  Convolutional sparse coding (CSC) can learn representative shift-invariant
patterns from multiple kinds of data. However, existing CSC methods can only
model noises from Gaussian distribution, which is restrictive and unrealistic.
In this paper, we propose a general CSC model capable of dealing with
complicated unknown noise. The noise is now modeled by Gaussian mixture model,
which can approximate any continuous probability density function. We use the
expectation-maximization algorithm to solve the problem and design an efficient
method for the weighted CSC problem in maximization step. The crux is to speed
up the convolution in the frequency domain while keeping the other computation
involving weight matrix in the spatial domain. Besides, we simultaneously
update the dictionary and codes by nonconvex accelerated proximal gradient
algorithm without bringing in extra alternating loops. The resultant method
obtains comparable time and space complexity compared with existing CSC
methods. Extensive experiments on synthetic and real noisy biomedical data sets
validate that our method can model noise effectively and obtain high-quality
filters and representation.
