signal reconstruction from multiscale edges

Signal reconstruction from multiscale edges

A wavelet based algorithm

Author

Yen-Ming Mark Lai

(ylai@amsc.umd.edu)

Advisor

Dr. Radu Balan

rvbalan@math.umd.edu

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 710-732, 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

edges edge type reconstruct+ =

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 waveletmother wavelet

dilate

2) Dilate mother wavelet

Convolve family with input

input

input

input

wavelet scale 1

wavelet scale 2

wavelet scale 4

=

=

=

Convolve “family”

input

input

input

wavelet scale 1

wavelet scale 2

wavelet scale 4

=

=

=

DWT

multiscale

What is DWT?

(mathematically)

How to dilate?

)2(

2

1)(

2 jj

xxj

mother wavelet

How to dilate?

)2(

2

1)(

2 jj

xxj

dyadic (powers of two)

How to dilate?

)2(

2

1)(

2 jj

xxj

scale

How to dilate?

)2(

2

1)(

2 jj

xxj 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.

)()2(

2

1)()(

2xf

xxfx

jjj

What is DWT?

Convolution of dilates of mother wavelets against original signal.

)()2(

2

1)()(

2xf

xxfx

jjj

convolution

What is DWT?

Convolution of dilates of mother wavelets against original signal.

)()2(

2

1)()(

2xf

xxfx

jjj

dilates

What is DWT?

Convolution of dilates of mother wavelets against original signal.

)()2(

2

1)()(

2xf

xxfx

jjj

original signal

What is convolution?(best match operation)

Discrete Wavelet

Transform

Input1)mother wavelet

2)dilation

3)convolution

Convolution (best match operator)

dtgftgf )()()(

dummy variable

Convolution (best match operator)

flip g around y axis

dtgftgf )()()(

dtgftgf )()()(

Convolution (best match operator)

shifts g by t

dtgftgf )()()(

do nothing to f

Convolution (best match operator)

dtgftgf )()()(

Convolution (best match operator)

pointwise multiplication

dtgftgf )()()(

Convolution (best match operator)

integrate over R

)7.7(gf

flip g and shift by 7.7

dgf )7.7()(

Convolution (one point)

)7.7(gf

do nothing to f

dgf )7.7()(

Convolution (one point)

)7.7(gf

multiply f and g pointwise

dgf )7.7()(

Convolution (one point)

)7.7(gf

integrate over R

dgf )7.7()(

Convolution (one point)

)7.7(gf

Convolution (one point)

scalar

Convolution of two boxes

f

Convolution of two boxes

45.1g

Convolution of two boxes

45.1t

Convolution of two boxes

dgfgf )45.1()()45.1( 0

Convolution of two boxes

f

Convolution of two boxes

5.0g

Convolution of two boxes

5.0t

Convolution of two boxes

dgfgf )5.0()()5.0( 5.0

Convolution of two boxes

Convolution of two boxes

gf

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

maxima best fit location

Where does exponential most look like box?

Where does exponential most look like box?

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)

Mother Wavelet (Haar)

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

input

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)

2'2

22:LnLnn ffnf

Find approximation (iterative)

2'2

22:LnLnn ffnf

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

2'2

22:LnLnn ffnf

Reconstruction

Find approximation(minimization

problem)

Inverse Wavelet

Transform

Output

Example

Input of 256 points

Input Signal (256 points)

Input Signal (256 points)

major edges

Input Signal (256 points)

minor edges (many)

Discrete Wavelet Transform

fW dj2

Dyadic (powers of 2)

= DWT of “f” at scale 2^j

DWT (9 scales, 256 points each)

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

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, 2011 - 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

TransformOutput

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

)()(22

jn

jn xfWxhW jj

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

Original Approximation

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?