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 represents 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 morphological 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 approximated 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 morphological 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.

This paper presents a deep network based on morphological filters for Gaussian denoising. The morphological filters can be applied with only addition, max, and min functions and require few computational resources. Therefore, the proposed network is suitable for implementation using a small microprocessor. Each layer of the proposed network consists of a top-hat transform, which extracts small peaks and valleys of noise components from the input image. Noise components are iteratively reduced in each layer by subtracting the noise components from the input image. In this paper, the extensions of opening and closing are introduced as linear combinations of the morphological filters for the top-hat transform of this deep network. Multiplications are only required for the linear combination of the morphological filters in the proposed network. Because almost all parameters of the network are structuring elements of the morphological filters, the feature maps and parameters can be represented in short bit-length integer form, which is suitable for implementation with single instructions, multiple data (SIMD) instructions. Denoising examples show that the proposed network obtains denoising results comparable to those of BM3D [1] without linear convolutions and with approximately one tenth the number of parameters of a full-scale deep convolutional neural network [2]. Moreover, the computational time of the proposed method using SIMD instructions of a microprocessor is also presented.