An extended Krylov subspace method for decoding edge-based compressed images by homogeneous diffusion

TL;DR

This paper tackles decoding edge- and dithering-based compressed images by solving a diffusion (heat) equation, reframing decoding as computing $\mathbf{y}(t)=e^{tA}\mathbf{b}$. It develops an extended Krylov subspace method that efficiently handles large time $t$ by using a Krylov basis built from $(\gamma I-A)^{-1}$, enabling accurate approximations ${\bf f}_m$ with very small $m$. The authors provide a rigorous error bound, a practical multigrid-based implementation, and demonstrate substantial speedups over standard time-stepping while maintaining high reconstruction quality on Kodak images, even on mobile devices. The approach yields significant practical impact for real-time or resource-constrained decoding in edge-based image compression and can extend to more general linear PDEs used in inpainting.

Abstract

The heat equation is often used in order to inpaint dropped data in inpainting-based lossy compression schemes. We propose an alternative way to numerically solve the heat equation by an extended Krylov subspace method. The method is very efficient with respect to the direct computation of the solution of the heat equation at large times. And this is exactly what is needed for decoding edge-compressed pictures by homogeneous diffusion.

Figures (13)

• Figure 1: Sketch of the compression scheme: in the encoding step the original picture is reduced to a subset, in the decoding phase the original is reconstructed from this subset.
• Figure 2: The mask has been chosen such that 10% of the pixels with the largest modulus of the Laplacian are contained.
• Figure 3: The mask has been chosen such that the modulus of the Laplacian has been scaled to $10\%$ and dithering has been applied.
• Figure 4: Example of image with dimension $3 \times 2$ and its row-major numbering
• Figure 5: Error bound of Theorem \ref{['Kryapproxext']} (black) solid line for the optimal choice of $\gamma$, error of the extended Krylov subspace approximation to the compressed all-white picture for the optimal $\gamma$ (green) circle-marked line, and error of the extended Krylov subspace with $\gamma$ fixed to one (red) diamond-marked line for $t=25$ on left-hand side, $t=10^2$ in the middle, $t=10^4$ on the right-hand side.

Theorems & Definitions (16)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof