Point-Value WENO Multiresolution Applications to Stable Image Compression

Abstract

The ideas that lead from ENO to Weighted ENO (WENO) reconstructions (i.e. cell-average “interpolators”), devised and extensively used for the design of highly accurate shock capturing schemes for conservation laws, are applied in this paper to obtain weighted essentially non-oscillatory point-value nonlinear interpolators that can generically achieve an order of accuracy of 2r, when using stencils of 2r points at regions where the interpolated function is smooth. This interpolatory technique can be used in Harten’s multiresolution framework for image compression applications.

More specifically, the nonlinear weights which the present interpolation is based upon are computed as proposed in Liu et al. (J. Comput. Phys. 115(1):200–212, 1994) and depend on smoothness indicators of the sub-stencils, defined in a way inspired by the smoothness indicators proposed in Jiang and Shu (J. Comput. Phys. 126(1):202–228, 1996), but through the corresponding Lagrange interpolators, instead of the cell-average interpolators.

We setup a unified framework that eases the consecution of the following results for any r, when using stencils of 2r points, by only using properties of the Lagrange interpolators: (1) the order of the interpolation is 2r at smooth regions, regardless of neighboring extrema; this is true even around points where successive derivatives of the function vanish; (2) the order of the interpolation is r+1, like the ENO interpolants, when the function has a discontinuity in the stencil of 2r points but it is smooth in at least one of the sub-stencils of r+1 points; (3) the optimal weights are obtained in closed form. All these results are obtained by a thorough study that highlights the importance of setting the parameter ε that appears in the definition of the weights to avoid null denominators to ε=h ².

The image compression capability of this interpolation is compared to other standard image compression techniques to conclude that its strength can be found in applications where images have relatively large regions of smoothness.