An extended Perona-Malik model based on probabilistic models

TL;DR

This paper extends the Perona-Malik image restoration model by integrating probabilistic Gaussian scale mixtures, leading to improved algorithms for image denoising and segmentation with better uncertainty modeling.

Contribution

It introduces a probabilistic reinterpretation of the Perona-Malik model using Gaussian scale mixtures, resulting in new algorithms with enhanced restoration and segmentation capabilities.

Findings

Modified lagged-diffusivity algorithm improves textured area restoration.

Probabilistic sampling procedure enables efficient computation of expectations.

Framework allows for a probabilistic Mumford-Shah segmentation model.

Abstract

The Perona-Malik model has been very successful at restoring images from noisy input. In this paper, we reinterpret the Perona-Malik model in the language of Gaussian scale mixtures and derive some extensions of the model. Specifically, we show that the expectation-maximization (EM) algorithm applied to Gaussian scale mixtures leads to the lagged-diffusivity algorithm for computing stationary points of the Perona-Malik diffusion equations. Moreover, we show how mean field approximations to these Gaussian scale mixtures lead to a modification of the lagged-diffusivity algorithm that better captures the uncertainties in the restoration. Since this modification can be hard to compute in practice we propose relaxations to the mean field objective to make the algorithm computationally feasible. Our numerical experiments show that this modified lagged-diffusivity algorithm often performs better at restoring textured areas and fuzzy edges than the unmodified algorithm. As a second application of the Gaussian scale mixture framework, we show how an efficient sampling procedure can be obtained for the probabilistic model, making the computation of the conditional mean and other expectations algorithmically feasible. Again, the resulting algorithm has a strong resemblance to the lagged-diffusivity algorithm. Finally, we show that a probabilistic version of the Mumford-Shah segementation model can be obtained in the same framework with a discrete edge-prior.