  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.
