Edge Direction Preserving Image Zooming: A Mathematical and Numerical Analysis

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Abstract
We focus in this paper on some reconstruction/restoration methods whose aim is to improve the resolution of digital images. The main point here is to study the ability of such methods to preserve one-dimensional (1D) structures. Indeed, such structures are important since they are often carried by the image edges. First we focus on linear methods, give a general framework to design them, and show that the preservation of 1D structures pleads in favor of the cancellation of the periodization of the image spectrum. More precisely, we show that preserving 1D structures implies the linear methods to be written as a convolution of the sinc interpolation. As a consequence, we cannot cope linearly with Gibbs effects, sharpness of the results, and the preservation of the 1D structure. Second, we study variational nonlinear methods and, in particular, the one based on total variation. We show that this latter permits us to avoid these shortcomings. We also prove the existence and consistency of an approximate solution to this variational problem. At last, this theoretical study is highlighted by experiments, both on synthetic and natural images, which show the effects of the described methods on images as well as on their spectrum.