bunwarpj
Class MathTools

java.lang.Object
  bunwarpj.MathTools

public class MathTools
extends java.lang.Object

This class has the math methods to deal with b-splines and images.

Field Summary

private static double	FLT_EPSILON float epsilon
private static int	MAX_SVD_ITERATIONS maximum number of iteration for the Singular Value Decomposition

Constructor Summary

MathTools()

Method Summary

static double[][]	antiSymmetricPadding(double[][] c, int extra)
static double[]	antiSymmetricPadding(double[] c, int n, int extra)
static float[]	antiSymmetricPadding(float[] c, int n, int extra)
static double	Bspline01(double x) B-spline 01.
static double	Bspline02(double x) B-spline 02.
static double	Bspline03(double x) B-spline 03.
static double	EuclideanNorm(double a, double b) Euclidean Norm.
static boolean	invertMatrixSVD(int Ydim, int Xdim, double[][] B, double[][] iB) Invert a matrix by the Singular Value Decomposition method.
static double[]	linearLeastSquares(double[][] A, double[] b) Gives the least-squares solution to (A * x = b) such that (A^T * A)^-1 * A^T * b = x is a vector of size (column), where A is a (line x column) matrix, and where b is a vector of size (line).
static double	nchoosek(int n, int k) N choose K.
static void	QRdecomposition(double[][] Q, double[][] R) Decomposes the (line x column) input matrix Q into an orthonormal output matrix Q of same size (line x column) and an upper-diagonal square matrix R of size (column x column), such that the matrix product (Q * R) gives the input matrix, and such that the matrix product (Q^T * Q) gives the identity.
static void	showMatrix(int Ydim, int Xdim, double[][] A) Method to display the matrix in the command line.
static void	singularValueBackSubstitution(double[][] U, double[] W, double[][] V, double[] B, double[] X) solve (U.W.Transpose(V)).X == B in terms of X {U, W, V} are given by SingularValueDecomposition by convention, set w[i,j]=0 to get (1/w[i,j])=0 the size of the input matrix U is (Lines x Columns) the size of the vector (1/W) of singular values is (Columns) the size of the untransposed orthogonal matrix V is (Columns x Columns) the size of the input vector B is (Lines) the size of the output vector X is (Columns)
static void	singularValueDecomposition(double[][] U, double[] W, double[][] V) Singular Value Decomposition.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

FLT_EPSILON

private static final double FLT_EPSILON

float epsilon

MAX_SVD_ITERATIONS

private static final int MAX_SVD_ITERATIONS

maximum number of iteration for the Singular Value Decomposition

See Also:

Constant Field Values

Constructor Detail

MathTools

public MathTools()

Method Detail

Bspline01

public static double Bspline01(double x)

B-spline 01.

Parameters:

x -

Bspline02

public static double Bspline02(double x)

B-spline 02.

Parameters:

x -

Bspline03

public static double Bspline03(double x)

B-spline 03.

Parameters:

x -

EuclideanNorm

public static double EuclideanNorm(double a,
                                   double b)

Euclidean Norm.

Parameters:

a -

b -

invertMatrixSVD

public static boolean invertMatrixSVD(int Ydim,
                                      int Xdim,
                                      double[][] B,
                                      double[][] iB)

Invert a matrix by the Singular Value Decomposition method.

Parameters:

Ydim - input, Y-dimension

Xdim - input, X-dimension

B - input, matrix to invert

iB - output, inverted matrix

Returns:

under-constrained flag

linearLeastSquares

public static double[] linearLeastSquares(double[][] A,
                                          double[] b)

Gives the least-squares solution to (A * x = b) such that (A^T * A)^-1 * A^T * b = x is a vector of size (column), where A is a (line x column) matrix, and where b is a vector of size (line). The result may differ from that obtained by a singular-value decomposition in the cases where the least-squares solution is not uniquely defined (SVD returns the solution of least norm, not QR).

Parameters:

A - An input matrix A[line][column] of size (line x column)

b - An input vector b[line] of size (line)

Returns:

An output vector x[column] of size (column)

nchoosek

public static double nchoosek(int n,
                              int k)

N choose K.

Parameters:

n -

k -

QRdecomposition

public static void QRdecomposition(double[][] Q,
                                   double[][] R)

Decomposes the (line x column) input matrix Q into an orthonormal output matrix Q of same size (line x column) and an upper-diagonal square matrix R of size (column x column), such that the matrix product (Q * R) gives the input matrix, and such that the matrix product (Q^T * Q) gives the identity.

Parameters:

Q - An in-place (line x column) matrix Q[line][column], which expects as input the matrix to decompose, and which returns as output an orthonormal matrix

R - An output (column x column) square matrix R[column][column]

showMatrix

public static void showMatrix(int Ydim,
                              int Xdim,
                              double[][] A)

Method to display the matrix in the command line.

Parameters:

Ydim - Y-dimension

Xdim - X-dimension

A - matrix to display

singularValueDecomposition

public static void singularValueDecomposition(double[][] U,
                                              double[] W,
                                              double[][] V)

Singular Value Decomposition.

Parameters:

U - input matrix

W - vector of singular values

V - untransposed orthogonal matrix

singularValueBackSubstitution

public static void singularValueBackSubstitution(double[][] U,
                                                 double[] W,
                                                 double[][] V,
                                                 double[] B,
                                                 double[] X)

solve (U.W.Transpose(V)).X == B in terms of X {U, W, V} are given by SingularValueDecomposition by convention, set w[i,j]=0 to get (1/w[i,j])=0 the size of the input matrix U is (Lines x Columns) the size of the vector (1/W) of singular values is (Columns) the size of the untransposed orthogonal matrix V is (Columns x Columns) the size of the input vector B is (Lines) the size of the output vector X is (Columns)

Parameters:

U - input matrix

W - vector of singular values

V - untransposed orthogonal matrix

B - input vector

X - returned solution

antiSymmetricPadding

public static double[][] antiSymmetricPadding(double[][] c,
                                              int extra)

antiSymmetricPadding

public static double[] antiSymmetricPadding(double[] c,
                                            int n,
                                            int extra)

antiSymmetricPadding

public static float[] antiSymmetricPadding(float[] c,
                                           int n,
                                           int extra)