  Speckle reduction is a prerequisite for many image processing tasks in
synthetic aperture radar (SAR) images, as well as all coherent images. In
recent years, predominant state-of-the-art approaches for despeckling are
usually based on nonlocal methods which mainly concentrate on achieving utmost
image restoration quality, with relatively low computational efficiency.
Therefore, in this study we aim to propose an efficient despeckling model with
both high computational efficiency and high recovery quality. To this end, we
exploit a newly-developed trainable nonlinear reaction diffusion(TNRD)
framework which has proven a simple and effective model for various image
restoration problems. {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 study. To solve this problem, we employed a new
architecture derived from the proximal gradient descent method.} {Taking into
account the speckle noise statistics, the diffusion process for the despeckling
task is derived. We then retrain all the model parameters in the presence of
speckle noise. Finally, optimized nonlinear diffusion filtering models are
obtained, which are specialized for despeckling with various noise levels.
Experimental results substantiate that the trained filtering models provide
comparable or even better results than state-of-the-art nonlocal approaches.
Meanwhile, our proposed model merely contains convolution of linear filters
with an image, which offers high level parallelism on GPUs. As a consequence,
for images of size $512 \times 512$, our GPU implementation takes less than 0.1
seconds to produce state-of-the-art despeckling performance.}
