Statistical compressive sensing based on convolutional Gaussian mixture model

Statistical compressive sensing needs to use the statistical description of source signal. By decomposing a whole image into a set of non-overlapping or overlapping patches, the Gaussian mixture model (GMM) has been used to statistically represent patches in an image. Compressive sensing, however, always imposes compression on the whole image. It is obvious that the entire image contains much richer information than the small patches. Extending from the small divided patches to an entire image, we propose a convolutional Gaussian mixture model (convGMM) to depict the statistics of an entire image and apply it to compressive sensing. We present the algorithm details by learning a convGMM from training images based on maximizing the marginal log-likelihood estimation. The learned convGMM is used to perform the model-based compressive sensing by using the convGMM as a model of the underlying image. In addition, aiming at the problem of high-dimensional image that makes learning, estimation and optimization suffer high computational complexity, all of the training and reconstruction process in our method can be fast and efficiently calculated in the frequency-domain by two-dimensional fast Fourier transforms. The performance of the convGMM on compressive sensing is demonstrated on several image sets.