001/*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements.  See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License.  You may obtain a copy of the License at
008 *
009 *      http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017package org.apache.commons.math3.transform;
018
019import java.io.Serializable;
020
021import org.apache.commons.math3.analysis.FunctionUtils;
022import org.apache.commons.math3.analysis.UnivariateFunction;
023import org.apache.commons.math3.complex.Complex;
024import org.apache.commons.math3.exception.MathIllegalArgumentException;
025import org.apache.commons.math3.exception.util.LocalizedFormats;
026import org.apache.commons.math3.util.ArithmeticUtils;
027import org.apache.commons.math3.util.FastMath;
028
029/**
030 * Implements the Fast Cosine Transform for transformation of one-dimensional
031 * real data sets. For reference, see James S. Walker, <em>Fast Fourier
032 * Transforms</em>, chapter 3 (ISBN 0849371635).
033 * <p>
034 * There are several variants of the discrete cosine transform. The present
035 * implementation corresponds to DCT-I, with various normalization conventions,
036 * which are specified by the parameter {@link DctNormalization}.
037 * <p>
038 * DCT-I is equivalent to DFT of an <em>even extension</em> of the data series.
039 * More precisely, if x<sub>0</sub>, &hellip;, x<sub>N-1</sub> is the data set
040 * to be cosine transformed, the extended data set
041 * x<sub>0</sub><sup>&#35;</sup>, &hellip;, x<sub>2N-3</sub><sup>&#35;</sup>
042 * is defined as follows
043 * <ul>
044 * <li>x<sub>k</sub><sup>&#35;</sup> = x<sub>k</sub> if 0 &le; k &lt; N,</li>
045 * <li>x<sub>k</sub><sup>&#35;</sup> = x<sub>2N-2-k</sub>
046 * if N &le; k &lt; 2N - 2.</li>
047 * </ul>
048 * <p>
049 * Then, the standard DCT-I y<sub>0</sub>, &hellip;, y<sub>N-1</sub> of the real
050 * data set x<sub>0</sub>, &hellip;, x<sub>N-1</sub> is equal to <em>half</em>
051 * of the N first elements of the DFT of the extended data set
052 * x<sub>0</sub><sup>&#35;</sup>, &hellip;, x<sub>2N-3</sub><sup>&#35;</sup>
053 * <br/>
054 * y<sub>n</sub> = (1 / 2) &sum;<sub>k=0</sub><sup>2N-3</sup>
055 * x<sub>k</sub><sup>&#35;</sup> exp[-2&pi;i nk / (2N - 2)]
056 * &nbsp;&nbsp;&nbsp;&nbsp;k = 0, &hellip;, N-1.
057 * <p>
058 * The present implementation of the discrete cosine transform as a fast cosine
059 * transform requires the length of the data set to be a power of two plus one
060 * (N&nbsp;=&nbsp;2<sup>n</sup>&nbsp;+&nbsp;1). Besides, it implicitly assumes
061 * that the sampled function is even.
062 *
063 * @version $Id: FastCosineTransformer.java 1385310 2012-09-16 16:32:10Z tn $
064 * @since 1.2
065 */
066public class FastCosineTransformer implements RealTransformer, Serializable {
067
068    /** Serializable version identifier. */
069    static final long serialVersionUID = 20120212L;
070
071    /** The type of DCT to be performed. */
072    private final DctNormalization normalization;
073
074    /**
075     * Creates a new instance of this class, with various normalization
076     * conventions.
077     *
078     * @param normalization the type of normalization to be applied to the
079     * transformed data
080     */
081    public FastCosineTransformer(final DctNormalization normalization) {
082        this.normalization = normalization;
083    }
084
085    /**
086     * {@inheritDoc}
087     *
088     * @throws MathIllegalArgumentException if the length of the data array is
089     * not a power of two plus one
090     */
091    public double[] transform(final double[] f, final TransformType type)
092      throws MathIllegalArgumentException {
093        if (type == TransformType.FORWARD) {
094            if (normalization == DctNormalization.ORTHOGONAL_DCT_I) {
095                final double s = FastMath.sqrt(2.0 / (f.length - 1));
096                return TransformUtils.scaleArray(fct(f), s);
097            }
098            return fct(f);
099        }
100        final double s2 = 2.0 / (f.length - 1);
101        final double s1;
102        if (normalization == DctNormalization.ORTHOGONAL_DCT_I) {
103            s1 = FastMath.sqrt(s2);
104        } else {
105            s1 = s2;
106        }
107        return TransformUtils.scaleArray(fct(f), s1);
108    }
109
110    /**
111     * {@inheritDoc}
112     *
113     * @throws org.apache.commons.math3.exception.NonMonotonicSequenceException
114     * if the lower bound is greater than, or equal to the upper bound
115     * @throws org.apache.commons.math3.exception.NotStrictlyPositiveException
116     * if the number of sample points is negative
117     * @throws MathIllegalArgumentException if the number of sample points is
118     * not a power of two plus one
119     */
120    public double[] transform(final UnivariateFunction f,
121        final double min, final double max, final int n,
122        final TransformType type) throws MathIllegalArgumentException {
123
124        final double[] data = FunctionUtils.sample(f, min, max, n);
125        return transform(data, type);
126    }
127
128    /**
129     * Perform the FCT algorithm (including inverse).
130     *
131     * @param f the real data array to be transformed
132     * @return the real transformed array
133     * @throws MathIllegalArgumentException if the length of the data array is
134     * not a power of two plus one
135     */
136    protected double[] fct(double[] f)
137        throws MathIllegalArgumentException {
138
139        final double[] transformed = new double[f.length];
140
141        final int n = f.length - 1;
142        if (!ArithmeticUtils.isPowerOfTwo(n)) {
143            throw new MathIllegalArgumentException(
144                LocalizedFormats.NOT_POWER_OF_TWO_PLUS_ONE,
145                Integer.valueOf(f.length));
146        }
147        if (n == 1) {       // trivial case
148            transformed[0] = 0.5 * (f[0] + f[1]);
149            transformed[1] = 0.5 * (f[0] - f[1]);
150            return transformed;
151        }
152
153        // construct a new array and perform FFT on it
154        final double[] x = new double[n];
155        x[0] = 0.5 * (f[0] + f[n]);
156        x[n >> 1] = f[n >> 1];
157        // temporary variable for transformed[1]
158        double t1 = 0.5 * (f[0] - f[n]);
159        for (int i = 1; i < (n >> 1); i++) {
160            final double a = 0.5 * (f[i] + f[n - i]);
161            final double b = FastMath.sin(i * FastMath.PI / n) * (f[i] - f[n - i]);
162            final double c = FastMath.cos(i * FastMath.PI / n) * (f[i] - f[n - i]);
163            x[i] = a - b;
164            x[n - i] = a + b;
165            t1 += c;
166        }
167        FastFourierTransformer transformer;
168        transformer = new FastFourierTransformer(DftNormalization.STANDARD);
169        Complex[] y = transformer.transform(x, TransformType.FORWARD);
170
171        // reconstruct the FCT result for the original array
172        transformed[0] = y[0].getReal();
173        transformed[1] = t1;
174        for (int i = 1; i < (n >> 1); i++) {
175            transformed[2 * i]     = y[i].getReal();
176            transformed[2 * i + 1] = transformed[2 * i - 1] - y[i].getImaginary();
177        }
178        transformed[n] = y[n >> 1].getReal();
179
180        return transformed;
181    }
182}