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On a collection of subsets of a space, fundamentally different metrics may be defined. In pattern matching, it is often required that a metric is invariant for a given transformation group. In addition, a pattern metric should be robust for defects in patterns caused by discretisation and unreliable feature detection. Furthermore, a pattern metric should have sufficient discriminative power. We formalise these properties by presenting five axioms. Finding invariant metrics without requiring such axioms is a trivial problem. Using our axioms, we analyse various pattern metrics, including the Hausdorff distance and the symmetric difference. Finally, we present the reflection metric. This metric is defined on finite unions of n − 1-dimensional hyper-surfaces in IR n. The reflection metric is affine invariant and satisfies our axioms.

Visibility partitions play an important role in computer vision and pattern matching. This paper studies a new type of visibility, reflection-visibility, with applications in affine pattern matching: it is used in the definition of the reflection metric between two patterns consisting of line segments. This metric is affine invariant, and robust against noise, deformation, blurring, and cracks. We present algorithms that compute the reflection visibility partition in O((n+k) log(n)+v) randomised time, where k is the number of visibility edges (at most O(n 2)), and v is the number of vertices in the partition (at most O(n 2 + k 2)). We use this partition to compute the reflection metric in O(r(nA + nB)) randomised time, for two line segment unions, with nA and nB line segments, separately, where r is the complexity of the overlay of two reflection-visibility partitions (at most O(nA 4 + nB 4)).

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