A Cell-Average Non-Separable Progressive Multivariate WENO Method for Image Processing Applications

TL;DR

The proposed method extends Harten's multiresolution framework through a non-linear WENO reconstruction adapted to the cell-average context, achieving high-order accuracy in smooth regions and stable, non-oscillatory behavior near discontinuities.

Abstract

Accurate and efficient reconstruction techniques are essential in multiresolution analysis and image compression, particularly when the data are represented as cell averages. In this work, we present a non-separable progressive multivariate Weighted Essentially Non-Oscillatory (WENO) scheme specifically designed for cell-average data, with applications to digital image processing. The proposed method extends Harten's multiresolution framework through a non-linear WENO reconstruction adapted to the cell-average context, achieving high-order accuracy in smooth regions and stable, non-oscillatory behavior near discontinuities. We also establish theoretical results regarding the consistency and approximation properties of the method. Finally, several numerical experiments on piecewise smooth functions and digital images are presented to demonstrate its performance and validate its effectiveness against the linear Lagrange reconstruction of the same order of accuracy.

Figures (11)

• Figure 1: Stencils used to get $p^{3}_{\mathbf{j}_1}(\mathbf{x}^*), \, \mathbf{j}_1\in \{0,1,2\}^2$. Each distinct color identifies one of the nine candidate $4 \times 4$ stencils spatially shifted within the global $6 \times 6$ domain, following the classical bivariate WENO framework arandigamuletrenau.
• Figure 2: Stencils used to get $p^4_{\mathbf{j}_0}(\mathbf{x}^*)$, $\mathbf{j}_0\in\{0,1\}^2$. The red and black colors highlight the four distinct intermediate stencils resulting from the first stage of the recursive Aitken–Neville procedure, acting as the transition toward the final degree 5 reconstruction.
• Figure 3: Contribution of each stencil of $p^{3}_{\mathbf{j}_1}(\mathbf{x}^*)$, $\mathbf{j}_1\in \mathbf{j}_1+\{0,1\}^2$ to the approximation of $p^4_{\mathbf{j}_0}(\mathbf{x}^*)$, $\mathbf{j}_0\in\{0,1\}^2$. The red and blue colors illustrate how four specific degree 3 stencils (shown in Figure \ref{['wenonormal']}) are recursively combined and overlapped to construct a single degree 4 stencil (shown in Figure \ref{['progresivo1']}).
• Figure 4: Three-dimensional representations of the polynomial test function \ref{['eq:gtilde']} defined on the domain $[-1,1]^2$. Panel (a) shows the original function exhibiting a discontinuity of magnitude $C=16$ in the lower half of the domain. Panels (b) and (c) display the reconstructed surfaces obtained with the linear and WENO-2D prediction schemes, respectively.
• Figure 5: Three-dimensional representations of the surface \ref{['surf2']} on $[-1,1]^2$. Panel (a) shows the surface decimated to half resolution. Panel (b) presents the reconstruction obtained with a linear scheme, while panel (c) shows the reconstruction obtained using the progressive WENO-2D method. The WENO reconstruction preserves the discontinuity along ${\color{black} x}+{\color{black} y}=0$, whereas the linear scheme smooths the jump.

Theorems & Definitions (4)

• Definition 1
• Proposition 2.1
• Proposition 2.2
• Theorem 3.1