Friday, March 25, 2016

Multiresolution Analysis of a Chirp with CDF(4, 4) and HWT: Programmatic Note 15 on Ripples in Mathematics




Problem
  
This is my programmatic note 15 on Ch. 04, p. 31-34, in "Ripples in Mathematics" by A. Jensen & A. la Cour-Harbo. These notes document my programmatic journey, sometimes easy, sometimes painful, but always joyful, through this great text. My notes are for those who want to use Java as a programming investigation tool of various wavelet algorithms and Octave for plotting the results. You can find my previous notes by searching my blog on "ripples in mathematics." If you are on the same journey, let me know of any bugs in my code or improvements you have found that let us all, fellow travelers on the wavelet road, get better and more beautiful results.

In this note, I cover my programmatic multiresolution analysis of a chirp, i.e., a signal obtained from sampling sin(t^2) in exercise 4.3 on p. 33. The multiresolution analysis is done on a chirp of size 512 with CDF44 and HWT obtained over 4 scales.

Multiresolution Analysis of Y=SIN(T^2)


The class that generates the range of sin(t^2) is defined in Ripples_F_ex_4_3_p33.java. Fig. 1 gives a graph of this signal on [0, 511] created by the Octave script ex_4_3_chirp_512_p33.m


Figure 1. Graph of y = sin(t^2) on [0, 511]
I added two static methods to RipplesInMathCh04.java  defined below. These methods do the multiresolution analysis with HWT and CDF(4, 4), respectively.

static void multires_ex_4_3_hwt_p33(String message, int range_start, int range_end, int signal_size, int num_scales) {
        double[] chirp_signal = generate_chirp(signal_size);
       
        ApplyDWT.forwardDWTForNumIters(chirp_signal, ApplyDWT.DWT.HWT, num_scales, range_start, range_end);
       
        double[] signal = new double[signal_size];
        for(int i = 0; i < signal_size; i++) {
            if ( i >= range_start && i <= range_end ) {
                signal[i] = chirp_signal[i];
            }
            else {
                signal[i] = 0;
            }
        }
       
        System.out.println("=========================");
        System.out.println(message);
        display_signal(signal);
        System.out.println("Inversed Signal");
        System.out.println("=========================");
        ApplyDWT.inverseDWTForNumIters(signal, ApplyDWT.DWT.HWT, num_scales);
        display_signal(signal);
        System.out.println("=========================");
    }


static void multires_ex_4_3_cdf44_p33(String message, int range_start, int range_end, int signal_size, int num_scales) {
        double[] chirp_signal = generate_chirp(signal_size);
       
        //CDF44.orderedDWTForNumIters(chirp_signal, num_scales, false);
        ApplyDWT.forwardDWTForNumIters(chirp_signal, ApplyDWT.DWT.CDF44, num_scales, range_start, range_end);
       
        double[] signal = new double[signal_size];
        for(int i = 0; i < signal_size; i++) {
            if ( i >= range_start && i <= range_end ) {
                signal[i] = chirp_signal[i];
            }
            else {
                signal[i] = 0;
            }
        }
       
        System.out.println("=========================");
        System.out.println(message);
        display_signal(signal);
        System.out.println("Inversed Signal");
        System.out.println("=========================");
        //CDF44.orderedInverseDWTForNumIters(signal, num_scales, false);
        ApplyDWT.inverseDWTForNumIters(signal, ApplyDWT.DWT.CDF44, num_scales);
        display_signal(signal);
        System.out.println("=========================");
    }  


Below are the graphed parts of the multiresolution analysis along with the Java methods from RipplesInMathCh04.java that generated their values. All graphs are generated by the Octave scripts in this repo with self-explanatory names, e.g., ex_4_3_chirp_512_CDF44_D8_p33.m. The graphs for D7 and D6 are skipped.

D8 Graphs

static void ex_4_3_chirp_512_cdf44_d8_p33() { multires_ex_4_3_cdf44_p33("Ex. 4.3, CDF44, D8, p. 33", D8_START_512, D8_END_512, 512, 4); }