Homeomorphic digitization, correction and compression of digital documents

One of the main tasks of digital image analysis is to recognize the properties of real objects based on their digital images. These images are obtained by some sampling device, like a CCD camera, and are represented as finite sets of points that are assigned some value in a gray-level or color scale. A fundamental question in image understanding is which features in the digital image correspond, under a given set of conditions, to certain properties of the underlying objects. In many practical applications this question is answered empirically by visually inspecting the digital images. In this paper, we present a mathematically comprehensive answer to this question with respect to topological properties. In particular, we derive conditions relating properties of real objects to the grid size of the sampling device which guarantee that a real object and its digital image are topologically equivalent. Moreover, we prove that a topologically-invariant digitization must result in well-composed or strongly-connected sets and that only certain local neighborhoods are realizable for such a digitization. Using the derived topological model of a well-composed digital image, we demonstrate the effectiveness of this model with respect to the digitization, thresholding, correction and compression of digital document images