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.math3.transform;

import java.io.Serializable;

import org.apache.commons.math3.analysis.FunctionUtils;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.complex.Complex;
import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.ArithmeticUtils;
import org.apache.commons.math3.util.FastMath;

/**
 * Implements the Fast Cosine Transform for transformation of one-dimensional
 * real data sets. For reference, see James S. Walker, <em>Fast Fourier
 * Transforms</em>, chapter 3 (ISBN 0849371635).
 * <p>
 * There are several variants of the discrete cosine transform. The present
 * implementation corresponds to DCT-I, with various normalization conventions,
 * which are specified by the parameter {@link DctNormalization}.
 * <p>
 * DCT-I is equivalent to DFT of an <em>even extension</em> of the data series.
 * More precisely, if x<sub>0</sub>, &hellip;, x<sub>N-1</sub> is the data set
 * to be cosine transformed, the extended data set
 * x<sub>0</sub><sup>&#35;</sup>, &hellip;, x<sub>2N-3</sub><sup>&#35;</sup>
 * is defined as follows
 * <ul>
 * <li>x<sub>k</sub><sup>&#35;</sup> = x<sub>k</sub> if 0 &le; k &lt; N,</li>
 * <li>x<sub>k</sub><sup>&#35;</sup> = x<sub>2N-2-k</sub>
 * if N &le; k &lt; 2N - 2.</li>
 * </ul>
 * <p>
 * Then, the standard DCT-I y<sub>0</sub>, &hellip;, y<sub>N-1</sub> of the real
 * data set x<sub>0</sub>, &hellip;, x<sub>N-1</sub> is equal to <em>half</em>
 * of the N first elements of the DFT of the extended data set
 * x<sub>0</sub><sup>&#35;</sup>, &hellip;, x<sub>2N-3</sub><sup>&#35;</sup>
 * <br/>
 * y<sub>n</sub> = (1 / 2) &sum;<sub>k=0</sub><sup>2N-3</sup>
 * x<sub>k</sub><sup>&#35;</sup> exp[-2&pi;i nk / (2N - 2)]
 * &nbsp;&nbsp;&nbsp;&nbsp;k = 0, &hellip;, N-1.
 * <p>
 * The present implementation of the discrete cosine transform as a fast cosine
 * transform requires the length of the data set to be a power of two plus one
 * (N&nbsp;=&nbsp;2<sup>n</sup>&nbsp;+&nbsp;1). Besides, it implicitly assumes
 * that the sampled function is even.
 *
 * @since 1.2
 */
public class FastCosineTransformer implements RealTransformer, Serializable {

    /** Serializable version identifier. */
    static final long serialVersionUID = 20120212L;

    /** The type of DCT to be performed. */
    private final DctNormalization normalization;

    /**
     * Creates a new instance of this class, with various normalization
     * conventions.
     *
     * @param normalization the type of normalization to be applied to the
     * transformed data
     */
    public FastCosineTransformer(final DctNormalization normalization) {
        this.normalization = normalization;
    }

    /**
     * {@inheritDoc}
     *
     * @throws MathIllegalArgumentException if the length of the data array is
     * not a power of two plus one
     */
    public double[] transform(final double[] f, final TransformType type)
      throws MathIllegalArgumentException {
        if (type == TransformType.FORWARD) {
            if (normalization == DctNormalization.ORTHOGONAL_DCT_I) {
                final double s = FastMath.sqrt(2.0 / (f.length - 1));
                return TransformUtils.scaleArray(fct(f), s);
            }
            return fct(f);
        }
        final double s2 = 2.0 / (f.length - 1);
        final double s1;
        if (normalization == DctNormalization.ORTHOGONAL_DCT_I) {
            s1 = FastMath.sqrt(s2);
        } else {
            s1 = s2;
        }
        return TransformUtils.scaleArray(fct(f), s1);
    }

    /**
     * {@inheritDoc}
     *
     * @throws org.apache.commons.math3.exception.NonMonotonicSequenceException
     * if the lower bound is greater than, or equal to the upper bound
     * @throws org.apache.commons.math3.exception.NotStrictlyPositiveException
     * if the number of sample points is negative
     * @throws MathIllegalArgumentException if the number of sample points is
     * not a power of two plus one
     */
    public double[] transform(final UnivariateFunction f,
        final double min, final double max, final int n,
        final TransformType type) throws MathIllegalArgumentException {

        final double[] data = FunctionUtils.sample(f, min, max, n);
        return transform(data, type);
    }

    /**
     * Perform the FCT algorithm (including inverse).
     *
     * @param f the real data array to be transformed
     * @return the real transformed array
     * @throws MathIllegalArgumentException if the length of the data array is
     * not a power of two plus one
     */
    protected double[] fct(double[] f)
        throws MathIllegalArgumentException {

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

        final int n = f.length - 1;
        if (!ArithmeticUtils.isPowerOfTwo(n)) {
            throw new MathIllegalArgumentException(
                LocalizedFormats.NOT_POWER_OF_TWO_PLUS_ONE,
                Integer.valueOf(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];
        // temporary variable for transformed[1]
        double t1 = 0.5 * (f[0] - f[n]);
        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;
        transformer = new FastFourierTransformer(DftNormalization.STANDARD);
        Complex[] y = transformer.transform(x, TransformType.FORWARD);

        // 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;
    }
}