Iterative Approach to Image Compression with Noise : Optimizing Spatial and Tonal Data

TL;DR

This paper introduces an iterative, shape-optimized image compression method that dynamically adjusts pixel data during inpainting to enhance denoising and reconstruction quality, supported by theoretical analysis and numerical experiments.

Contribution

It develops a novel iterative approach combining shape optimization and PDE modeling for improved noisy image compression and inpainting.

Findings

The method effectively enhances image quality in noisy conditions.

Shape-based analysis guides optimal pixel set selection.

Numerical results confirm the efficiency of the proposed algorithms.

Abstract

We consider some iterative methods for finding the best interpolation data in the images compression with noise. The interpolation data consists of the set of pixels and their grey/color values. The aim in the iterative approach is to allow the change of the data dynamically during the inpainting process for a reconstruction of the image that includes the enhancement and denoising effects. The governing PDE model of this approach is a fully parabolic problem where the set of stored pixels is time dependent. We consider the semi-discrete dynamical system associated to the model which gives rise to an iterative method where the stored data are modified during the iterations for best outcomes. Finding the compression sets follows from a shape-based analysis within the $\Gamma$-convergence tools, in particular well suited topological asymptotic and a ``fat pixels'' approach are considered to obtain an analytic characterization of the optimal sets in the sense of shape optimization theory. We perform the analysis and derive several iterative algorithms that we implement and compare to obtain the most efficient strategies of compression and inpainting for noisy images. Some numerical computations are presented to confirm the theoretical findings. Finally, we propose a modified model that allows the inpainting data to change with the iteration and compare the resulting new method to the ``probabilistic'' ones from the state-of-the-art.