Signal reconstruction from multiscale edges A wavelet based algorithm

Algorithm

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

Reconstruction Find approximation Inverse Wavelet Transform Output local extrema contained two bugs

What is Discrete Wavelet Transform?

Discrete Wavelet Transform (1 level) input Detail Coefficients Approximation Coefficients

input Detail Level 1 Detail Level 2 Detail Level 3 Approximation Level 3

Input + Details (3 Levels)

Input + Approximation (3 Levels)

DWT + IDWT input Detail Coefficients Approximation Coefficients + output

Perfect Reconstruction Property + XX =

+ XX = Bug: Incorrect coefficient given in paper

Perfect Reconstruction Property + XX = Given Correct

input Detail Coefficients Approximation Coefficients + output + XX =

Detail (L3) Approx. (L3) + + Detail (L2) + Detail (L1) output

Detail (L3) Approx. (L3) + + Detail (L2) + Detail (L1) output Bug: cascade not implemented correctly

Validation

Interpolation Space  check for interpolation DWT Space  check for perfect reconstruction DWT -> IDWT = identity

Results

Step Edge (Length 32)

Details (5 levels)

Approximation (5 levels)

All coefficients discarded since last level zero

Detail (L3) Approx. (L3) + + Detail (L2) + Detail (L1) output

Extrema (20/32 points, 63% compression)

Find approximation (iterative) Alternate projections between two spaces

Reconstructed (Iteration 1, 11, 21, 31)

l2 error (31 iterations)

Previous Result with Bug (Iteration 1, 11, 21)

Previous Result with Bug

Gaussian (Length 32)

Details (5 Levels)

Approximation (5 Levels)

Extrema (19/32 points, 59% compression)

Reconstructed (Iteration 1, 11, 21, 31)

l2 error (31 iterations)

Previous Result with Bug (Iteration 1, 11, 21)

Previous Result with Bug

Dirac Delta (Length 32)

Details (5 Levels)

Approximation (5 Levels)

Extrema (20/32 points, 62% compression)

Reconstructed (Iteration 1, 11, 21, 31)

l2 error (31 iterations)

Sinusoid (Length 32)

Details (5 Levels)

Approximation (5 Levels)

Extrema (22/32 points, 69% compression)

Reconstructed (Iteration 1, 11, 21, 31)

l2 error (31 iterations)

Triangle (Length 32)

Details (5 Levels)

Approximation (5 Levels)

Extrema (33/32 points,- 3% compression)

Reconstructed (Iteration 1, 11, 21, 31)

l2 error (31 iterations)

Random Noise (Length 32)

Details (5 Levels)

Approximation (5 Levels)

Extrema (40/32 points,- 25% compression)

Reconstructed (Iteration 1, 11, 21, 31)

l2 error (31 iterations)

Audio Signals

Source: Zoolander (2001)

Reconstructed Signal

Zoom 1

Reconstructed Signal

Zoom 2

Reconstructed Signal

Zoom 3

WindowExtremaWindowExtremaWindowExtremaWindowExtrema Extrema Saved For Each Window

Extrema versus Level (Window 9)

Input

Reconstructed

1)Do not do maximum DWT decomposition Lowest approximation coefficient will not be constant 2)Interpolate approximation coefficients instead Future Work

input Detail Level 1 Detail Level 2 Detail Level 3 Approximation Level 3

input Detail Level 1 Detail Level 2 Approximation Level 2

Questions?

Supplemental Slides

How to find approximation?

What do we know about saved points? 1) Local extrema 2) Come from wavelet transform

Find approximation (iterative) Alternate projections between two spaces

Find approximation (iterative) sequences of functionswhose H1 normis finite

Find approximation (iterative)

sequences of functions: interpolate input signal’s wavelet extrema

Sample Element from Γ

Find approximation (iterative) What happens when element is projected onto Γ? 1) Saved points are interpolated 2) H1 norm minimized

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

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

Find approximation (iterative) How to project onto V space (DWT space)?

Project onto V space (DWT space) IDWT input (interpolated functions) DWT output (DWT of l2 function) l2 function not necessarily DWT of l2 function

Find approximation (iterative) start at zero element

What do we know about saved points? 1) Local extrema 2) Come from wavelet transform

Find approximation (iterative) intersection = space of solutions interpolation space wavelet transform space

Error Analysis (Step Edge)

Step Edge (Length 32)

Discrete Wavelet Transform (3 levels)

Save Points (Extrema + Ends)

Alternate Projection (Initialization) zero out all wavelet values

Alternate Projection (Iteration 1a) project onto Γ space (interpolate points)

DWT after Γ Projection (1 iteration) extrema points interpolated

DWT after Γ Projection (1 iteration) same concavity

Alternate Projection (Iteration 1b) project onto V space (DWT space)

false edge(periodic padding)

save end points(interpolate end intervals)

Discrete Wavelet Transform input edge detection (scale 1) edge detection (scale 2) edge detection (scale 4) = = =