Deep Unrolling of Non-Linear Diffusion with Extended Morphological Laplacian

This paper presents a deep network based on unrolling the diffusion process with the morphological Laplacian. The diffusion process is an iterative algorithm that can solve the diffusion equation and repre-sents time evolution with Laplacian. The diffusion process is applied to smoothing of images and has been extended with non-linear operators for various image processing tasks. In this study, we introduce the morpholog-ical Laplacian to the basic diffusion process and unwrap to deep networks. The morphological filters are non-linear operators with parameters that are referred to as structuring elements. The discrete Laplacian can be approxi-mated with the morphological filters without multiplications. Owing to the non-linearity of the morphological filter with trainable structuring elements, the training uses error back propagation and the network of the morphology can be adapted to specific image processing applications. We introduce two extensions of the morphological Laplacian for deep networks. Since the morphological filters are realized with addition, max, and min, the error caused by the limited bit-length is not amplified. Consequently, the mor-phological parts of the network are implemented in unsigned 8-bit integer with single instruction multiple data set (SIMD) to achieve fast computation on small devices. We applied the proposed network to image completion and Gaussian denoising. The results and computational time are compared with other denoising algorithm and deep networks.