| /* |
| * 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.math.analysis.interpolation; |
| |
| import org.apache.commons.math.DimensionMismatchException; |
| import org.apache.commons.math.FunctionEvaluationException; |
| import org.apache.commons.math.analysis.BivariateRealFunction; |
| import org.apache.commons.math.exception.NoDataException; |
| import org.apache.commons.math.exception.OutOfRangeException; |
| import org.apache.commons.math.util.MathUtils; |
| |
| /** |
| * Function that implements the |
| * <a href="http://en.wikipedia.org/wiki/Bicubic_interpolation"> |
| * bicubic spline interpolation</a>. |
| * |
| * @version $Revision$ $Date$ |
| * @since 2.1 |
| */ |
| public class BicubicSplineInterpolatingFunction |
| implements BivariateRealFunction { |
| /** |
| * 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 BivariateRealFunction[][][] 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 org.apache.commons.math.exception.NonMonotonousSequenceException |
| * 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 { |
| 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); |
| } |
| |
| MathUtils.checkOrder(x); |
| MathUtils.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) { |
| 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. |
| * @since 2.2 |
| */ |
| public double partialDerivativeX(double x, double y) { |
| 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. |
| * @since 2.2 |
| */ |
| public double partialDerivativeY(double x, double y) { |
| 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. |
| * @since 2.2 |
| */ |
| public double partialDerivativeXX(double x, double y) { |
| 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. |
| * @since 2.2 |
| */ |
| public double partialDerivativeYY(double x, double y) { |
| 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. |
| * @since 2.2 |
| */ |
| public double partialDerivativeXY(double x, double y) { |
| 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 FunctionEvaluationException |
| */ |
| private double partialDerivative(int which, double x, double y) { |
| 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]); |
| |
| try { |
| return partialDerivatives[which][i][j].value(xN, yN); |
| } catch (FunctionEvaluationException fee) { |
| // this should never happen |
| throw new RuntimeException(fee); |
| } |
| |
| } |
| |
| /** |
| * Compute all partial derivatives. |
| */ |
| private void computePartialDerivatives() { |
| final int lastI = xval.length - 1; |
| final int lastJ = yval.length - 1; |
| partialDerivatives = new BivariateRealFunction[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 end 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: |
| * <ul> |
| * <li>f(0,0)</li> |
| * <li>f(1,0)</li> |
| * <li>f(0,1)</li> |
| * <li>f(1,1)</li> |
| * <li>f<sub>x</sub>(0,0)</li> |
| * <li>f<sub>x</sub>(1,0)</li> |
| * <li>f<sub>x</sub>(0,1)</li> |
| * <li>f<sub>x</sub>(1,1)</li> |
| * <li>f<sub>y</sub>(0,0)</li> |
| * <li>f<sub>y</sub>(1,0)</li> |
| * <li>f<sub>y</sub>(0,1)</li> |
| * <li>f<sub>y</sub>(1,1)</li> |
| * <li>f<sub>xy</sub>(0,0)</li> |
| * <li>f<sub>xy</sub>(1,0)</li> |
| * <li>f<sub>xy</sub>(0,1)</li> |
| * <li>f<sub>xy</sub>(1,1)</li> |
| * </ul> |
| * 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 $Revision$ $Date$ |
| */ |
| class BicubicSplineFunction |
| implements BivariateRealFunction { |
| |
| /** Number of points. */ |
| private static final short N = 4; |
| |
| /** Coefficients */ |
| private final double[][] a; |
| |
| /** First partial derivative along x. */ |
| private BivariateRealFunction partialDerivativeX; |
| |
| /** First partial derivative along y. */ |
| private BivariateRealFunction partialDerivativeY; |
| |
| /** Second partial derivative along x. */ |
| private BivariateRealFunction partialDerivativeXX; |
| |
| /** Second partial derivative along y. */ |
| private BivariateRealFunction partialDerivativeYY; |
| |
| /** Second crossed partial derivative. */ |
| private BivariateRealFunction 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 BivariateRealFunction partialDerivativeX() { |
| if (partialDerivativeX == null) { |
| computePartialDerivatives(); |
| } |
| |
| return partialDerivativeX; |
| } |
| /** |
| * @return the partial derivative wrt {@code y}. |
| */ |
| public BivariateRealFunction partialDerivativeY() { |
| if (partialDerivativeY == null) { |
| computePartialDerivatives(); |
| } |
| |
| return partialDerivativeY; |
| } |
| /** |
| * @return the second partial derivative wrt {@code x}. |
| */ |
| public BivariateRealFunction partialDerivativeXX() { |
| if (partialDerivativeXX == null) { |
| computePartialDerivatives(); |
| } |
| |
| return partialDerivativeXX; |
| } |
| /** |
| * @return the second partial derivative wrt {@code y}. |
| */ |
| public BivariateRealFunction partialDerivativeYY() { |
| if (partialDerivativeYY == null) { |
| computePartialDerivatives(); |
| } |
| |
| return partialDerivativeYY; |
| } |
| /** |
| * @return the second partial cross-derivative. |
| */ |
| public BivariateRealFunction 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 BivariateRealFunction() { |
| 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 BivariateRealFunction() { |
| 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 BivariateRealFunction() { |
| 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 BivariateRealFunction() { |
| 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 BivariateRealFunction() { |
| 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); |
| } |
| }; |
| } |
| } |