Signal reconstruction from multiscale edges

Signal reconstruction from multiscale edges

A wavelet based algorithm

Author

Yen-Ming Mark Lai

([email protected])

Advisor

Dr. Radu Balan

[email protected]

CSCAMM, MATH

Motivation

Save edges

Motivation

Save edge type

sharp one-sided edge

sharp two-sided edge

“noisy” edges

Motivation

edges edge type reconstruct+ =

Algorithm

Decomposition + Reconstruction

Decomposition

Discrete Wavelet

Transform

Save edges e.g. local extrema

Input

“edges+edge type”

Decomposition

input

input

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 functions

n: nf

whose H1 norm

2'2

22LnLn ff

is finite

Find approximation (iterative)

0k

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)

0

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

Take DWT

[1,-1]( ,

Convolution in Matlab

*[0,0,0,0,1,1,1,1]conv )

next

1 current 1+ =next-current

*

Convolution in Matlab

next-current =0 next-current =-1

*

Convolution in Matlab

=

next-current = 0

next-current = 0

next-current = 0

next-current = 0

next-current = 1

next-current = 0

next-current = 0

next-current = 0

next-current = -1

Save Local Extrema

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

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