Let and be two finite point-sets in . Given some class of transformations, such as the class of affine, conformal affine, rigid, or projective transformations, the problem of point-pattern matching is to find a transformation in that class which maps to well enough.
Ideally, one would give an as a parameter such that a transformation is good enough if the image of each point has a unique nearest neighbor in its -neighborhood (under some chosen metric). The is called the matching distance. If there is no solution, then the algorithm should report so, and if there are multiple such solutions, then any of them will do. However, in practical problems both and can be missing some points, or have some extraneous points. Thus the condition for a good transformation is relaxed by requiring only a percentage of the points of to satisfy the -criterion.