Procedure that compute and decompose the covariance matrix of the neighbors positions in \(3d\). More...

#include <covarianceFit.h>

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

Collaboration diagram for Ponca::CovarianceFitBase< 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.

using	MatrixType = typename DataPoint::MatrixType
	Alias to matrix type.

using	Solver = Eigen::SelfAdjointEigenSolver< MatrixType >
	Solver used to analyse the covariance matrix.

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

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

void	init (const VectorType &_evalPos)
	Set the evaluation position and reset the internal states.

bool	addLocalNeighbor (Scalar w, const VectorType &localQ, const DataPoint &attributes)
	Add a neighbor to perform the fit.

FIT_RESULT	finalize ()
	Finalize the procedure.

Scalar	surfaceVariation () const
	Implements [12] surface variation.

const Solver &	solver () const
	Reading access to the Solver used to analyse the covariance matrix.

Protected Types
enum	{ Check = Base::PROVIDES_MEAN_POSITION , PROVIDES_POSITION_COVARIANCE }

using	Base = T
	Base class of the procedure.

Protected Attributes
MatrixType	m_cov {MatrixType::Zero()}
	Covariance matrix.

Solver	m_solver
	Solver used to analyse the covariance matrix.

Detailed Description

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

Procedure that compute and decompose the covariance matrix of the neighbors positions in \(3d\).

This process is commonly used for plane fitting and local variance analysis. It is often called Principal Component Analysis (PCA) of the neighborhood, and used in Geometry Processing and Computer Vision.

See also: CovariancePlaneFit which use a similar approach for Plane estimation

Computation details

Standard PCA algorithm involves a two-steps process where the barycenter \(\mathbf{b}\) is first computed, and then the covariance matrix \(\text{C}\) (in the following, the weights are ignored for clarity but without loss of generality):

\begin{align} \mathbf{b} &= \frac{1}{n}\sum_i\mathbf{p}_i \\ \text{C} &= \frac{1}{n}\sum_i(\mathbf{p}_i-\mathbf{b})(\mathbf{p}_i-\mathbf{b})^T \end{align}

This class implements a single-pass version, where the first formulation is re-expressed as follows:

\begin{align} \text{C} &= \frac{1}{n}\sum_i (\mathbf{p}_i\mathbf{p}_i^T - \mathbf{b}\mathbf{p}_i^T - \mathbf{p}_i\mathbf{b}^T + \mathbf{b}\mathbf{b}^T) \\ &= \frac{1}{n}\sum_i (\mathbf{p}_i\mathbf{p}_i^T) - \frac{1}{n}\sum_i(\mathbf{b}\mathbf{p}_i^T) - \frac{1}{n}\sum_i(\mathbf{p}_i\mathbf{b}^T) + \frac{1}{n}\sum_i (\mathbf{b}\mathbf{b}^T) \\ &= \frac{1}{n}\sum_i (\mathbf{p}_i\mathbf{p}_i^T) - \mathbf{b}\frac{1}{n}\sum_i(\mathbf{p}_i^T) - \frac{1}{n}\sum_i(\mathbf{p}_i)\mathbf{b}^T + \frac{1}{n}\sum_i(1) \mathbf{b}\mathbf{b}^T \\ &= \frac{1}{n}\sum_i (\mathbf{p}_i\mathbf{p}_i^T) - \mathbf{b}\mathbf{b}^T - \mathbf{b}\mathbf{b}^T + \mathbf{b}\mathbf{b}^T \end{align}

Leading to a single pass where \(\text{C}\) is express by substracting two terms that can be computed independently in one run:

\[ \text{C} = \frac{1}{n}\sum_i (\mathbf{p}_i\mathbf{p}_i^T) - \mathbf{b}\mathbf{b}^T \]

Warning: This class is valid only in 3D.

Definition at line 50 of file covarianceFit.h.

Member Typedef Documentation

◆ Base

template<class DataPoint , class _WFunctor , typename T >

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

protected

Base class of the procedure.

Definition at line 52 of file covarianceFit.h.

◆ MatrixType

template<class DataPoint , class _WFunctor , typename T >

using Ponca::CovarianceFitBase< DataPoint, _WFunctor, T >::MatrixType = typename DataPoint::MatrixType

Alias to matrix type.

Definition at line 62 of file covarianceFit.h.

◆ Scalar

template<class DataPoint , class _WFunctor , typename T >

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

Alias to scalar type.

Definition at line 52 of file covarianceFit.h.

◆ Solver

template<class DataPoint , class _WFunctor , typename T >

using Ponca::CovarianceFitBase< DataPoint, _WFunctor, T >::Solver = Eigen::SelfAdjointEigenSolver<MatrixType>

Solver used to analyse the covariance matrix.

Definition at line 64 of file covarianceFit.h.

◆ VectorType

template<class DataPoint , class _WFunctor , typename T >

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

Alias to vector type.

Definition at line 52 of file covarianceFit.h.

◆ WFunctor

template<class DataPoint , class _WFunctor , typename T >

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

Alias to weight function.

Definition at line 52 of file covarianceFit.h.

Member Enumeration Documentation

◆ anonymous enum

template<class DataPoint , class _WFunctor , typename T >

anonymous enum

protected

Definition at line 55 of file covarianceFit.h.

Member Function Documentation

◆ addLocalNeighbor()

template<class DataPoint , class _WFunctor , typename T >

bool CovarianceFitBase::addLocalNeighbor	(	Scalar	w,
		const VectorType &	localQ,
		const DataPoint &	attributes
	)

inline

Add a neighbor to perform the fit.

Returns: false if param nei is not a valid neighbour (weight = 0)

Definition at line 21 of file covarianceFit.hpp.

◆ covarianceFit() [1/2]

template<class DataPoint , class _WFunctor , typename T >

CovarianceFitBase< DataPoint, _WFunctor, T > & Ponca::CovarianceFitBase< DataPoint, _WFunctor, T >::covarianceFit ( )

inline

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

Definition at line 72 of file covarianceFit.h.

◆ covarianceFit() [2/2]

template<class DataPoint , class _WFunctor , typename T >

const CovarianceFitBase< DataPoint, _WFunctor, T > & Ponca::CovarianceFitBase< DataPoint, _WFunctor, T >::covarianceFit ( ) const

inline

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

Definition at line 72 of file covarianceFit.h.

◆ finalize()

template<class DataPoint , class _WFunctor , typename T >

FIT_RESULT CovarianceFitBase::finalize ( )

inline

Finalize the procedure.

Returns: Fitting Status

Warning: Must be called be for any use of the fitting output

Definition at line 35 of file covarianceFit.hpp.

◆ init()

template<class DataPoint , class _WFunctor , typename T >

void CovarianceFitBase::init ( const VectorType & _evalPos )

inline

Set the evaluation position and reset the internal states.

Warning: Must be called be for any computation (but after #setWeightFunc)

Definition at line 13 of file covarianceFit.hpp.

◆ solver()

template<class DataPoint , class _WFunctor , typename T >

const Solver & Ponca::CovarianceFitBase< DataPoint, _WFunctor, T >::solver ( ) const

inline

Reading access to the Solver used to analyse the covariance matrix.

Definition at line 82 of file covarianceFit.h.

◆ surfaceVariation()

template<class DataPoint , class _WFunctor , typename T >

CovarianceFitBase< DataPoint, _WFunctor, T >::Scalar CovarianceFitBase::surfaceVariation ( ) const

inline

Implements [12] surface variation.

It computes the ratio \( d \frac{\lambda_0}{\sum_i \lambda_i} \) with d the dimension of the ambient space.

Returns: 0 for invalid fits

Definition at line 60 of file covarianceFit.hpp.

Member Data Documentation

◆ m_cov

template<class DataPoint , class _WFunctor , typename T >

MatrixType Ponca::CovarianceFitBase< DataPoint, _WFunctor, T >::m_cov {MatrixType::Zero()}

protected

Covariance matrix.

Definition at line 68 of file covarianceFit.h.

◆ m_solver

template<class DataPoint , class _WFunctor , typename T >

Solver Ponca::CovarianceFitBase< DataPoint, _WFunctor, T >::m_solver

protected

Solver used to analyse the covariance matrix.

Definition at line 69 of file covarianceFit.h.

Public Types

Public Member Functions

Protected Types

Protected Attributes

Detailed Description

Computation details

Member Typedef Documentation

◆ Base

◆ MatrixType

◆ Scalar

◆ Solver

◆ VectorType

◆ WFunctor

Member Enumeration Documentation

◆ anonymous enum

Member Function Documentation

◆ addLocalNeighbor()

◆ covarianceFit() [1/2]

◆ covarianceFit() [2/2]

◆ finalize()

◆ init()

◆ solver()

◆ surfaceVariation()

Member Data Documentation

◆ m_cov

◆ m_solver