Recently, a tight frame, called convolution framelets (CFs) and constructed by convolving local and non-local bases, is proposed to provide valuable insights in understanding the patch-based processing approaches in the viewpoint of sparse representation (SR). However, it is still unclear how to represent signals with energy concentration guarantee in its lifted space and how to optimise the local base for a given non-local base. To address these issues, the equivalence between the signal space and its lifted space is established by the Hankel operator. In the lifted space, the energy concentration property of signals is measured by the sparsity instead of the Euclidean norm and the rank. With the new objective function, an optimisation model is built to train the optimal local base for a given nonlocal base from the training samples, motivating us to propose the optimal matching convolution framelets (OMCFs). In addition, a numerical algorithm is also designed to solve the proposed model using alternating optimisation strategy. The OMCF for SR is tested on the speech signals, and the comparisons with traditional popular SR tools, such as discrete cosine transform (DCT) and Haar wavelets, demonstrate its better performance.

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Sparse representation via optimal matching convolution framelets

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