Predicting Wavelet Coefficients Over Edges Using Estimates Based on Nonlinear Approximants

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Predicting Wavelet Coefficients Over Edges Using
Estimates Based on Nonlinear Approximants

• Onur G. Guleryuz
• oguleryuz_at_erd.epson.com
• Epson Palo Alto Laboratory
• Palo Alto, CA
• google Onur Guleryuz

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Overview
Topic Wavelet compression of piecewise smooth
signals with edges.
(piecewise sparse)
Benchmark scenario
Piecewise smooth signal
Erase all high frequency wavelet coefficients
Predict erased data
mse?
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Notes
Q What are edges? (Vague and loose) A Edges
are localized singularities that separate
statistically uniform regions of a nonstationary
process.

• Caveats
• This method is not
• edge/singularity detection,
• convex (and therefore not POCS),
• solving inverse problems under additive noise
  (wavelet-vaguelette),
• an explicit edge/singularity model.

No amount of looking at one side helps predict
the other side.

• This method is
• a systematic way of constructing adaptive linear
  estimators,
• an adaptive sparse reconstruction,
• based on sparse nonlinear approximants
  (non-convex by design),
• a model for non-edges (sparsity/predictable
  detection).

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Wavelet Compression in 1-D and 2-D
1-D
Wavelets of compact support achieve sparse
decompositions
A. Cohen, I. Daubechies, O. G. Guleryuz, and M.
T. Orchard, On the importance of combining
wavelet-based nonlinear approximation with coding
strategies,'' IEEE Trans. Info. Theory, vol.
48, no. 7, pp. 1895-1921, July 2002.
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Current Approaches
1 Modeling higher order dependencies over
edges in wavelet domain.

• F. Arandiga, A. Cohen, M. Doblas, and B. Matei,
  Edge Adapted Nonlinear Multiscale Transforms
  for Compact Image Representation ,' Proc. IEEE
  Int. Conf. Image Proc., Barcelona, Spain, 2003.

• H. F. Ates and M. T. Orchard, Nonlinear
  Modeling of Wavelet Coefficients around Edges,'
  Proc. IEEE Int. Conf. Image Proc., Barcelona,
  Spain, 2003.

(Reduce by prediction)
2 New Representations.

• J. Starck, E. J. Candes, and D. L. Donoho, The
  Curvelet Transform for Image Denoising,' IEEE
  Trans. on Image Proc., vol. 11, pp. 670-684, 2002.

• M. Wakin, J. Romberg, C. Hyeokho, and R.
  Baraniuk, Rate-distortion optimized image
  compression using wedgelets,' Proc. IEEE Int.
  Conf. Image Proc. June 2002.

• P.L. Dragotti and M. Vetterli, Wavelet
  footprints theory, algorithms, and
  applications,' IEEE Trans. on Sig. Proc., vol.
  51, pp. 1306-1323, 2003.

(Dont create too many)
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Q What are Overcomplete Transforms?
Example Translation invariant, overcomplete
transforms

• Spatial DCT tilings of an Image

image-wide, orthonormal transform
Image arranged in a (Nx1) vector x,
are (NxN)
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Sparse Decompositions and Overcomplete Transforms
No single orthonormal transform in the
overcomplete set provides a very sparse
decomposition.
sparse portions
nonsparse portions
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Issues with Overcomplete Trfs
Compression angle
Thresholding based Denoising
sparse portions
nonsparse portions
image (x)
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DCC02
http//eeweb.poly.edu/onur
Onur G. Guleryuz, "Nonlinear Approximation Based
Image Recovery Using Adaptive Sparse
Reconstructions and Iterated Denoising Part I -
Theory, Part II Adaptive Algorithms, IEEE
Transactions on Image Processing, in review.
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Nonlinear Approximation and Nonconvex Image Models
Assume single transform
missing sample
available sample
Recovery transform coordinates
Sample coordinates for a two sample signal
Find the missing data to minimize
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Underlying Estimation Method
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Modeling Non-Edges (Sparse Regions)
DCT1
DCT2shift(DCT1)
DCTM
edge
smooth
smooth
I dont care how badly the transform I am using
does over the edges. I determine non-edges
aggressively.
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Algorithm
Fill missing information (high frequency wavelet
coefficients) with initial values (0), TT .
0
Denoise image with hard-threshold T.
Enforce available information (low frequency
wavelet coefficients).
TT-dT
I use DCTs and a simple but good denoising
technique
http//eeweb.poly.edu/onur
Onur G. Guleryuz, Weighted Overcomplete
Denoising, Proc. Asilomar Conference on Signals
and Systems, Pacific Grove, CA, Nov. 2003.
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Test Images
Graphics (512x512)
Bubbles (512x512)
Cross (512x512)
Pattern (512x512)
I admit, you can do edge detection on this one
Teapot (960x1280)
Lena (512x512)
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Implementation
1 l-level wavelet transform (l1, l2)
2 All high frequency coefficients set to zero
(l1 half resolution, l2 quarter resolution)
3 Predict missing information
4 Report PSNR10log10(255255/mse)
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Results on Graphics
Graphics, l1
Graphics, l2
30.48dB to 51dB
27.15dB to 37.44dB
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Results on Bubbles
Bubbles, l1
Bubbles, l2
33.10dB to 35.10dB
29.03dB to 30.14dB
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Bubbles crop, l1
magnitude info. location info
Unproc. 30.41dB
Predicted 33.00dB
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Bubbles crop, l2
Unproc. 26.92dB
Predicted 28.20dB
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Pattern crop, l1
Holder exponent extrapolation, step edge
assumption, edge detection, etc., arent going
to work well here.
still a jump
Unproc. 25.94dB
Predicted 26.63dB
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Cross crop, l1
Holder exponent extrapolation, step edge
assumption, edge detection, etc., arent going
to work well here.
Unproc. 18.52dB
Predicted 18.78dB
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PSNR over 3 and 5 pixel neighborhood of edges
(l1)
21 dB
21 dB
2 dB
4 dB
0.5 dB
2 dB
0 dB
1.5 dB
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Comments and Conclusion

• I will show a few more results.
• Around edges, magnitude and location
  distortions.
• Instead of trying to model many different types
  of edges, model non-edges as sparse (same
  algorithm handles all varieties).
• Early work 1 Interpolation in pixel domain may
  give misleading PSNR numbers for two reasons.
• Early work 2 Hemamis group and Vetterlis
  group have wavelet domain results (based on
  Holder exponents), but not on same scale.
• You can implement this for your own
  transform/filter bank
• (denoise, available info, reduce
  threshold, ).

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Results on Teapot
Teapot, l1
Teapot, l2
36.17dB to 41.81dB
32.54dB to 35.93dB
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Teapot crop, l1
Unproc. 28.38dB
Predicted 34.78dB
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Teapot crop, l2
Unproc. 25.10dB
Predicted ??.??dB
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Results on Lena
Lena, l1
Lena, l2
35.26dB to 35.65dB
29.58dB to 30.04dB
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Lena crop, l1
Unproc. 34.42dB
Predicted 35.03dB
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Lena crop, l2
Unproc. 27.79dB
Predicted 29.83dB