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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.analysis.interpolation;
import org.apache.commons.math3.analysis.BivariateFunction;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.NoDataException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.NonMonotonicSequenceException;
import org.apache.commons.math3.util.MathArrays;
/**
* Function that implements the
*
* bicubic spline interpolation.
*
* @since 2.1
* @version $Id: BicubicSplineInterpolatingFunction.java 1379904 2012-09-01 23:54:52Z erans $
*/
public class BicubicSplineInterpolatingFunction
implements BivariateFunction {
/**
* Matrix to compute the spline coefficients from the function values
* and function derivatives values
*/
private static final double[][] AINV = {
{ 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
{ 0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0 },
{ -3,3,0,0,-2,-1,0,0,0,0,0,0,0,0,0,0 },
{ 2,-2,0,0,1,1,0,0,0,0,0,0,0,0,0,0 },
{ 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0 },
{ 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0 },
{ 0,0,0,0,0,0,0,0,-3,3,0,0,-2,-1,0,0 },
{ 0,0,0,0,0,0,0,0,2,-2,0,0,1,1,0,0 },
{ -3,0,3,0,0,0,0,0,-2,0,-1,0,0,0,0,0 },
{ 0,0,0,0,-3,0,3,0,0,0,0,0,-2,0,-1,0 },
{ 9,-9,-9,9,6,3,-6,-3,6,-6,3,-3,4,2,2,1 },
{ -6,6,6,-6,-3,-3,3,3,-4,4,-2,2,-2,-2,-1,-1 },
{ 2,0,-2,0,0,0,0,0,1,0,1,0,0,0,0,0 },
{ 0,0,0,0,2,0,-2,0,0,0,0,0,1,0,1,0 },
{ -6,6,6,-6,-4,-2,4,2,-3,3,-3,3,-2,-1,-2,-1 },
{ 4,-4,-4,4,2,2,-2,-2,2,-2,2,-2,1,1,1,1 }
};
/** Samples x-coordinates */
private final double[] xval;
/** Samples y-coordinates */
private final double[] yval;
/** Set of cubic splines patching the whole data grid */
private final BicubicSplineFunction[][] splines;
/**
* Partial derivatives
* The value of the first index determines the kind of derivatives:
* 0 = first partial derivatives wrt x
* 1 = first partial derivatives wrt y
* 2 = second partial derivatives wrt x
* 3 = second partial derivatives wrt y
* 4 = cross partial derivatives
*/
private BivariateFunction[][][] partialDerivatives = null;
/**
* @param x Sample values of the x-coordinate, in increasing order.
* @param y Sample values of the y-coordinate, in increasing order.
* @param f Values of the function on every grid point.
* @param dFdX Values of the partial derivative of function with respect
* to x on every grid point.
* @param dFdY Values of the partial derivative of function with respect
* to y on every grid point.
* @param d2FdXdY Values of the cross partial derivative of function on
* every grid point.
* @throws DimensionMismatchException if the various arrays do not contain
* the expected number of elements.
* @throws NonMonotonicSequenceException if {@code x} or {@code y} are
* not strictly increasing.
* @throws NoDataException if any of the arrays has zero length.
*/
public BicubicSplineInterpolatingFunction(double[] x,
double[] y,
double[][] f,
double[][] dFdX,
double[][] dFdY,
double[][] d2FdXdY)
throws DimensionMismatchException,
NoDataException,
NonMonotonicSequenceException {
final int xLen = x.length;
final int yLen = y.length;
if (xLen == 0 || yLen == 0 || f.length == 0 || f[0].length == 0) {
throw new NoDataException();
}
if (xLen != f.length) {
throw new DimensionMismatchException(xLen, f.length);
}
if (xLen != dFdX.length) {
throw new DimensionMismatchException(xLen, dFdX.length);
}
if (xLen != dFdY.length) {
throw new DimensionMismatchException(xLen, dFdY.length);
}
if (xLen != d2FdXdY.length) {
throw new DimensionMismatchException(xLen, d2FdXdY.length);
}
MathArrays.checkOrder(x);
MathArrays.checkOrder(y);
xval = x.clone();
yval = y.clone();
final int lastI = xLen - 1;
final int lastJ = yLen - 1;
splines = new BicubicSplineFunction[lastI][lastJ];
for (int i = 0; i < lastI; i++) {
if (f[i].length != yLen) {
throw new DimensionMismatchException(f[i].length, yLen);
}
if (dFdX[i].length != yLen) {
throw new DimensionMismatchException(dFdX[i].length, yLen);
}
if (dFdY[i].length != yLen) {
throw new DimensionMismatchException(dFdY[i].length, yLen);
}
if (d2FdXdY[i].length != yLen) {
throw new DimensionMismatchException(d2FdXdY[i].length, yLen);
}
final int ip1 = i + 1;
for (int j = 0; j < lastJ; j++) {
final int jp1 = j + 1;
final double[] beta = new double[] {
f[i][j], f[ip1][j], f[i][jp1], f[ip1][jp1],
dFdX[i][j], dFdX[ip1][j], dFdX[i][jp1], dFdX[ip1][jp1],
dFdY[i][j], dFdY[ip1][j], dFdY[i][jp1], dFdY[ip1][jp1],
d2FdXdY[i][j], d2FdXdY[ip1][j], d2FdXdY[i][jp1], d2FdXdY[ip1][jp1]
};
splines[i][j] = new BicubicSplineFunction(computeSplineCoefficients(beta));
}
}
}
/**
* {@inheritDoc}
*/
public double value(double x, double y)
throws OutOfRangeException {
final int i = searchIndex(x, xval);
if (i == -1) {
throw new OutOfRangeException(x, xval[0], xval[xval.length - 1]);
}
final int j = searchIndex(y, yval);
if (j == -1) {
throw new OutOfRangeException(y, yval[0], yval[yval.length - 1]);
}
final double xN = (x - xval[i]) / (xval[i + 1] - xval[i]);
final double yN = (y - yval[j]) / (yval[j + 1] - yval[j]);
return splines[i][j].value(xN, yN);
}
/**
* @param x x-coordinate.
* @param y y-coordinate.
* @return the value at point (x, y) of the first partial derivative with
* respect to x.
* @throws OutOfRangeException if {@code x} (resp. {@code y}) is outside
* the range defined by the boundary values of {@code xval} (resp.
* {@code yval}).
*/
public double partialDerivativeX(double x, double y)
throws OutOfRangeException {
return partialDerivative(0, x, y);
}
/**
* @param x x-coordinate.
* @param y y-coordinate.
* @return the value at point (x, y) of the first partial derivative with
* respect to y.
* @throws OutOfRangeException if {@code x} (resp. {@code y}) is outside
* the range defined by the boundary values of {@code xval} (resp.
* {@code yval}).
*/
public double partialDerivativeY(double x, double y)
throws OutOfRangeException {
return partialDerivative(1, x, y);
}
/**
* @param x x-coordinate.
* @param y y-coordinate.
* @return the value at point (x, y) of the second partial derivative with
* respect to x.
* @throws OutOfRangeException if {@code x} (resp. {@code y}) is outside
* the range defined by the boundary values of {@code xval} (resp.
* {@code yval}).
*/
public double partialDerivativeXX(double x, double y)
throws OutOfRangeException {
return partialDerivative(2, x, y);
}
/**
* @param x x-coordinate.
* @param y y-coordinate.
* @return the value at point (x, y) of the second partial derivative with
* respect to y.
* @throws OutOfRangeException if {@code x} (resp. {@code y}) is outside
* the range defined by the boundary values of {@code xval} (resp.
* {@code yval}).
*/
public double partialDerivativeYY(double x, double y)
throws OutOfRangeException {
return partialDerivative(3, x, y);
}
/**
* @param x x-coordinate.
* @param y y-coordinate.
* @return the value at point (x, y) of the second partial cross-derivative.
* @throws OutOfRangeException if {@code x} (resp. {@code y}) is outside
* the range defined by the boundary values of {@code xval} (resp.
* {@code yval}).
*/
public double partialDerivativeXY(double x, double y)
throws OutOfRangeException {
return partialDerivative(4, x, y);
}
/**
* @param which First index in {@link #partialDerivatives}.
* @param x x-coordinate.
* @param y y-coordinate.
* @return the value at point (x, y) of the selected partial derivative.
* @throws OutOfRangeException if {@code x} (resp. {@code y}) is outside
* the range defined by the boundary values of {@code xval} (resp.
* {@code yval}).
*/
private double partialDerivative(int which, double x, double y)
throws OutOfRangeException {
if (partialDerivatives == null) {
computePartialDerivatives();
}
final int i = searchIndex(x, xval);
if (i == -1) {
throw new OutOfRangeException(x, xval[0], xval[xval.length - 1]);
}
final int j = searchIndex(y, yval);
if (j == -1) {
throw new OutOfRangeException(y, yval[0], yval[yval.length - 1]);
}
final double xN = (x - xval[i]) / (xval[i + 1] - xval[i]);
final double yN = (y - yval[j]) / (yval[j + 1] - yval[j]);
return partialDerivatives[which][i][j].value(xN, yN);
}
/**
* Compute all partial derivatives.
*/
private void computePartialDerivatives() {
final int lastI = xval.length - 1;
final int lastJ = yval.length - 1;
partialDerivatives = new BivariateFunction[5][lastI][lastJ];
for (int i = 0; i < lastI; i++) {
for (int j = 0; j < lastJ; j++) {
final BicubicSplineFunction f = splines[i][j];
partialDerivatives[0][i][j] = f.partialDerivativeX();
partialDerivatives[1][i][j] = f.partialDerivativeY();
partialDerivatives[2][i][j] = f.partialDerivativeXX();
partialDerivatives[3][i][j] = f.partialDerivativeYY();
partialDerivatives[4][i][j] = f.partialDerivativeXY();
}
}
}
/**
* @param c Coordinate.
* @param val Coordinate samples.
* @return the index in {@code val} corresponding to the interval
* containing {@code c}, or {@code -1} if {@code c} is out of the
* range defined by the boundary values of {@code val}.
*/
private int searchIndex(double c, double[] val) {
if (c < val[0]) {
return -1;
}
final int max = val.length;
for (int i = 1; i < max; i++) {
if (c <= val[i]) {
return i - 1;
}
}
return -1;
}
/**
* Compute the spline coefficients from the list of function values and
* function partial derivatives values at the four corners of a grid
* element. They must be specified in the following order:
*
* - f(0,0)
* - f(1,0)
* - f(0,1)
* - f(1,1)
* - fx(0,0)
* - fx(1,0)
* - fx(0,1)
* - fx(1,1)
* - fy(0,0)
* - fy(1,0)
* - fy(0,1)
* - fy(1,1)
* - fxy(0,0)
* - fxy(1,0)
* - fxy(0,1)
* - fxy(1,1)
*
* where the subscripts indicate the partial derivative with respect to
* the corresponding variable(s).
*
* @param beta List of function values and function partial derivatives
* values.
* @return the spline coefficients.
*/
private double[] computeSplineCoefficients(double[] beta) {
final double[] a = new double[16];
for (int i = 0; i < 16; i++) {
double result = 0;
final double[] row = AINV[i];
for (int j = 0; j < 16; j++) {
result += row[j] * beta[j];
}
a[i] = result;
}
return a;
}
}
/**
* 2D-spline function.
*
* @version $Id: BicubicSplineInterpolatingFunction.java 1379904 2012-09-01 23:54:52Z erans $
*/
class BicubicSplineFunction
implements BivariateFunction {
/** Number of points. */
private static final short N = 4;
/** Coefficients */
private final double[][] a;
/** First partial derivative along x. */
private BivariateFunction partialDerivativeX;
/** First partial derivative along y. */
private BivariateFunction partialDerivativeY;
/** Second partial derivative along x. */
private BivariateFunction partialDerivativeXX;
/** Second partial derivative along y. */
private BivariateFunction partialDerivativeYY;
/** Second crossed partial derivative. */
private BivariateFunction partialDerivativeXY;
/**
* Simple constructor.
* @param a Spline coefficients
*/
public BicubicSplineFunction(double[] a) {
this.a = new double[N][N];
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++) {
this.a[i][j] = a[i + N * j];
}
}
}
/**
* {@inheritDoc}
*/
public double value(double x, double y) {
if (x < 0 || x > 1) {
throw new OutOfRangeException(x, 0, 1);
}
if (y < 0 || y > 1) {
throw new OutOfRangeException(y, 0, 1);
}
final double x2 = x * x;
final double x3 = x2 * x;
final double[] pX = {1, x, x2, x3};
final double y2 = y * y;
final double y3 = y2 * y;
final double[] pY = {1, y, y2, y3};
return apply(pX, pY, a);
}
/**
* Compute the value of the bicubic polynomial.
*
* @param pX Powers of the x-coordinate.
* @param pY Powers of the y-coordinate.
* @param coeff Spline coefficients.
* @return the interpolated value.
*/
private double apply(double[] pX, double[] pY, double[][] coeff) {
double result = 0;
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++) {
result += coeff[i][j] * pX[i] * pY[j];
}
}
return result;
}
/**
* @return the partial derivative wrt {@code x}.
*/
public BivariateFunction partialDerivativeX() {
if (partialDerivativeX == null) {
computePartialDerivatives();
}
return partialDerivativeX;
}
/**
* @return the partial derivative wrt {@code y}.
*/
public BivariateFunction partialDerivativeY() {
if (partialDerivativeY == null) {
computePartialDerivatives();
}
return partialDerivativeY;
}
/**
* @return the second partial derivative wrt {@code x}.
*/
public BivariateFunction partialDerivativeXX() {
if (partialDerivativeXX == null) {
computePartialDerivatives();
}
return partialDerivativeXX;
}
/**
* @return the second partial derivative wrt {@code y}.
*/
public BivariateFunction partialDerivativeYY() {
if (partialDerivativeYY == null) {
computePartialDerivatives();
}
return partialDerivativeYY;
}
/**
* @return the second partial cross-derivative.
*/
public BivariateFunction partialDerivativeXY() {
if (partialDerivativeXY == null) {
computePartialDerivatives();
}
return partialDerivativeXY;
}
/**
* Compute all partial derivatives functions.
*/
private void computePartialDerivatives() {
final double[][] aX = new double[N][N];
final double[][] aY = new double[N][N];
final double[][] aXX = new double[N][N];
final double[][] aYY = new double[N][N];
final double[][] aXY = new double[N][N];
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++) {
final double c = a[i][j];
aX[i][j] = i * c;
aY[i][j] = j * c;
aXX[i][j] = (i - 1) * aX[i][j];
aYY[i][j] = (j - 1) * aY[i][j];
aXY[i][j] = j * aX[i][j];
}
}
partialDerivativeX = new BivariateFunction() {
public double value(double x, double y) {
final double x2 = x * x;
final double[] pX = {0, 1, x, x2};
final double y2 = y * y;
final double y3 = y2 * y;
final double[] pY = {1, y, y2, y3};
return apply(pX, pY, aX);
}
};
partialDerivativeY = new BivariateFunction() {
public double value(double x, double y) {
final double x2 = x * x;
final double x3 = x2 * x;
final double[] pX = {1, x, x2, x3};
final double y2 = y * y;
final double[] pY = {0, 1, y, y2};
return apply(pX, pY, aY);
}
};
partialDerivativeXX = new BivariateFunction() {
public double value(double x, double y) {
final double[] pX = {0, 0, 1, x};
final double y2 = y * y;
final double y3 = y2 * y;
final double[] pY = {1, y, y2, y3};
return apply(pX, pY, aXX);
}
};
partialDerivativeYY = new BivariateFunction() {
public double value(double x, double y) {
final double x2 = x * x;
final double x3 = x2 * x;
final double[] pX = {1, x, x2, x3};
final double[] pY = {0, 0, 1, y};
return apply(pX, pY, aYY);
}
};
partialDerivativeXY = new BivariateFunction() {
public double value(double x, double y) {
final double x2 = x * x;
final double[] pX = {0, 1, x, x2};
final double y2 = y * y;
final double[] pY = {0, 1, y, y2};
return apply(pX, pY, aXY);
}
};
}
}