Discontinuous Functions

We have desiged an adaptive ENO-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the one-side information idea from Essentially Non-Oscillatory (ENO) schemes for numerical shock capturing to standard wavelet transforms. This transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies and having a multiresolution framework and fast algorithms, all without any edge artifacts. Furthermore, we have obtained a rigorous uniform approximation error bound regardless of the presence of discontinuities. We will show some numerical examples and some applications to image compression. It is well known that wavelet linear approximation (i.e. truncating the high frequencies) can approximate smooth functions very efficiently but cannot achieve similar results for piecewise continuous functions, especially functions with large jumps. Several problems arise near jumps, primarily caused by the well-known Gibb’s phenomenon. The jumps generate large high frequency wavelet coefficients and thus linear approximations cannot get the same high accuracy near discontinuties as in the smooth region. To overcome these problems within the standard wavelet transform framework, non-linear data-dependent approximations, which selectively retain certain high frequency coefficients, are often used, e.g. hard and soft thresholding techniques, see [Don95],[Mal98]. Another way is to construct orthonormal basis to represent the discontinuities, such as Donoho’s wedgelets [Don97], rigdelets [CD99b], and curvelets [CD99a], and Mallat’s bandelets [Mal00]. A different aproach is to modify the wavelet transform to not generate large wavelet coefficients near jumps. Claypoole, Davis, Sweldens and Baraniuk [PCB99] proposed an adaptive lifting scheme which lowers the order of approximation near jumps, thus minimizing the Gibbs’ effect. We use a different approach in developing our ENO-wavelet transforms by borrowing the well developed Essentially NonOscillatory (ENO) technique for shock capturing in computational fluid dynamics (e.g. see [AHC87]) to modify the standard wavelet transform near discontinuities so that the Gibbs’ phenomenon can be completely removed. ENO schemes are systematic ways of adaptively defining piecewise polynomial approximations of the given functions according to their smoothness. A crucial point in designing ENO schemes is to use one-sided information near jumps, and never differencing across the discontinuities. Combining the ENO idea with the multiresolution data representation is a natural