A Phase Field Approach for a Modified Mumford-Shah Functional Based on Fractional Derivatives
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Recovering the true image, or true signal, from a corrupted one is by no means a trivial task, and we will in this paper study some approaches to finding good approximations to this unknown true image. The main focus will be on a minimization method based on wavelet decompositions of images and finding a solution which is sparse with respect to the wavelet basis. We then argue that knowledge of the position of edges between objects in the image could be utilized with the aim to improve the edge preserving capabilities of the method. Therefore, we present an edge detection approach and a subsequent edge dependent weighting of the coefficients in the minimization method. If this method does in fact improve the appearance of the solution, we shall seek a more sophisticated and efficient approach. Borrowing a phase field approach used by Ambrosio and Tortorelli to approximate a solution to the Mumford-Shah functional, we construct a modified functional where the edge dependent weighting is found based on the phase field. After developing an alternating minimization method, we test the different approaches on a blurred image, and study the differences in the results.