Transcript Slide 1

Image Compression with
a Geometrical Entropy
Coder
Onur G. Guleryuz and Arthur L. da Cunha
DoCoMo USA Labs
Palo Alto, CA
University of Illinois at
Urbana-Champaign,
Champaign, IL
1
Overview
In a nutshell: This work is about compressing images with
geometrical singularities (~ edges along curves). We utilize a
known suboptimal transform (the wavelet transform) and try to
improve its performance on such images by adding singularity
intelligence (~directional prediction).

Summary of transform compression.






Optimality in 1-D vs. in 2-D.
Illustration of the problems for wavelet transforms in 2-D.
Our approach with examples.
Algorithm outline.
Results
Conclusion
2
Transform Compression of Signals
Type of signal:
Stationary, Gaussian signals
Optimal Transform Compression:
Karhunen-Loeve transform
+ “classical” coefficient coder
[Gersho&Gray]
1-D Piecewise smooth with
point singularities at random
locations
Wavelet transform
2-D Piecewise smooth with
singularities along “geometric”
curves
Curvelet/Contourlet/... transform
Classical Coefficient Coder Syntax:
Send number of “significant” coefficients
Send each significant coefficient
+ “modern” coefficient coder
[Falzon&Mallat, Donoho-Vetterli-DeVore-Daubechies, CohenDaubechies-Guleryuz-Orchard]
+ modern coefficient coder
[Candes&Donoho, Do&Vetterli, Peyre&Mallat, Cohen, Simoncelli
and others]
Modern Coefficient Coder Syntax:
Send number of significant coefficients
Send which coefficients are significant.
Send each significant coefficient
[JPEG,SPIHT,JPEG2000,...]
3
Practical Algorithms in TwoDimensions
 There are many practical difficulties that newly designed transforms must surmount.
 New, compression targeted image-adaptive representations and transforms try to
overcome these issues
 Wavelet footprints, Bandelettes, Directionlets, h.264/AVC INTRA mode,
directional lifting based approaches …
 Simple Observation: While wavelet based algorithms are not good over geometric
singularities, they are very good away from geometric singularities (it is difficult to beat
wavelet coders on images with few singularities).
 Our approach: Take something that works well (wavelets) and add singularity
intelligence to it (directional prediction using geometric flow). Try to get the best of
both worlds.
4
Why?
Practical issues that motivate this work:
 General design steps for new transform compression algorithms:
• Design new representation
• Design associated modern coefficient coder
• Test against the state-of-the-art (SPIHT, JPEG2000, ...)
 Our approach:
 Keep old representation but add geometric prediction
 Optimally (R-D) incorporate inside state-of-the-art, modern
coefficient coder (SPIHT in our case)
 Always do equal or better than the state-of-the art
Hard
Hard

Hard
Easy

Conceptual issues that motivate this work:
 Can we exploit regularity in wavelet domain using simple, first principal
approaches rather than using sophisticated mathematical constructs (such
as approaches based on complex wavelets, Alpert wavelets, etc.) ?
5
Simple Illustration of the Problem
Faced by the Wavelet Transform
TOO MANY SIGNIFICANT
COEFFICIENTS!
2-D Discrete Wavelet
Transform (DWT)
LHLH
HLHL
HHHH
LH Band
Legend for wavelet transform
image:
gray : zero valued coefficient,
light : large positive value,
dark : large negative value
HL Band
HH Band
6
Our Solution: New Geometrical
Representation
Image
Wavelet
Transform
Wavelet
Coefficients
Image adaptive
geometrical
transform
New
Coefficients
Image adaptive
“Flow”
New Syntax:
Send geometric flow.
Send number of “significant” coefficients
Send which coefficients are significant.
Send each coefficient.
(i.e., send flow then use a modern coefficient
coder to code new coefficients.)
7
What is a “Flow”
The flow field is a field along which a function is regular (~ two points connected by a
flow do not have a singularity between them along the flow.)
Image
Image with
superimposed flow
...
8
Algorithm Outline & Properties
• Calculate wavelet transform coefficients.
• Compute a set of helper variables to use in flow based prediction.
• Compute optimal flow (each wavelet band has its own flow).
• Encode flow.
• Generate a new set of coefficients conditioned on the flow (do directional
prediction in wavelet domain based on the flow and generate prediction errors).
• Encode the new coefficients with a modern coefficient coder (SPIHT).
9
Example
• Our algorithm amounts to doing directional prediction in wavelet domain.
Computed flow on LH band
...
LHLH
LH Band
New LHLH
New LH Band
Too easy!
10
Directional prediction in wavelet
domain is hard!
11
UDWT as Helper in Prediction
• We causally compute undecimated wavelet transform coefficients and use
them as helper variables in prediction.
DWT (NxN)
(Flow direction is unclear)
UDWT (2Nx2N)
(Flow direction is clear)
12
Harder Example
LH Band
Computed flow on LH Band
(superimposed on the image)
New LH Band
(~4dB better in R-D)
13
The picture faced by our predictor
We generate UDWT coefficients, i.e., auxiliary variables
(128x128)
(256x256)
Difficult prediction
Easier prediction
14
Views of Data to be Predicted
(Profile depends on wavelet
filter and singularity
structure)
 We optimize over a discretized set of directions using simple interpolation.
 We use AR prediction.
15
Example Flows Computed on LH
Band
16
Algorithm Details
 Separate flow for each band. Flow on a quad-tree. Optimized using bottomup dynamic programming.
 Causal UDWT computation (this is detailed, please look a the paper if
interested):
 Two buffers,
spatial domain buffer of current
image approximation
UDWT buffer
 Encode next DWT coefficient. Update spatial buffer. Update UDWT buffer.
(Decoder does the same).
 For each DWT coefficient:
 Form a 1D causal sequence of coefficient values along the flow (UDWT).
 Use these to predict DWT coefficient
17
Two Coders
“Geometrical Entropy Coder”
Image
DPCM
Image
Wavelet
Transform
Wavelet
Coefficients
Wavelet
Transform
Wavelet
Coefficients
Quantization
Quantization +
Flow based
prediction
Quantized
Coefficients
(Invertible)
New
Coefficients
Flow based
prediction
New
Coefficients
(The utilized flow is the one that is
rate optimized for the geometric
entropy coder.)
18
Some Rate-Distortion Results
(Computer generated image)
Better
SPIHT mode (better by ~ 1+ dB)
First order entropy (better by ~ 2+
dB)
19
Some Rate-Distortion Results
SPIHT mode (better by ~ 1 dB)
First order entropy (better by ~ 220dB)
Some Rate-Distortion Results
SPIHT mode (better by ~ 0.6 dB)
First order entropy (better by ~ 1.521dB)
Conclusion
• Flow bits < %5 (we need to encode the flow better to be more competitive).
• Current state of our work:
• Currently “up to” ~ %40 rate improvements on toy images and ~ %10 rate
improvements on some natural images.
• Lena, boat, etc., ~ small improvements
• Better prediction and interpolation will improve our results.
• Issues with bit-plane coders (255=128).
• Our technique keeps working at low rates (the results in the paper stop predicting
after a certain DWT level) .
• Advantages of our work:
• We do directional prediction along lines but we can also do so along curves easily.
• Our work can be used to design sophisticated context based arithmetic coders.
22
[1] E. J. Cand`es and D. L. Donoho, “New tight frames of curvelets and optimal
representations of objects with piecewise C2 singularities,” Comm. Pure and Appl.
Math, vol. 57, no. 2, pp. 219–266, February 2004.
[2] E. Le Pennec and S. Mallat, “Sparse geometric image representation with bandelets,”
IEEE Trans. Image Proc., vol. 14, no. 4, pp. 423–438, April 2005.
[3] M. B. Wakin, J. K. Romberg, H. Choi, and R. G. Baraniuk, “Wavelet-domain approximation
and compression of piecewise smooth images,” in IEEE Trans. Image Proc., to appear, 2005.
[4] R. Shukla, P. L. Dragotti, M. N. Do, and M. Vetterli, “Rate-distortion optimized
tree structured compression algorithms for piecewise smooth images,” IEEE Transactions
on Image Processing, vol. 14, pp. 342–359, 2005.
[5] A. Said and W.A. Pearlman, “A new fast and efficient image codec based on set
partitioning in hierachical trees,” IEEE Trans. on Circuits and Systems for Video
technology, vol. 6, no. 3, pp. 243-250, June 1996.
…
(please contact me if you would like more references)
23