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Convolutional sparse coding (CSC) can learn representative shift-invariant patterns from data. However, existing CSC methods assume the Gaussian noise, which can be restrictive in some challenging applications. In this paper, we propose a generalized CSC model capable of handling complicated unknown noise. The noise is modeled by the Gaussian mixture model, which can approximate any continuous probability density function. The Expectation-Maximization algorithm is used to solve the resultant learning problem. For efficient optimization, the crux is to speed up the convolution in the frequency domain while keeping the other computations involving the weight matrix in the spatial domain. We design an efficient solver for the weighted CSC problem in the M-step. The dictionary and codes are updated simultaneously by an efficient nonconvex accelerated proximal gradient algorithm. The resultant procedure, called generalized convolutional sparse coding (GCSC), obtains the same space complexity and a smaller running time than existing CSC methods (which are limited to the Gaussian noise). Extensive experiments on synthetic and real-world noisy data sets validate that GCSC can model the noise effectively and obtain high-quality filters and representations. [ABSTRACT FROM AUTHOR]

A novel scheme of edge detection based on the physical law of diffusion is presented in this paper. Though the most current studies are using data based methods such as deep neural networks, these methods on machine learning need big data of labeled ground truth as well as a large amount of resources for training. On the other hand, the widely used traditional methods are based on the gradient of the grayscale or color of images with using different sorts of mathematical tools to accomplish the mission. Instead of treating the outline of an object in an image as a kind of gradient of grayscale or color, our scheme deals with the edge detection as a character of an energy diffusing in the space of media such as charge-coupled device. By using the characteristic function of diffusion, the information of the energy will be extracted. The scheme preserves the structural information of images very well. Because it comes from the inhere law of images’ physical property, it has a unified mathematical framework for images’ edge detection under different conditions, for example, multiscales, diferent light conditions, and so on. Moreover, it has low computational complexity. [ABSTRACT FROM AUTHOR]