Learning overcomplete sparsifying transforms with block cosparsity

The sparsity of images in a transform domain or dictionary has been widely exploited in image processing. Compared to the synthesis dictionary model, sparse coding in the (single) transform model is cheap. However, natural images typically contain diverse textures that cannot be sparsified well by a single transform. Hence, we propose a union of sparsifying transforms model, which is equivalent to an overcomplete transform model with block cosparsity (OC-TOBOS). Our alternating algorithm for transform learning involves simple closed-form updates. When applied to images, our algorithm learns a collection of well-conditioned transforms, and a good clustering of the patches or textures. Our learnt transforms provide better image representations than learned square transforms. We also show the promising denoising performance and speedups provided by the proposed method compared to synthesis dictionary-based denoising.