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     */
017    package org.apache.commons.math3.transform;
018    
019    import java.io.Serializable;
020    
021    import org.apache.commons.math3.analysis.FunctionUtils;
022    import org.apache.commons.math3.analysis.UnivariateFunction;
023    import org.apache.commons.math3.exception.MathIllegalArgumentException;
024    import org.apache.commons.math3.exception.util.LocalizedFormats;
025    import org.apache.commons.math3.util.ArithmeticUtils;
026    
027    /**
028     * Implements the <a href="http://www.archive.chipcenter.com/dsp/DSP000517F1.html">Fast Hadamard Transform</a> (FHT).
029     * Transformation of an input vector x to the output vector y.
030     * <p>
031     * In addition to transformation of real vectors, the Hadamard transform can
032     * transform integer vectors into integer vectors. However, this integer transform
033     * cannot be inverted directly. Due to a scaling factor it may lead to rational results.
034     * As an example, the inverse transform of integer vector (0, 1, 0, 1) is rational
035     * vector (1/2, -1/2, 0, 0).
036     *
037     * @version $Id: FastHadamardTransformer.java 1385310 2012-09-16 16:32:10Z tn $
038     * @since 2.0
039     */
040    public class FastHadamardTransformer implements RealTransformer, Serializable {
041    
042        /** Serializable version identifier. */
043        static final long serialVersionUID = 20120211L;
044    
045        /**
046         * {@inheritDoc}
047         *
048         * @throws MathIllegalArgumentException if the length of the data array is
049         * not a power of two
050         */
051        public double[] transform(final double[] f, final TransformType type) {
052            if (type == TransformType.FORWARD) {
053                return fht(f);
054            }
055            return TransformUtils.scaleArray(fht(f), 1.0 / f.length);
056        }
057    
058        /**
059         * {@inheritDoc}
060         *
061         * @throws org.apache.commons.math3.exception.NonMonotonicSequenceException
062         *   if the lower bound is greater than, or equal to the upper bound
063         * @throws org.apache.commons.math3.exception.NotStrictlyPositiveException
064         *   if the number of sample points is negative
065         * @throws MathIllegalArgumentException if the number of sample points is not a power of two
066         */
067        public double[] transform(final UnivariateFunction f,
068            final double min, final double max, final int n,
069            final TransformType type) {
070    
071            return transform(FunctionUtils.sample(f, min, max, n), type);
072        }
073    
074        /**
075         * Returns the forward transform of the specified integer data set.The
076         * integer transform cannot be inverted directly, due to a scaling factor
077         * which may lead to double results.
078         *
079         * @param f the integer data array to be transformed (signal)
080         * @return the integer transformed array (spectrum)
081         * @throws MathIllegalArgumentException if the length of the data array is not a power of two
082         */
083        public int[] transform(final int[] f) {
084            return fht(f);
085        }
086    
087        /**
088         * The FHT (Fast Hadamard Transformation) which uses only subtraction and
089         * addition. Requires {@code N * log2(N) = n * 2^n} additions.
090         *
091         * <h3>Short Table of manual calculation for N=8</h3>
092         * <ol>
093         * <li><b>x</b> is the input vector to be transformed,</li>
094         * <li><b>y</b> is the output vector (Fast Hadamard transform of <b>x</b>),</li>
095         * <li>a and b are helper rows.</li>
096         * </ol>
097         * <table align="center" border="1" cellpadding="3">
098         * <tbody align="center">
099         * <tr>
100         *     <th>x</th>
101         *     <th>a</th>
102         *     <th>b</th>
103         *     <th>y</th>
104         * </tr>
105         * <tr>
106         *     <th>x<sub>0</sub></th>
107         *     <td>a<sub>0</sub> = x<sub>0</sub> + x<sub>1</sub></td>
108         *     <td>b<sub>0</sub> = a<sub>0</sub> + a<sub>1</sub></td>
109         *     <td>y<sub>0</sub> = b<sub>0</sub >+ b<sub>1</sub></td>
110         * </tr>
111         * <tr>
112         *     <th>x<sub>1</sub></th>
113         *     <td>a<sub>1</sub> = x<sub>2</sub> + x<sub>3</sub></td>
114         *     <td>b<sub>0</sub> = a<sub>2</sub> + a<sub>3</sub></td>
115         *     <td>y<sub>0</sub> = b<sub>2</sub> + b<sub>3</sub></td>
116         * </tr>
117         * <tr>
118         *     <th>x<sub>2</sub></th>
119         *     <td>a<sub>2</sub> = x<sub>4</sub> + x<sub>5</sub></td>
120         *     <td>b<sub>0</sub> = a<sub>4</sub> + a<sub>5</sub></td>
121         *     <td>y<sub>0</sub> = b<sub>4</sub> + b<sub>5</sub></td>
122         * </tr>
123         * <tr>
124         *     <th>x<sub>3</sub></th>
125         *     <td>a<sub>3</sub> = x<sub>6</sub> + x<sub>7</sub></td>
126         *     <td>b<sub>0</sub> = a<sub>6</sub> + a<sub>7</sub></td>
127         *     <td>y<sub>0</sub> = b<sub>6</sub> + b<sub>7</sub></td>
128         * </tr>
129         * <tr>
130         *     <th>x<sub>4</sub></th>
131         *     <td>a<sub>0</sub> = x<sub>0</sub> - x<sub>1</sub></td>
132         *     <td>b<sub>0</sub> = a<sub>0</sub> - a<sub>1</sub></td>
133         *     <td>y<sub>0</sub> = b<sub>0</sub> - b<sub>1</sub></td>
134         * </tr>
135         * <tr>
136         *     <th>x<sub>5</sub></th>
137         *     <td>a<sub>1</sub> = x<sub>2</sub> - x<sub>3</sub></td>
138         *     <td>b<sub>0</sub> = a<sub>2</sub> - a<sub>3</sub></td>
139         *     <td>y<sub>0</sub> = b<sub>2</sub> - b<sub>3</sub></td>
140         * </tr>
141         * <tr>
142         *     <th>x<sub>6</sub></th>
143         *     <td>a<sub>2</sub> = x<sub>4</sub> - x<sub>5</sub></td>
144         *     <td>b<sub>0</sub> = a<sub>4</sub> - a<sub>5</sub></td>
145         *     <td>y<sub>0</sub> = b<sub>4</sub> - b<sub>5</sub></td>
146         * </tr>
147         * <tr>
148         *     <th>x<sub>7</sub></th>
149         *     <td>a<sub>3</sub> = x<sub>6</sub> - x<sub>7</sub></td>
150         *     <td>b<sub>0</sub> = a<sub>6</sub> - a<sub>7</sub></td>
151         *     <td>y<sub>0</sub> = b<sub>6</sub> - b<sub>7</sub></td>
152         * </tr>
153         * </tbody>
154         * </table>
155         *
156         * <h3>How it works</h3>
157         * <ol>
158         * <li>Construct a matrix with {@code N} rows and {@code n + 1} columns,
159         * {@code hadm[n+1][N]}.<br/>
160         * <em>(If I use [x][y] it always means [row-offset][column-offset] of a
161         * Matrix with n rows and m columns. Its entries go from M[0][0]
162         * to M[n][N])</em></li>
163         * <li>Place the input vector {@code x[N]} in the first column of the
164         * matrix {@code hadm}.</li>
165         * <li>The entries of the submatrix {@code D_top} are calculated as follows
166         *     <ul>
167         *         <li>{@code D_top} goes from entry {@code [0][1]} to
168         *         {@code [N / 2 - 1][n + 1]},</li>
169         *         <li>the columns of {@code D_top} are the pairwise mutually
170         *         exclusive sums of the previous column.</li>
171         *     </ul>
172         * </li>
173         * <li>The entries of the submatrix {@code D_bottom} are calculated as
174         * follows
175         *     <ul>
176         *         <li>{@code D_bottom} goes from entry {@code [N / 2][1]} to
177         *         {@code [N][n + 1]},</li>
178         *         <li>the columns of {@code D_bottom} are the pairwise differences
179         *         of the previous column.</li>
180         *     </ul>
181         * </li>
182         * <li>The consputation of {@code D_top} and {@code D_bottom} are best
183         * understood with the above example (for {@code N = 8}).
184         * <li>The output vector {@code y} is now in the last column of
185         * {@code hadm}.</li>
186         * <li><em>Algorithm from <a href="http://www.archive.chipcenter.com/dsp/DSP000517F1.html">chipcenter</a>.</em></li>
187         * </ol>
188         * <h3>Visually</h3>
189         * <table border="1" align="center" cellpadding="3">
190         * <tbody align="center">
191         * <tr>
192         *     <td></td><th>0</th><th>1</th><th>2</th><th>3</th>
193         *     <th>&hellip;</th>
194         *     <th>n + 1</th>
195         * </tr>
196         * <tr>
197         *     <th>0</th>
198         *     <td>x<sub>0</sub></td>
199         *     <td colspan="5" rowspan="5" align="center" valign="middle">
200         *         &uarr;<br/>
201         *         &larr; D<sub>top</sub> &rarr;<br/>
202         *         &darr;
203         *     </td>
204         * </tr>
205         * <tr><th>1</th><td>x<sub>1</sub></td></tr>
206         * <tr><th>2</th><td>x<sub>2</sub></td></tr>
207         * <tr><th>&hellip;</th><td>&hellip;</td></tr>
208         * <tr><th>N / 2 - 1</th><td>x<sub>N/2-1</sub></td></tr>
209         * <tr>
210         *     <th>N / 2</th>
211         *     <td>x<sub>N/2</sub></td>
212         *     <td colspan="5" rowspan="5" align="center" valign="middle">
213         *         &uarr;<br/>
214         *         &larr; D<sub>bottom</sub> &rarr;<br/>
215         *         &darr;
216         *     </td>
217         * </tr>
218         * <tr><th>N / 2 + 1</th><td>x<sub>N/2+1</sub></td></tr>
219         * <tr><th>N / 2 + 2</th><td>x<sub>N/2+2</sub></td></tr>
220         * <tr><th>&hellip;</th><td>&hellip;</td></tr>
221         * <tr><th>N</th><td>x<sub>N</sub></td></tr>
222         * </tbody>
223         * </table>
224         *
225         * @param x the real data array to be transformed
226         * @return the real transformed array, {@code y}
227         * @throws MathIllegalArgumentException if the length of the data array is not a power of two
228         */
229        protected double[] fht(double[] x) throws MathIllegalArgumentException {
230    
231            final int n     = x.length;
232            final int halfN = n / 2;
233    
234            if (!ArithmeticUtils.isPowerOfTwo(n)) {
235                throw new MathIllegalArgumentException(
236                        LocalizedFormats.NOT_POWER_OF_TWO,
237                        Integer.valueOf(n));
238            }
239    
240            /*
241             * Instead of creating a matrix with p+1 columns and n rows, we use two
242             * one dimension arrays which we are used in an alternating way.
243             */
244            double[] yPrevious = new double[n];
245            double[] yCurrent  = x.clone();
246    
247            // iterate from left to right (column)
248            for (int j = 1; j < n; j <<= 1) {
249    
250                // switch columns
251                final double[] yTmp = yCurrent;
252                yCurrent  = yPrevious;
253                yPrevious = yTmp;
254    
255                // iterate from top to bottom (row)
256                for (int i = 0; i < halfN; ++i) {
257                    // Dtop: the top part works with addition
258                    final int twoI = 2 * i;
259                    yCurrent[i] = yPrevious[twoI] + yPrevious[twoI + 1];
260                }
261                for (int i = halfN; i < n; ++i) {
262                    // Dbottom: the bottom part works with subtraction
263                    final int twoI = 2 * i;
264                    yCurrent[i] = yPrevious[twoI - n] - yPrevious[twoI - n + 1];
265                }
266            }
267    
268            return yCurrent;
269    
270        }
271    
272        /**
273         * Returns the forward transform of the specified integer data set. The FHT
274         * (Fast Hadamard Transform) uses only subtraction and addition.
275         *
276         * @param x the integer data array to be transformed
277         * @return the integer transformed array, {@code y}
278         * @throws MathIllegalArgumentException if the length of the data array is not a power of two
279         */
280        protected int[] fht(int[] x) throws MathIllegalArgumentException {
281    
282            final int n     = x.length;
283            final int halfN = n / 2;
284    
285            if (!ArithmeticUtils.isPowerOfTwo(n)) {
286                throw new MathIllegalArgumentException(
287                        LocalizedFormats.NOT_POWER_OF_TWO,
288                        Integer.valueOf(n));
289            }
290    
291            /*
292             * Instead of creating a matrix with p+1 columns and n rows, we use two
293             * one dimension arrays which we are used in an alternating way.
294             */
295            int[] yPrevious = new int[n];
296            int[] yCurrent  = x.clone();
297    
298            // iterate from left to right (column)
299            for (int j = 1; j < n; j <<= 1) {
300    
301                // switch columns
302                final int[] yTmp = yCurrent;
303                yCurrent  = yPrevious;
304                yPrevious = yTmp;
305    
306                // iterate from top to bottom (row)
307                for (int i = 0; i < halfN; ++i) {
308                    // Dtop: the top part works with addition
309                    final int twoI = 2 * i;
310                    yCurrent[i] = yPrevious[twoI] + yPrevious[twoI + 1];
311                }
312                for (int i = halfN; i < n; ++i) {
313                    // Dbottom: the bottom part works with subtraction
314                    final int twoI = 2 * i;
315                    yCurrent[i] = yPrevious[twoI - n] - yPrevious[twoI - n + 1];
316                }
317            }
318    
319            // return the last computed output vector y
320            return yCurrent;
321    
322        }
323    
324    }