Image Compression and Wavelet Applications

* Compression
* 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
< Figures by Hao-Min Zhou : Links to .PS files >

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
Reports on Wavelets

People
Tony Chan , Stanley Osher , Christopher Thiele , Jianhong Shen , Hao-Min Zhou