Synthetic aperture radar (SAR) images are inevitably corrupted by speckle noise, due to constructive and destructive electromagnetic wave interference during image acquisition. With fleets of satellites delivering a huge number of images, automatic analysis tools are essential for remote sensing major applications. Therefore, the quality of source images should be sufficient such that it is easy to extract information. However, the speckle noise vi- sually degrades the appearance of images and therefore hinders automatic scene analysis and information extraction [6, 20]. For instance, speckle is the main obstacle towards the development of an effective optical-SAR fusion [4]. Hence, speckle reduction is a necessary preprocessing step in SAR image processing. Despeckling techniques have been extensively studied for almost 30 years [33, 31], and new algorithms are continuously proposed to provide better and better performance. Up to now, the despeckling techniques fall broadly into four categories: filtering based methods in (1) spatial domain or (2) a transform domain, e.g., wavelet domain; (3) nonlocal filtering; and (4) variational methods. As a comprehensive review of the despeckling algo- rithms is beyond the scope of this paper, we only provide a brief introduction for these methods. For more details, we refer the reader to [6].

The multi-look technique is a traditional spatial approach. It amounts to incoherently averaging independent observations of the same resolution cell, thus reducing the noise intensity. However, this simple averaging approach results in a clear loss in image resolution. To overcome this deficiency, a great deal of research has been conducted to develop suitable spatial filters which can reduce the noise, yet preserve details and edges [26] [37]. Filters of this kind include Lee filter proposed in [33] [34] which was developed under the minimum-mean-square-error (MMSE) criterion, and Kuan filter [31] as well as the Γ-Map filter [36] which are based on the more sophisticated maximum a posteriori (MAP) approach.

Anisotropic diffusion [43] based method is also a type of widely exploited spatial filtering technology for despeckling. Anisotropic diffusion is a popular technique in the image processing community, that aims at reducing image noise without removing significant parts of the image content. A few re- lated works that apply anisotropic diffusion filtering for the despeckling task include speckle reducing anisotropic diffusion (SRAD) [53] and detail pre- serving anisotropic diffusion (DPAD) [3]. SRAD exploits the instantaneous coefficient of variation and it leads to better performance than the conven- tional anisotropic diffusion method in terms of mean preservation, variance reduction and edge localization. DPAD modifies the SRAD filter to rely on the Kuan filter [31] rather than the Lee filter. DPAD estimates the local statistics using a larger neighborhood, instead of the four direct neighbors used by SRAD. However, the despeckling methods based on anisotropic dif- fusion fell out of favor in recent years mainly because of limited performance. It is clear that there is a despeckling quality gap between the diffusion based approaches and state-of-the-art nonlocal algorithms.

Image filtering in the domain of wavelet has also been widely exploited for despeckling. Most of the wavelet-based despeckling techniques employ the statistical wavelet shrinkage technique with MAP Bayesian approach, e.g.[44, 1, 8, 45, 5]. In general, the wavelet-based methods guarantee a superior ability to preserve signal resolution in comparison with conventional spatial filters. However, they often suffer from isolated patterns in flat areas, or ringing effects near the edges of the images, leading to visually unappealing results.

Recently, incorporation with the modern denoising methods, e.g., nonlo- cal mean (NLM) [10], block-matching 3-D (BM3D) [17] and K-SVD [23], sev- eral nonlocal despeckling approaches have been proposed [18], [50], [42], [29]. Originated from the NLM algorithm, the probabilistic patch-based (PPB) filter [18] provides promising results by developing an effective similarity measure well suited to SAR images. A drawback of the PPB filter is the suppression of thin and dark details in the regularized images. As an exten- sion of BM3D, Parrilli et al. [42] derived a SAR-oriented version of BM3D by taking into account the peculiar features of SAR images. It exhibits an objective performance comparable or superior to other techniques on sim- ulated speckled images, and guarantees a very good subjective quality on real SAR images. Typical artifacts of the nonlocal methods are in the form of structured signal-like patches in flat areas, originated from the random- ness of speckle and reinforced through the patch selection process. Generally speaking, most of these techniques mainly concentrate on achieving utmost despeckling quality, with relatively low computational efficiency. An notable exception is BM3D with its improved version [15]. However, the BM3D-based method involves a block matching process, which is challenging for parallel computation on GPUs, alluding to the fact that it is not straightforward to accelerate BM3D algorithm on parallel architectures.

The fourth category, i.e., variational methods [48, 7, 25, 12, 22, 30, 11, 24, 21, 28], minimizes some appropriate energy functionals consisted of an image prior regularizer and a data fitting term. As a well-known regularizer, total variation (TV) has been widely used for the despeckling task [48, 7, 21, 28]. For instance, in [21] a new variational model based on a hybrid data term and the widely used TV regularizer is proposed for restoring blurred images with multiplicative noise. Moreover, [28] proposes a two-step approach to solve the problem of restoring images degraded by multiplicative noise and blurring, where the multiplicative noise is first reduced by nonlocal filters and then a convex variational model is adopted to obtain the final restored images. Solutions of variational problems with TV regularization admit many desirable properties, most notably the appearance of sharp edges.

However, TV-based methods generate the so-called staircasing artifact. To remedy the staircasing artifact, [25] incorporates the total generalized variation (TGV) [9] penalty into the existing data fidelity term for speckle removal, and develops two novel variation despeckling models. By involving and balancing higher-order derivatives of the image, the TGV-based despeck- ling method outperforms the traditional TV methods by reducing the stair- casing artifact. Recently, different from hand-crafted regularizers, such as TV and TGV models, [12] proposes a novel variational approach for speckle re- moval, which combines an image prior model named Fields of Experts (FoE) [46] and a recently proposed efficient non-convex optimization algorithm – iPiano [40]. The proposed method in [12] can obtain strongly competitive despeckling performance w.r.t. the state-of-the-art method – SAR-BM3D, meanwhile, preserve the property of computational efficiency.

1.1. Our Contribution

Traditional despeckling approaches based on anisotropic diffusion are handcrafted models which include elaborate selections of diffusivity coeffi- cient, or optimal stopping time or proper reaction force term. In order to improve the capacity of the traditional diffusion-based despeckling models, we employ the newly-developed anisotropic diffusion based model with trainable filters and influence functions. Instead of the first order gradient operator in previous diffusion-based despeckling models, we explore more filters of larger kernel size targeted for despeckling. On the other hand, different influence functions are considered and trained for different filters, rather than an unique function in the traditional diffusion model. Moreover, the parameters of each iteration can vary across diffusion steps.

As shown in [13], the optimized nonlinear diffusion model has broad ap- plicability to a variety of image restoration problems, and achieves recovery results of high quality surpassing recent state-of-the-arts. Furthermore, it only involves a small number of explicit filtering steps, and hence is highly computationally efficient, especially with parallel computation on GPUs.

In this paper, we intend to apply the TNRD framework [13] to the task of multiplicative noise reduction. However, a direct use of the original TNRD model is not feasible, as we have to make a few modifications oriented to this specific task:

• We need to redesign the diffusion process by taking into consideration the peculiarity of multiplicative noise statistics.
• Based on the new diffusion process specialized for multiplicative noise reduction, we need to recalculate the gradients required for the training phase.
• In the original TNRD applications, the diffusion network is usually derived based on the direct gradient descent scheme. However, this approach will encounter some problem for the task of multiplicative noise reduction exploited in this work. The reason is explained in detail in Section 3.1 and the experimental part 4.1. To solve this problem, we employed a new architecture as shown in Fig. 2, which is derived from the proximal gradient descent method. Comparing Fig. 2 and Fig. 1, we can see that the structure of the proposed diffusion process Fig. 2 in our study is quite different from the original TNRD model Fig. 1.Then, the model parameters in the diffusion process need to be trained by taking into account the Speckle noise statistics, including the linear filters and influence functions. Eventually, we reach a nonlinear reaction diffusion based approach for despeckling, which leads to state-of-the-art performance, meanwhile gains high computationally efficiency. Experimental results show that the proposed despeckling approach with optimized nonlinear diffusion filtering leads to state-of-the-art performance, meanwhile gains high computationally efficiency.

1.2. Organization

The remainder of the paper is organized as follows. Section II presents a general review of the speckle noise and the trainable nonlinear reaction diffusion process, which is required to derive the optimized diffusion process for despeckling. In the subsequent section III, we propose the optimized non- linear diffusion process for despeckling. Subsequently, Section IV describes comprehensive experiment results for the proposed model. The concluding remarks are drawn in the final Section.