A Variational Approach for Joint Image Recovery and Feature Extraction Based on Spatially-Varying Generalised Gaussian Models

TL;DR

This work proposes a novel nonsmooth and nonconvex variational formulation of the joint problem of reconstruction / feature extraction and introduces a versatile generalised Gaussian prior whose parameters, including its exponent, are space-variant.

Abstract

The joint problem of reconstruction / feature extraction is a challenging task in image processing. It consists in performing, in a joint manner, the restoration of an image and the extraction of its features. In this work, we firstly propose a novel nonsmooth and non-convex variational formulation of the problem. For this purpose, we introduce a versatile generalised Gaussian prior whose parameters, including its exponent, are space-variant. Secondly, we design an alternating proximal-based optimisation algorithm that efficiently exploits the structure of the proposed non-convex objective function. We also analyse the convergence of this algorithm. As shown in numerical experiments conducted on joint deblurring/segmentation tasks, the proposed method provides high-quality results.

Figures (3)

• Figure 1: Probabilistic dependence graph of our model. Hyperparameters are represented as diamonds, and variables as ellipses: $a$ and $b$ are the lower and the upper bound for the interval appearing in the uniform distribution of the shape parameter, $p$ is the shape parameter, $\alpha$ is the original scale parameter, $\beta$ is the reparameterised scale parameter with mean $\mu_{\beta}$ and standard deviation $\sigma_\beta$, $x$ is the sought signal, $y$ is the observed one, $K$ is the linear operator, and $\omega$ is the additive Gaussian noise with standard deviation $\sigma$.
• Figure 2: First and Second lines: B--mode of Simu1 and Simu2. The B--mode image is the most common type of ultrasound image, displaying the acoustic impedance of a 2-dimensional cross section of the considered tissue. All images are presented in the same scale [0,1]. Third and Fourth lines: Segmentation of the shape parameter for Simu1 and Simu2: reference $p$, estimated $\hat{p}$ and quantised $\bar{p}$.
• Figure 3: Decay of the objective value along 500 iterations for Simu1 (a) and Simu2 (b). We considered ten random sampling for $p^0$ and $\beta^0$. The continuous line in the plot represents the mean objective value at each iteration, and the shaded area, highlighted in the zoomed region at the centre spanning over 20 iterations, corresponds to the confidence interval related to the standard deviation. Logarithmic plot of the relative distance from the iterates $\zeta^\ell$ to the solution $\zeta^{\infty}$ over 1500 iterations for Simu1 (c) and Simu2 (d).

Theorems & Definitions (28)

• Definition 1: Subgradient of a convex function
• Definition 2: Limiting Subdifferential
• Definition 3: Critical point
• Definition 4
• Definition 5
• Definition 6
• Remark 1
• Definition 7: KŁ property
• Remark 2
• Proposition 1