Feature Matching and Signal Recognition using Wavelet Analysis Dr. Robert Barsanti, Edwin Spencer, James Cares, Lucas Parobek

Overview Introduction Normalized Cross Correlation Noise Removal in the Wavelet Domain Feature Matching Using Wavelets Simulations and Results Summary

CROSS CORRELATION The squared Euclidean distance measure Assuming that both the first and second terms (representing the signal and feature energies) are constant, then last term, the cross-correlation is a measure of the similarity between the signal and the feature.

NORMALIZED CROSS CORRELATION The normalized cross-correlation is given by

The NCC Algorithm Remove Mean Compare Threshold Remove Mean s1s1 s2s2 Compute Energy Compute Energy Compute NCC ρ(n) Peak Detect

Wavelets 1. Can be defined by a wavelet function (Morlet & Mexican hat) 2. Can be defined by an FIR Filter Function (Haar, D4, S8) Some S8 Symmlets at Various Scales and Locations time index k Scale j

Symmlet Wavelet vs. Time and Frequency

Noise Removal Separate the signal from the noise TRANSFORMATION Noisy Signal Signal Noise

Wavelet Based Filtering THREE STEP DENOISING 1. PERFORM DWT 2. THRESHOLD COEFFICIENTS 3. PERFORM INVERSE DWT

Calculating a Threshold Let the DWT coefficient be a series of noisy observations y(n) then the following parameter estimation problem exists: y(n) = f(n) + s z(n), n = 1,2,…. z ~N(0,1) and  = noise std.  is estimated from the data by analysis of the coefficients at the lowest scale.  = E/ where E is the absolute median deviation [Kenny]

Thresholding Techniques * Hard Thresholding (keep or kill) * Soft Thresholding (reduce all by Threshold) The Threshold Value is determined as a multiple of the noise standard deviation, eg., T = m  where typically 2< m <5

Hard vs. Soft Thresholds

FEATURE MATCHING USING WAVELETS (1)transform feature signal into the wavelet domain and pre-stored DWT coefficients (2)transform data into the wavelet domain via the DWT, (3)apply a non-linear threshold to the DWT coefficients (to remove noise), (4)correlation of the noise free DWT coefficients of the signal, and the pre-stored DWT coefficients of the template feature.

The WDC Algorithm Remove Mean Compare Threshold Remove Mean s1s1 s2s2 Compute Energy Compute Energy Compute NCC ρ(n) Peak Detect DWT Wavelet De-noise

Simulation and Results Two signals were tested 500 Monte Carlo Runs at each SNR 25 SNR’s between -10 and +15 dB Symmlet 4 wavelet & soft threshold Four correlators compared –NCC –NCC with Roth pre-filter –NCC with Phat pre-filter –WDC

Signals of Interest

Comparison of Peak NCC vs. SNR for Signal 1

Comparison of Peak NCC vs. SNR for Signal 2

Summary (1)Algorithm for signal feature matching in the presence of AWGN. (2) Uses the normalized cross- correlation between DWT coefficients (3)Procedure is enhanced by using standard wavelet noise removal techniques (4)Simulations of the performance of the proposed algorithm were presented.

Wavelet Filtering

EFFECTIVENESS OF WAVELET ANALYSIS Wavelets are adjustable and adaptable by virtue of large number of possible wavelet bases. DWT well suited to digital implementation. ~O (N) Ideally suited for analysis non-stationary signals [ Strang, 1996] Has been shown to be a viable denoising technique for transients [Donoho, 1995] Has been shown to be a viable detection technique for transients [Carter, 1994] Has been shown to be a viable TDOA technique for transients [Wu, 1997]

Wavelet Implementation HP Filter LP Filter X(n) LPF HPF Frequency Response F/2 Pair of Half Band Quadrature Mirror Filters (QMF) [Vetterli, 1995] Details Averages

Signal Reconstruction Two Channel Perfect Reconstruction QMF Bank Analysis + Synthesis = LTI system

Wavelet Implementation [Mallat, 1989] LP HP LPLPLP J = 4 LPLPHP J = 3 LPHP J = 2 HP J = 1 HP LPHP LPLPHP LPLPLP 2 J samples LP