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

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.

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

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

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

  3. 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.

  4. NORMALIZED CROSS CORRELATION The normalized cross-correlation is given by

  5. s1 Remove Mean Compute Energy Compute NCC ρ(n) Peak Detect Compare Threshold Compute Energy Remove Mean s2 The NCC Algorithm

  6. Wavelets Some S8 Symmlets at Various Scales and Locations 9 8 7 6 5 Scale j 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 time index k 1. Can be defined by a wavelet function (Morlet & Mexican hat) 2. Can be defined by an FIR Filter Function (Haar, D4, S8)

  7. Symmlet Wavelet vs. Time and Frequency

  8. Signal TRANSFORMATION Noisy Signal Noise Noise Removal • Separate the signal from the noise

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

  10. 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 s = noise std. s is estimated from the data by analysis of the coefficients at the lowest scale. s = E/0.6475 where E is the absolute median deviation [Kenny]

  11. 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 = ms where typically 2< m <5

  12. Hard vs. Soft Thresholds

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

  14. Wavelet De-noise s1 Remove Mean DWT Compute Energy Compute NCC ρ(n) Peak Detect Compare Threshold Compute Energy Remove Mean DWT s2 The WDC Algorithm

  15. 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

  16. Signals of Interest

  17. Comparison of Peak NCC vs. SNR for Signal 1

  18. Comparison of Peak NCC vs. SNR for Signal 2

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

  20. Wavelet Filtering

  21. 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]

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

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

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

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