Generalized Convolutional Sparse Coding With Unknown Noise

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.