Adaptive ENO-wavelets for Image Compression

Friday, October 20, 2000 - 4:00pm - 4:30pm
Lind 409
Haomin Zhou (University of California, Los Angeles)
We have designed an 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 from the discontinuities. We will show some numerical examples to illustrate this error estimate.