Fast optimization for multichannel total variation minimization with non-quadratic fidelity

Abstract

Total variation (TV) has been proved very successful in image processing, and it has been combined with various non-quadratic fidelities for non-Gaussian noise removal. However, these models are hard to solve because TV is non-differentiable and nonlinear, and non-quadratic fidelity term is also nonlinear and even non-differentiable for some special cases. This prevents their widespread use in practical applications. Very recently, it was found that the augmented Lagrangian method is extremely efficient for this kind of models. However, only the single-channel case (e.g., gray images) is considered. In this paper, we propose a general computational framework based on augmented Lagrangian method for multichannel TV minimization with non-quadratic fidelity, and then show how to apply it to two special cases: L¹ and Kullback–Leibler (KL) fidelities, two common and important data terms for blurry images corrupted by impulsive noise or Poisson noise, respectively. For these typical fidelities, we show that the sub-problems either can be fast solved by FFT or have closed form solutions. The experiments demonstrate that our algorithm can fast restore high quality images.

Introduction

In image processing and signal processing areas, many problems can be casted as an energy minimization formulation, whose objective energy functional often contains two terms, one is the regularization term modeling some a priori information about the original data, and a fidelity term measuring some types of deviation of the original data from the observed data. One of the most used and successful regularization term is total variation, which was first put forward in [1], and has been successfully applied in many image processing areas, such as image restoration [1], [2], [3], [4], [5], [6], image segmentation, image compression, image reconstruction and surface reconstruction [5], [7], [8]. Its success is based on the fact that the gradient is sparse in most images and total variation catches this property, like the basis pursuit problem [9] in compressive sensing [10], [11].

As for the fidelity term, different problems use different formulation to suit its applications. For image restoration, the most common and popular fidelity term is the L² type, which has shown great success for recovering blurry images corrupted by Gaussian noise [1], [12], [2], [13]. However, for many other important applications, the L² type fidelity term is not suitable any more and we need to use a non-quadratic fidelity term to depict the relation between the observed data and the original data. For example, it has been shown that we can get high restoration results by using TV-L¹ model to recover images corrupted by blur and impulsive noise [14], [3]. Another typical example is the TV-KL model for Poisson noise [4].

Although TV regularization has many good properties, it is also well-known to be hard to solve because of its non-differentiability and nonlinear properties. A lot of effort has been contributed to design fast solvers [15], [16], [17], [18], [19], [20], [21], [22] in the past few years, however, all of these works only consider TV minimization with squared L² fidelity term. Compared with TV-L² model, TV with non-quadratic fidelity models are more complicated because the first order variations of these fidelities are no longer linear. Several algorithms have been put forward to overcome these difficulties. For example, for TV-L¹ model, there are gradient descent method [23], LAD method [24], splitting-and-penalty based method [25], and primal-dual method [26] based on semi-smooth Newton algorithm [27], alternating direction methods [28], as well as augmented Lagrangian method [29] for gray images. The same problem exists for TV-KL model, which is effective to deblur images corrupted by Poisson noise, but is quite hard to minimize. Popular methods to solve this TV-KL model are gradient descent [4], multilevel method [30], the scaled gradient projection method [31], and EM-TV alternative minimization [32], as well as variable splitting and convex optimization based methods [33], [34].

In this paper, we put forward a general computational framework for multichannel TV (MTV) minimization with non-quadratic fidelity. Our algorithm is derived from the well-known variable-splitting and augmented Lagrangian method in optimization, which is an unconstrained function based on augmenting the Lagrangian function with a quadratic penalty term. Another benefit of this formulation is that two of three subproblems either can be fast solved by FFT or has closed form solution for general fidelity terms, and the remaining subproblem has closed-form formula for some typical fidelities. We will see that our algorithm dramatically improves the computation speed because of the FFT implementation and closed form solutions. Besides, our different parameters for different auxiliary variables strategy is also much more effective than the one in [28], [33], [34]. Except the MTV regularization, our algorithm can also be extended to effectively handle weighted MTV, as well as high-order regularization terms, like the strategy used in [35]. In general, we present a very efficient algorithm for a quite general type of models which is composed of MTV-like regularization term and non-quadratic fidelity term. As shown in Section 4, our algorithm makes the edge-preserving variational color image restoration models to be practically viable technologies.

The rest of the paper is organized as follows. In the next section, we will first give an introduction about multichannel total variation. Augmented Lagrangian method based numerical solver for multichannel total variation with non-quadratic fidelity terms will be given in Section 3, and its application to two special cases: blurry images corrupted by impulsive noise or Poisson noise will also be presented in this section. Experimental results are shown in Section 4 and we conclude the paper in Section 5.