Closed-form solutions within sparsifying transform learning

Many applications in signal processing benefit from the sparsity of signals in a certain transform domain or dictionary. Synthesis sparsifying dictionaries that are directly adapted to data have been popular in applications such as image denoising, and medical image reconstruction. In this work, we focus specifically on the learning of orthonormal as well as well-conditioned square sparsifying transforms. The proposed algorithms alternate between a sparse coding step, and a transform update step. We derive the exact analytical solution for each of these steps. Adaptive well-conditioned transforms are shown to perform better in applications compared to adapted orthonormal ones. Moreover, the closed form solution for the transform update step achieves the global minimum in that step, and also provides speedups over iterative solutions involving conjugate gradients. We also present examples illustrating the promising performance and significant speed-ups of transform learning over synthesis K-SVD in image denoising.