Image Compression and Wavelet Applications
* ENO-Wavelet Transforms
* Heatlets, and Wavelets-Diffusion Interaction
Compression
We propose using Partial Differential Equation techniques in wavelet image
processing to reduce edge artifacts generated by wavelet approximations.
We have been exploring in two directions : the first is to apply the main
idea of the well developed ENO schemes for numerical shock capturing to
modify standard wavelet transforms so that a fewer number of large high
frequency coefficients are thresholded. Another direction is to use
minimization techniques, in particular the minimization to total variation,
to select and modify the retained standard wavelet coefficients so that
reconstructed images have less oscillations near edges. Numerical
experiments show that both approaches improve the reconstructed image quality
in wavelet compression and in denoising. (from CAM 00-21 Abstract, June 2000)
* Eno-Wavelet Compression < Figures by
Hao-Min Zhou : Links to .PS files >
 Initial Image |
 Haar |
 Haar, Hard Thresholding |
 ENO-Haar |
 ENO-Haar, Hard Thresholding |
* TV improved wavelet thresholding < Figures by Hao-Min Zhou : Links to .PS files>
 Observed Image |
 Wavelet Hard Thresholding |
 TV Wavelet Compression |
ENO-Wavelet Transforms
We have designed as adaptive ENO-Wavelet transform for approximating
discontinuous functions without oscillations near the discontinuities.
Our approach is to apply the main idea from Essentially Non-Oscillatory
(ENO) schemes for numerical shock capturing to standard wavelet
transforms. The crucial point is that the wavelet coefficients are
computed without differencing function values across jumps. However, we
accomplish this in a different way than in the standard ENO-schemes.
Whereas in the standard ENO schemes, the stencils are adaptively chosen,
in the ENO-wavelet transforms, we adaptively change the function and
use the same uniform stencils. The ENO-wavelet transform retains the
essential properties and advantages of standard wavelet transforms
such as concentrating the energy to the low frequencies, obtaining
arbitrary high order accuracy uniformly and having a multiresolution
framework and fast algorithms, all without any edge artifacts.
We have obtained a rigorous approximation error bound which shows that
the error in the ENO-wavelet approximation depends only on the size of
the derivative of the function away form the discontinuities.
(from CAM 99-21 Abstract, June 2000)
Report by Chan and Zhou
Initial Image |
Haar |
Haar, Hard Thresholding |
ENO-Haar |
ENO-Haar, Hard Thresholding |
Heatlets, and Wavelets-Diffusion Interaction
We present an application of wavelet theory in partial differential
equations. We study the wavelet fundamental solutions to the heat
equation. The heat evolution of an initial wavelet state is called
a heatlet. Like wavelets for the L2 space, heatlets are
"atomic" heat evolutions in the sense that any general heat evolution
can be "assembled" from one single heatlet according to some simple
algebraic rules. We study the basic properties and algorithms of
heatlets and related functions. This work appeared in J. Diff.
Eqns. (Abstract from CAM 99-18, May 99). [A recent follow-up paper is "
A note on
wavelets and diffusions" by
J. Shen, in J. Comp.
Anal. Appl. , vol. 5, no. 1, pp. 147-159, 2003.]
CAM report by Shen and Strang.
[Appeared in J. Diff. Eqn., 161(2), 403-421, 2000]
Reports on Compressions
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