Topology-Preserving Rigid Transformation of 2D Digital Images

We provide conditions under which 2D digital images preserve their topological properties under rigid transformations. We consider the two most common digital topology models, namely dual adjacency and well-composedness. This paper leads to the proposal of optimal preprocessing strategies that ensure the topological invariance of images under arbitrary rigid transformations. These results and methods are proved to be valid for various kinds of images (binary, gray-level, label), thus providing generic and efficient tools, which can be used in particular in the context of image registration and warping.

I. Introduction

In image computing, the preservation of topological properties is a crucial issue. Indeed, topological properties provide useful information and descriptors when performing image processing and analysis tasks such as segmentation [1], classification [2], registration [3] or tracking [4]. This concerns not only 3D (e.g., in medical imaging [5]) but also 2D data (e.g., in remote sensing [6] or computer vision [7]). In particular, topology preservation-pioneered nearly fifty years ago [8], [9]-has been investigated in the context of image transformation, both from the viewpoints of registration [10] and warping [11]. To this end, efforts have been mainly devoted to handling complex transformations, while simpler ones have remained largely under-considered.