Weighting function based on the euclidean distance between a query and a reference position. More...

#include <weightFunc.h>

Collaboration diagram for Ponca::DistWeightFunc< DataPoint, WeightKernel >:

Public Types
using	Scalar = typename DataPoint::Scalar
	Scalar type from DataPoint.

using	VectorType = typename DataPoint::VectorType
	Vector type from DataPoint.

using	MatrixType = typename DataPoint::MatrixType
	Matrix type from DataPoint.

using	WeightReturnType = std::pair< Scalar, VectorType >
	Return type of the method w()

Public Member Functions
	DistWeightFunc (const Scalar &_t=Scalar(1.))
	Constructor that defines the current evaluation scale.

void	init (const VectorType &_evalPos)
	Initialization method, called by the fitting procedure.

const VectorType &	basisCenter () const
	Get access to basis center location in global coordinate system.

VectorType	convertToGlobalBasis (const VectorType &_q) const
	Convert position from local to global coordinate system.

VectorType	convertToLocalBasis (const VectorType &_q) const
	Convert query from global to local coordinate system (used internally(.

WeightReturnType	w (const VectorType &_q, const DataPoint &) const
	Compute the weight of the given query with respect to its coordinates.

VectorType	spacedw (const VectorType &_q, const DataPoint &) const
	First order derivative in space (for each spatial dimension \(\mathsf{x})\).

MatrixType	spaced2w (const VectorType &_q, const DataPoint &) const
	Second order derivative in space (for each spatial dimension \(\mathsf{x})\).

Scalar	scaledw (const VectorType &_q, const DataPoint &) const
	First order derivative in scale \(t\).

Scalar	scaled2w (const VectorType &_q, const DataPoint &) const
	Second order derivative in scale \(t\).

VectorType	scaleSpaced2w (const VectorType &_q, const DataPoint &) const
	Cross derivative in scale \(t\) and in space (for each spatial dimension \(\mathsf{x})\).

Scalar	evalScale () const
	Access to the evaluation scale set during the initialization.

const VectorType &	evalPos () const
	Access to the evaluation position set during the initialization.

Protected Attributes
Scalar	m_t
	Evaluation scale.

WeightKernel	m_wk
	1D function applied to weight queries

VectorType	m_p
	basis center

Detailed Description

template<class DataPoint, class WeightKernel>
class Ponca::DistWeightFunc< DataPoint, WeightKernel >

Weighting function based on the euclidean distance between a query and a reference position.

The evaluation position is set using the init method. All the queries are expressed in global system, and the weighting function convert them to local coordinates (ie. relatively to the evaluation position).

It can be specialized for any DataPoint and uses a generic 1D BaseWeightKernel.

Warning: it assumes that the evaluation scale t is strictly positive

Definition at line 28 of file weightFunc.h.

Member Typedef Documentation

◆ MatrixType

template<class DataPoint , class WeightKernel >

using Ponca::DistWeightFunc< DataPoint, WeightKernel >::MatrixType = typename DataPoint::MatrixType

Matrix type from DataPoint.

Definition at line 36 of file weightFunc.h.

◆ Scalar

template<class DataPoint , class WeightKernel >

using Ponca::DistWeightFunc< DataPoint, WeightKernel >::Scalar = typename DataPoint::Scalar

Scalar type from DataPoint.

Definition at line 32 of file weightFunc.h.

◆ VectorType

template<class DataPoint , class WeightKernel >

using Ponca::DistWeightFunc< DataPoint, WeightKernel >::VectorType = typename DataPoint::VectorType

Vector type from DataPoint.

Definition at line 34 of file weightFunc.h.

◆ WeightReturnType

template<class DataPoint , class WeightKernel >

using Ponca::DistWeightFunc< DataPoint, WeightKernel >::WeightReturnType = std:: pair <Scalar, VectorType>

Return type of the method w()

Definition at line 38 of file weightFunc.h.

Constructor & Destructor Documentation

◆ DistWeightFunc()

template<class DataPoint , class WeightKernel >

Ponca::DistWeightFunc< DataPoint, WeightKernel >::DistWeightFunc ( const Scalar & _t = Scalar(1.) )

inline

Constructor that defines the current evaluation scale.

Warning: t > 0

Definition at line 44 of file weightFunc.h.

Member Function Documentation

◆ basisCenter()

template<class DataPoint , class WeightKernel >

const VectorType & Ponca::DistWeightFunc< DataPoint, WeightKernel >::basisCenter ( ) const

inline

Get access to basis center location in global coordinate system.

Definition at line 62 of file weightFunc.h.

◆ convertToGlobalBasis()

template<class DataPoint , class WeightKernel >

DistWeightFunc< DataPoint, WeightKernel >::VectorType DistWeightFunc::convertToGlobalBasis ( const VectorType & _q ) const

inline

Convert position from local to global coordinate system.

Parameters

_q	Position in local coordinate

Returns: Position expressed independently of the local basis center

Definition at line 10 of file weightFunc.hpp.

◆ convertToLocalBasis()

template<class DataPoint , class WeightKernel >

DistWeightFunc< DataPoint, WeightKernel >::VectorType DistWeightFunc::convertToLocalBasis ( const VectorType & _q ) const

inline

Convert query from global to local coordinate system (used internally(.

Parameters

_q	Query in global coordinate

Returns: Query expressed relatively to the basis center

Definition at line 17 of file weightFunc.hpp.

◆ evalPos()

template<class DataPoint , class WeightKernel >

const VectorType & Ponca::DistWeightFunc< DataPoint, WeightKernel >::evalPos ( ) const

inline

Access to the evaluation position set during the initialization.

Definition at line 202 of file weightFunc.h.

◆ evalScale()

template<class DataPoint , class WeightKernel >

Scalar Ponca::DistWeightFunc< DataPoint, WeightKernel >::evalScale ( ) const

inline

Access to the evaluation scale set during the initialization.

Definition at line 199 of file weightFunc.h.

◆ init()

template<class DataPoint , class WeightKernel >

void Ponca::DistWeightFunc< DataPoint, WeightKernel >::init ( const VectorType & _evalPos )

inline

Initialization method, called by the fitting procedure.

Parameters

_evalPos Basis center

Definition at line 56 of file weightFunc.h.

◆ scaled2w()

template<class DataPoint , class WeightKernel >

DistWeightFunc< DataPoint, WeightKernel >::Scalar DistWeightFunc::scaled2w	(	const VectorType &	_q,
		const DataPoint &
	)		const

inline

Second order derivative in scale \(t\).

Parameters

_q	Query in global coordinate

\( \frac{\delta^2 \frac{\left|\mathbf{q}\right|}{t}}{\delta t^2} \nabla w(\frac{\left|\mathbf{q}\right|}{t}) + \left(\frac{\delta \frac{\left|\mathbf{q}\right|}{t}}{\delta t}\right)^2 \nabla^2 w(\frac{\left|\mathbf{q}\right|}{t}) = \frac{2\left|\mathbf{q}\right|}{t^3} \nabla{w(\frac{\left|\mathbf{q}\right|}{t})} + \frac{\left|\mathbf{q}\right|^2}{t^4} \nabla^2{w(\frac{\left|\mathbf{q}\right|}{t})} \)

where \( \left|\mathbf{q}\right| \) represents the norm of the query coordinates expressed in centered basis.

Warning: Requires \(\nabla^2 w(x)\) to be valid

Definition at line 76 of file weightFunc.hpp.

◆ scaledw()

template<class DataPoint , class WeightKernel >

DistWeightFunc< DataPoint, WeightKernel >::Scalar DistWeightFunc::scaledw	(	const VectorType &	_q,
		const DataPoint &
	)		const

inline

First order derivative in scale \(t\).

Parameters

_q	Query in global coordinate

\( \frac{\delta \frac{\left|\mathbf{q}\right|}{t}}{\delta t} \nabla w(\frac{\left|\mathbf{q}\right|}{t}) = - \frac{\left|\mathbf{q}\right|}{t^2} \nabla{w(\frac{\left|\mathbf{q}\right|}{t})} \)

where \( \left|\mathbf{q}\right| \) represents the norm of the query coordinates expressed in centered basis.

Warning: Requires \(\nabla w(x)\) to be valid

Definition at line 66 of file weightFunc.hpp.

◆ scaleSpaced2w()

template<class DataPoint , class WeightKernel >

DistWeightFunc< DataPoint, WeightKernel >::VectorType DistWeightFunc::scaleSpaced2w	(	const VectorType &	_q,
		const DataPoint &
	)		const

inline

Cross derivative in scale \(t\) and in space (for each spatial dimension \(\mathsf{x})\).

Parameters

_q	Query in global coordinate

\( \frac{\delta^2 \frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}}{\delta t\ \delta \mathsf{x}} \nabla w(\frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}) + \frac{\delta \frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}}{\delta \mathsf{x}} \frac{\delta \frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}}{\delta t} \nabla^2 w(\frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}) = -\frac{\mathbf{q}_\mathsf{x}}{t^2} \left( \frac{1}{\left|\mathbf{q}_\mathsf{x}\right|}\nabla w(\frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}) + \frac{1}{t}\nabla^2 w(\frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}) \right)\)

where \( \left|\mathbf{q}_\mathsf{x}\right| \) represents the norm of the query coordinates expressed in centered basis.

Warning: Requires \(\nabla^2 w(x)\) to be valid

Definition at line 88 of file weightFunc.hpp.

◆ spaced2w()

template<class DataPoint , class WeightKernel >

DistWeightFunc< DataPoint, WeightKernel >::MatrixType DistWeightFunc::spaced2w	(	const VectorType &	_q,
		const DataPoint &
	)		const

inline

Second order derivative in space (for each spatial dimension \(\mathsf{x})\).

Parameters

_q	Query in global coordinate

\( \frac{\delta^2 \frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}}{\delta \mathsf{x}^2} \nabla w(\frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}) + \left(\frac{\delta \frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}}{\delta \mathsf{x}}\right)^2 \nabla^2 w(\frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}) = \frac{1}{t\left|\mathbf{q}_\mathsf{x}\right|} \left( I_d - \frac{\mathbf{q}_\mathsf{x}\mathbf{q}_\mathsf{x}^T}{\left|\mathbf{q}_\mathsf{x}\right|^2}\right) \nabla w(\frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}) + \frac{\mathbf{q}_\mathsf{x}\mathbf{q}_\mathsf{x}^T}{t^2\left|\mathbf{q}_\mathsf{x}\right|^2} \nabla^2 w(\frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}) \)

where \( \left|\mathbf{q}_\mathsf{x}\right| \) represents the norm of the query coordinates expressed in centered basis, for each spatial dimensions \( \mathsf{x}\).

Warning: Requires \(\nabla^2 w(x)\) to be valid

Definition at line 47 of file weightFunc.hpp.

◆ spacedw()

template<class DataPoint , class WeightKernel >

DistWeightFunc< DataPoint, WeightKernel >::VectorType DistWeightFunc::spacedw	(	const VectorType &	_q,
		const DataPoint &
	)		const

inline

First order derivative in space (for each spatial dimension \(\mathsf{x})\).

Parameters

_q	Query in global coordinate

\( \frac{\delta \frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}}{\delta \mathsf{x}} \nabla w(\frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}) = \frac{\mathbf{q}}{t\left|q\right|} \nabla{w(\frac{\left|\mathbf{q}_\mathsf{x}\right|}{t})} \)

where \( \left|\mathbf{q}_\mathsf{x}\right| \) represents the norm of the query coordinates expressed in centered basis, for each spatial dimensions \( \mathsf{x}\).

Warning: Requires \(\nabla w(x)\) to be valid

Definition at line 34 of file weightFunc.hpp.

◆ w()

template<class DataPoint , class WeightKernel >

DistWeightFunc< DataPoint, WeightKernel >::WeightReturnType DistWeightFunc::w	(	const VectorType &	_q,
		const DataPoint &
	)		const

inline

Compute the weight of the given query with respect to its coordinates.

Parameters

_q	Query in global coordinate

As the query \(\mathbf{q}\) is expressed in global coordinate, it is first converted to the centered basis. Then, the WeightKernel is directly applied to the norm of its coordinates with respect to the current scale \( t \) :

\( w(\frac{\left|\mathbf{q}_\mathsf{x}\right|}{t}) \)

See also: convertToLocalBasis

Returns: The computed weight + the point expressed in local basis

Definition at line 24 of file weightFunc.hpp.

Member Data Documentation

◆ m_p

template<class DataPoint , class WeightKernel >

VectorType Ponca::DistWeightFunc< DataPoint, WeightKernel >::m_p

protected

basis center

Definition at line 207 of file weightFunc.h.

◆ m_t

template<class DataPoint , class WeightKernel >

Scalar Ponca::DistWeightFunc< DataPoint, WeightKernel >::m_t

protected

Evaluation scale.

Definition at line 205 of file weightFunc.h.

◆ m_wk

template<class DataPoint , class WeightKernel >

WeightKernel Ponca::DistWeightFunc< DataPoint, WeightKernel >::m_wk

protected

1D function applied to weight queries

Definition at line 206 of file weightFunc.h.

Public Types

Public Member Functions

Protected Attributes

Detailed Description

Member Typedef Documentation

◆ MatrixType

◆ Scalar

◆ VectorType

◆ WeightReturnType

Constructor & Destructor Documentation

◆ DistWeightFunc()

Member Function Documentation

◆ basisCenter()

◆ convertToGlobalBasis()

◆ convertToLocalBasis()

◆ evalPos()

◆ evalScale()

◆ init()

◆ scaled2w()

◆ scaledw()

◆ scaleSpaced2w()

◆ spaced2w()

◆ spacedw()

◆ w()

Member Data Documentation

◆ m_p

◆ m_t

◆ m_wk