Measuring Resemblance of Complex Patterns

Abstract

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.

A New Visibility Partition for Affine Pattern Matching

“…The reflection metric, introduced in [16,14], defines a distance between finite unions of algebraic curve segments in the plane. If A and B are such unions, the reflection distance is denoted as d R (A, B).…”

“…The reflection metric can be generalised to finite complexes of d − 1 dimensional algebraic hyper-surface patches in d dimensions. For this, we refer to [16]. Here, we focus at the computation of the reflection metric for finite unions of segments in the plane.…”

Measuring Similarity Among Various Shapes Based on Geometric Matching

“…In addition, the continuity of a similarity index indicates that it would vary continually with changes in the shape of two geographical objects (Veltkamp 2001). A very small change in shape results in only a minor alteration in the value of a similarity index (Hagedoorn and Veltkamp 1999a, b; Veltkamp 2001). Accordingly, the smaller the shape difference between two spatial objects, the larger the similarity index, and vice versa.…”