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Image registration has become a crucial step in a wide range of imaging domains, from computer vision to computer graphics. The core of image registration consists of determining the transformation that induces the best mapping between two images. This problem is ill-posed; it is also difficult to handle, due to the high size of the images and the high dimension of the transformation parameter spaces. Computing an actually optimal solution is practically impossible when transformations are assumed continuous (i.e., defined on R n ). In this article, we initiate the exploration of a new way of considering image registration. Since digital images are basically defined in a discrete framework (i.e., in Z n ), the transformation spaces–despite a potentially high complexity–actually remain finite, allowing for the development of explicit exploration of the parameter space via discrete optimization schemes. We propose an analysis of the very basis of registration, by considering rigid registration between 2D images. We show, in particular, how this problem can be handled in a fully discrete fashion, by computing locally the combinatorial structure of the parameter space of discrete rigid transformations, and by navigating on-the-flight within this space via gradient descent paradigms. This registration framework is applied in real imaging cases, emphasizing the relevance of our approach, and the potential usefulness of its further extension to higher dimension images and richer transformations. [ABSTRACT FROM AUTHOR]