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Signal reconstruction from multiscale edges A wavelet based algorithm

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Author Yen-Ming Mark Lai Advisor Dr. Radu Balan CSCAMM, MATH

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Motivation Save edges

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Motivation Save edge type sharp one- sided edge sharp two- sided edge “noisy” edges

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Motivation edgesedge typereconstruct +=

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Algorithm Decomposition + Reconstruction

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Decomposition Discrete Wavelet Transform Save edges e.g. local extrema Input “edges+edge type”

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Decomposition input edge detection (scale 1) edge detection (scale 2) edge detection (scale 4) = = =

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Reconstruction Find approximation Inverse Wavelet Transform Output local extrema “edges+edge type”

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How to find approximation? Find approximation local extrema “edges+edge type”

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Find approximation (iterative) Alternate projections between two spaces

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Find approximation (iterative) sequences of functionswhose H1 normis finite

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Find approximation (iterative)

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sequences of functions: 1) interpolate input signal’s wavelet extrema 2) have minimal H1 norm

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Q: Why minimize over H1 norm? A: Interpolation points act like local extrema

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Numerical Example algorithm interpolates between points unclear what to do outside interpolation points

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Find approximation (iterative)

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dyadic wavelet transforms of L^2 functions

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Find approximation (iterative) intersection = space of solutions

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Find approximation (iterative) Start at zero element to minimize solution’s norm

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Preliminary Results

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Step Edge (length 8)

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Quadratic Spline Wavelet

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Take DWT

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[1,-1](, Convolution in Matlab * [0,0,0,0,1,1,1,1]conv) next current +=next-current

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* Convolution in Matlab next-current =0 next-current=-1

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* Convolution in Matlab = next-current= 0next-current= 0next-current= 0next-current= 0next-current= 1next-current= 0next-current= 0next-current= 0 next-current= -1

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Save Local Extrema

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Interpolate DWT (Level 1) interpolation to minimize H1 norm unclear what to do outside interpolation points

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error Original DWT – Level 1 Interpolated DWT – Level 1

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error Original DWT – Level 2 Interpolated DWT – Level 2

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error Original DWT – Level 3 Interpolated DWT – Level 3

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matrix inversion failed Original DWT – Level 4 Interpolated DWT – Level 4

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Interpolated DWT

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Take IDWT to Recover Signal

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Recovered Signal (Red) and Original Step Edge (Blue)

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Summary

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Choose Input

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Take DWT

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Save Local Extrema of DWT

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Interpolate Local Extrema of DWT

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Take IDWT

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Issues Convolution detects false edges What to do with values outside interpolations points? What to do when matrix inversion fails?

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Timeline Dec – write up mid-year report Jan– code local extrema search Oct/Nov – code Alternate Projections (90%) (85%) (100%)

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Timeline February/March – test and debug entire system (8 weeks) April – run code against database (4 weeks) May – write up final report (2 weeks)

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Questions?

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Supplemental Slides

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Input Signal (256 points) Which points to save?

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Compressed Signal (37 points) What else for reconstruction?

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Compressed Signal (37 points) sharp one-sided edge

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Compressed Signal (37 points) sharp two-sided edge

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Compressed Signal (37 points) “noisy” edges

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Calculation Reconstruction: edges edge type information Original:(256 points) (37 points) (x points)

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37 Compression edges edge type + x<256

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Summary Save edges

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Summary Save edge type sharp one- sided edge sharp two- sided edge “noisy” edges

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Summary edgesedge typereconstruct +=

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Algorithm Decomposition + Reconstruction

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Decomposition Discrete Wavelet Transform Save edges e.g. local extrema Input “edges+edge type”

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Reconstruction Find approximation Inverse Wavelet Transform Output local extrema “edges+edge type”

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What is Discrete Wavelet Transform? Discrete Wavelet Transform Input

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What is DWT? 1)Choose mother wavelet 2)Dilate mother wavelet 3)Convolve family with input DWT

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1) Choose mother wavelet

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2) Dilate mother wavelet mother wavelet dilate

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2) Dilate mother wavelet

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Convolve family with input input wavelet scale 1 wavelet scale 2 wavelet scale 4 = = =

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Convolve “family” input wavelet scale 1 wavelet scale 2 wavelet scale 4 = = = DWT multiscale

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What is DWT? (mathematically)

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How to dilate? mother wavelet

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How to dilate? dyadic (powers of two)

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How to dilate? scale

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How to dilate? z halve amplitude double support

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Mother Wavelet (Haar) scale 1, j=0

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Mother Wavelet (Haar) scale 2, j=1

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Mother Wavelet (Haar) scale 4, j=2

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What is DWT? Convolution of dilates of mother wavelets against original signal.

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What is DWT? Convolution of dilates of mother wavelets against original signal. convolution

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What is DWT? Convolution of dilates of mother wavelets against original signal. dilates

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What is DWT? Convolution of dilates of mother wavelets against original signal. original signal

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What is convolution? (best match operation) Discrete Wavelet Transform Input 1)mother wavelet 2)dilation 3)convolution

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Convolution (best match operator) dummy variable

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Convolution (best match operator) flip g around y axis

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Convolution (best match operator) shifts g by t

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do nothing to f Convolution (best match operator)

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pointwise multiplication

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Convolution (best match operator) integrate over R

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flip g and shift by 7.7 Convolution (one point)

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do nothing to f Convolution (one point)

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multiply f and g pointwise Convolution (one point)

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integrate over R Convolution (one point)

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scalar

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Convolution of two boxes

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Why convolution? Location of maximum best fit

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Where does red box most look like blue box?

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Why convolution? Location of maximum best fit maximum

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Why convolution? Location of maximum best fit maximabest fit location

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Where does exponential most look like box?

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maximum

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Where does exponential most look like box? maximum best fit location

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So what? If wavelet is an edge, convolution detects location of edges

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Mother Wavelet (Haar)

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What is edge? Local extrema of wavelet transform

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Summary of Decomposition Discrete Wavelet Transform Save “edges” e.g. local extrema Input “edges+edge type”

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Summary of Decomposition input edge detection (scale 1) edge detection (scale 2) edge detection (scale 4) = = =

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How to find approximation? Find approximation local extrema “edges+edge type”

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Find approximation (iterative) Alternate projections between two spaces

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Find approximation (iterative)

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H_1 Sobolev Norm

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Find approximation (iterative) functions that interpolate given local maxima points

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Find approximation (iterative) dyadic wavelet transforms of L^2 functions

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Find approximation (iterative) intersection = space of solutions

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Find approximation (iterative) Start at zero element to minimize solution’s norm

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Q: Why minimize over K? A: Interpolation points act like local extrema

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Reconstruction Find approximation (minimization problem) Inverse Wavelet Transform Output

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Example Input of 256 points

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Input Signal (256 points)

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major edges

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Input Signal (256 points) minor edges (many)

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Discrete Wavelet Transform Dyadic (powers of 2) = DWT of “f” at scale 2^j

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DWT (9 scales, 256 points each)

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major edges

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Input Signal (256 points) major edges

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DWT (9 scales, 256 points each) minor edges (many)

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Input Signal (256 points) minor edges (many)

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Decomposition Discrete Wavelet Transform Save “edges” e.g. local extrema Input

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DWT (9 scales, 256 points each)

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Save Local Maxima

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Local Maxima of Transform

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low scale most sensitive

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Mother Wavelet (Haar)

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Local Maxima of Transform high scale least sensitive

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Mother Wavelet (Haar)

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Decomposition Discrete Wavelet Transform Save “edges” e.g. local extrema Input

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Local Maxima of Transform

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Find approximation (iterative) Alternate projections between two spaces

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Reconstruction Find approximation (minimization problem) Inverse Wavelet Transform Output

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Mallat’s Reconstruction (20 iterations)

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original reconstruction (20 iterations)

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Implementation Language: MATLAB –Matlab wavelet toolbox Complexity: convergence criteria

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Databases Baseline signals –sinusoids, Gaussians, step edges, Diracs Audio signals

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Validation Unit testing of components –DWT/IDWT –Local extrema search –Projection onto interpolation space (\Gamma)

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Testing L2 norm of the error (sum of squares) versus iterations Saturation point in iteration (knee)

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Schedule (Coding) October/November – code Alternate Projections (8 weeks) December – write up mid-year report (2 weeks) January – code local extrema search (1 week)

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Schedule (Testing) February/March – test and debug entire system (8 weeks) April – run code against database (4 weeks) May – write up final report (2 weeks)

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Milestones December 1, 2010 – Alternate Projections code passes unit test February 1, 2011 – local extrema search code passes unit test April 1, codes passes system test

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Deliverables Documented MATLAB code Testing results (reproducible) Mid-year report/Final report

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