We address the problem of searching for a two-dimensional pattern in a two-dimensional text (or image), such that the pattern can be found even if it appears rotated and it is brighter or darker than its occurrence. Furthermore, we consider approximate matching under several tolerance models. We obtain algorithms that are almost optimal both in the worst and the average cases simultaneously. The complexities we obtain are very close to the best current results for the case where only rotations, but not lighting invariance, are supported. These are the first results for this problem under a combinatorial approach. (c) 2007 Elsevier Inc. All rights reserved.