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

d

2 (

n

)  

m

[

s

1 (

m

) 

s

2 (

m



n

)] 2 , ( 1 )

d

2 (

n

)  

m

[

s

1 2 (

m

) 

s

2 2 (

n



m

)  2

s

1 (

m

)

s

2 (

m



n

)] .

( 2 ) Assuming that both the first and second terms (representing the signal and feature energies) are constant, then last term, the cross-correlation

c

(

n

)  

m s

1 (

m

)

s

2 (

n



m

) ( 3 ) is a measure of the similarity between the signal and the feature.

NORMALIZED CROSS CORRELATION

The normalized cross-correlation is given by  (

n

)  

m

[

s

1 (

m

) 

s

1 ][

s

2 (

m



n

) 

s

2 ] 

m

[

s

1 (

m

) 

s

1 ] 2 

m

[

s

2 (

m



n

) 

s

2 ] 2 , ( 4 )

The NCC Algorithm

s 1 s 2 Remove Mean Compute Energy Compute Energy Remove Mean Compute NCC ρ(n) Peak Detect Compare Threshold

Wavelets

0.2

0.1

0 -0.1

-0.2

0.2

0.1

0 -0.1

-0.2

H aar W avelet 0.5

C 3 Coiflet 0.5

0.2

0.1

0 -0.1

-0.2

D 4 Wavelet 0.2

0 -0.2

-0.4

0.5

S8 Symmlet 0.5

9 8 7 6 5 4 3 2 1 0 0 Some S8 Symmlets at Various Scales and Locations 0.2

0.4

0.6

time index k 0.8

1 1. Can be defined by a wavelet function (Morlet & Mexican hat) 2. Can be defined by an FIR Filter Function (Haar, D4, S8)

Symmlet Wavelet vs. Time and Frequency

Noise Removal

• Separate the signal from the noise

Noisy Signal TRANSFORMATION Signal Noise

Wavelet Based Filtering

THREE STEP DENOISING

1. PERFORM DWT 2. THRESHOLD COEFFICIENTS 3. PERFORM INVERSE DWT 10 S1 Signal + Noise 5 0 -5 0 0.5

S2 Denoised Signal 2 0 6 4 -2 -4 0 0.5

1 1 0 -2 -4 -6 -8 -10 0 0 -2 -4 -6 -8 -10 0 DWT of S1 0.5

DWT of S2 0.5

1 1

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]

Thresholding Techniques

* Hard Thresholding (keep or kill) 

H

(

C j

,

k

)  

C

0

j

,

k

|

C j

,

k

| 

T

|

C j

,

k

| 

T

, * Soft Thresholding (reduce all by Threshold) 

S

(

C j

,

k

)

sign

(

C

  0

j

,

k

)[|

C j

,

k

| 

T

] |

C j

,

k

| 

T

|

C j

,

k

| 

T

, The Threshold Value is determined as a multiple of the noise standard deviation, eg., T = m s 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

s 1 s 2 Remove Mean DWT Compute Energy Wavelet De-noise Compute NCC ρ(n) Compute Energy Peak Detect Remove Mean DWT Compare Threshold

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 Signal 1 1 0.8

0.6

0.4

0.2

0 -0.2

-0.4

0 1 0.5

0 -0.5

-1 -1.5

0 0.1

0.2

0.3

0.4

0.5

Signal 2 0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1

Comparison of Peak NCC vs. SNR for Signal 1 1 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 -10 TDC WDC PHAT ROTH -5 Peak NCC vs. SNR for Signal 1 0 SNR dB 5 10 15

Comparison of Peak NCC vs. SNR for Signal 2 1 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 -10 TDC WDC PHAT ROTH -5 Peak NCC vs. SNR for Signal 2 0 SNR dB 5 10 15

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

Signal Reconstruction

x(n) -----Analysis Section------- LPF 2 > > -----Synthesis Section------ 2 LPF HPF 2 > > 2 HPF x(n) Two Channel Perfect Reconstruction QMF Bank Analysis + Synthesis = LTI system

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