Introduction To Wavelets Transforming Problem Representation Gene A

Introduction To Wavelets Transforming Problem Representation Gene A. Tagliarini

Why Representation Is Key? l l The curse of dimensionality Dealing with distractors Signal enhancement goals such as increasing low contrast Feature identification Image 1 2 Gene A. Tagliarini 2/20/2021

Sobel Filtering of Raw Image Data l l l 3 Sobel filtered raw data image Note speckling Median filters can reduce speckle (this image is not median filtered) Image 2 Gene A. Tagliarini 2/20/2021

Another Spatial Representation l l l One level of a twodimensional wavelet transform Image resized for comparison Contrast is enhanced Image 3 4 Gene A. Tagliarini 2/20/2021

Sobel Filtering Of Transformed Image l l 5 Source (Image 3) is ¼ the size of Images 1 and 2 Blocks arise from rescaling Image 4 Gene A. Tagliarini 2/20/2021

Sobel Filtering Of Raw Image Data l l l 6 Sobel filtered raw data image Note speckling Median filters can reduce speckle (this image is not median filtered) Image 2 Gene A. Tagliarini 2/20/2021

One-dimensional Signal l 7 Gene A. Tagliarini A clean signal Short duration (circa 200 m. Sec) Typical of sonar transients 2/20/2021

Noisy One-dimensional Signal l 8 Gene A. Tagliarini The transient occurs slightly after the center time displayed Detection-determining that a transient occurred Classificationdetermining which transient occurred 2/20/2021

Common Wavelet Applications l Requiring only decomposition – – – l Requiring decomposition and reconstruction – – 9 Signal processing Image filtering Feature extraction Data transmission using compression Matrix multiplication Gene A. Tagliarini 2/20/2021

Wavelet Transform Properties l Analogous to Fourier Transforms—but different – – – l 10 Constant ratio of scale versus constant difference of frequencies Open choice of basis functions versus fixed choice of basis functions (sines and cosines) Compact support versus non-compact support Provides a decomposition for square integrable Localize in time and scale versus time and frequency Readily computable Gene A. Tagliarini 2/20/2021

The Basic Dilation Equation Two simple solutions: • c 0 = 2, implies f(x) = d(x) • c 0=c 1=1, implies f(x) = f(2 x) + f(2 x-1) and f(x) = c([0, 1]) 11 Gene A. Tagliarini 2/20/2021

A Graphical Example of Dilation f(2 x) + f(2 x-1) f(x) = c([0, 1)) = 0 12 1 0 Gene A. Tagliarini 0. 5 1 2/20/2021

The Basic Wavelet Equation l l 13 Uses differences and the scaling function The Haar wavelet (based on the box function) is given by W(x)= f(2 x) - f(2 x-1) where f(x) = c([0, 1)) Gene A. Tagliarini 2/20/2021

The Box Function And The Basic Haar Wavelet f(2 x) + f(2 x-1) f(2 x) - f(2 x-1) Haar Wavelet Box Function 14 Gene A. Tagliarini 2/20/2021

A Normalized Dilation Equation 15 Gene A. Tagliarini 2/20/2021

The Goal Of The Transformation Process l Write one function (the signal S) as a linear combination of the scaling and wavelet functions where the wj, k are the wavelet scalars and Wj, k(x) are translated and dilated wavelets 16 Gene A. Tagliarini 2/20/2021

Matrix Representation Of The Haar Wavelet Transform 17 Gene A. Tagliarini 2/20/2021

Inverting The Effects Of The Haar Wavelet Transform 18 Gene A. Tagliarini 2/20/2021

The Inverse Matrix Is A Scaled Version Of The Transpose 19 Gene A. Tagliarini 2/20/2021

A Sample Signal And Its Decomposition (One Level) 20 Gene A. Tagliarini 2/20/2021

WT Computation: Low-pass Filter Output And Down-sampling c 1 c 2 c 3 c 0 c 1 lp 0 c 2 c 3 c 0 c 1 lp 1 c 2 c 3 c 0 c 1 c 2 c 3 s 6 s 7 s 8 s 9 … lp 2 lp 3 … c 0 s 0 21 s 2 s 3 s 4 s 5 Gene A. Tagliarini 2/20/2021

WT Computation: High-pass Filter Output And Down-sampling d 0 d 1 d 2 d 3 hp 1 d 0 d 1 d 2 d 3 hp 2 d 0 d 1 d 2 d 3 … hp 3 s 0 22 s 1 s 2 s 3 s 4 s 5 s 6 Gene A. Tagliarini s 7 s 8 s 9 … 2/20/2021

For Two-Dimensional Signals (Using Separable Wavelets) Original l 23 LP HP LL LH HL HH Process each row, storing LP results on the left and HP results on the right Process each column of the previous, storing LP results at the top and HP at the bottom One level of a 2 -D transform Gene A. Tagliarini 2/20/2021

A Two-level Two-dimensional Example l l l 24 Images from the TRIM-2 data set acquired from the Night Vision and Electronic Sensors Directorate, Night Vision Laboratories, Ft. Belvoir Images are in ARF format using 480 x 640 pixels, each byte representing one of 256 possible gray-scale levels Terrain board images simulate scenes viewed in infrared Gene A. Tagliarini 2/20/2021

So, Where Is The Compression? l l 25 Suppose the signal is represented with N samples Low-pass filtering produces n/2 values High-pass filtering produces n/2 values Compression arises from quantization and encoding of HP filter output Gene A. Tagliarini 2/20/2021

How Can One Generate Wavelet Filter Coefficients? l Rigorous mathematical analysis – l Daubechies coefficients Exploit parameterizations – – Te. Kolste Pollen l l l 26 C 0 = [(1 + cos a + sin a)(1 – cos b – sin b) + 2 sin b cos a]/4 C 1 = [(1 - cos a + sin a)(1 + cos b – sin b) - 2 sin b cos a]/4 C 2 = [1 + cos(a – b) + sin(a – b)]/2 C 3 = [1 + cos(a – b) - sin(a – b)]/2 C 4 = 1 – c 2 – c 0 C 5 = 1 – c 3 – c 1 Gene A. Tagliarini 2/20/2021

What Might One Ask Using The Pollen Parameterization? l l l What wavelet provides the best basis for compression? Can one minimize variability or magnitudes in the high-pass filter output? What wavelet might emerge using a small set of samples from a piece-wise linear function? – – 27 32 samples Corresponds to image regions having smooth gradients or homogeneous contents Gene A. Tagliarini 2/20/2021

Resulting Coefficients Proc. SPIE, Vol. 2762, p. 89 28 Gene A. Tagliarini 2/20/2021

Concluding Comments l l Wavelets provide an approach to transforming problems from the time domain to a scale domain The transform can be tailored to meet processing objectives Optimization techniques can lead to results that correspond to (possibly difficult to obtain) analytical results There is much more to say about things like: – – – 29 Orthogonality, bi-orthogonality, and orthonormality Separability and nonseparability Super-wavelets Gene A. Tagliarini 2/20/2021