Signal reconstruction from multiscale edges A wavelet based algorithm

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

Reference “Characterization of Signals from Multiscale Edges” Stephane Mallat and Sifen Zhong IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, pp , July 1992

Motivation

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

Summary Discrete Wavelet Transform Save edges e.g. local extrema Input Find approximation Inverse Wavelet Transform Output Questions?

Supplemental Slides

Similar Idea to JPEG Discrete Fourier Transform Save largest coefficients Inverse Discrete Fourier Transform

Comparison of algorithms DFT Save largest coefficients IDFT DWT (Redundant) Save local maxima on each scale Find approximation IDWT

Problem Definition Given: –positions –values of local maxima of |W_2^j f(x)| at each scale Find: –Approximation h(x) of f(x) –or equivalently W_{2^j} h(x)

Maxima Constraint I At each scale 2^j, for each local maximum located at x_n^j, e.g. W h(x) = W f(x) at given set of local maxima points (interpolation problem) at each scale

Maxima Constraint II At each scale 2^j, the local maxima of |W_2^j h(x)| are located at the abscissa (x_n^j)_n \in Z e.g Local maxima of |W h(x)| are local maxima of |W f(x)| at each scale

Maxima Constraint II Constraint not convex (difficult to analyze) Use convex constraint instead: local maxima of |W h(x)| at certain points |W h(x)|^2 and |d W h(x)|^2 small as possible on average OriginalApproximation

Maxima Constraint “II” Minimize |W h(x)|^2  creates local maxima at specified positions Minimize |d W h|^2  minimize modulus maxima outside specified positions

Maxima Constraint “II” Solve minimization problem |||h|||^2= \sum_j ( ||W_{2^j} h ||^2 + 2^{2j} || dW_{2^j}h/ dx ||^2)

Solve minimization problem Let K be the space of all sequences of fucntions (g_j^(x) )_j \in Z such that |(g_j(x))_j \in Z|^2 = \sum_j ( \| g_j\| ^2 + 2^{2j} \| \frac{dg_j}{dx} \|^2 < \infty (inspired by condition “II”)

V space Let V be the space of all dyadic wavelet tranforms of L^2(R). V \subset K

\Gamma space Let \Gamma be the affine space of sequences of functions (g_j(x))_j \in Z \in K such that for any index j and all maxima positions x_n^j g_j(x_n^j)= W_{2^j} f(x_n^j) (inspired by Condition I)

\Gamma space One can prove that \Gamma is closed in K

Recap K: sequence of functions whose sum of –norm of each element –norm of each element’s derivative is finite V: dyadic wavelet transform of L^2 \Gamma: sequences of functions whose value match those of |W f| at local maxima

Condition I in K Dyadic wavelet transforms that satisfy Condition I \Lambda = V \cap \Gamma

Condition I + “II” in K Find element of \Lambda = V \cap \Gamma whose norm is minimum Use alternate projections on V and \Gamma

Projection on V P_V = W \circ W^{-1} In other words, First: W^-1  dyadic inverse wavelet transform Then: W  dyadic wavelet transform

Projection on \Gamma At each scale 2^j, add discrete signal e_j^d = ( e_j(n))_{1 \leq n \leq N} that is computed from e_j(x) = \alpha e^{2^{-jx}} + \beta e^{-2^{-jx}} e.g. Add piecewise exponential curves to each function of sequence

How to compute coefficients? Solve system of equations given by equation (110)

Where to start? Answer: zero element of K e.g. g_j(x) = 0 for all j \in Z

Converge to what? Answer: Alternate projections converge to –element of \Lambda (Condition I) –whose norm is minimum (Condition “II”)

How fast is convergence? Answer: If (\sqrt{2^j} \psi_{2^j}(x_n^j-x)) is frame and there exists constant 0< D <=1 such that at all scales 2^j the distances between any two consecutive maxima satisfy |x_n^j – x_{n-1}^j | \geq D2^j then, convergence is exponential

Convex constraint?