Files
file	WlzBasisFn.c
	Functions for creating and manipulating basis functions.

Data Structures
struct	_WlzBasisFnMapData3D
	Used to memorise the nodes of a mesh element between 3D distance function evaluation calls. More...

struct	_WlzBasisFn
	A basis function. Typedef: WlzBasisFn. More...

Typedefs
typedef WlzPixelV(*	Wlz3DProjectionIntFn) (WlzPixelP, int, int, void , WlzErrorNum )
	Callback function for the WlzGetProjectionFromObject(). More...

typedef double(*	WlzBasisEvalFn) (void *, double)
	An alternative basis function evaluation function that may may be called. More...

typedef double(*	WlzBasisDistFn) (void , int, WlzVertex, void )
	An alternative basis function distance function that may may be called. More...

typedef int(*	WlzThreshCbFn) (WlzObject , void , WlzThreshCbStr *)
	Callback function for the WlzCbThreshold() More...

Enumerations
enum	_WlzFnType { WLZ_FN_BASIS_2DGAUSS, WLZ_FN_BASIS_3DGAUSS, WLZ_FN_BASIS_2DIMQ, WLZ_FN_BASIS_3DIMQ, WLZ_FN_BASIS_2DPOLY, WLZ_FN_BASIS_3DPOLY, WLZ_FN_BASIS_2DMQ, WLZ_FN_BASIS_3DMQ, WLZ_FN_BASIS_2DTPS, WLZ_FN_BASIS_3DTPS, WLZ_FN_BASIS_2DCONF_POLY, WLZ_FN_BASIS_3DCONF_POLY, WLZ_FN_BASIS_3DMOS, WLZ_FN_BASIS_SCALAR_3DMOS, WLZ_FN_SCALAR_MOD, WLZ_FN_SCALAR_EXP, WLZ_FN_SCALAR_LOG, WLZ_FN_SCALAR_SQRT, WLZ_FN_SCALAR_INVSQRT, WLZ_FN_SCALAR_SQR, WLZ_FN_COUNT }
	The types of function. Typedef: WlzFnType. More...

Functions
WlzErrorNum	WlzBasisFnFree (WlzBasisFn *basisFn)
	Free's the given basis function. More...

WlzDVertex2	WlzBasisFnValuePoly2D (WlzBasisFn *basisFn, WlzDVertex2 srcVx)
	Calculates the value for the given vertex using a 2D polynomial basis function. This is not a basis function just a polynomial, but it makes sense for it to be incuded in the basis functions since they have a polynomial component. More...

WlzDVertex2	WlzBasisFnValueGauss2D (WlzBasisFn *basisFn, WlzDVertex2 srcVx)
	Calculates the value for the given vertex using a 2D Gaussian basis function. More...

WlzDVertex2	WlzBasisFnValueMQ2D (WlzBasisFn *basisFn, WlzDVertex2 srcVx)
	Calculates the value for the given vertex using a 2D multiquadric basis function. More...

WlzDVertex3	WlzBasisFnValueMQ3D (WlzBasisFn *basisFn, WlzDVertex3 srcVx)
	Calculates the displacement value for the given vertex using a 3D multiquadric basis function. More...

WlzDVertex2	WlzBasisFnValueIMQ2D (WlzBasisFn *basisFn, WlzDVertex2 srcVx)
	Calculates the value for the given vertex using a 2D inverse multiquadric basis function. More...

WlzDVertex3	WlzBasisFnValueIMQ3D (WlzBasisFn *basisFn, WlzDVertex3 srcVx)
	Calculates the displacement value for the given vertex using a 3D inverse multiquadric basis function. More...

WlzDVertex2	WlzBasisFnValueTPS2D (WlzBasisFn *basisFn, WlzDVertex2 srcVx)
	Calculates the value for the given vertex using a 2D thin plate spline basis function. More...

WlzDVertex2	WlzBasisFnValueConf2D (WlzBasisFn *basisFn, WlzDVertex2 srcVx)
	Calculates the value for the given vertex using a 2D conformal polynomial basis function. More...

WlzDVertex3	WlzBasisFnValueMOS3D (WlzBasisFn *basisFn, WlzDVertex3 srcVx)
	Calculates the value for the given vertex using a 3D multiorder basis function: \[ \phi(r) = \frac{1}{4 \pi \delta^2 r} (1 - \frac{w}{w - v} e^{- \sqrt{v} r} + \frac{v}{w - v} e^{- \sqrt{w} r}) \] \[ v = \frac{1 + \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \] \[ w = \frac{1 - \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \] . More...

double	WlzBasisFnValueScalarMOS3D (WlzBasisFn *basisFn, WlzDVertex3 srcVx)
	Calculates the value for the given vertex using a scalar 3D multiorder basis function: \[ \phi(r) = \frac{1}{4 \pi \delta^2 r} (1 - \frac{w}{w - v} e^{- \sqrt{v} r} + \frac{v}{w - v} e^{- \sqrt{w} r}) \] \[ v = \frac{1 + \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \] \[ w = \frac{1 - \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \] . More...

double	WlzBasisFnValueMOSPhi (double r, double delta, double tau)
	Computes the value of a single multiorder radial basis function: \[ \phi(r) = \frac{1}{4 \pi \delta^2 r} (1 - \frac{w}{w - v} e^{- \sqrt{v} r} + \frac{v}{w - v} e^{- \sqrt{w} r}) \] \[ v = \frac{1 + \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \] \[ w = \frac{1 - \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \] . More...

WlzBasisFn *	WlzBasisFnGauss2DFromCPts (int nPts, WlzDVertex2 dPts, WlzDVertex2 sPts, double delta, WlzBasisFn prvBasisFn, WlzCMesh2D mesh, WlzErrorNum *dstErr)
	Creates a new Gaussian basis function. More...

WlzBasisFn *	WlzBasisFnPoly2DFromCPts (int nPts, int order, WlzDVertex2 dPts, WlzDVertex2 sPts, WlzErrorNum *dstErr)
	Creates a new polynomial basis function. More...

WlzBasisFn *	WlzBasisFnConf2DFromCPts (int nPts, int order, WlzDVertex2 dPts, WlzDVertex2 sPts, WlzErrorNum *dstErr)
	Creates a new conformal basis function. More...

WlzBasisFn *	WlzBasisFnMQ2DFromCPts (int nPts, WlzDVertex2 dPts, WlzDVertex2 sPts, double delta, WlzBasisFn prvBasisFn, WlzCMesh2D mesh, WlzErrorNum *dstErr)
	Creates a new multiquadric basis function. More...

WlzBasisFn *	WlzBasisFnMQ3DFromCPts (int nPts, WlzDVertex3 dPts, WlzDVertex3 sPts, double delta, WlzBasisFn prvBasisFn, WlzCMesh3D mesh, WlzErrorNum *dstErr)
	Creates a new multiquadric basis function. More...

WlzBasisFn *	WlzBasisFnIMQ2DFromCPts (int nPts, WlzDVertex2 dPts, WlzDVertex2 sPts, double delta, WlzBasisFn prvBasisFn, WlzCMesh2D mesh, WlzErrorNum *dstErr)
	Creates a new inverse-multiquadric basis function. More...

WlzBasisFn *	WlzBasisFnIMQ3DFromCPts (int nPts, WlzDVertex3 dPts, WlzDVertex3 sPts, double delta, WlzBasisFn prvBasisFn, WlzCMesh3D mesh, WlzErrorNum *dstErr)
	Creates a new inverse-multiquadric basis function. More...

WlzBasisFn *	WlzBasisFnTPS2DFromCPts (int nPts, WlzDVertex2 dPts, WlzDVertex2 sPts, WlzBasisFn prvBasisFn, WlzCMesh2D mesh, WlzErrorNum *dstErr)
	Creates a new thin plate spline basis function. More...

WlzBasisFn *	WlzBasisFnMOS3DFromCPts (int nPts, WlzDVertex2 dPts, WlzDVertex2 sPts, double alpha, double param, WlzErrorNum *dstErr)
	Creates a new 3D multi order spline basis function: \[ \phi(r) = \frac{1}{4 \pi \delta^2 r} (1 - \frac{w}{w - v} e^{- \sqrt{v} r} + \frac{v}{w - v} e^{- \sqrt{w} r}) \] \[ v = \frac{1 + \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \] \[ w = \frac{1 - \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \] \[ f(\mathbf{x}) = P(\mathbf{x}) + \sum_{i=1}^{n}{\lambda_i \phi(r_i)} \] \[ r_i = \|\mathbf{x} - \mathbf{x'_i}\| \] The multi order spline may either be an exact interpolating function or one which approximates the given control points. In the approximating case values for the regularization parameters should be given, with the regularization parameters \(\alpha_i\) given by: \[ H[f] = \beta[f] + \sum_{i=1}^{n}{\frac{(y_i - f(x))^2}{\alpha_i}} \] Giving the design equation: \[ \left( \begin{array}{cccccccc} 0 & 0 & 0 & 0 & 1 & 1 & \cdots & 1 \\ 0 & 0 & 0 & 0 & x_1 & x_2 & \cdots & x_n \\ 0 & 0 & 0 & 0 & y_1 & y_2 & \cdots & y_n \\ 0 & 0 & 0 & 0 & z_1 & z_2 & \cdots & z_n \\ 1 & x_1 & y_1 & z_1 & \phi(r_{11}) + \alpha_1 & \phi(r_{12}) & \cdots & \phi(r_{1n}) \\ \dotfill \\ 1 & x_n & y_n & z_n & \phi(r_{n1}) & \phi(r_{n2}) & \cdots & \phi(r_{nn} + \alpha_n) \end{array} \right) \left( \begin{array}{c} a_o \\ a_1 \\ a_2 \\ a_3 \\ \lambda_1 \\ \lambda_1 \\ \vdots \\ \lambda_n \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ f(\mathbf{x_1}) \\ f(\mathbf{x_1}) \\ \vdots \\ f(\mathbf{x_n}) \end{array} \right) \] The vertex values with components \(x\), \(y \) and \(z\) are used to solve for the four polynomial coefficients and the three basis function weights \(\lambda\), \(\mu\) and \(\nu\). The given values of \(\lambda\) and \(\tau\) are constrained by \(\lambda > 0, \tau > 0\) and \(\lambda \tau < 0.5\). More...

WlzBasisFn *	WlzBasisFnScalarMOS3DFromCPts (int nPts, WlzDVertex3 cPts, double cVal, double alpha, double param, WlzErrorNum *dstErr)
	Computes a new 3D multi order spline basis function which either interpolates or approximates the given scalar values. See WlzBasisFnMOS3DFromCPts() for details of the multi order spline. The given values of \(\lambda\) and \(\tau\) are constrained by \(\lambda > 0, \tau > 0\) and \(\lambda \tau < 0.5\). More...

Detailed Description

Typedef Documentation

◆ Wlz3DProjectionIntFn

Wlz3DProjectionIntFn

Callback function for the WlzGetProjectionFromObject().

◆ WlzBasisEvalFn

WlzBasisEvalFn

An alternative basis function evaluation function that may may be called.

◆ WlzBasisDistFn

WlzBasisDistFn

An alternative basis function distance function that may may be called.

◆ WlzThreshCbFn

WlzThreshCbFn

Callback function for the WlzCbThreshold()

Enumeration Type Documentation

◆ _WlzFnType

enum _WlzFnType

The types of function. Typedef: WlzFnType.

Enumerator
WLZ_FN_BASIS_2DGAUSS	Gaussian basis function.
WLZ_FN_BASIS_3DGAUSS
WLZ_FN_BASIS_2DIMQ	Inverse multiquadric basis function.
WLZ_FN_BASIS_3DIMQ
WLZ_FN_BASIS_2DPOLY	Polynomial basis function.
WLZ_FN_BASIS_3DPOLY
WLZ_FN_BASIS_2DMQ	Multiquadric basis function.
WLZ_FN_BASIS_3DMQ
WLZ_FN_BASIS_2DTPS	Thin plate spline basis fn.
WLZ_FN_BASIS_3DTPS
WLZ_FN_BASIS_2DCONF_POLY	2D Conformal polynomial basis function.
WLZ_FN_BASIS_3DCONF_POLY
WLZ_FN_BASIS_3DMOS	3D Multi-order spline.
WLZ_FN_BASIS_SCALAR_3DMOS	3D Multi-order spline with scalar values.
WLZ_FN_SCALAR_MOD	Modulus (abs() or fabs()).
WLZ_FN_SCALAR_EXP	Exponential (exp()).
WLZ_FN_SCALAR_LOG	Logarithm (log()).
WLZ_FN_SCALAR_SQRT	Square root (x^-1/2).
WLZ_FN_SCALAR_INVSQRT	Inverse square root (x^-1/2).
WLZ_FN_SCALAR_SQR	Square (x * x).
WLZ_FN_COUNT	Not a function but the number of functions. Keep this the last of the enums!

Function Documentation

◆ WlzBasisFnFree()

WlzErrorNum WlzBasisFnFree ( WlzBasisFn * basisFn )

Free's the given basis function.

Returns: Woolz error number.

Parameters

basisFn Given basis function, may be NULL.

References AlcFree(), _WlzBasisFn::basis, _WlzBasisFn::distMap, _WlzBasisFn::evalData, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::poly, _WlzBasisFn::sVertices, _WlzVertexP::v, _WlzBasisFn::vertices, WLZ_ERR_NONE, and WlzFreeHistogramDomain().

Referenced by WlzBasisFnFreeTransform(), and WlzBasisFnTPS2DChangeCPtsParam().

◆ WlzBasisFnValuePoly2D()

WlzDVertex2 WlzBasisFnValuePoly2D	(	WlzBasisFn *	basisFn,
		WlzDVertex2	srcVx
	)

Calculates the value for the given vertex using a 2D polynomial basis function. This is not a basis function just a polynomial, but it makes sense for it to be incuded in the basis functions since they have a polynomial component.

Returns: New vertex value.

Parameters

basisFn	Basis function.
srcVx	Source vertex.

References _WlzVertexP::d2, _WlzBasisFn::nPoly, _WlzBasisFn::poly, _WlzDVertex2::vtX, and _WlzDVertex2::vtY.

Referenced by WlzBasisFnSetMesh(), and WlzBasisFnTransformVertexD().

◆ WlzBasisFnValueGauss2D()

WlzDVertex2 WlzBasisFnValueGauss2D	(	WlzBasisFn *	basisFn,
		WlzDVertex2	srcVx
	)

Calculates the value for the given vertex using a 2D Gaussian basis function.

Returns: New vertex value.

Parameters

basisFn	Basis function.
srcVx	Source vertex.

References _WlzBasisFn::basis, _WlzVertexP::d2, _WlzVertex::d2, _WlzBasisFn::distFn, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::vertices, _WlzDVertex2::vtX, and _WlzDVertex2::vtY.

Referenced by WlzBasisFnSetCMesh2D(), WlzBasisFnSetMesh(), and WlzBasisFnTransformVertexD().

◆ WlzBasisFnValueMQ2D()

WlzDVertex2 WlzBasisFnValueMQ2D	(	WlzBasisFn *	basisFn,
		WlzDVertex2	srcVx
	)

Calculates the value for the given vertex using a 2D multiquadric basis function.

Returns: New vertex value.

Parameters

basisFn	Basis function.
srcVx	Source vertex.

References _WlzBasisFn::basis, _WlzVertexP::d2, _WlzVertex::d2, _WlzBasisFn::distFn, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::vertices, _WlzDVertex2::vtX, and _WlzDVertex2::vtY.

Referenced by WlzBasisFnSetCMesh2D(), WlzBasisFnSetMesh(), and WlzBasisFnTransformVertexD().

◆ WlzBasisFnValueMQ3D()

WlzDVertex3 WlzBasisFnValueMQ3D	(	WlzBasisFn *	basisFn,
		WlzDVertex3	srcVx
	)

Calculates the displacement value for the given vertex using a 3D multiquadric basis function.

Returns: New vertex value.

Parameters

basisFn	Basis function.
srcVx	Source vertex.

References _WlzBasisFn::basis, _WlzVertexP::d3, _WlzVertex::d3, _WlzBasisFn::distFn, _WlzBasisFnMapData3D::elmIdx, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::vertices, _WlzDVertex3::vtX, _WlzDVertex3::vtY, and _WlzDVertex3::vtZ.

Referenced by WlzBasisFnSetCMesh3D().

◆ WlzBasisFnValueIMQ2D()

WlzDVertex2 WlzBasisFnValueIMQ2D	(	WlzBasisFn *	basisFn,
		WlzDVertex2	srcVx
	)

Calculates the value for the given vertex using a 2D inverse multiquadric basis function.

Returns: New vertex value.

Parameters

basisFn	Basis function.
srcVx	Source vertex.

References _WlzBasisFn::basis, _WlzVertexP::d2, _WlzVertex::d2, _WlzBasisFn::distFn, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::vertices, _WlzDVertex2::vtX, and _WlzDVertex2::vtY.

Referenced by WlzBasisFnSetCMesh2D(), WlzBasisFnSetMesh(), and WlzBasisFnTransformVertexD().

◆ WlzBasisFnValueIMQ3D()

WlzDVertex3 WlzBasisFnValueIMQ3D	(	WlzBasisFn *	basisFn,
		WlzDVertex3	srcVx
	)

Calculates the displacement value for the given vertex using a 3D inverse multiquadric basis function.

Returns: New vertex value.

Parameters

basisFn	Basis function.
srcVx	Source vertex.

References _WlzBasisFn::basis, _WlzVertexP::d3, _WlzVertex::d3, _WlzBasisFn::distFn, _WlzBasisFnMapData3D::elmIdx, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::vertices, _WlzDVertex3::vtX, _WlzDVertex3::vtY, and _WlzDVertex3::vtZ.

Referenced by WlzBasisFnSetCMesh3D().

◆ WlzBasisFnValueTPS2D()

WlzDVertex2 WlzBasisFnValueTPS2D	(	WlzBasisFn *	basisFn,
		WlzDVertex2	srcVx
	)

Calculates the value for the given vertex using a 2D thin plate spline basis function.

Returns: New vertex value.

Parameters

basisFn	Basis function.
srcVx	Source vertex.

References _WlzBasisFn::basis, _WlzVertexP::d2, _WlzVertex::d2, _WlzBasisFn::distFn, _WlzBasisFn::nVtx, _WlzBasisFn::vertices, _WlzDVertex2::vtX, and _WlzDVertex2::vtY.

Referenced by WlzBasisFnSetCMesh2D(), WlzBasisFnSetMesh(), and WlzBasisFnTransformVertexD().

◆ WlzBasisFnValueConf2D()

WlzDVertex2 WlzBasisFnValueConf2D	(	WlzBasisFn *	basisFn,
		WlzDVertex2	srcVx
	)

Calculates the value for the given vertex using a 2D conformal polynomial basis function.

Returns: New vertex value.

Parameters

basisFn	Basis function.
srcVx	Source vertex.

References AlgCMult(), _WlzVertexP::d2, _ComplexD::im, _WlzBasisFn::nPoly, _WlzBasisFn::poly, _ComplexD::re, _WlzDVertex2::vtX, and _WlzDVertex2::vtY.

Referenced by WlzBasisFnSetMesh(), and WlzBasisFnTransformVertexD().

◆ WlzBasisFnValueMOS3D()

WlzDVertex3 WlzBasisFnValueMOS3D	(	WlzBasisFn *	basisFn,
		WlzDVertex3	srcVx
	)

Calculates the value for the given vertex using a 3D multiorder basis function:

\[ \phi(r) = \frac{1}{4 \pi \delta^2 r} (1 - \frac{w}{w - v} e^{- \sqrt{v} r} + \frac{v}{w - v} e^{- \sqrt{w} r}) \]

\[ v = \frac{1 + \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \]

\[ w = \frac{1 - \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \]

.

Returns: New vertex value.

Parameters

basisFn	Basis function.
srcVx	Source vertex.

References ALG_M_PI, _WlzBasisFn::basis, _WlzVertexP::d3, _WlzBasisFn::evalFn, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::vertices, _WlzDVertex3::vtX, _WlzDVertex3::vtY, _WlzDVertex3::vtZ, WLZ_VTX_3_LENGTH, and WLZ_VTX_3_SUB.

◆ WlzBasisFnValueScalarMOS3D()

double WlzBasisFnValueScalarMOS3D	(	WlzBasisFn *	basisFn,
		WlzDVertex3	srcVx
	)

Calculates the value for the given vertex using a scalar 3D multiorder basis function:

\[ \phi(r) = \frac{1}{4 \pi \delta^2 r} (1 - \frac{w}{w - v} e^{- \sqrt{v} r} + \frac{v}{w - v} e^{- \sqrt{w} r}) \]

\[ v = \frac{1 + \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \]

\[ w = \frac{1 - \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \]

.

Returns: New scalar value.

Parameters

basisFn	Basis function.
srcVx	Source vertex.

References ALG_M_PI, _WlzBasisFn::basis, _WlzVertexP::d3, _WlzBasisFn::evalFn, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzVertexP::v, _WlzBasisFn::vertices, WLZ_VTX_3_LENGTH, and WLZ_VTX_3_SUB.

◆ WlzBasisFnValueMOSPhi()

double WlzBasisFnValueMOSPhi	(	double	r,
		double	delta,
		double	tau
	)

Computes the value of a single multiorder radial basis function:

\[ \phi(r) = \frac{1}{4 \pi \delta^2 r} (1 - \frac{w}{w - v} e^{- \sqrt{v} r} + \frac{v}{w - v} e^{- \sqrt{w} r}) \]

\[ v = \frac{1 + \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \]

\[ w = \frac{1 - \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \]

.

Returns: The value of a single multiorder radial basis function.

Parameters

r	Radial distance, \((r > 0)\).
delta	The 1st order smoothness parameter \(\delta\), \((\delta > 0)\).
tau	The 3rd order smoothness parameter \(\tau\), \((\tau > 0)\).

References ALG_M_PI.

◆ WlzBasisFnGauss2DFromCPts()

WlzBasisFn* WlzBasisFnGauss2DFromCPts	(	int	nPts,
		WlzDVertex2 *	dPts,
		WlzDVertex2 *	sPts,
		double	delta,
		WlzBasisFn *	prvBasisFn,
		WlzCMesh2D *	mesh,
		WlzErrorNum *	dstErr
	)

Creates a new Gaussian basis function.

Returns

New basis function.

            This function may be used either to compute a new Gaussian
            radial basis function, or if given a previous Gaussian
            radial basis function, to recompute a Gaussian basis
            recycling the buffers (and distance map if used) from
            the prevoius basis function when the control points
            have changed. If recycling the buffers a new basis
            function is created which uses the (reallocated)
            buffers of the previous basis function. After this function
            has been called the previous basis function should be freed.

Parameters

nPts	Number of control point pairs.
dPts	Destination control points.
sPts	Source control points.
delta	Normalized delta value in range [> 0.0 , < 1.0 ].
prvBasisFn	Previous basis function to be recycled, may be NULL. Arrays will be used for the new basis function. Must be freed after recycling.
mesh	Used to compute constrained distances. If NULL Euclidean distances will be used.
dstErr	Destination error pointer, may be NULL.

References AlcCalloc(), AlcFree(), AlcMalloc(), AlcRealloc(), ALG_MIN, _WlzBasisFn::basis, _AlgMatrix::core, _WlzVertexP::d2, _WlzBasisFn::distFn, _WlzBasisFn::distMap, _WlzCMeshP::m2, _WlzBasisFn::maxVx, _WlzBasisFn::mesh, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::poly, _WlzBasisFn::sVertices, _WlzCMesh2D::type, _WlzBasisFn::type, _WlzVertexP::v, _WlzCMeshP::v, _WlzBasisFn::vertices, WLZ_CMESH_2D, WLZ_ERR_DOMAIN_TYPE, WLZ_ERR_MEM_ALLOC, WLZ_ERR_NONE, WLZ_FN_BASIS_2DGAUSS, and WlzGeomCmpVtx2D().

Referenced by WlzBasisFnTPS2DChangeCPtsParam(), and WlzBasisFnTrFromCPts2DParam().

◆ WlzBasisFnPoly2DFromCPts()

WlzBasisFn* WlzBasisFnPoly2DFromCPts	(	int	nPts,
		int	order,
		WlzDVertex2 *	dPts,
		WlzDVertex2 *	sPts,
		WlzErrorNum *	dstErr
	)

Creates a new polynomial basis function.

Returns: New basis function.

Parameters

nPts	Number of control point pairs.
order	Order of polynomial.
dPts	Destination control points.
sPts	Source control points.
dstErr	Destination error pointer, may be NULL.

References AlcCalloc(), AlcMalloc(), AlgMatrixRectNew(), AlgMatrixSVDecomp(), _AlgMatrixRect::array, _AlgMatrix::core, _WlzBasisFn::nBasis, _WlzBasisFn::nPoly, _WlzBasisFn::nVtx, _WlzBasisFn::poly, _AlgMatrix::rect, _WlzBasisFn::type, _WlzVertexP::v, _WlzDVertex2::vtX, _WlzDVertex2::vtY, WLZ_ERR_MEM_ALLOC, WLZ_ERR_NONE, WLZ_ERR_PARAM_DATA, WLZ_FN_BASIS_2DPOLY, and WlzErrorFromAlg().

Referenced by WlzBasisFnTPS2DChangeCPtsParam(), and WlzBasisFnTrFromCPts2DParam().

◆ WlzBasisFnConf2DFromCPts()

WlzBasisFn* WlzBasisFnConf2DFromCPts	(	int	nPts,
		int	order,
		WlzDVertex2 *	dPts,
		WlzDVertex2 *	sPts,
		WlzErrorNum *	dstErr
	)

Creates a new conformal basis function.

Returns: New basis function.

Parameters

nPts	Number of control point pairs.
order	Order of conformal poly.
dPts	Destination control points.
sPts	Source control points.
dstErr	Destination error pointer, may be NULL.

References AlcCalloc(), AlcMalloc(), AlgCMult(), AlgMatrixRectNew(), AlgMatrixSVDecomp(), _AlgMatrixRect::array, _AlgMatrix::core, _ComplexD::im, _WlzBasisFn::nBasis, _WlzBasisFn::nPoly, _WlzBasisFn::nVtx, _WlzBasisFn::poly, _ComplexD::re, _AlgMatrix::rect, _WlzBasisFn::type, _WlzVertexP::v, _WlzDVertex2::vtX, _WlzDVertex2::vtY, WLZ_ERR_MEM_ALLOC, WLZ_ERR_NONE, WLZ_ERR_PARAM_DATA, WLZ_FN_BASIS_2DCONF_POLY, and WlzErrorFromAlg().

Referenced by WlzBasisFnTPS2DChangeCPtsParam(), and WlzBasisFnTrFromCPts2DParam().

◆ WlzBasisFnMQ2DFromCPts()

WlzBasisFn* WlzBasisFnMQ2DFromCPts	(	int	nPts,
		WlzDVertex2 *	dPts,
		WlzDVertex2 *	sPts,
		double	delta,
		WlzBasisFn *	prvBasisFn,
		WlzCMesh2D *	mesh,
		WlzErrorNum *	dstErr
	)

Creates a new multiquadric basis function.

Returns

New basis function.

    To improve the design matix condition number the problem
    is rescaled when a distance map is not being used.

            This function may be used either to compute a new MQ
            radial basis function, or if given a previous MQ
            radial basis function, to recompute a MQ basis
            recycling the buffers (and distance map if used) from
            the prevoius basis function when the control points
            have changed. If recycling the buffers a new basis
            function is created which uses the (reallocated)
            buffers of the previous basis function. After this function
            has been called the previous basis function should be freed.

Parameters

nPts	Number of control point pairs.
dPts	Destination control points.
sPts	Source control points.
delta	Normalized delta value in range [> 0.0 , < 1.0 ].
prvBasisFn	Previous basis function to be recycled, may be NULL. Arrays will be used for the new basis function. Must be freed after recycling.
mesh	Used to compute constrained distances. If NULL Euclidean distances will be used.
dstErr	Destination error pointer, may be NULL.

References AlcCalloc(), AlcFree(), AlcMalloc(), AlcRealloc(), ALG_MIN, _WlzBasisFn::basis, _AlgMatrix::core, _WlzVertexP::d2, _WlzBasisFn::distFn, _WlzBasisFn::distMap, _WlzCMeshP::m2, _WlzBasisFn::maxVx, _WlzBasisFn::mesh, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::poly, _WlzBasisFn::sVertices, _WlzCMesh2D::type, _WlzBasisFn::type, _WlzVertexP::v, _WlzCMeshP::v, _WlzBasisFn::vertices, WLZ_CMESH_2D, WLZ_ERR_DOMAIN_TYPE, WLZ_ERR_MEM_ALLOC, WLZ_ERR_NONE, WLZ_FN_BASIS_2DMQ, and WlzGeomCmpVtx2D().

Referenced by WlzBasisFnTPS2DChangeCPtsParam(), and WlzBasisFnTrFromCPts2DParam().

◆ WlzBasisFnMQ3DFromCPts()

WlzBasisFn* WlzBasisFnMQ3DFromCPts	(	int	nPts,
		WlzDVertex3 *	dPts,
		WlzDVertex3 *	sPts,
		double	delta,
		WlzBasisFn *	prvBasisFn,
		WlzCMesh3D *	mesh,
		WlzErrorNum *	dstErr
	)

Creates a new multiquadric basis function.

Returns: New basis function.

Parameters

nPts	Number of control point pairs.
dPts	Destination control points.
sPts	Source control points.
delta	Normalized delta value in range [> 0.0 , < 1.0 ].
prvBasisFn	Previous basis function to be recycled, may be NULL. Arrays will be used for the new basis function. Must be freed after recycling.
mesh	Used to compute constrained distances. If NULL Euclidean distances will be used.
dstErr	Destination error pointer, may be NULL.

References AlcCalloc(), AlcFree(), AlcMalloc(), AlcRealloc(), ALG_MIN, _WlzBasisFn::basis, _AlgMatrix::core, _WlzVertexP::d3, _WlzBasisFn::distFn, _WlzBasisFn::distMap, _WlzCMeshP::m3, _WlzBasisFn::maxVx, _WlzBasisFn::mesh, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::poly, _WlzBasisFn::sVertices, _WlzCMesh3D::type, _WlzBasisFn::type, _WlzVertexP::v, _WlzCMeshP::v, _WlzBasisFn::vertices, WLZ_CMESH_3D, WLZ_ERR_DOMAIN_TYPE, WLZ_ERR_MEM_ALLOC, WLZ_ERR_NONE, WLZ_FN_BASIS_3DMQ, and WlzGeomCmpVtx3D().

Referenced by WlzBasisFnTrFromCPts3DParam().

◆ WlzBasisFnIMQ2DFromCPts()

WlzBasisFn* WlzBasisFnIMQ2DFromCPts	(	int	nPts,
		WlzDVertex2 *	dPts,
		WlzDVertex2 *	sPts,
		double	delta,
		WlzBasisFn *	prvBasisFn,
		WlzCMesh2D *	mesh,
		WlzErrorNum *	dstErr
	)

Creates a new inverse-multiquadric basis function.

Returns

New basis function.

    To improve the design matix condition number the problem
    is rescaled when a distance map is not being used.

            This function may be used either to compute a new IMQ
            radial basis function, or if given a previous IMQ
            radial basis function, to recompute a IMQ basis
            recycling the buffers (and distance map if used) from
            the prevoius basis function when the control points
            have changed. If recycling the buffers a new basis
            function is created which uses the (reallocated)
            buffers of the previous basis function. After this function
            has been called the previous basis function should be freed.

Parameters

nPts	Number of control point pairs.
dPts	Destination control points.
sPts	Source control points.
delta	Normalized delta value in range [> 0.0 , < 1.0 ].
prvBasisFn	Previous basis function to be recycled, may be NULL. Arrays will be used for the new basis function. Must be freed after recycling.
mesh	Used to compute constrained distances. If NULL Euclidean distances will be used.
dstErr	Destination error pointer, may be NULL.

References AlcCalloc(), AlcFree(), AlcMalloc(), AlcRealloc(), ALG_MIN, _WlzBasisFn::basis, _AlgMatrix::core, _WlzVertexP::d2, _WlzBasisFn::distFn, _WlzBasisFn::distMap, _WlzCMeshP::m2, _WlzBasisFn::maxVx, _WlzBasisFn::mesh, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::poly, _WlzBasisFn::sVertices, _WlzCMesh2D::type, _WlzBasisFn::type, _WlzVertexP::v, _WlzCMeshP::v, _WlzBasisFn::vertices, WLZ_CMESH_2D, WLZ_ERR_DOMAIN_TYPE, WLZ_ERR_MEM_ALLOC, WLZ_ERR_NONE, WLZ_FN_BASIS_2DIMQ, and WlzGeomCmpVtx2D().

Referenced by WlzBasisFnTPS2DChangeCPtsParam(), and WlzBasisFnTrFromCPts2DParam().

◆ WlzBasisFnIMQ3DFromCPts()

WlzBasisFn* WlzBasisFnIMQ3DFromCPts	(	int	nPts,
		WlzDVertex3 *	dPts,
		WlzDVertex3 *	sPts,
		double	delta,
		WlzBasisFn *	prvBasisFn,
		WlzCMesh3D *	mesh,
		WlzErrorNum *	dstErr
	)

Creates a new inverse-multiquadric basis function.

Returns: New basis function.

Parameters

nPts	Number of control point pairs.
dPts	Destination control points.
sPts	Source control points.
delta	Normalized delta value in range [> 0.0 , < 1.0 ].
prvBasisFn	Previous basis function to be recycled, may be NULL. Arrays will be used for the new basis function. Must be freed after recycling.
mesh	Used to compute constrained distances. If NULL Euclidean distances will be used.
dstErr	Destination error pointer, may be NULL.

References AlcCalloc(), AlcFree(), AlcMalloc(), AlcRealloc(), ALG_MIN, _WlzBasisFn::basis, _AlgMatrix::core, _WlzVertexP::d3, _WlzBasisFn::distFn, _WlzBasisFn::distMap, _WlzCMeshP::m3, _WlzBasisFn::maxVx, _WlzBasisFn::mesh, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::poly, _WlzBasisFn::sVertices, _WlzCMesh3D::type, _WlzBasisFn::type, _WlzVertexP::v, _WlzCMeshP::v, _WlzBasisFn::vertices, WLZ_CMESH_3D, WLZ_ERR_DOMAIN_TYPE, WLZ_ERR_MEM_ALLOC, WLZ_ERR_NONE, WLZ_FN_BASIS_3DIMQ, and WlzGeomCmpVtx3D().

Referenced by WlzBasisFnTrFromCPts3DParam().

◆ WlzBasisFnTPS2DFromCPts()

WlzBasisFn* WlzBasisFnTPS2DFromCPts	(	int	nPts,
		WlzDVertex2 *	dPts,
		WlzDVertex2 *	sPts,
		WlzBasisFn *	prvBasisFn,
		WlzCMesh2D *	mesh,
		WlzErrorNum *	dstErr
	)

Creates a new thin plate spline basis function.

Returns: New basis function.

Parameters

nPts	Number of control point pairs.
dPts	Destination control points.
sPts	Source control points.
prvBasisFn	Previous basis function, which if not NULL, may be used to avoid recomputing values (eg distance map).
mesh	Mesh which is used to compute constrained distances. If non NULL and the mesh type is WLZ_CMESH_2D then constrained distances are used and these are computed using the mesh. If NULL or the transform is some other type then Euclidean distances are used. indexed using the node index.
dstErr	Destination error pointer, may be NULL.

References AlcCalloc(), AlcFree(), AlcMalloc(), AlcRealloc(), ALG_MIN, _WlzBasisFn::basis, _AlgMatrix::core, _WlzVertexP::d2, _WlzBasisFn::distFn, _WlzBasisFn::distMap, _WlzCMeshP::m2, _WlzBasisFn::maxVx, _WlzBasisFn::mesh, _WlzBasisFn::nVtx, _WlzBasisFn::param, _WlzBasisFn::poly, _WlzBasisFn::sVertices, _WlzCMesh2D::type, _WlzBasisFn::type, _WlzVertexP::v, _WlzCMeshP::v, _WlzBasisFn::vertices, WLZ_CMESH_2D, WLZ_ERR_DOMAIN_TYPE, WLZ_ERR_MEM_ALLOC, WLZ_ERR_NONE, WLZ_FN_BASIS_2DTPS, and WlzGeomCmpVtx2D().

Referenced by WlzBasisFnTPS2DChangeCPtsParam(), and WlzBasisFnTrFromCPts2DParam().

◆ WlzBasisFnMOS3DFromCPts()

WlzBasisFn* WlzBasisFnMOS3DFromCPts	(	int	nPts,
		WlzDVertex2 *	dPts,
		WlzDVertex2 *	sPts,
		double *	alpha,
		double *	param,
		WlzErrorNum *	dstErr
	)

Creates a new 3D multi order spline basis function:

\[ \phi(r) = \frac{1}{4 \pi \delta^2 r} (1 - \frac{w}{w - v} e^{- \sqrt{v} r} + \frac{v}{w - v} e^{- \sqrt{w} r}) \]

\[ v = \frac{1 + \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \]

\[ w = \frac{1 - \sqrt{1 - 4 \pi \tau^2 \delta^2}}{2 \tau^2} \]

\[ f(\mathbf{x}) = P(\mathbf{x}) + \sum_{i=1}^{n}{\lambda_i \phi(r_i)} \]

\[ r_i = |\mathbf{x} - \mathbf{x'_i}| \]

The multi order spline may either be an exact interpolating function or one which approximates the given control points. In the approximating case values for the regularization parameters should be given, with the regularization parameters \(\alpha_i\) given by:

\[ H[f] = \beta[f] + \sum_{i=1}^{n}{\frac{(y_i - f(x))^2}{\alpha_i}} \]

Giving the design equation:

\[ \left( \begin{array}{cccccccc} 0 & 0 & 0 & 0 & 1 & 1 & \cdots & 1 \\ 0 & 0 & 0 & 0 & x_1 & x_2 & \cdots & x_n \\ 0 & 0 & 0 & 0 & y_1 & y_2 & \cdots & y_n \\ 0 & 0 & 0 & 0 & z_1 & z_2 & \cdots & z_n \\ 1 & x_1 & y_1 & z_1 & \phi(r_{11}) + \alpha_1 & \phi(r_{12}) & \cdots & \phi(r_{1n}) \\ \dotfill \\ 1 & x_n & y_n & z_n & \phi(r_{n1}) & \phi(r_{n2}) & \cdots & \phi(r_{nn} + \alpha_n) \end{array} \right) \left( \begin{array}{c} a_o \\ a_1 \\ a_2 \\ a_3 \\ \lambda_1 \\ \lambda_1 \\ \vdots \\ \lambda_n \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ f(\mathbf{x_1}) \\ f(\mathbf{x_1}) \\ \vdots \\ f(\mathbf{x_n}) \end{array} \right) \]

The vertex values with components \(x\), \(y \) and \(z\) are used to solve for the four polynomial coefficients and the three basis function weights \(\lambda\), \(\mu\) and \(\nu\). The given values of \(\lambda\) and \(\tau\) are constrained by \(\lambda > 0, \tau > 0\) and \(\lambda \tau < 0.5\).

Returns: New basis function.

Todo:: WlzBasisFnMOS3DFromCPts() has yet to be implemented.

Parameters

nPts	Number of control point pairs.
dPts	Destination control points.
sPts	Source control points.
alpha	Regularization parameter for the control points. If NULL all regularization parameters are set to zero for exact interpolation.
param	Smoothness parameters \(\delta\) and \(\tau\) in that order, but may be NULL in which case default parameter values will be used.
dstErr	Destination error pointer, may be NULL.

References WLZ_ERR_UNIMPLEMENTED.

◆ WlzBasisFnScalarMOS3DFromCPts()

WlzBasisFn* WlzBasisFnScalarMOS3DFromCPts	(	int	nPts,
		WlzDVertex3 *	cPts,
		double *	cVal,
		double *	alpha,
		double *	param,
		WlzErrorNum *	dstErr
	)

Computes a new 3D multi order spline basis function which either interpolates or approximates the given scalar values. See WlzBasisFnMOS3DFromCPts() for details of the multi order spline. The given values of \(\lambda\) and \(\tau\) are constrained by \(\lambda > 0, \tau > 0\) and \(\lambda \tau < 0.5\).

Returns: New basis function.

Parameters

nPts	Number of values.
cPts	Positions of the control points.
cVal	Values at the control points.
alpha	Regularization parameter for the control points. If NULL all regularization parameters are set to zero for exact interpolation.
param	Smoothness parameters \(\delta\) and \(\tau\) in that order, but may be NULL in which case default parameter values will be used.
dstErr	Destination error pointer, may be NULL.

References _AlgMatrix::core, WLZ_ERR_NONE, and WLZ_ERR_PARAM_DATA.

Files

Data Structures

Typedefs

Enumerations

Functions

Detailed Description

Typedef Documentation

◆ Wlz3DProjectionIntFn

◆ WlzBasisEvalFn

◆ WlzBasisDistFn

◆ WlzThreshCbFn

Enumeration Type Documentation

◆ _WlzFnType

Function Documentation

◆ WlzBasisFnFree()

◆ WlzBasisFnValuePoly2D()

◆ WlzBasisFnValueGauss2D()

◆ WlzBasisFnValueMQ2D()

◆ WlzBasisFnValueMQ3D()

◆ WlzBasisFnValueIMQ2D()

◆ WlzBasisFnValueIMQ3D()

◆ WlzBasisFnValueTPS2D()

◆ WlzBasisFnValueConf2D()

◆ WlzBasisFnValueMOS3D()

◆ WlzBasisFnValueScalarMOS3D()

◆ WlzBasisFnValueMOSPhi()

◆ WlzBasisFnGauss2DFromCPts()

◆ WlzBasisFnPoly2DFromCPts()

◆ WlzBasisFnConf2DFromCPts()

◆ WlzBasisFnMQ2DFromCPts()

◆ WlzBasisFnMQ3DFromCPts()

◆ WlzBasisFnIMQ2DFromCPts()

◆ WlzBasisFnIMQ3DFromCPts()

◆ WlzBasisFnTPS2DFromCPts()

◆ WlzBasisFnMOS3DFromCPts()

◆ WlzBasisFnScalarMOS3DFromCPts()