The problem of two-dimensional pattern matching invariant under a given class of admissible transformations F\mathcal{F} is to find in text T matches of transformed versions f(P) of the pattern P, for all f in F\mathcal{F} . In this paper, pattern matching invariant under compositions of real scaling and rotation are investigated. We give a new discretization technique for this class of transformations and prove sharp lower and upper bounds on the number of different possibilities to transform a pattern in this way. Subsequently, we present the first efficient pattern matching algorithm invariant under compositions of scaling and rotation. The algorithm works in time O(m ² n ⁶) for patterns of size m ² and texts of size n ². Our method can also be applied to the image matching problem, the well known issue in the image processing research.