Ioannis Kakadaris, U of Houston Wavelets ? Raghu Machiraju Contributions: Robert Moorhead, James Fowler, David Thompson, Mississippi State University Ioannis Kakadaris, U of Houston

State-Of-Affairs Concurrent Presentation Retrospective Analysis Simulations, scanners Concurrent Presentation Retrospective Analysis Representation November 12, 2018

Why Wavelets? We are generating and measuring larger datasets every year We can not store all the data we create (too much, too fast) We can not look at all the data (too busy, too hard) We need to develop techniques to store the data in better formats November 12, 2018

Data Analysis Frequency spectrum correctly shows a spike at 10 Hz Spike not narrow - significant component at between 5 and 15 Hz. Leakage - discrete data acquisition does not stop at exactly the same phase in the sine wave as it started. November 12, 2018

QuickFix November 12, 2018

Windowing &Filtering November 12, 2018

Image Example 8x8 Blocked Window (Cosine) Transform Each DCT basis waveform represents a fixed frequency in two orthogonal directions frequency spacing in each direction is an integer multiple of a base frequency November 12, 2018

Cannot resolve all features at all instants Windowing & Filtering Windows – fixed in space and frequencies Cannot resolve all features at all instants November 12, 2018

Linear Scale Space input s = 1 s = 16 s = 24 s = 32 November 12, 2018

Successive Smoothing November 12, 2018

Sub-sampled Images Keep 1 of 4 values from 2x2 blocks This naive approach and introduces aliasing Sub-samples are bad representatives of area Little spatial correlation November 12, 2018

Image Pyramid November 12, 2018

Image Pyramid – MIP MAP Average over a 2x2 block This is a rather straight forward approach This reduces aliasing and is a better representation However, this produces 11% expansion in the data November 12, 2018

Image Pyramid – Another Twist November 12, 2018

Time Frequency Diagram November 12, 2018

Create new signal G such that ||F-G|| = e Ideally ! Create new signal G such that ||F-G|| = e November 12, 2018

Wavelet Analysis A1 D1 D2 D3 D3 D2 D1 A1 November 12, 2018

Why Wavelets? Because … We need to develop techniques to analyze data better through noise discrimination Wavelets can be used to detect features and to compare features Wavelets can provide compressed representations Wavelet Theory provides a unified framework for data processing November 12, 2018

Scale-Coherent Structures Coherent structure - frequencies at all scales Examples - edges, peaks, ridges Locate extent and assign saliency November 12, 2018

Wavelets – Analysis November 12, 2018

Wavelets – DeNoising November 12, 2018

Wavelets – Compression Original 50:1 November 12, 2018

Wavelets – Compression Original 50:1 November 12, 2018

Wavelets – Compression November 12, 2018

Yet Another Example 50% 7% November 12, 2018

Final Example 2% 50% 100% 1% November 12, 2018

Information Rate Curve 0.0 0.2 0.4 0.6 0.8 1.0 normalized rate n o r m a l i z e d f t ( E ) density u momentum v momentum w momentum energy Energy Compaction – Few coefficients can efficiently represent functions The Curve should be as vertical as possible near 0 rate November 12, 2018

Filter Bank Implementation G: High Pass Filter H: Low Pass Filter November 12, 2018

Synthesis Bank November 12, 2018

Successive Approximations November 12, 2018

Successive Details November 12, 2018

Wavelet Representation November 12, 2018

Coefficients November 12, 2018

Lossey Compression November 12, 2018

Lossey Compression November 12, 2018

Image Example A Frame Another Frame November 12, 2018

Image Example Average Difference November 12, 2018

Wavelet Transform November 12, 2018

Frequency Support November 12, 2018

Image Example LvLh LvHh HvHh HvLh November 12, 2018

Image Example LvLh LvHh HvHh HvLh November 12, 2018

How Does One Do This ? November 12, 2018

Dilations Rescaling Operation t --> 2t Down Sampling, n --> 2n Halve function support Double frequency content Octave division of spectrum- Gives rise to different scales and resolutions Mother wavelet! - basic function gives rise to differing versions November 12, 2018

Dilations November 12, 2018

Successive Approximations November 12, 2018

Translations Covers space-frequency diagram Versions are November 12, 2018

Wavelet Decomposition Induced functional Space - Wj. Related to Vjs Space Wj+1 is orthogonal to Vj+1 Also J-level wavelet decomposition - November 12, 2018

Successive Differences November 12, 2018

Wavelet Expansion Wavelet expansion (Tiling- j: scale, k: translates), Synthesis Orthogonal transformation, Coarsest level of resolution - J Smoothing function - f, Detail function - y Analysis: Commonly used wavelets are Haar, Daubechies and Coiflets November 12, 2018

Scaling Functions Compact support Bandlimited - cut-off frequency Cannot achieve both DC value (or the average) is defined Translates of f are orthogonal November 12, 2018

Scaling Functions Nested smooth spaces Dilation Equation - Haar Generally – Frequency Domain November 12, 2018

Wavelet Functions Wavelet Equation - Haar System: G Filter Generally November 12, 2018

Perfect Reconstruction Synthesis and Analysis Filter Banks Synthesis Filters - Transpose of Analysis filters For compact scaling function November 12, 2018

Orthogonal Filter Banks Alternating Flip Not symmetric - h is even length! Example Orthogonality conditions November 12, 2018

Examples Haar Daubechies(2) November 12, 2018

Approximation: Vanishing Moments Property Function is smooth - Taylor Series expansion Wavelets with m vanishing moments Function with m derivatives can be accurately represented! November 12, 2018

Design of Compact Orthogonal Wavelets Compute scaling function Use Refinement Equation N vanishing moments property - H(w) has a zero of order N at w=p P(y) is pth order polynomial (Daubechies 1992) Maxflat filter November 12, 2018

Example N=4 November 12, 2018

Example N=16 November 12, 2018

Noise Uncorrelated Gaussian noise is correlated Region of correlation is small at coarse scale Smooth versions - no noise Orthogonal transform - uncorrelated November 12, 2018

Noise Across Scales November 12, 2018

Denoising Statistical thresholding methods [Donohoe] Assuming Gaussian Noise Universal Threshold Smoothness guaranteed Hard Soft Works for additive noise since wavelet transform is linear November 12, 2018

November 12, 2018

Discontinuity November 12, 2018

Multi-scale Edges Mallat and Hwang Location - maximas (edges) of wavelet coefficients at all scales Maxima chains for each edge Ranking - compute Lipschitz coefficient at all points Representation - store maximas Reconstruction- approximate but works in practice November 12, 2018

Bi-Orthogonal Filter Banks Analysis/synthesis different Aliasing - overlap in spectras Alias cancellation Distortion Free (phase shift l) Alternating Flip condition valid Can be odd length, symmetric November 12, 2018

Bi-Orthogonal Wavelets Governing equations Spline Wavelets - Many choices of either H0 or H1 Choose H 0 as spline and solve equations to generate H1 November 12, 2018

Bi-orthogonal: Lifting Scheme Lazy wavelet transform: split data in 2 parts Keep even part; predict (linear/cubic) odd part Lifting - update lj+1 with gj+1: Maintain properties (moments, avg.) Synthesis is just flip of analysis November 12, 2018

Summary Wavelets have good representation property They improve on image pyramid schemes Orthogonal and biorthogonal filter bank implementations are efficient Wavelets can filter signals They can efficiently denoise signals The presence of singularities can be detected from the magnitude of wavelet coefficients and their behavior across scales November 12, 2018