math/transform/FastCosineTransformer.java

/*
 * 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.transform;

import org.apache.commons.math.FunctionEvaluationException;
import org.apache.commons.math.MathRuntimeException;
import org.apache.commons.math.analysis.UnivariateRealFunction;
import org.apache.commons.math.complex.Complex;
import org.apache.commons.math.exception.util.LocalizedFormats;
import org.apache.commons.math.util.FastMath;

/**
 * Implements the <a href="http://documents.wolfram.com/v5/Add-onsLinks/
 * StandardPackages/LinearAlgebra/FourierTrig.html">Fast Cosine Transform</a>
 * for transformation of one-dimensional data sets. For reference, see
 * <b>Fast Fourier Transforms</b>, ISBN 0849371635, chapter 3.
 * <p>
 * FCT is its own inverse, up to a multiplier depending on conventions.
 * The equations are listed in the comments of the corresponding methods.</p>
 * <p>
 * Different from FFT and FST, FCT requires the length of data set to be
 * power of 2 plus one. Users should especially pay attention to the
 * function transformation on how this affects the sampling.</p>
 * <p>As of version 2.0 this no longer implements Serializable</p>
 *
 * @version $Revision:670469 $ $Date:2008-06-23 10:01:38 +0200 (lun., 23 juin 2008) $
 * @since 1.2
 */
public class FastCosineTransformer implements RealTransformer {

    /**
     * Construct a default transformer.
     */
    public FastCosineTransformer() {
        super();
    }

    /**
     * Transform the given real data set.
     * <p>
     * The formula is F<sub>n</sub> = (1/2) [f<sub>0</sub> + (-1)<sup>n</sup> f<sub>N</sub>] +
     *                        &sum;<sub>k=1</sub><sup>N-1</sup> f<sub>k</sub> cos(&pi; nk/N)
     * </p>
     *
     * @param f the real data array to be transformed
     * @return the real transformed array
     * @throws IllegalArgumentException if any parameters are invalid
     */
    public double[] transform(double f[]) throws IllegalArgumentException {
        return fct(f);
    }

    /**
     * Transform the given real function, sampled on the given interval.
     * <p>
     * The formula is F<sub>n</sub> = (1/2) [f<sub>0</sub> + (-1)<sup>n</sup> f<sub>N</sub>] +
     *                        &sum;<sub>k=1</sub><sup>N-1</sup> f<sub>k</sub> cos(&pi; nk/N)
     * </p>
     *
     * @param f the function to be sampled and transformed
     * @param min the lower bound for the interval
     * @param max the upper bound for the interval
     * @param n the number of sample points
     * @return the real transformed array
     * @throws FunctionEvaluationException if function cannot be evaluated
     * at some point
     * @throws IllegalArgumentException if any parameters are invalid
     */
    public double[] transform(UnivariateRealFunction f,
                              double min, double max, int n)
        throws FunctionEvaluationException, IllegalArgumentException {
        double data[] = FastFourierTransformer.sample(f, min, max, n);
        return fct(data);
    }

    /**
     * Transform the given real data set.
     * <p>
     * The formula is F<sub>n</sub> = &radic;(1/2N) [f<sub>0</sub> + (-1)<sup>n</sup> f<sub>N</sub>] +
     *                        &radic;(2/N) &sum;<sub>k=1</sub><sup>N-1</sup> f<sub>k</sub> cos(&pi; nk/N)
     * </p>
     *
     * @param f the real data array to be transformed
     * @return the real transformed array
     * @throws IllegalArgumentException if any parameters are invalid
     */
    public double[] transform2(double f[]) throws IllegalArgumentException {

        double scaling_coefficient = FastMath.sqrt(2.0 / (f.length-1));
        return FastFourierTransformer.scaleArray(fct(f), scaling_coefficient);
    }

    /**
     * Transform the given real function, sampled on the given interval.
     * <p>
     * The formula is F<sub>n</sub> = &radic;(1/2N) [f<sub>0</sub> + (-1)<sup>n</sup> f<sub>N</sub>] +
     *                        &radic;(2/N) &sum;<sub>k=1</sub><sup>N-1</sup> f<sub>k</sub> cos(&pi; nk/N)
     *
     * </p>
     *
     * @param f the function to be sampled and transformed
     * @param min the lower bound for the interval
     * @param max the upper bound for the interval
     * @param n the number of sample points
     * @return the real transformed array
     * @throws FunctionEvaluationException if function cannot be evaluated
     * at some point
     * @throws IllegalArgumentException if any parameters are invalid
     */
    public double[] transform2(UnivariateRealFunction f,
                               double min, double max, int n)
        throws FunctionEvaluationException, IllegalArgumentException {

        double data[] = FastFourierTransformer.sample(f, min, max, n);
        double scaling_coefficient = FastMath.sqrt(2.0 / (n-1));
        return FastFourierTransformer.scaleArray(fct(data), scaling_coefficient);
    }

    /**
     * Inversely transform the given real data set.
     * <p>
     * The formula is f<sub>k</sub> = (1/N) [F<sub>0</sub> + (-1)<sup>k</sup> F<sub>N</sub>] +
     *                        (2/N) &sum;<sub>n=1</sub><sup>N-1</sup> F<sub>n</sub> cos(&pi; nk/N)
     * </p>
     *
     * @param f the real data array to be inversely transformed
     * @return the real inversely transformed array
     * @throws IllegalArgumentException if any parameters are invalid
     */
    public double[] inversetransform(double f[]) throws IllegalArgumentException {

        double scaling_coefficient = 2.0 / (f.length - 1);
        return FastFourierTransformer.scaleArray(fct(f), scaling_coefficient);
    }

    /**
     * Inversely transform the given real function, sampled on the given interval.
     * <p>
     * The formula is f<sub>k</sub> = (1/N) [F<sub>0</sub> + (-1)<sup>k</sup> F<sub>N</sub>] +
     *                        (2/N) &sum;<sub>n=1</sub><sup>N-1</sup> F<sub>n</sub> cos(&pi; nk/N)
     * </p>
     *
     * @param f the function to be sampled and inversely transformed
     * @param min the lower bound for the interval
     * @param max the upper bound for the interval
     * @param n the number of sample points
     * @return the real inversely transformed array
     * @throws FunctionEvaluationException if function cannot be evaluated at some point
     * @throws IllegalArgumentException if any parameters are invalid
     */
    public double[] inversetransform(UnivariateRealFunction f,
                                     double min, double max, int n)
        throws FunctionEvaluationException, IllegalArgumentException {

        double data[] = FastFourierTransformer.sample(f, min, max, n);
        double scaling_coefficient = 2.0 / (n - 1);
        return FastFourierTransformer.scaleArray(fct(data), scaling_coefficient);
    }

    /**
     * Inversely transform the given real data set.
     * <p>
     * The formula is f<sub>k</sub> = &radic;(1/2N) [F<sub>0</sub> + (-1)<sup>k</sup> F<sub>N</sub>] +
     *                        &radic;(2/N) &sum;<sub>n=1</sub><sup>N-1</sup> F<sub>n</sub> cos(&pi; nk/N)
     * </p>
     *
     * @param f the real data array to be inversely transformed
     * @return the real inversely transformed array
     * @throws IllegalArgumentException if any parameters are invalid
     */
    public double[] inversetransform2(double f[]) throws IllegalArgumentException {
        return transform2(f);
    }

    /**
     * Inversely transform the given real function, sampled on the given interval.
     * <p>
     * The formula is f<sub>k</sub> = &radic;(1/2N) [F<sub>0</sub> + (-1)<sup>k</sup> F<sub>N</sub>] +
     *                        &radic;(2/N) &sum;<sub>n=1</sub><sup>N-1</sup> F<sub>n</sub> cos(&pi; nk/N)
     * </p>
     *
     * @param f the function to be sampled and inversely transformed
     * @param min the lower bound for the interval
     * @param max the upper bound for the interval
     * @param n the number of sample points
     * @return the real inversely transformed array
     * @throws FunctionEvaluationException if function cannot be evaluated at some point
     * @throws IllegalArgumentException if any parameters are invalid
     */
    public double[] inversetransform2(UnivariateRealFunction f,
                                      double min, double max, int n)
        throws FunctionEvaluationException, IllegalArgumentException {

        return transform2(f, min, max, n);
    }

    /**
     * Perform the FCT algorithm (including inverse).
     *
     * @param f the real data array to be transformed
     * @return the real transformed array
     * @throws IllegalArgumentException if any parameters are invalid
     */
    protected double[] fct(double f[])
        throws IllegalArgumentException {

        final double transformed[] = new double[f.length];

        final int n = f.length - 1;
        if (!FastFourierTransformer.isPowerOf2(n)) {
            throw MathRuntimeException.createIllegalArgumentException(
                    LocalizedFormats.NOT_POWER_OF_TWO_PLUS_ONE,
                    f.length);
        }
        if (n == 1) {       // trivial case
            transformed[0] = 0.5 * (f[0] + f[1]);
            transformed[1] = 0.5 * (f[0] - f[1]);
            return transformed;
        }

        // construct a new array and perform FFT on it
        final double[] x = new double[n];
        x[0] = 0.5 * (f[0] + f[n]);
        x[n >> 1] = f[n >> 1];
        double t1 = 0.5 * (f[0] - f[n]);   // temporary variable for transformed[1]
        for (int i = 1; i < (n >> 1); i++) {
            final double a = 0.5 * (f[i] + f[n-i]);
            final double b = FastMath.sin(i * FastMath.PI / n) * (f[i] - f[n-i]);
            final double c = FastMath.cos(i * FastMath.PI / n) * (f[i] - f[n-i]);
            x[i] = a - b;
            x[n-i] = a + b;
            t1 += c;
        }
        FastFourierTransformer transformer = new FastFourierTransformer();
        Complex y[] = transformer.transform(x);

        // reconstruct the FCT result for the original array
        transformed[0] = y[0].getReal();
        transformed[1] = t1;
        for (int i = 1; i < (n >> 1); i++) {
            transformed[2 * i]     = y[i].getReal();
            transformed[2 * i + 1] = transformed[2 * i - 1] - y[i].getImaginary();
        }
        transformed[n] = y[n >> 1].getReal();

        return transformed;
    }
}