Cartoon-like image reconstruction via constrained ℓ _p- minimization

This paper considers the problem of reconstructing images from only a few measurements. A method is proposed that is based on the theory of Compressive Sensing. We introduce a new prior that combines an ℓ _p-pseudo-norm approximation of the image gradient and the bounded range of the original signal. Ultimately, this leads to a reconstruction algorithm that works particularly well for Cartoon-like images that commonly occur in medical imagery. The arising optimization task is solved by a Conjugate Gradient method that is capable of dealing with large scale problems and easily adapts to extensions of the prior. To overcome the none differentiability of the ℓ _p-pseudo-norm we employ a Huber-loss term like approximation together with a continuation of the smoothing parameter. Numerical results and a comparison with the state-of-the-art methods show the effectiveness of the proposed algorithm.