Algebraic Sphere primitive. More...

#include <algebraicSphere.h>

Inheritance diagram for Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >:

Collaboration diagram for Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >:

Public Types
using	Scalar = typename DataPoint::Scalar
	Alias to scalar type.

using	VectorType = typename Base::VectorType
	Alias to vector type.

using	WFunctor = typename Base::WFunctor
	Alias to weight function.

Public Member Functions
AlgebraicSphere< DataPoint, _WFunctor, T > &	algebraicSphere ()
	Explicit conversion to AlgebraicSphere , to access methods potentially hidden by heritage.

const AlgebraicSphere< DataPoint, _WFunctor, T > &	algebraicSphere () const
	Explicit conversion to AlgebraicSphere , to access methods potentially hidden by heritage.

void	init (const VectorType &_basisCenter=VectorType::Zero())
	Set the scalar field values to 0 and reset the isNormalized() status.

bool	isValid () const
	Tell if the sphere as been correctly set. Used to set CONFLICT_ERROR_FOUND during fitting.

bool	operator== (const AlgebraicSphere< DataPoint, WFunctor, T > &other) const
	Comparison operator.

bool	operator!= (const AlgebraicSphere< DataPoint, WFunctor, T > &other) const
	Comparison operator, convenience function.

void	changeBasis (const VectorType &newbasis)
	Express the scalar field relatively to a new basis.

Scalar	prattNorm () const
	compute the Pratt norm of the implicit scalar field.

Scalar	prattNorm2 () const
	compute the squared Pratt norm of the implicit scalar field.

bool	applyPrattNorm ()
	Normalize the scalar field by the Pratt norm.

Scalar	radius () const
	return the estimated radius of the sphere

VectorType	center () const
	return the estimated center of the sphere

bool	isNormalized () const
	State indicating when the sphere has been normalized.

Scalar	potential (const VectorType &_q) const
	Value of the scalar field at the location \( \mathbf{q}\).

Scalar	potential () const
	Value of the scalar field at the evaluation point.

VectorType	project (const VectorType &_q) const
	Project a point on the algebraic hypersphere.

VectorType	projectDescent (const VectorType &_q, int nbIter=16) const
	Project a point on the algebraic hypersphere using Gradient Descent This projection is realized by following the gradient of the hypersphere scalar field.

VectorType	primitiveGradient (const VectorType &_q) const
	Approximation of the scalar field gradient at \( \mathbf{q}\).

const VectorType &	primitiveGradient () const
	Approximation of the scalar field gradient at the evaluation point.

bool	isPlane () const
	Used to know if the fitting result to a plane.

Public Attributes
Scalar	m_uc {0}
	Constant parameter of the Algebraic hyper-sphere.

Scalar	m_uq {0}
	Quadratic parameter of the Algebraic hyper-sphere.

VectorType	m_ul {VectorType::Zero()}
	Linear parameter of the Algebraic hyper-sphere.

Protected Types
enum	{ check = Base::PROVIDES_PRIMITIVE_BASE , PROVIDES_ALGEBRAIC_SPHERE }

using	Base = T
	Base class of the procedure.

Protected Attributes
bool	m_isNormalized
	Is the implicit scalar field normalized using Pratt.

Detailed Description

template<class DataPoint, class _WFunctor, typename T>
class Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >

Algebraic Sphere primitive.

Method published in [6]

An algebraic hyper-sphere is defined as the \(0\)-isosurface of the scalar field

\( s_\mathbf{u}(\mathbf{x}) = \left[ 1 \; \mathbf{x}^T \; \mathbf{x}^T\mathbf{x}\right]^T \cdot \mathbf{u} \)

with \( \mathbf{u} \left[ u_c \; \mathbf{u_l} \; u_q\right]^T \) is the vector of the constant, linear and quadratic parameters.

Note: If internally the scalar fields are stored in a local frame defined by the evaluation position, the public methods involving a query (such as project, potential, gradient) have to be defined in global coordinates (e.g. you don't need to convert your query in the current locale frame).

This primitive provides:

PROVIDES_ALGEBRAIC_SPHERE

Definition at line 46 of file algebraicSphere.h.

Member Typedef Documentation

◆ Base

template<class DataPoint , class _WFunctor , typename T >

using Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::Base = T

protected

Base class of the procedure.

Definition at line 48 of file algebraicSphere.h.

◆ Scalar

template<class DataPoint , class _WFunctor , typename T >

using Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::Scalar = typename DataPoint::Scalar

Alias to scalar type.

Definition at line 48 of file algebraicSphere.h.

◆ VectorType

template<class DataPoint , class _WFunctor , typename T >

using Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::VectorType = typename Base::VectorType

Alias to vector type.

Definition at line 48 of file algebraicSphere.h.

◆ WFunctor

template<class DataPoint , class _WFunctor , typename T >

using Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::WFunctor = typename Base::WFunctor

Alias to weight function.

Definition at line 48 of file algebraicSphere.h.

Member Enumeration Documentation

◆ anonymous enum

template<class DataPoint , class _WFunctor , typename T >

anonymous enum

protected

Enumerator
check	Requires PrimitiveBase.
PROVIDES_ALGEBRAIC_SPHERE	Provides Algebraic Sphere.

Definition at line 51 of file algebraicSphere.h.

Member Function Documentation

◆ algebraicSphere() [1/2]

template<class DataPoint , class _WFunctor , typename T >

AlgebraicSphere< DataPoint, _WFunctor, T > & Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::algebraicSphere ( )

inline

Explicit conversion to AlgebraicSphere , to access methods potentially hidden by heritage.

Definition at line 68 of file algebraicSphere.h.

◆ algebraicSphere() [2/2]

template<class DataPoint , class _WFunctor , typename T >

const AlgebraicSphere< DataPoint, _WFunctor, T > & Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::algebraicSphere ( ) const

inline

Explicit conversion to AlgebraicSphere , to access methods potentially hidden by heritage.

Definition at line 68 of file algebraicSphere.h.

◆ applyPrattNorm()

template<class DataPoint , class _WFunctor , typename T >

bool Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::applyPrattNorm ( )

inline

Normalize the scalar field by the Pratt norm.

Returns: false when the normalization fails (sphere is already normalized)

Definition at line 164 of file algebraicSphere.h.

◆ center()

template<class DataPoint , class _WFunctor , typename T >

VectorType Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::center ( ) const

inline

return the estimated center of the sphere

Warning: return Vector inf if the fitted surface is planar

Definition at line 204 of file algebraicSphere.h.

◆ changeBasis()

template<class DataPoint , class _WFunctor , typename T >

void Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::changeBasis ( const VectorType & newbasis )

inline

Express the scalar field relatively to a new basis.

The sphere in dimension \(d\) is parametrized in the original basis by \(\mathbf{u} = \left[u_c, \mathbf{u}_l^T, u_q\right]^T \in \mathbb{R}^{d+2}\). The sphere equation is given in this basis by:

\begin{equation} u_c + \mathbf{u}_l^T.\mathbf{x} + u_q \mathbf{x}^T.\mathbf{x}= 0 \end{equation}

The same equation written in a different basis is: \begin{equation} u_c + \mathbf{u}_l^T.(\mathbf{x}-\mathbf{\Delta}) + u_q (\mathbf{x}-\mathbf{\Delta})^T.(\mathbf{x}-\mathbf{\Delta})= 0 \end{equation} where \(\mathbf{\Delta}\) is the difference vector between the two basis: \(\mathbf{\Delta} = \mathbf{b}_\mathrm{original}-\mathbf{b}_\mathrm{new}\). It corresponds to a translation of \(\Delta\) of the sphere. This develops in: \begin{equation} \left[u_c - \mathbf{u}_l^T.\mathbf{\Delta} + u_q \mathbf{\Delta}^T.\mathbf{\Delta}\right] + \left[\mathbf{u}_l - 2 u_q \mathbf{\Delta}\right]^T.\mathbf{x} + u_q \mathbf{x}^T.\mathbf{x} = 0 \end{equation}

By identification, the parametrization \(\left[u_c', \mathbf{u}_l'^T, u_q'\right]^T\) of the sphere in the new basis is given by

\(u_c' = u_c - \mathbf{u}_l^T.\mathbf{\Delta} + u_q \mathbf{\Delta}^T.\mathbf{\Delta}\)

\(\mathbf{u}_l' = \mathbf{u}_l - 2 u_q \mathbf{\Delta}\)

\(u_q'=u_q\)

Definition at line 136 of file algebraicSphere.h.

◆ init()

template<class DataPoint , class _WFunctor , typename T >

void Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::init ( const VectorType & _basisCenter = VectorType::Zero() )

inline

Set the scalar field values to 0 and reset the isNormalized() status.

Warning: Set m_ul to Zero(), which leads to nans in OrientedSphere::normal()

Definition at line 74 of file algebraicSphere.h.

◆ isNormalized()

template<class DataPoint , class _WFunctor , typename T >

bool Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::isNormalized ( ) const

inline

State indicating when the sphere has been normalized.

Definition at line 221 of file algebraicSphere.h.

◆ isPlane()

template<class DataPoint , class _WFunctor , typename T >

bool Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::isPlane ( ) const

inline

Used to know if the fitting result to a plane.

Returns: true if finalize() have been called and the fitting result to a plane

Definition at line 262 of file algebraicSphere.h.

◆ isValid()

template<class DataPoint , class _WFunctor , typename T >

bool Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::isValid ( ) const

inline

Tell if the sphere as been correctly set. Used to set CONFLICT_ERROR_FOUND during fitting.

Returns: false when called straight after init. Should be true after fitting

Definition at line 88 of file algebraicSphere.h.

◆ operator!=()

template<class DataPoint , class _WFunctor , typename T >

bool Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::operator!= ( const AlgebraicSphere< DataPoint, WFunctor, T > & other ) const

inline

Comparison operator, convenience function.

Definition at line 103 of file algebraicSphere.h.

◆ operator==()

template<class DataPoint , class _WFunctor , typename T >

bool Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::operator== ( const AlgebraicSphere< DataPoint, WFunctor, T > & other ) const

inline

Comparison operator.

Warning: Assume that other shares the same basis

See also: changeBasis()

Definition at line 93 of file algebraicSphere.h.

◆ potential() [1/2]

template<class DataPoint , class _WFunctor , typename T >

Scalar Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::potential ( ) const

inline

Value of the scalar field at the evaluation point.

See also: method #isSigned of the fit to check if the sign is reliable

Definition at line 229 of file algebraicSphere.h.

◆ potential() [2/2]

template<class DataPoint , class _WFunctor , typename T >

AlgebraicSphere< DataPoint, _WFunctor, T >::Scalar AlgebraicSphere::potential ( const VectorType & _q ) const

inline

Value of the scalar field at the location \( \mathbf{q}\).

See also: method #isSigned of the fit to check if the sign is reliable

Definition at line 61 of file algebraicSphere.hpp.

◆ prattNorm()

template<class DataPoint , class _WFunctor , typename T >

Scalar Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::prattNorm ( ) const

inline

compute the Pratt norm of the implicit scalar field.

Definition at line 148 of file algebraicSphere.h.

◆ prattNorm2()

template<class DataPoint , class _WFunctor , typename T >

Scalar Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::prattNorm2 ( ) const

inline

compute the squared Pratt norm of the implicit scalar field.

Definition at line 155 of file algebraicSphere.h.

◆ primitiveGradient() [1/2]

template<class DataPoint , class _WFunctor , typename T >

const VectorType & Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::primitiveGradient ( ) const

inline

Approximation of the scalar field gradient at the evaluation point.

Warning: The gradient is not normalized by default

Definition at line 256 of file algebraicSphere.h.

◆ primitiveGradient() [2/2]

template<class DataPoint , class _WFunctor , typename T >

AlgebraicSphere< DataPoint, _WFunctor, T >::VectorType AlgebraicSphere::primitiveGradient ( const VectorType & _q ) const

inline

Approximation of the scalar field gradient at \( \mathbf{q}\).

Warning: The gradient is not normalized by default

Definition at line 72 of file algebraicSphere.hpp.

◆ project()

template<class DataPoint , class _WFunctor , typename T >

AlgebraicSphere< DataPoint, _WFunctor, T >::VectorType AlgebraicSphere::project ( const VectorType & _q ) const

inline

Project a point on the algebraic hypersphere.

This projection is realized in closed-form: the algebraic hypersphere is converted to a geometrical representation (hyperplane or hypersphere), and _q is orthogonally projected on the primtive.

Note: This function is in most cases more accurate and faster than projectDescent

Definition at line 9 of file algebraicSphere.hpp.

◆ projectDescent()

template<class DataPoint , class _WFunctor , typename T >

AlgebraicSphere< DataPoint, _WFunctor, T >::VectorType AlgebraicSphere::projectDescent	(	const VectorType &	_q,
		int	nbIter = `16`
	)		const

inline

Project a point on the algebraic hypersphere using Gradient Descent This projection is realized by following the gradient of the hypersphere scalar field.

Warning: This function is in most cases slower and less accurate than project.

Parameters

_q	Starting point
nbIter	Number of iterations (default = 16)

Definition at line 34 of file algebraicSphere.hpp.

◆ radius()

template<class DataPoint , class _WFunctor , typename T >

Scalar Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::radius ( ) const

inline

return the estimated radius of the sphere

Warning: return inf if the fitted surface is planar

Definition at line 182 of file algebraicSphere.h.

Member Data Documentation

◆ m_isNormalized

template<class DataPoint , class _WFunctor , typename T >

bool Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::m_isNormalized

protected

Is the implicit scalar field normalized using Pratt.

Definition at line 59 of file algebraicSphere.h.

◆ m_uc

template<class DataPoint , class _WFunctor , typename T >

Scalar Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::m_uc {0}

Constant parameter of the Algebraic hyper-sphere.

Definition at line 63 of file algebraicSphere.h.

◆ m_ul

template<class DataPoint , class _WFunctor , typename T >

VectorType Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::m_ul {VectorType::Zero()}

Linear parameter of the Algebraic hyper-sphere.

Definition at line 65 of file algebraicSphere.h.

◆ m_uq

template<class DataPoint , class _WFunctor , typename T >

Scalar Ponca::AlgebraicSphere< DataPoint, _WFunctor, T >::m_uq {0}

Quadratic parameter of the Algebraic hyper-sphere.

Definition at line 64 of file algebraicSphere.h.

Public Types

Public Member Functions

Public Attributes

Protected Types

Protected Attributes

Detailed Description

Member Typedef Documentation

◆ Base

◆ Scalar

◆ VectorType

◆ WFunctor

Member Enumeration Documentation

◆ anonymous enum

Member Function Documentation

◆ algebraicSphere() [1/2]

◆ algebraicSphere() [2/2]

◆ applyPrattNorm()

◆ center()

◆ changeBasis()

◆ init()

◆ isNormalized()

◆ isPlane()

◆ isValid()

◆ operator!=()

◆ operator==()

◆ potential() [1/2]

◆ potential() [2/2]

◆ prattNorm()

◆ prattNorm2()

◆ primitiveGradient() [1/2]

◆ primitiveGradient() [2/2]

◆ project()

◆ projectDescent()

◆ radius()

Member Data Documentation

◆ m_isNormalized

◆ m_uc

◆ m_ul

◆ m_uq