**Problem**

This is my programmatic note 18 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.

As I was working on Ex. 4.4, p. 34, I started thinking about how one can measure the quality of noise removal in a signal. In particular, suppose there is a signal with some added noise to which the multi-resolution analysis is applied for a specific number of scales. How close is the signal reconstructed from the scalar coefficients (S8, S7, S6, S6, etc) to the signal without any noise.

One method I thought of using is to measure the difference between the original (w/o any noise) and reconstructed signals with dynamic time warping. The exact steps of the method are as follows: 1) do the multi-resolution analysis with forward DWT (CDF or HWT) for a given number of scales on the noisy signal; 2) reconstruct the signal from the appropriate scalar coefficients with inverse DWT; 3) measure the difference between the original and reconstructed signal with dynamic time warping.

One method I thought of using is to measure the difference between the original (w/o any noise) and reconstructed signals with dynamic time warping. The exact steps of the method are as follows: 1) do the multi-resolution analysis with forward DWT (CDF or HWT) for a given number of scales on the noisy signal; 2) reconstruct the signal from the appropriate scalar coefficients with inverse DWT; 3) measure the difference between the original and reconstructed signal with dynamic time warping.

**Dynamic Time Warping**

A repo with my Java implementation of DTW is here. The class TwoDCurveComparator has two methods for comparing two curves with DWT.

public static double compareTwoDCurve(double[] curve_one, double[] curve_two, IDTWSimilarity isim, IWarpingWindow iww) {

return DTW.dpDTW2ColDoubleCost(curve_one, curve_two, isim, iww);

}

public static double compareTwoDCurve(String curveFilePathOne, String curveFilePathTwo, IDTWSimilarity isim, IWarpingWindow iww) {

double[] curve_one = TwoDCurveComparator.fileToPrimitiveDoubleArray(curveFilePathOne);

double[] curve_two = TwoDCurveComparator.fileToPrimitiveDoubleArray(curveFilePathTwo);

final double sim = DTW.dpDTW2ColDoubleCost(curve_two, curve_two, isim, iww);

curve_one = null;

curve_two = null;

return sim;

}

Here is how one can compare two curves using squared euclidian similarity (SquaredEquclideanSim.java), where the first two arguments are the arrays of doubles with self-explanatory names.

double cdf44_dtw_se = DTW.dpDTW2ColDoubleCost(ex_4_4_512_no_noise_p34, ex_4_4_512_with_noise_cdf44_s5_p34,

new SquaredEuclideanSim(), DTW.MaxWW);

double hwt_dtw_se = DTW.dpDTW2ColDoubleCost(ex_4_4_512_no_noise_p34, ex_4_4_512_with_noise_hwt_s5_p34,

new SquaredEuclideanSim(), DTW.MaxWW);

Below are some signal graphs.

public static double compareTwoDCurve(double[] curve_one, double[] curve_two, IDTWSimilarity isim, IWarpingWindow iww) {

return DTW.dpDTW2ColDoubleCost(curve_one, curve_two, isim, iww);

}

public static double compareTwoDCurve(String curveFilePathOne, String curveFilePathTwo, IDTWSimilarity isim, IWarpingWindow iww) {

double[] curve_one = TwoDCurveComparator.fileToPrimitiveDoubleArray(curveFilePathOne);

double[] curve_two = TwoDCurveComparator.fileToPrimitiveDoubleArray(curveFilePathTwo);

final double sim = DTW.dpDTW2ColDoubleCost(curve_two, curve_two, isim, iww);

curve_one = null;

curve_two = null;

return sim;

}

Here is how one can compare two curves using squared euclidian similarity (SquaredEquclideanSim.java), where the first two arguments are the arrays of doubles with self-explanatory names.

double cdf44_dtw_se = DTW.dpDTW2ColDoubleCost(ex_4_4_512_no_noise_p34, ex_4_4_512_with_noise_cdf44_s5_p34,

new SquaredEuclideanSim(), DTW.MaxWW);

double hwt_dtw_se = DTW.dpDTW2ColDoubleCost(ex_4_4_512_no_noise_p34, ex_4_4_512_with_noise_hwt_s5_p34,

new SquaredEuclideanSim(), DTW.MaxWW);

**Some Graphs**Below are some signal graphs.

**Figures 1**and**2**give the original and noisy signals, respectively.**Figures 3**-**6**show various reconstructions.Figure 1. Signal from ex. 4.4., p. 34 |

Figure 2. Signal from ex. 4.4, p. 34, with a noise spike added at 0.4 |

Figure 3. CDF44 reconstruction from S8 |

Figure 4. HWT reconstruction from S8 |

Figure 5. CDF44 reconstruction from S6 |

Figure 6. HWT signal reconstruction from S6. |

**Tests**
The
class TwoDCurveComparator.java has the following methods that measure
the quality of noise removal with difference numbers of scales from the
signal in Ex. 4.4 on p. 34: test_ex_4_4_p34_s8(), test_ex_4_4_p34_s7(),
test_ex_4_4_p34_s6(), and test_ex_4_4_p34_s5(). These methods display
the values of various similarity metrics between the original and
reconstructed signals. Below are some DWT values between the original and reconstructed signals. In the notation below, CDF(4, 4) or HWT denote the DWT, S8, S7, and S6 denote the scalar values at a specific scale, the number to the right side of the equality sign is the value of the DTW between the reconstructed signal and the signal in

**Fig. 2**.
CDF(4, 4), S8 == 3.9685421518395367

HWT, S8 == 7.684719899677848

CDF(4, 4), S7 == 1.688302564009318

HWT, S7 == 3.4639739360892845

CDF(4, 4), S6 == 1.7377716142440158

HWT, S6 == 2.0367401581624174