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/* * 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 * <a href="http://en.wikipedia.org/wiki/Bicubic_interpolation"> * bicubic spline interpolation</a>. * * @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: * <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 $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); } }; } }