MathTools (Fiji Javadocs 0-SNAPSHOT API)

java.lang.Object
- bunwarpj.MathTools

```
public class MathTools
extends Object
```
This class has the math methods to deal with b-splines and images.

Constructor Summary

Constructors
Constructor and Description

MathTools()

Constructors
Constructor and Description
`MathTools()`

Method Summary

All Methods Static Methods Concrete Methods
Modifier and Type	Method and Description
`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

- 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)
```

Class MathTools

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

MathTools

Method Detail

Bspline01

Bspline02

Bspline03

EuclideanNorm

invertMatrixSVD

linearLeastSquares

nchoosek

QRdecomposition

showMatrix

singularValueDecomposition

singularValueBackSubstitution

antiSymmetricPadding

antiSymmetricPadding

antiSymmetricPadding