Analysis Operator Learning and Its Application to Image Reconstruction
This work addresses the need for effective analysis operators in image reconstruction, which is important for applications like medical imaging or photography, but it is incremental as it builds on existing analysis model frameworks.
The authors tackled the problem of learning an analysis operator for image reconstruction by developing an algorithm based on ℓp-norm minimization and manifold optimization, and they achieved competitive performance in denoising, inpainting, and super-resolution compared to specialized state-of-the-art methods.
Exploiting a priori known structural information lies at the core of many image reconstruction methods that can be stated as inverse problems. The synthesis model, which assumes that images can be decomposed into a linear combination of very few atoms of some dictionary, is now a well established tool for the design of image reconstruction algorithms. An interesting alternative is the analysis model, where the signal is multiplied by an analysis operator and the outcome is assumed to be the sparse. This approach has only recently gained increasing interest. The quality of reconstruction methods based on an analysis model severely depends on the right choice of the suitable operator. In this work, we present an algorithm for learning an analysis operator from training images. Our method is based on an $\ell_p$-norm minimization on the set of full rank matrices with normalized columns. We carefully introduce the employed conjugate gradient method on manifolds, and explain the underlying geometry of the constraints. Moreover, we compare our approach to state-of-the-art methods for image denoising, inpainting, and single image super-resolution. Our numerical results show competitive performance of our general approach in all presented applications compared to the specialized state-of-the-art techniques.