BIAS::SVD Class Reference
[Algorithms]

computes and holds the singular value decomposition of a rectangular (not necessarily quadratic) Matrix A More...

#include <MathAlgo/SVD.hh>

Inheritance diagram for BIAS::SVD:

Collaboration diagram for BIAS::SVD:

List of all members.

Public Member Functions

const int AbsLeftNullspaceDim () const

returns the dim of the matrix's left nullspace, using absolute threshold criterion

const int AbsNullspaceDim () const

returns dim of nullspace using the absolute threshold criterion

const int AbsRightNullspaceDim () const

returns the dim of the matrix's right nullspace, using absolute threshold criterion

void AddDebugLevel (const std::string &name)

void AddDebugLevel (const long int lv)

int compute (const Matrix< double > &M, double ZeroThreshold=DEFAULT_DOUBLE_ZERO_THRESHOLD)

use our naming convention

int Compute (const Matrix< double > &M, double ZeroThreshold=DEFAULT_DOUBLE_ZERO_THRESHOLD)

set a new matrix and compute its decomposition.

bool DebugLevelIsSet (const std::string &name) const

bool DebugLevelIsSet (const long int lv) const

int General_Eigenproblem_GeneralMatrix_Lapack (const Matrix< double > &M)

solve the general (non-special) eigenvalue/eigenvector problem of a general (non-symmetric) matrix M calls the extern liblapack routine dgesvd .

int GetDebugLevel () const

void GetDebugStream (std::ostream &os) const

std::ostream & GetDebugStream () const

BIAS::Vector< double > GetEigenValues ()

Call this after Compute(), returns the eigenvalues of the matrix in ascending order (smallest first), which are NOT the singular values! Works only, if the original Matrix was symmetric.

Vector< double > GetLeftNullvector (const int last_offset=0)

same as above but returning vector

bool GetLeftNullvector (Vector< double > &nv, const int last_offset=0)

Return one of the left nullvectors.

bool GetNullvector (Vector< double > &NullVec, const int last_offset=0)

Returns one of the nullvectors in argument and true if Nullvector exists.

Vector< double > GetNullvector (const int last_offset=0)

return one of the nullvectors.

const Vector< double > & GetS () const

return S which is a vector of the singular values of A in descending order.

const double GetSingularValue (int index) const

return one singular value (which may be zero).

const Matrix< double > & GetU () const

return U U is a m x m orthogonal matrix

Matrix< double > GetV () const

return V

const Matrix< double > & GetVT () const

return VT (=transposed(V))

double GetZeroThreshold () const

return zerothresh currently used

Matrix< double > Invert (Matrix< double > A)

as above, but compute new svd for a

Matrix< double > Invert (int rank)

returns pseudoinverse of A = U * S * V^T A^+ = V * S^+ * U^T the first "rank" elements of S are inverted the others are set to zero

Matrix< double > Invert ()

returns pseudoinverse of A = U * S * V^T A^+ = V * S^+ * U^T

const int LeftNullspaceDim () const

returns the dim of the matrix's left nullspace

long int Name2DebugLevel (const std::string &name) const

looks up a debuglevel in the internal map, returns 0 if not found

long int NewDebugLevel (const std::string &name)

creates a new debuglevel

const int NullspaceDim () const

return the dim of the nullspace.

void PrintDebugLevel (std::ostream &os=std::cout) const

unsigned int Rank ()

returns the rank of A_

const int RelLeftNullspaceDim () const

const int RelNullspaceDim () const

compare singular values against greatest, not absolute

const int RelRightNullspaceDim () const

void RemoveDebugLevel (const std::string &name)

void RemoveDebugLevel (const long int lv)

const int RightNullspaceDim () const

returns the dim of the matrix's right nullspace

void SetDebugLevel (const std::string &name)

void SetDebugLevel (const long int lv)

void SetDebugStream (const std::ostream &os)

void ShowDebugLevel (std::ostream &os=std::cout) const

prints all internally known debuglevels

int size () const

return the length of the singular value vector inline const int size() const

int Solve (Matrix< double > &A, Vector< double > &B, Vector< double > &X)

solves the overdetermined linear system $AX=B$ with the unknown $X$, where $A$ is an $m x n$ matrix ($m>n$)and B is a vector of size $m$.

Vector< double > solve (const Vector< double > &y) const

use our naming convention

Vector< double > Solve (const Vector< double > &y) const

Matrix< double > Sqrt (const Matrix< double > &A)

as above, but compute new svd for a

Matrix< double > Sqrt ()

returns the square root of a symmetric positive definite matrix M A = sqrt(M) = U * sqrt(S) * V_T, such that A*A = M.

Matrix< double > SqrtT (const Matrix< double > &A)

as above, but compute new svd for a

Matrix< double > SqrtT ()

returns the square root of a symmetric positive definite matrix M A = sqrt(M) = U * sqrt(S), such that A*A^T = M.

SVD (bool AbsoluteZeroThreshold=true)

creates an empty svd without automatically decomposing any matrix

SVD (const Matrix< double > &M, double ZeroThreshold=DEFAULT_DOUBLE_ZERO_THRESHOLD, bool AbsoluteZeroThreshold=true)

solve the general eigenproblem of the rectangular matrix M or with other words, make the U*S*V^T decomposition

virtual ~SVD ()

Static Public Member Functions

static long int GetGlobalDebugLevel ()

static void SetGlobalDebugLevel (long int lev)

Protected Member Functions

long ConsumeNextFreeDebuglevel_ ()

returns the next available debuglevel

Protected Attributes

long int _liDebugLevel

long int _liNextDebugLevel

new concept, debuglevel are managed here in the debug class

std::map< std::string, long int > _String2Debuglevel

Matrix< double > A_

data members: original matrix (to be decomposed)

bool AbsoluteZeroThreshold_

determines whether we compare singular-values directly (absolute) to the zero threshold or whether we compare the ratio of the sing-value under inspection and the largest one to the threshold (relative)

bool Decomposed_

flag for holding decomposed matrix

Vector< double > S_

contains the singular values of A_ corresponding to the i'th column in descending order.

Matrix< double > U_

contains columnwise the left singular vectors of A_ corresponding to the i'th singular value

Matrix< double > VT_

contains the right singular vectors of A- in rows [beause VT is transpose(V) ]

double ZeroThreshold_

values below this threshold are treated as zero

Static Protected Attributes

static std::ostream _zDebugStream

static long int GlobalDebugLevel = 0

Detailed Description

computes and holds the singular value decomposition of a rectangular (not necessarily quadratic) Matrix A

M = U S V't.

The singular vectors vi and (right) singular values sigma_i satisfy the equation: A vi = sigma_i vi

The Vector S_ stores the singular values of A in decreasing order (smallest is last)

The columns of U are the left singular vectors of A and form an orthonormal basis of A which corresponds to the nonzero singular values. while the columns of U corresponding to (nearly) zero values are the nullspace.

This function is a base class and not a member in Matrix because:

holding the U,S,V matrices allows repeated queries of the same SVD results. -avoid namespace clutter in the Matrix class. There are many other decompositions that might be of interest, and adding them all would make for a very large Matrix class with extra members which are irrelevant for the Matrix itself.

You can create the svd object either in relative or in absolute mode: if AbsoluteZeroThreshold is true, a singular value is compared to the threshold "absolutely", otherwise the ratio of the largest and the current singular value is compared against the threshold. Usually you want AbsoluteZeroThreshold = false, but for the sake of interface stability/downwards compatibility, the default is true

Author:: Jan Woetzel

Definition at line 90 of file SVD.hh.

Constructor & Destructor Documentation

SVD::SVD	(	const Matrix< double > &	M,
		double	ZeroThreshold = `DEFAULT_DOUBLE_ZERO_THRESHOLD`,
		bool	AbsoluteZeroThreshold = `true`
	)

solve the general eigenproblem of the rectangular matrix M or with other words, make the U*S*V^T decomposition

Parameters:

	M	matrix to decompose
	ZeroThreshold	for nullspace determination, if a singular value is lower than ZeroThreshold, define the corresponding vector in V^T to be a null-vector (also sing.vals are treated this way)
	AbsoluteZeroThreshold	if true, a singular value is compared to the threshold "absoultely", otherwise the ratio of the largest and the current singular value is compared against the threshold. Usually you want AbsoluteZeroThreshold = false, but for the sake of interface stability/downwards compatibility, the default is true

Author:: Jan Woetzel

Definition at line 40 of file SVD.cpp.

References AbsoluteZeroThreshold_, Compute(), and Decomposed_.

BIAS::SVD::SVD ( bool AbsoluteZeroThreshold = true ) [inline]

creates an empty svd without automatically decomposing any matrix

Parameters:

AbsoluteZeroThreshold

if true, a singular value is compared to the threshold "absolutely", otherwise the ratio of the largest and the current singular value is compared against the threshold. Usually you want AbsoluteZeroThreshold = false, but for the sake of interface stability/downwards compatibility, the default is true

Definition at line 115 of file SVD.hh.

References DEFAULT_DOUBLE_ZERO_THRESHOLD.

virtual BIAS::SVD::~SVD ( ) [inline, virtual]

Definition at line 121 of file SVD.hh.

Member Function Documentation

const int SVD::AbsLeftNullspaceDim ( ) const

returns the dim of the matrix's left nullspace, using absolute threshold criterion

Author:: woelk 04 2003

Definition at line 380 of file SVD.cpp.

References A_, TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), S_, TNT::Vector< T >::size(), and ZeroThreshold_.

const int SVD::AbsNullspaceDim ( ) const

returns dim of nullspace using the absolute threshold criterion

For rectangular matrices this is the number of singular values which are (about) zero. The nullspace is spaned by the singular vectors corresponding to the zero singular values. woelk 4 2003: changed API, diffentiate between Left and Right Nullspace

Author:: Jan Woetzel 04/04/2002

Definition at line 328 of file SVD.cpp.

References A_, TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), S_, TNT::Vector< T >::size(), and ZeroThreshold_.

Referenced by AbsRightNullspaceDim().

const int SVD::AbsRightNullspaceDim ( ) const

returns the dim of the matrix's right nullspace, using absolute threshold criterion

Author:: woelk 04 2003

Definition at line 372 of file SVD.cpp.

References AbsNullspaceDim().

void BIAS::Debug::AddDebugLevel ( const std::string & name ) [inline, inherited]

Definition at line 354 of file Debug.hh.

void BIAS::Debug::AddDebugLevel ( const long int lv ) [inline, inherited]

Definition at line 347 of file Debug.hh.

Referenced by BIAS::Triangulation::CorrectCorrespondences(), BIAS::FilterBase< InputStorageType, OutputStorageType >::FilterBase(), main(), BIAS::PMDImageProc::PreparePolyDepthCalibration(), and BIAS::RANSAC< SolutionType >::RANSAC().

int BIAS::SVD::compute	(	const Matrix< double > &	M,
		double	ZeroThreshold = `DEFAULT_DOUBLE_ZERO_THRESHOLD`
	)			`[inline]`

use our naming convention

Definition at line 129 of file SVD.hh.

Referenced by BIAS::PMatrix::MakeSVD_().

int SVD::Compute	(	const Matrix< double > &	M,
		double	ZeroThreshold = `DEFAULT_DOUBLE_ZERO_THRESHOLD`
	)

set a new matrix and compute its decomposition.

solves the general eigenproblem of the rectangular matrix M

Author:: Jan Woetzel 05/142002)

Reimplemented in BIAS::SVD3x3.

Definition at line 48 of file SVD.cpp.

References Decomposed_, General_Eigenproblem_GeneralMatrix_Lapack(), and ZeroThreshold_.

Referenced by BIAS::CovMatrix3x3::_Decompose(), BIAS::ImageAlignment::Align(), BIAS::HMatrix::FactorizeAffineMatrixRLeft(), BIAS::HMatrix::FactorizeAffineMatrixRRight(), BIAS::EMatrix::GetRotationTranslation(), BIAS::Interpolator::InitSpline3(), Invert(), main(), Solve(), BIAS::LeastSquaresSVD::Solve(), Sqrt(), SqrtT(), and SVD().

long BIAS::Debug::ConsumeNextFreeDebuglevel_ ( ) [inline, protected, inherited]

returns the next available debuglevel

Author:: woelk 09/2006

Definition at line 513 of file Debug.hh.

References ABORT.

bool BIAS::Debug::DebugLevelIsSet ( const std::string & name ) const [inline, inherited]

Definition at line 342 of file Debug.hh.

bool BIAS::Debug::DebugLevelIsSet ( const long int lv ) const [inline, inherited]

Definition at line 333 of file Debug.hh.

int SVD::General_Eigenproblem_GeneralMatrix_Lapack ( const Matrix< double > & M )

solve the general (non-special) eigenvalue/eigenvector problem of a general (non-symmetric) matrix M calls the extern liblapack routine dgesvd .

computes the genral singular value decomposition with M = U * Sigma(S) * VT with Sigma(S) is the qudratic, symmetric matrix which contains the singular vales in the diagonal and zero and zero else. U, VT as described in the data mambers.

Author:: Jan Woetzel (04/04/2002)

Definition at line 56 of file SVD.cpp.

References A_, Decomposed_, General_singular_value_decomposition(), TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), res, S_, TNT::Vector< T >::size(), U_, and VT_.

Referenced by Compute().

int BIAS::Debug::GetDebugLevel ( ) const [inline, inherited]

Definition at line 324 of file Debug.hh.

Referenced by BIAS::MSAC< SolutionType >::_GenerateSamplesRandom(), BIAS::ImageAlignment::Align(), BIAS::ExampleDebug::func(), BIAS::FMatrix::GetEpipoles(), BIAS::listenthread(), main(), BIAS::GuiGTK::MouseMotionCallback_(), BIAS::Image< StorageType >::operator=(), BIAS::ImageBase::operator=(), BIAS::ExampleDebug::ResetDebugLevelFromGlobal(), BIAS::TrackerBaseAffine2< StorageType >::TrackAffine_(), and BIAS::TrackerBaseAffine< StorageType >::TrackAffine_().

void BIAS::Debug::GetDebugStream ( std::ostream & os ) const [inline, inherited]

Definition at line 406 of file Debug.hh.

std::ostream& BIAS::Debug::GetDebugStream ( ) const [inline, inherited]

Definition at line 397 of file Debug.hh.

Referenced by BIAS::MSAC< SolutionType >::_GenerateSamplesRandom(), BIAS::RANSAC< SolutionType >::GenerateSamplesRandom(), and BIAS::GuiGTK::MouseMotionCallback_().

BIAS::Vector< double > SVD::GetEigenValues ( )

Call this after Compute(), returns the eigenvalues of the matrix in ascending order (smallest first), which are NOT the singular values! Works only, if the original Matrix was symmetric.

Author:: grest, July 2006

Definition at line 486 of file SVD.cpp.

References A_, BIAS::Matrix< T >::GetRow(), BIAS::Matrix< T >::GetRows(), and VT_.

Referenced by BIAS::CamPoseCalib::SolveLM_().

static long int BIAS::Debug::GetGlobalDebugLevel ( ) [inline, static, inherited]

Definition at line 423 of file Debug.hh.

Referenced by BIAS::ExampleDebug::func(), and BIAS::ExampleDebug::ResetDebugLevelFromGlobal().

Vector< double > BIAS::SVD::GetLeftNullvector ( const int last_offset = 0 ) [inline]

same as above but returning vector

Definition at line 419 of file SVD.hh.

References BIASERR, GetLeftNullvector(), TNT::Matrix< T >::num_rows(), and VT_.

bool BIAS::SVD::GetLeftNullvector	(	Vector< double > &	nv,
		const int	last_offset = `0`
	)			`[inline]`

Return one of the left nullvectors.

If last_offset=0 then the last nullvector (corresponding to the smallest singular value) is returned, if last_offset=1 then the last but one nullvector is returned, and so on. The last_offset must be in [0..NullspaceDim-1] otherwise 0 is returned

Author:: JMF/koeser

Definition at line 428 of file SVD.hh.

References BIASERR, TNT::Vector< T >::newsize(), NullspaceDim(), TNT::Matrix< T >::num_rows(), S_, U_, and VT_.

Referenced by BIAS::TrifocalTensor::ComputeEpipoles(), BIAS::TrifocalTensor::GetEpipolarLines(), BIAS::FMatrix::GetEpipoles(), and GetLeftNullvector().

bool BIAS::SVD::GetNullvector	(	Vector< double > &	NullVec,
		const int	last_offset = `0`
	)			`[inline]`

Returns one of the nullvectors in argument and true if Nullvector exists.

if last_offset=0 then the last nullvector (corresponding to the smallest singular value) is returned, if last_offset=1 then the last but one nullvector is returned, and so on. The last_offset must be in [0..NullspaceDim-1].

Author:: Jan Woetzel (08/08/2002), JMF

Definition at line 398 of file SVD.hh.

References TNT::Vector< T >::newsize(), NullspaceDim(), TNT::Matrix< T >::num_cols(), and VT_.

Vector< double > BIAS::SVD::GetNullvector ( const int last_offset = 0 ) [inline]

return one of the nullvectors.

if last_offset=0 then the last nullvector (corresponding to the smallest singular value) is returned, if last_offset=1 then the last but one nullvector is returned, and so on. The last_offset must be in [0..NullspaceDim-1].

Author:: Jan Woetzel(08/08/2002), JMF

Definition at line 389 of file SVD.hh.

References TNT::Matrix< T >::num_cols(), and VT_.

Referenced by calc_Regressionsebene(), BIAS::TrifocalTensor::ComputeEpipoles(), BIAS::TrifocalTensor::GetEpipolarLines(), BIAS::FMatrix::GetEpipoles(), BIAS::PMatrix::GetNullVector(), BIAS::Conic2D::GetPoint(), BIAS::PlaneRANSAC::RefineSolution(), and BIAS::FMatrixEstimation::SevenPoint().

const Vector<double>& BIAS::SVD::GetS ( ) const [inline]

return S which is a vector of the singular values of A in descending order.

The diagonalelements of Sigma are the singular values of A.

Author:: Jan Woetzel 04/04/2002

Definition at line 154 of file SVD.hh.

Referenced by BIAS::CovMatrix3x3::_Decompose(), BIAS::ImageAlignment::Align(), BIAS::PMatrixLinear::Compute(), BIAS::PMatrixEstimation::ComputeFromFDirect(), BIAS::FMatrixEstimation::EnforceMaxRank2(), BIAS::HMatrix::FactorizeAffineMatrixRLeft(), BIAS::HMatrix::FactorizeAffineMatrixRRight(), GetBaseLevel(), BIAS::FMatrix::GetEpipoles(), BIAS::PMatrix::GetPseudoInverse(), BIAS::Quadric3D::GetSignature(), BIAS::Conic2D::GetSignature(), BIAS::PMatrix::GetSVD(), BIAS::EMatrix::InitFromF(), Lapack_LLS_QR_linear_solve(), main(), Print(), BIAS::Quadric3D::Quadric3D(), Solve(), and BIAS::LeastSquaresSVD::Solve().

const double BIAS::SVD::GetSingularValue ( int index ) const [inline]

return one singular value (which may be zero).

inline implementations

the index runs from 0.. (size-1)

Author:: Jan Woetzel 04/04/2002

Definition at line 371 of file SVD.hh.

References BIASASSERT, S_, TNT::Vector< T >::size(), and size().

Referenced by BIAS::CamPoseCalib::outputPose_(), and RelLeftNullspaceDim().

const Matrix<double>& BIAS::SVD::GetU ( ) const [inline]

return U U is a m x m orthogonal matrix

Author:: Jan Woetzel (04/04/2002)

Definition at line 174 of file SVD.hh.

Referenced by BIAS::CovMatrix3x3::_Decompose(), BIAS::FMatrixEstimation::EnforceMaxRank2(), BIAS::RMatrix::EnforceRConstraints(), BIAS::HMatrix::FactorizeAffineMatrixRLeft(), BIAS::HMatrix::FactorizeAffineMatrixRRight(), BIAS::Conic2D::GetDualConic(), BIAS::FMatrix::GetEpipoles(), BIAS::PMatrix::GetPseudoInverse(), BIAS::EMatrix::GetRotationTranslation(), BIAS::PMatrix::GetSVD(), BIAS::EMatrix::InitFromF(), main(), Print(), BIAS::Quadric3D::Quadric3D(), Solve(), and BIAS::LeastSquaresSVD::Solve().

Matrix< double > BIAS::SVD::GetV ( ) const [inline]

return V

Author:: Jan Woetzel (04/04/2002)

Definition at line 381 of file SVD.hh.

References BIAS::Matrix< T >::Transpose(), and VT_.

Referenced by BIAS::Conic2D::GetDualConic(), BIAS::PMatrix::GetPseudoInverse(), Print(), and BIAS::Quadric3D::Quadric3D().

const Matrix<double>& BIAS::SVD::GetVT ( ) const [inline]

double BIAS::SVD::GetZeroThreshold ( ) const [inline]

return zerothresh currently used

Author:: koeser

Definition at line 147 of file SVD.hh.

Referenced by BIAS::PMatrixLinear::Compute().

Matrix< double > SVD::Invert ( Matrix< double > A )

as above, but compute new svd for a

Definition at line 181 of file SVD.cpp.

References Compute(), and Invert().

Matrix< double > SVD::Invert ( int rank )

returns pseudoinverse of A = U * S * V^T A^+ = V * S^+ * U^T the first "rank" elements of S are inverted the others are set to zero

Author:: Christian Beder

Definition at line 156 of file SVD.cpp.

References Decomposed_, TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), res, S_, BIAS::Matrix< T >::SetZero(), TNT::Vector< T >::size(), BIAS::Matrix< T >::Transpose(), U_, and VT_.

Matrix< double > SVD::Invert ( )

returns pseudoinverse of A = U * S * V^T A^+ = V * S^+ * U^T

Author:: F Woelk

Definition at line 127 of file SVD.cpp.

References AbsoluteZeroThreshold_, Decomposed_, TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), res, S_, BIAS::Matrix< T >::SetZero(), TNT::Vector< T >::size(), BIAS::Matrix< T >::Transpose(), U_, VT_, and ZeroThreshold_.

Referenced by BIAS::ImageAlignment::Align(), BIAS::TextureTransformAffine::ComposeWithInverseDeltaP(), BIAS::ImageBlender::ComputeCylCamGeometry(), BIAS::GaussHelmert::ComputeNormalSystem_(), BIAS::ContourBSplineShapeMatrix::ComputeRegularisationMatrix(), BIAS::ColorFilter< InputStorageType, OutputStorageType >::FindWaterExp(), BIAS::ContourDetectorBSpline< StorageType >::Fit_(), BIAS::PMatrix::GetCanonicalH(), BIAS::Triangulation::GetCovarianceProjective(), BIAS::CamPoseCalib::GetCoVarMatrix(), BIAS::Conic2D::GetDualConic(), BIAS::Quadric3D::GetDualQuadric(), BIAS::Interpolator::InitSpline3(), Invert(), main(), BIAS::GaussHelmert::Solve(), BIAS::TrackerBaseAffine2< StorageType >::TrackAffine_(), BIAS::TrackerBaseAffine< StorageType >::TrackAffine_(), BIAS::Kalman::Update(), and BIAS::ExtendedKalman::Update().

const int BIAS::SVD::LeftNullspaceDim ( ) const [inline]

returns the dim of the matrix's left nullspace

Author:: woelk 04 2003

Definition at line 210 of file SVD.hh.

long int BIAS::Debug::Name2DebugLevel ( const std::string & name ) const [inline, inherited]

looks up a debuglevel in the internal map, returns 0 if not found

Author:: woelk 09/2006

Definition at line 446 of file Debug.hh.

References it.

long int BIAS::Debug::NewDebugLevel ( const std::string & name ) [inline, inherited]

creates a new debuglevel

Author:: woelk 09/2006

Definition at line 466 of file Debug.hh.

References ABORT, it, and res.

Referenced by BIAS::Condensation::Condensation(), BIAS::Histogram::Histogram(), BIAS::MonteCarloTransform::MonteCarloTransform(), myclass::myclass(), BIAS::PreemptiveRANSAC< SolutionType >::PreemptiveRANSAC(), BIAS::RANSAC< SolutionType >::RANSAC(), BIAS::RANSACPreKnowledge< SolutionType >::RANSACPreKnowledge(), BIAS::ROI::ROI(), BIAS::Triangulation::Triangulation(), and BIAS::UnscentedTransform::UnscentedTransform().

const int BIAS::SVD::NullspaceDim ( ) const [inline]

return the dim of the nullspace.

For rectangular matrices this is the number of singular values which are (about) zero. The nullspace is spaned by the singular vectors corresponding to the zero singular values. woelk 4 2003: changed API, diffentiate between Left and Right Nullspace koeser 01/2005: rel/abs handled by flag: AbsoluteZeroThreshold_ determines whther we compare absolute or relative singular values against the threshold

Author:: Jan Woetzel 04/04/2002

Definition at line 196 of file SVD.hh.

Referenced by BIAS::FMatrix::GetEpipoles(), GetLeftNullvector(), GetNullvector(), BIAS::FMatrix::IsValid(), main(), Rank(), and RelLeftNullspaceDim().

void BIAS::Debug::PrintDebugLevel ( std::ostream & os = std::cout ) const [inline, inherited]

Definition at line 375 of file Debug.hh.

References WS.

Referenced by main().

unsigned int SVD::Rank ( )

returns the rank of A_

Definition at line 419 of file SVD.cpp.

References A_, BIASDOUT, D_SVD_RANK, NullspaceDim(), TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), S_, and BIAS::Vector< T >::Size().

Referenced by Lapack_LLS_QR_linear_solve(), and BIAS::PlaneRANSAC::RefineSolution().

const int SVD::RelLeftNullspaceDim ( ) const

Definition at line 398 of file SVD.cpp.

References A_, GetSingularValue(), NullspaceDim(), TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), S_, TNT::Vector< T >::size(), and ZeroThreshold_.

const int SVD::RelNullspaceDim ( ) const

compare singular values against greatest, not absolute

Author:: Christian Buck changed criterion for insignificant singular values according to hartley/zisserman from s_i<ZeroThreshold to s_i/s_0<ZeroThreshold. fallback to old criterion iff s_0=0.

Definition at line 346 of file SVD.cpp.

References A_, TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), S_, TNT::Vector< T >::size(), and ZeroThreshold_.

Referenced by BIAS::PMatrixLinear::Compute(), BIAS::Conic2D::GetDualConic(), and RelRightNullspaceDim().

const int SVD::RelRightNullspaceDim ( ) const

Definition at line 376 of file SVD.cpp.

References RelNullspaceDim().

void BIAS::Debug::RemoveDebugLevel ( const std::string & name ) [inline, inherited]

Definition at line 368 of file Debug.hh.

void BIAS::Debug::RemoveDebugLevel ( const long int lv ) [inline, inherited]

Definition at line 361 of file Debug.hh.

const int BIAS::SVD::RightNullspaceDim ( ) const [inline]

returns the dim of the matrix's right nullspace

Author:: woelk 04 2003

Definition at line 203 of file SVD.hh.

void BIAS::Debug::SetDebugLevel ( const std::string & name ) [inline, inherited]

Definition at line 317 of file Debug.hh.

void BIAS::Debug::SetDebugLevel ( const long int lv ) [inline, inherited]

Definition at line 310 of file Debug.hh.

Referenced by main(), BIAS::ExampleDebug::ResetDebugLevelFromGlobal(), BIAS::MDcam::ScanBus(), BIAS::MDcam::SetDebugLevelDcam(), BIAS::VideoSource_Disk::VideoSource_Disk(), BIAS::VideoSource_DV2::VideoSource_DV2(), BIAS::VideoSource_DV::VideoSource_DV_Internal_Init_(), and BIAS::VideoSource_V4L::VideoSource_V4L().

void BIAS::Debug::SetDebugStream ( const std::ostream & os ) [inline, inherited]

Definition at line 390 of file Debug.hh.

Referenced by main().

static void BIAS::Debug::SetGlobalDebugLevel ( long int lev ) [inline, static, inherited]

Definition at line 416 of file Debug.hh.

Referenced by main().

void BIAS::Debug::ShowDebugLevel ( std::ostream & os = std::cout ) const [inline, inherited]

prints all internally known debuglevels

Author:: woelk 09/2006

Definition at line 488 of file Debug.hh.

References it.

Referenced by main().

int BIAS::SVD::size ( ) const [inline]

return the length of the singular value vector inline const int size() const

Definition at line 179 of file SVD.hh.

Referenced by GetSingularValue().

int SVD::Solve	(	Matrix< double > &	A,
		Vector< double > &	B,
		Vector< double > &	X
	)

solves the overdetermined linear system $AX=B$ with the unknown $X$, where $A$ is an $m x n$ matrix ($m>n$)and B is a vector of size $m$.

Author:: woelk 07/2004

Definition at line 429 of file SVD.cpp.

References AbsoluteZeroThreshold_, BIASCDOUT, BIASDOUT, Compute(), D_SVD_SOLVE, GetS(), BIAS::Matrix< T >::GetSystemMatrix(), GetU(), GetVT(), TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), res, S_, BIAS::Matrix< T >::Transpose(), and ZeroThreshold_.

Vector<double> BIAS::SVD::solve ( const Vector< double > & y ) const [inline]

use our naming convention

Definition at line 280 of file SVD.hh.

Vector< double > SVD::Solve ( const Vector< double > & y ) const

Bug:: bug in this function (woelk)

Definition at line 81 of file SVD.cpp.

References BEXCEPTION, BIASERR, Decomposed_, TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), S_, TNT::Vector< T >::size(), BIAS::Matrix< T >::Transpose(), U_, VT_, and ZeroThreshold_.

Referenced by BIAS::CamPoseCalib::Estimate(), BIAS::PolynomialSolve::FitPolynomial(), FitPolynomialWithoutLinearMonomial(), BIAS::PMatrix::GetCanonicalH(), BIAS::UnVignette::PrepareLeastSquaresMin_(), BIAS::PMDImageProc::PreparePolyDepthCalibration(), BIAS::GaussHelmert::Solve(), BIAS::CamPoseCalib::Solve_(), and BIAS::TrackerBaseAffine< StorageType >::TrackAffine_().

Matrix< double > SVD::Sqrt ( const Matrix< double > & A )

as above, but compute new svd for a

Definition at line 250 of file SVD.cpp.

References Compute(), and Sqrt().

Matrix< double > SVD::Sqrt ( )

returns the square root of a symmetric positive definite matrix M A = sqrt(M) = U * sqrt(S) * V_T, such that A*A = M.

The result is only valid when the M *is* symmetric positive definite.

Author:: woelk 01/2006

Definition at line 189 of file SVD.cpp.

References A_, AbsoluteZeroThreshold_, BEXCEPTION, BIASERR, Compute(), Decomposed_, BIAS::Equal(), BIAS::Matrix< T >::GetCols(), BIAS::Matrix< T >::GetRows(), BIAS::GreaterEqual(), BIAS::MatrixZero, num, TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), r, res, S_, TNT::Vector< T >::size(), U_, Upper_symmetric_eigenvalue_solve(), VT_, and ZeroThreshold_.

Referenced by MatrixSqrt(), and Sqrt().

Matrix< double > SVD::SqrtT ( const Matrix< double > & A )

as above, but compute new svd for a

Definition at line 320 of file SVD.cpp.

References Compute(), and SqrtT().

Matrix< double > SVD::SqrtT ( )

returns the square root of a symmetric positive definite matrix M A = sqrt(M) = U * sqrt(S), such that A*A^T = M.

The result is only valid when the M *is* symmetric positive definite.

Author:: woelk 01/2006

Definition at line 257 of file SVD.cpp.

References A_, AbsoluteZeroThreshold_, BEXCEPTION, Compute(), Decomposed_, BIAS::Equal(), BIAS::Matrix< T >::GetCols(), BIAS::Matrix< T >::GetRows(), BIAS::GreaterEqual(), BIAS::MatrixZero, num, TNT::Matrix< T >::num_cols(), TNT::Matrix< T >::num_rows(), r, S_, TNT::Vector< T >::size(), U_, Upper_symmetric_eigenvalue_solve(), VT_, and ZeroThreshold_.

Referenced by BIAS::UnscentedTransform::ComputeSigmaPoints_(), and SqrtT().

Member Data Documentation

long int BIAS::Debug::_liDebugLevel [protected, inherited]

Definition at line 502 of file Debug.hh.

Referenced by BIAS::GuiGTK::DeleteEvent_(), BIAS::GuiGTK::Destroy_(), BIAS::GuiGTK::DrawingAreaExposeCallback_(), BIAS::GuiGTK::GtkMain(), BIAS::GuiGTK::KeyPressEventCallback_(), BIAS::GuiGTK::LabelExposeCallback_(), BIAS::GuiGTK::MouseButtonPre


Public Member Functions
const int	AbsLeftNullspaceDim () const
	returns the dim of the matrix's left nullspace, using absolute threshold criterion
const int	AbsNullspaceDim () const
	returns dim of nullspace using the absolute threshold criterion
const int	AbsRightNullspaceDim () const
	returns the dim of the matrix's right nullspace, using absolute threshold criterion
void	AddDebugLevel (const std::string &name)
void	AddDebugLevel (const long int lv)
int	compute (const Matrix< double > &M, double ZeroThreshold=DEFAULT_DOUBLE_ZERO_THRESHOLD)
	use our naming convention
int	Compute (const Matrix< double > &M, double ZeroThreshold=DEFAULT_DOUBLE_ZERO_THRESHOLD)
	set a new matrix and compute its decomposition.
bool	DebugLevelIsSet (const std::string &name) const
bool	DebugLevelIsSet (const long int lv) const
int	General_Eigenproblem_GeneralMatrix_Lapack (const Matrix< double > &M)
	solve the general (non-special) eigenvalue/eigenvector problem of a general (non-symmetric) matrix M calls the extern liblapack routine dgesvd .
int	GetDebugLevel () const
void	GetDebugStream (std::ostream &os) const
std::ostream &	GetDebugStream () const
BIAS::Vector< double >	GetEigenValues ()
	Call this after Compute(), returns the eigenvalues of the matrix in ascending order (smallest first), which are NOT the singular values! Works only, if the original Matrix was symmetric.
Vector< double >	GetLeftNullvector (const int last_offset=0)
	same as above but returning vector
bool	GetLeftNullvector (Vector< double > &nv, const int last_offset=0)
	Return one of the left nullvectors.
bool	GetNullvector (Vector< double > &NullVec, const int last_offset=0)
	Returns one of the nullvectors in argument and true if Nullvector exists.
Vector< double >	GetNullvector (const int last_offset=0)
	return one of the nullvectors.
const Vector< double > &	GetS () const
	return S which is a vector of the singular values of A in descending order.
const double	GetSingularValue (int index) const
	return one singular value (which may be zero).
const Matrix< double > &	GetU () const
	return U U is a m x m orthogonal matrix
Matrix< double >	GetV () const
	return V
const Matrix< double > &	GetVT () const
	return VT (=transposed(V))
double	GetZeroThreshold () const
	return zerothresh currently used
Matrix< double >	Invert (Matrix< double > A)
	as above, but compute new svd for a
Matrix< double >	Invert (int rank)
	returns pseudoinverse of A = U * S * V^T A^+ = V * S^+ * U^T the first "rank" elements of S are inverted the others are set to zero
Matrix< double >	Invert ()
	returns pseudoinverse of A = U * S * V^T A^+ = V * S^+ * U^T
const int	LeftNullspaceDim () const
	returns the dim of the matrix's left nullspace
long int	Name2DebugLevel (const std::string &name) const
	looks up a debuglevel in the internal map, returns 0 if not found
long int	NewDebugLevel (const std::string &name)
	creates a new debuglevel
const int	NullspaceDim () const
	return the dim of the nullspace.
void	PrintDebugLevel (std::ostream &os=std::cout) const
unsigned int	Rank ()
	returns the rank of A_
const int	RelLeftNullspaceDim () const
const int	RelNullspaceDim () const
	compare singular values against greatest, not absolute
const int	RelRightNullspaceDim () const
void	RemoveDebugLevel (const std::string &name)
void	RemoveDebugLevel (const long int lv)
const int	RightNullspaceDim () const
	returns the dim of the matrix's right nullspace
void	SetDebugLevel (const std::string &name)
void	SetDebugLevel (const long int lv)
void	SetDebugStream (const std::ostream &os)
void	ShowDebugLevel (std::ostream &os=std::cout) const
	prints all internally known debuglevels
int	size () const
	return the length of the singular value vector inline const int size() const
int	Solve (Matrix< double > &A, Vector< double > &B, Vector< double > &X)
	solves the overdetermined linear system $AX=B$ with the unknown $X$, where $A$ is an $m x n$ matrix ($m>n$)and B is a vector of size $m$.
Vector< double >	solve (const Vector< double > &y) const
	use our naming convention
Vector< double >	Solve (const Vector< double > &y) const
Matrix< double >	Sqrt (const Matrix< double > &A)
	as above, but compute new svd for a
Matrix< double >	Sqrt ()
	returns the square root of a symmetric positive definite matrix M A = sqrt(M) = U * sqrt(S) * V_T, such that A*A = M.
Matrix< double >	SqrtT (const Matrix< double > &A)
	as above, but compute new svd for a
Matrix< double >	SqrtT ()
	returns the square root of a symmetric positive definite matrix M A = sqrt(M) = U * sqrt(S), such that A*A^T = M.
	SVD (bool AbsoluteZeroThreshold=true)
	creates an empty svd without automatically decomposing any matrix
	SVD (const Matrix< double > &M, double ZeroThreshold=DEFAULT_DOUBLE_ZERO_THRESHOLD, bool AbsoluteZeroThreshold=true)
	solve the general eigenproblem of the rectangular matrix M or with other words, make the USV^T decomposition
virtual	~SVD ()
Static Public Member Functions
static long int	GetGlobalDebugLevel ()
static void	SetGlobalDebugLevel (long int lev)
Protected Member Functions
long	ConsumeNextFreeDebuglevel_ ()
	returns the next available debuglevel
Protected Attributes
long int	_liDebugLevel
long int	_liNextDebugLevel
	new concept, debuglevel are managed here in the debug class
std::map< std::string, long int >	_String2Debuglevel
Matrix< double >	A_
	data members: original matrix (to be decomposed)
bool	AbsoluteZeroThreshold_
	determines whether we compare singular-values directly (absolute) to the zero threshold or whether we compare the ratio of the sing-value under inspection and the largest one to the threshold (relative)
bool	Decomposed_
	flag for holding decomposed matrix
Vector< double >	S_
	contains the singular values of A_ corresponding to the i'th column in descending order.
Matrix< double >	U_
	contains columnwise the left singular vectors of A_ corresponding to the i'th singular value
Matrix< double >	VT_
	contains the right singular vectors of A- in rows [beause VT is transpose(V) ]
double	ZeroThreshold_
	values below this threshold are treated as zero
Static Protected Attributes
static std::ostream	_zDebugStream
static long int	GlobalDebugLevel = 0

BIAS::SVD Class Reference [Algorithms]

Public Member Functions

Static Public Member Functions

Protected Member Functions

Protected Attributes

Static Protected Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

BIAS::SVD Class Reference
[Algorithms]