Edge Direction Preserving Image Zooming : A Mathematical andNumerical

We focus in this paper on some reconstruction/restoration methods which aim is to improve the resolution of digital images. The main point is here to study the ability of such methods to preserve 1D structures. Indeed such structures are important since they are often carried by the image "edges". We rst 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 can not cope linearly with Gibbs eeects, sharpness of the results and the preservation of the 1D structure. Secondly, we study variational non-linear methods and in particular the one based on total variation. We show that this latter permits 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 eeects of the described methods on images as well as on their spectrum.