On the Complexity of Affine Image Matching

The problem of image matching is to find for two given digital images A and B an admissible transformation that converts image A as close as possible to B. This problem becomes hard if the space of admissible transformations is too complex. Consequently, in many real applications, like the ones allowing nonlinear elastic transformations, the known algorithms solving the problem either work in exponential worst-case time or can only guarantee to find a local optimum. Recently Keysers and Unger have proved that the image matching problem for this class of transformations is NP-complete, thus giving evidence that the known exponential time algorithms are justified. On the other hand, allowing only such transformations as translations, rotations, or scalings the problem becomes tractable. In this paper we analyse the computational complexity of image matching for a larger space of admissible transformations, namely for all affine transformations. In signal processing there are no efficient algorithms known for this class. Similarly, the research in combinatorial pattern matching does not cover this set of transformations neither providing efficient algorithms nor proving intractability of the problem, although it is a basic one and of high practical importance. The main result of this paper is that the image matching problem can be solved in polynomial time even allowing all affine transformations.