Learning sparsifying transforms for image processing

The sparsity of signals and images in a certain analytically defined transform domain or dictionary such as discrete cosine transform or wavelets has been exploited in many applications in signal and image processing. Recently, the idea of learning a dictionary for sparse representation of data has become popular. However, while there has been extensive research on learning synthesis dictionaries, the idea of learning analysis sparsifying transforms has received only little attention. We propose a novel problem formulation and an alternating algorithm for learning well-conditioned square sparsifying transforms from data. We show the superiority of our approach for image representation over analytical sparsifying transforms such as the DCT. We also show promise in image denoising. Denoising using the learnt analysis transforms is not only better than by synthesis dictionaries learnt using the K-SVD algorithm but also faster.