImageType = typename Superclass::ImageType |
using | IndexType = typename Superclass::IndexType |
using | InputCovariantVectorType = typename Superclass::InputCovariantVectorType |
using | InputPointType = typename Superclass::InputPointType |
using | InputVectorType = typename Superclass::InputVectorType |
using | InputVnlVectorType = typename Superclass::InputVnlVectorType |
using | InverseJacobianPositionType = typename Superclass::InverseJacobianPositionType |
using | JacobianPositionType = typename Superclass::JacobianPositionType |
using | JacobianType = typename Superclass::JacobianType |
using | MeshSizeType = typename Superclass::MeshSizeType |
using | NumberOfParametersType = typename Superclass::NumberOfParametersType |
using | OriginType = typename Superclass::OriginType |
using | OutputCovariantVectorType = typename Superclass::OutputCovariantVectorType |
using | OutputPointType = typename Superclass::OutputPointType |
using | OutputVectorType = typename Superclass::OutputVectorType |
using | OutputVnlVectorType = typename Superclass::OutputVnlVectorType |
using | ParameterIndexArrayType = typename Superclass::ParameterIndexArrayType |
using | ParametersType = typename Superclass::ParametersType |
using | ParametersValueType = typename Superclass::ParametersValueType |
using | PhysicalDimensionsType = typename Superclass::SpacingType |
using | PixelType = typename Superclass::PixelType |
using | Pointer = SmartPointer< Self > |
using | RegionType = typename Superclass::RegionType |
using | ScalarType = typename Superclass::ScalarType |
using | Self = BSplineTransform |
using | SizeType = typename Superclass::SizeType |
using | SpacingType = typename Superclass::SpacingType |
using | Superclass = BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder > |
using | WeightsFunctionType = typename Superclass::WeightsFunctionType |
using | WeightsType = typename Superclass::WeightsType |
Public Types inherited from itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder > | |
using | CoefficientImageArray = FixedArray< ImagePointer, NDimensions > |
using | ConstPointer = SmartPointer< const Self > |
using | ContinuousIndexType = typename WeightsFunctionType::ContinuousIndexType |
using | DerivativeType = typename Superclass::DerivativeType |
using | DirectionType = typename ImageType::DirectionType |
using | FixedParametersType = typename Superclass::FixedParametersType |
using | ImagePointer = typename ImageType::Pointer |
using | ImageType = Image< ParametersValueType, Self::SpaceDimension > |
using | IndexType = typename RegionType::IndexType |
using | InputCovariantVectorType = CovariantVector< TParametersValueType, Self::SpaceDimension > |
using | InputPointType = Point< TParametersValueType, Self::SpaceDimension > |
using | InputVectorType = Vector< TParametersValueType, Self::SpaceDimension > |
using | InputVnlVectorType = vnl_vector_fixed< TParametersValueType, SpaceDimension > |
using | InverseJacobianPositionType = typename Superclass::InverseJacobianPositionType |
using | JacobianPositionType = typename Superclass::JacobianPositionType |
using | JacobianType = typename Superclass::JacobianType |
using | MeshSizeType = SizeType |
using | NumberOfParametersType = typename Superclass::NumberOfParametersType |
using | OriginType = typename ImageType::PointType |
using | OutputCovariantVectorType = CovariantVector< TParametersValueType, Self::SpaceDimension > |
using | OutputPointType = Point< TParametersValueType, Self::SpaceDimension > |
using | OutputVectorType = Vector< TParametersValueType, Self::SpaceDimension > |
using | OutputVnlVectorType = vnl_vector_fixed< TParametersValueType, SpaceDimension > |
using | ParameterIndexArrayType = Array< unsigned long > |
using | ParametersType = typename Superclass::ParametersType |
using | ParametersValueType = typename ParametersType::ValueType |
using | PhysicalDimensionsType = typename ImageType::SpacingType |
using | PixelType = typename ImageType::PixelType |
using | Pointer = SmartPointer< Self > |
using | RegionType = ImageRegion< Self::SpaceDimension > |
using | ScalarType = typename Superclass::ScalarType |
using | Self = BSplineBaseTransform |
using | SizeType = typename RegionType::SizeType |
using | SpacingType = typename ImageType::SpacingType |
using | Superclass = Transform< TParametersValueType, NDimensions, NDimensions > |
using | TransformCategoryType = typename Superclass::TransformCategoryType |
using | WeightsFunctionType = BSplineInterpolationWeightFunction< ScalarType, Self::SpaceDimension, Self::SplineOrder > |
using | WeightsType = typename WeightsFunctionType::WeightsType |
Public Types inherited from itk::Transform< TParametersValueType, NDimensions, NDimensions > | |
using | ConstPointer = SmartPointer< const Self > |
using | DerivativeType = Array< ParametersValueType > |
using | DirectionChangeMatrix = Matrix< double, Self::OutputSpaceDimension, Self::InputSpaceDimension > |
using | FixedParametersType = typename Superclass::FixedParametersType |
using | FixedParametersValueType = typename Superclass::FixedParametersValueType |
using | InputCovariantVectorType = CovariantVector< TParametersValueType, NInputDimensions > |
using | InputDiffusionTensor3DType = DiffusionTensor3D< TParametersValueType > |
using | InputDirectionMatrix = Matrix< double, Self::InputSpaceDimension, Self::InputSpaceDimension > |
using | InputPointType = Point< TParametersValueType, NInputDimensions > |
using | InputSymmetricSecondRankTensorType = SymmetricSecondRankTensor< TParametersValueType, NInputDimensions > |
using | InputVectorPixelType = VariableLengthVector< TParametersValueType > |
using | InputVectorType = Vector< TParametersValueType, NInputDimensions > |
using | InputVnlVectorType = vnl_vector_fixed< TParametersValueType, NInputDimensions > |
using | InverseJacobianPositionType = vnl_matrix_fixed< ParametersValueType, NInputDimensions, NOutputDimensions > |
using | InverseTransformBasePointer = typename InverseTransformBaseType::Pointer |
using | InverseTransformBaseType = Transform< TParametersValueType, NOutputDimensions, NInputDimensions > |
using | JacobianPositionType = vnl_matrix_fixed< ParametersValueType, NOutputDimensions, NInputDimensions > |
using | JacobianType = Array2D< ParametersValueType > |
using | MatrixType = Matrix< TParametersValueType, Self::OutputSpaceDimension, Self::InputSpaceDimension > |
using | NumberOfParametersType = typename Superclass::NumberOfParametersType |
using | OutputCovariantVectorType = CovariantVector< TParametersValueType, NOutputDimensions > |
using | OutputDiffusionTensor3DType = DiffusionTensor3D< TParametersValueType > |
using | OutputDirectionMatrix = Matrix< double, Self::OutputSpaceDimension, Self::OutputSpaceDimension > |
using | OutputPointType = Point< TParametersValueType, NOutputDimensions > |
using | OutputSymmetricSecondRankTensorType = SymmetricSecondRankTensor< TParametersValueType, NOutputDimensions > |
using | OutputVectorPixelType = VariableLengthVector< TParametersValueType > |
using | OutputVectorType = Vector< TParametersValueType, NOutputDimensions > |
using | OutputVnlVectorType = vnl_vector_fixed< TParametersValueType, NOutputDimensions > |
using | ParametersType = typename Superclass::ParametersType |
using | ParametersValueType = typename Superclass::ParametersValueType |
using | Pointer = SmartPointer< Self > |
using | ScalarType = ParametersValueType |
using | Self = Transform |
using | Superclass = TransformBaseTemplate< TParametersValueType > |
using | TransformCategoryType = typename Superclass::TransformCategoryType |
Public Types inherited from itk::TransformBaseTemplate< TParametersValueType > | |
using | ConstPointer = SmartPointer< const Self > |
using | FixedParametersType = OptimizerParameters< FixedParametersValueType > |
using | FixedParametersValueType = double |
using | NumberOfParametersType = IdentifierType |
using | ParametersType = OptimizerParameters< ParametersValueType > |
using | ParametersValueType = TParametersValueType |
using | Pointer = SmartPointer< Self > |
using | Self = TransformBaseTemplate |
using | Superclass = Object |
enum | TransformCategoryType { UnknownTransformCategory =0, Linear =1, BSpline =2, Spline =3, DisplacementField =4, VelocityField =5 } |
Public Types inherited from itk::Object | |
using | ConstPointer = SmartPointer< const Self > |
using | Pointer = SmartPointer< Self > |
using | Self = Object |
using | Superclass = LightObject |
Public Types inherited from itk::LightObject | |
using | ConstPointer = SmartPointer< const Self > |
using | Pointer = SmartPointer< Self > |
using | Self = LightObject |
Static Public Member Functions | |
static Pointer | New () |
Static Public Member Functions inherited from itk::Object | |
static bool | GetGlobalWarningDisplay () |
static void | GlobalWarningDisplayOff () |
static void | GlobalWarningDisplayOn () |
static Pointer | New () |
static void | SetGlobalWarningDisplay (bool flag) |
Static Public Member Functions inherited from itk::LightObject | |
static void | BreakOnError () |
static Pointer | New () |
Static Public Attributes | |
static constexpr unsigned int | SpaceDimension = NDimensions |
static constexpr unsigned int | SplineOrder = VSplineOrder |
Static Public Attributes inherited from itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder > | |
static constexpr unsigned int | SpaceDimension = NDimensions |
static constexpr unsigned int | SplineOrder = VSplineOrder |
Static Public Attributes inherited from itk::Transform< TParametersValueType, NDimensions, NDimensions > | |
static constexpr unsigned int | InputSpaceDimension |
static constexpr unsigned int | OutputSpaceDimension |
Private Member Functions | |
bool | InsideValidRegion (ContinuousIndexType &) const override |
void | SetCoefficientImageInformationFromFixedParameters () override |
void | SetFixedParametersFromCoefficientImageInformation () |
void | SetFixedParametersFromTransformDomainInformation (const OriginType &meshOrigin, const PhysicalDimensionsType &meshPhysical, const DirectionType &meshDirection, const MeshSizeType &meshSize) |
virtual void | SetFixedParametersGridSizeFromTransformDomainInformation () const override |
virtual void | SetFixedParametersGridOriginFromTransformDomainInformation () const override |
virtual void | SetFixedParametersGridSpacingFromTransformDomainInformation () const override |
virtual void | SetFixedParametersGridDirectionFromTransformDomainInformation () const override |
Additional Inherited Members | |
Protected Attributes inherited from itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder > | |
CoefficientImageArray | m_CoefficientImages |
ParametersType | m_InternalParametersBuffer |
WeightsFunctionType::Pointer | m_WeightsFunction |
Protected Attributes inherited from itk::Transform< TParametersValueType, NDimensions, NDimensions > | |
DirectionChangeMatrix | m_DirectionChange |
FixedParametersType | m_FixedParameters |
ParametersType | m_Parameters |
Protected Attributes inherited from itk::LightObject | |
std::atomic< int > | m_ReferenceCount |
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::CoefficientImageArray = typename Superclass::CoefficientImageArray |
Definition at line 195 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::ConstPointer = SmartPointer<const Self> |
Definition at line 117 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::ContinuousIndexType = typename Superclass::ContinuousIndexType |
Definition at line 223 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::DirectionType = typename Superclass::DirectionType |
Definition at line 216 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::FixedParametersType = typename Superclass::FixedParametersType |
Definition at line 137 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::FixedParametersValueType = typename Superclass::FixedParametersValueType |
Definition at line 138 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::ImagePointer = typename Superclass::ImagePointer |
Definition at line 194 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::ImageType = typename Superclass::ImageType |
Parameters as SpaceDimension number of images.
Definition at line 193 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::IndexType = typename Superclass::IndexType |
Definition at line 213 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::InputCovariantVectorType = typename Superclass::InputCovariantVectorType |
Standard covariant vector type for this class.
Definition at line 153 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::InputPointType = typename Superclass::InputPointType |
Standard coordinate point type for this class.
Definition at line 161 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::InputVectorType = typename Superclass::InputVectorType |
Standard vector type for this class.
Definition at line 149 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::InputVnlVectorType = typename Superclass::InputVnlVectorType |
Standard vnl_vector type for this class.
Definition at line 157 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::InverseJacobianPositionType = typename Superclass::InverseJacobianPositionType |
Definition at line 143 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::JacobianPositionType = typename Superclass::JacobianPositionType |
Definition at line 142 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::JacobianType = typename Superclass::JacobianType |
Standard Jacobian container.
Definition at line 141 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::MeshSizeType = typename Superclass::MeshSizeType |
Definition at line 253 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::NumberOfParametersType = typename Superclass::NumberOfParametersType |
The number of parameters defininig this transform.
Definition at line 146 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::OriginType = typename Superclass::OriginType |
Definition at line 217 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::OutputCovariantVectorType = typename Superclass::OutputCovariantVectorType |
Definition at line 154 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::OutputPointType = typename Superclass::OutputPointType |
Definition at line 162 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::OutputVectorType = typename Superclass::OutputVectorType |
Definition at line 150 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::OutputVnlVectorType = typename Superclass::OutputVnlVectorType |
Definition at line 158 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::ParameterIndexArrayType = typename Superclass::ParameterIndexArrayType |
Parameter index array type.
Definition at line 226 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::ParametersType = typename Superclass::ParametersType |
Standard parameters container.
Definition at line 135 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::ParametersValueType = typename Superclass::ParametersValueType |
Definition at line 136 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::PhysicalDimensionsType = typename Superclass::SpacingType |
Definition at line 250 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::PixelType = typename Superclass::PixelType |
Definition at line 251 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::Pointer = SmartPointer<Self> |
Definition at line 116 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::RegionType = typename Superclass::RegionType |
Typedefs for specifying the extent of the grid.
Definition at line 211 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::ScalarType = typename Superclass::ScalarType |
Standard scalar type for this class.
Definition at line 132 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::Self = BSplineTransform |
Standard class type aliases.
Definition at line 114 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::SizeType = typename Superclass::SizeType |
Definition at line 214 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::SpacingType = typename Superclass::SpacingType |
Definition at line 215 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::Superclass = BSplineBaseTransform<TParametersValueType,NDimensions,VSplineOrder> |
Definition at line 115 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::WeightsFunctionType = typename Superclass::WeightsFunctionType |
Interpolation weights function type.
Definition at line 220 of file itkBSplineTransform.h.
using itk::BSplineTransform< TParametersValueType, NDimensions, VSplineOrder >::WeightsType = typename Superclass::WeightsType |
Definition at line 222 of file itkBSplineTransform.h.
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Compute the Jacobian in one position.
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
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Create an object from an instance, potentially deferring to a factory. This method allows you to create an instance of an object that is exactly the same type as the referring object. This is useful in cases where an object has been cast back to a base class.
Reimplemented from itk::Object.
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Run-time type information (and related methods).
Reimplemented from itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
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Return the number of parameters that completely define the Transfom.
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
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Return the number of parameters per dimension.
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
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Function to retrieve the transform domain direction.
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Function to retrieve the transform domain mesh size.
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Function to retrieve the transform domain origin.
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Function to retrieve the transform domain physical dimensions.
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Generate a platform independent name
Implements itk::TransformBaseTemplate< TParametersValueType >.
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Check if a continuous index is inside the valid region.
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
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New macro for creation of through the object factory.
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Print contents of an BSplineTransform.
Reimplemented from itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
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Construct control point grid size from transform domain information in the fixed parameters.
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
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Set the array of coefficient images.
This is an alternative API for setting the BSpline coefficients as an array of SpaceDimension images. The fixed parameters are taken from the first image. It is assumed that the buffered region of all the subsequent images are the same as the first image. Note that no error checking is done.
Warning: use either the SetParameters() or SetCoefficientImages() API. Mixing the two modes may results in unexpected results.
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
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This method sets the fixed parameters of the transform. For a BSpline deformation transform, the fixed parameters are the following: grid size, grid origin, grid spacing, and grid direction. However, all of these are set via the much more intuitive SetTransformDomainXXX() functions
The fixed parameters are the three times the size of the templated dimensions. This function has the effect of make the following non- existing functional calls: transform->SetGridSpacing( spacing ); transform->SetGridOrigin( origin ); transform->SetGridDirection( direction ); transform->SetGridRegion( bsplineRegion );
With recent updates to this transform, however, all these parameters are set indirectly by setting the transform domain parameters unless the user sets them with SetFixedParameters().
This function was added to allow the transform to work with the itkTransformReader/Writer I/O filters.
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
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Methods have empty implementations
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
Definition at line 295 of file itkBSplineTransform.h.
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Methods have empty implementations
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
Definition at line 293 of file itkBSplineTransform.h.
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Methods have empty implementations
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
Definition at line 292 of file itkBSplineTransform.h.
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Methods have empty implementations
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
Definition at line 294 of file itkBSplineTransform.h.
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Function to specify the transform domain direction.
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Function to specify the transform domain mesh size.
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Function to specify the transform domain origin.
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Function to specify the transform domain physical dimensions.
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Transform points by a BSpline deformable transformation. On return, weights contains the interpolation weights used to compute the deformation and indices of the x (zeroth) dimension coefficient parameters in the support region used to compute the deformation. Parameter indices for the i-th dimension can be obtained by adding ( i * this->GetNumberOfParametersPerDimension() ) to the indices array.
Implements itk::BSplineBaseTransform< TParametersValueType, NDimensions, VSplineOrder >.
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Dimension of the domain space.
Definition at line 126 of file itkBSplineTransform.h.
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The BSpline order.
Definition at line 129 of file itkBSplineTransform.h.