A fast adaptive reweighted residual-feedback iterative algorithm for fractional-order total variation regularized multiplicative noise removal of partly-textured images
Graphical abstract
Introduction
Multiplicative noise removal is central to the study of coherent imaging systems and appears in many applications, e.g. in synthetic aperture radar (SAR) where the noise is assumed to follow Gamma distribution, and in positron emission tomography where the Poisson noise appears. Given a noisy image . with an open and bounded domain, the problem is to extract the noise free image u from f. In most practical applications, two important factors should be considered seriously: one is the preservation of the details such as edges and textures of the image; the other is the computational efficiency.
For the edge-preserving, the total variation (TV) regularized energy minimization [1] is one of the most well-known techniques. It performs well for preserving edges while removing noise and has been used for multiplicative noise removal [2], [3], [4], [5] in recent years. However, TV regularized term favors piecewise constant solution, and therefore it goes against texture preservation and often causes staircase effect. To improve the performance of texture preservation, the non-local methods have been used widely in recent years [6], [7], [8], [9], [10]. However, these time-consuming non-local methods cannot eliminate the staircase effect and even cause extra blocky artifacts. For the staircase effect elimination, high-order derivative based models [11], [12], [13] perform well, but they often cause blurring of the edges. As a compromise between the first-order TV regularized models and the high-order derivative based models, some fractional-order derivative based models have been introduced in [14], [15] for additive noise removal and subsequently used for image restoration [16] and super-resolution [17]. They can ease the conflict between staircase elimination and edge preservation by choosing the order of derivative properly. Moreover, the fractional-order derivative operator has a “non-local” behavior because the fractional-order derivative at a point depends upon the characteristics of the entire function and not just the values in the vicinity of the point [18], which is beneficial to improve the performance of texture preservation. The numerical results in literatures [14], [15], [16], [17] have demonstrated that the fractional-order derivative performs well for eliminating staircase effect and preserving textures. Motivated by these works, we propose to consider fractional-order total variation regularized models with different fidelity terms for multiplicative noise removal, which can be written uniformly as follows:where is the set of functions of fractional-order bounded variation with the fractional-order ( order) total variation of u [15]. Here, and are the order Grünwald-Letnikov fractional-order partial derivatives [18] with respect to the variable x and y respectively, which are defined bywhere is the generalized binomial coefficient with the Gamma function. Especially when , we have , and for any , and therefore is the total variation of u as usual, and therefore the fractional-order total variation can be cast as an extension of the traditional total variation.
The term in Eq. (1) is a fidelity term related to the prior information of the noise and is the regularization parameter. For instance, when we consider the multiplicative Gamma noise removal, the AA model [3] with and the I-divergence model [5] with are two most commonly used TV regularized models. However, the computation of these two models suffers from serious difficulties caused by the non-linearity and non-differentiability of the fidelity terms. To simplify the computation, Huang et al. [20] proposed a splitting-and-penalty based algorithm so-called the HNM algorithm to solve the AA model, and Steidl et al. [5] proposed the Douglas–Rachford splitting (DRS) algorithm to solve the I-divergence model. However, each iteration loop of these two algorithms consists of solving two nonlinear equations. For the HNM algorithm, one equation can be solved by using Chambolle's algorithm [19], but the other one should be solved by using Newton-like iterative method which leads to slow convergence speed and high computational cost. For the DRS algorithm, though one equation has a close-form solution, the other one should be linearized firstly and solved by using iterative method to solve the linear equation. Moreover, once we replace the TV regularized model, it is difficult to extend these two algorithms to solve the improved fractional-order total variation regularized models due to the complexity of the fractional-order derivative.
To solve these fractional-order TV regularized models with complex fidelity terms, we introduce a reweighted residual-feedback iterative (RRFI) algorithm in this paper. This new algorithm can be cast as an extension of the attractive iterative regularization method proposed by Osher et al. in [21] which has led to the popularity of the Bregman iteration in signal and image processing. As an application, we utilize the RRFI algorithm to solve model (1) with the fidelity terms used in the AA model and the I-divergence model. Moreover, we propose a strategy to update the parameters adaptively, and then develop a fast adaptive RRFI algorithm for multiplicative noise removal.
The rest of this paper is organized as follows: In Section 2, we introduce the RRFI algorithm and analyze its convergence. In Section 3, we propose a strategy to update the weighting matrix adaptively and obtain the adaptive RRFI algorithm for multiplicative noise removal. In Section 4, numerical examples and comments are given to show the efficiency of the new algorithm. Finally, the paper is concluded in Section 5.
Section snippets
Reweighted residual-feedback iteration (RRFI)
In [21], Osher et al. have proposed an attractive iterative regularization method to solve the TV regularized model for additive noise removal. Each iteration loop of this algorithm consists of two steps: one is to generate a new modified noisy image by feeding a residual image obtained in the last iteration loop back to the original noisy image; the other is to remove the noise from the modified noisy image again. The convergence speed of this algorithm is very fast. However, the denoised
Adaptive RRFI algorithm for multiplicative noise removal
The RRFI algorithm provides a general framework to solve the TV and fractional-order TV regularized models with different fidelity terms. In this paper, as an application, we propose to use the RRFI algorithm to solve the TV and fractional-order TV regularized models with certain fidelity terms for multiplicative noise removal.
Numerical experiments
In this section, we illustrate the performance of the RRFI algorithm for multiplicative Gamma noise removal of partly-textured images. The test noise free images are showed in Fig. 1. There are high presences of textures combined with non-textured parts in these images. We use different models and algorithms to remove Gamma distributed multiplicative noise with different variances (the noisy images with noise variance are showed in Fig. 1). The models and algorithms compared in this
Conclusions
In this paper, we propose a simple adaptive reweighted residual-feedback iterative algorithm for fractional-order total variation regularized multiplicative noise removal of partly-textured images. The main contributions of this paper are as follows:
- (i)
We proposed a reweighted residual-feedback iterative algorithm to solve the fractional-order TV regularized models with different fidelity terms. It is a general framework to solve the TV regularized and fractional-order TV regularized models with
Acknowledgments
This work is supported by the National Natural Science Fund of China (Nos. 61101198 and 61071146), the Natural Science Fund of Jiangsu Province of China (No. BK2012800), the China Postdoctoral Science Fund (No. 2012M511281) and the Jiangsu Planned Projects for Postdoctoral Research Fund (No. 1102064C).
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