Signal reconstruction from multiscale edges A wavelet based algorithm
Author Yen-Ming Mark Lai Advisor Dr. Radu Balan CSCAMM, MATH
Motivation Save edges
Motivation Save edge type sharp one- sided edge sharp two- sided edge “noisy” edges
Motivation edgesedge typereconstruct +=
Algorithm Decomposition + Reconstruction
Decomposition Discrete Wavelet Transform Save edges e.g. local extrema Input “edges+edge type”
Decomposition input edge detection (scale 1) edge detection (scale 2) edge detection (scale 4) = = =
Reconstruction Find approximation Inverse Wavelet Transform Output local extrema “edges+edge type”
How to find approximation? Find approximation local extrema “edges+edge type”
Find approximation (iterative) Alternate projections between two spaces
Find approximation (iterative) sequences of functionswhose H1 normis finite
Find approximation (iterative)
sequences of functions: 1) interpolate input signal’s wavelet extrema 2) have minimal H1 norm
Q: Why minimize over H1 norm? A: Interpolation points act like local extrema
Numerical Example algorithm interpolates between points unclear what to do outside interpolation points
Find approximation (iterative)
dyadic wavelet transforms of L^2 functions
Find approximation (iterative) intersection = space of solutions
Find approximation (iterative) Start at zero element to minimize solution’s norm
Preliminary Results
Step Edge (length 8)
Quadratic Spline Wavelet
Take DWT
[1,-1](, Convolution in Matlab * [0,0,0,0,1,1,1,1]conv) next current +=next-current
* Convolution in Matlab next-current =0 next-current=-1
* Convolution in Matlab = next-current= 0next-current= 0next-current= 0next-current= 0next-current= 1next-current= 0next-current= 0next-current= 0 next-current= -1
Save Local Extrema
Interpolate DWT (Level 1) interpolation to minimize H1 norm unclear what to do outside interpolation points
error Original DWT – Level 1 Interpolated DWT – Level 1
error Original DWT – Level 2 Interpolated DWT – Level 2
error Original DWT – Level 3 Interpolated DWT – Level 3
matrix inversion failed Original DWT – Level 4 Interpolated DWT – Level 4
Interpolated DWT
Take IDWT to Recover Signal
Recovered Signal (Red) and Original Step Edge (Blue)
Summary
Choose Input
Take DWT
Save Local Extrema of DWT
Interpolate Local Extrema of DWT
Take IDWT
Issues Convolution detects false edges What to do with values outside interpolations points? What to do when matrix inversion fails?
Timeline Dec – write up mid-year report Jan– code local extrema search Oct/Nov – code Alternate Projections (90%) (85%) (100%)
Timeline February/March – test and debug entire system (8 weeks) April – run code against database (4 weeks) May – write up final report (2 weeks)
Questions?
Supplemental Slides
Input Signal (256 points) Which points to save?
Compressed Signal (37 points) What else for reconstruction?
Compressed Signal (37 points) sharp one-sided edge
Compressed Signal (37 points) sharp two-sided edge
Compressed Signal (37 points) “noisy” edges
Calculation Reconstruction: edges edge type information Original:(256 points) (37 points) (x points)
37 Compression edges edge type + x<256
Summary Save edges
Summary Save edge type sharp one- sided edge sharp two- sided edge “noisy” edges
Summary edgesedge typereconstruct +=
Algorithm Decomposition + Reconstruction
Decomposition Discrete Wavelet Transform Save edges e.g. local extrema Input “edges+edge type”
Reconstruction Find approximation Inverse Wavelet Transform Output local extrema “edges+edge type”
What is Discrete Wavelet Transform? Discrete Wavelet Transform Input
What is DWT? 1)Choose mother wavelet 2)Dilate mother wavelet 3)Convolve family with input DWT
1) Choose mother wavelet
2) Dilate mother wavelet mother wavelet dilate
2) Dilate mother wavelet
Convolve family with input input wavelet scale 1 wavelet scale 2 wavelet scale 4 = = =
Convolve “family” input wavelet scale 1 wavelet scale 2 wavelet scale 4 = = = DWT multiscale
What is DWT? (mathematically)
How to dilate? mother wavelet
How to dilate? dyadic (powers of two)
How to dilate? scale
How to dilate? z halve amplitude double support
Mother Wavelet (Haar) scale 1, j=0
Mother Wavelet (Haar) scale 2, j=1
Mother Wavelet (Haar) scale 4, j=2
What is DWT? Convolution of dilates of mother wavelets against original signal.
What is DWT? Convolution of dilates of mother wavelets against original signal. convolution
What is DWT? Convolution of dilates of mother wavelets against original signal. dilates
What is DWT? Convolution of dilates of mother wavelets against original signal. original signal
What is convolution? (best match operation) Discrete Wavelet Transform Input 1)mother wavelet 2)dilation 3)convolution
Convolution (best match operator) dummy variable
Convolution (best match operator) flip g around y axis
Convolution (best match operator) shifts g by t
do nothing to f Convolution (best match operator)
pointwise multiplication
Convolution (best match operator) integrate over R
flip g and shift by 7.7 Convolution (one point)
do nothing to f Convolution (one point)
multiply f and g pointwise Convolution (one point)
integrate over R Convolution (one point)
scalar
Convolution of two boxes
Why convolution? Location of maximum best fit
Where does red box most look like blue box?
Why convolution? Location of maximum best fit maximum
Why convolution? Location of maximum best fit maximabest fit location
Where does exponential most look like box?
maximum
Where does exponential most look like box? maximum best fit location
So what? If wavelet is an edge, convolution detects location of edges
Mother Wavelet (Haar)
What is edge? Local extrema of wavelet transform
Summary of Decomposition Discrete Wavelet Transform Save “edges” e.g. local extrema Input “edges+edge type”
Summary of Decomposition input edge detection (scale 1) edge detection (scale 2) edge detection (scale 4) = = =
How to find approximation? Find approximation local extrema “edges+edge type”
Find approximation (iterative) Alternate projections between two spaces
Find approximation (iterative)
H_1 Sobolev Norm
Find approximation (iterative) functions that interpolate given local maxima points
Find approximation (iterative) dyadic wavelet transforms of L^2 functions
Find approximation (iterative) intersection = space of solutions
Find approximation (iterative) Start at zero element to minimize solution’s norm
Q: Why minimize over K? A: Interpolation points act like local extrema
Reconstruction Find approximation (minimization problem) Inverse Wavelet Transform Output
Example Input of 256 points
Input Signal (256 points)
major edges
Input Signal (256 points) minor edges (many)
Discrete Wavelet Transform Dyadic (powers of 2) = DWT of “f” at scale 2^j
DWT (9 scales, 256 points each)
major edges
Input Signal (256 points) major edges
DWT (9 scales, 256 points each) minor edges (many)
Input Signal (256 points) minor edges (many)
Decomposition Discrete Wavelet Transform Save “edges” e.g. local extrema Input
DWT (9 scales, 256 points each)
Save Local Maxima
Local Maxima of Transform
low scale most sensitive
Mother Wavelet (Haar)
Local Maxima of Transform high scale least sensitive
Mother Wavelet (Haar)
Decomposition Discrete Wavelet Transform Save “edges” e.g. local extrema Input
Local Maxima of Transform
Find approximation (iterative) Alternate projections between two spaces
Reconstruction Find approximation (minimization problem) Inverse Wavelet Transform Output
Mallat’s Reconstruction (20 iterations)
original reconstruction (20 iterations)
Implementation Language: MATLAB –Matlab wavelet toolbox Complexity: convergence criteria
Databases Baseline signals –sinusoids, Gaussians, step edges, Diracs Audio signals
Validation Unit testing of components –DWT/IDWT –Local extrema search –Projection onto interpolation space (\Gamma)
Testing L2 norm of the error (sum of squares) versus iterations Saturation point in iteration (knee)
Schedule (Coding) October/November – code Alternate Projections (8 weeks) December – write up mid-year report (2 weeks) January – code local extrema search (1 week)
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)
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
Deliverables Documented MATLAB code Testing results (reproducible) Mid-year report/Final report