Invariant Pattern Representations and Lie Groups Theory

Abstract

This chapter discusses invariant pattern recognition and the problem of object recognition. The ability of any visual system to perform invariant to a given transformation is determined by the way the visual input is encoded, or internally represented, by the system. Thus, “invariant coding” deals precisely with the problem of finding a representation of the pattern—that is, invariant under certain transformations preserves the uniqueness of the representation. Two requirements are necessary for the invariant pattern recognition: the invariance of the encoding ensures recognition of the pattern even though it is transformed with respect to the prototype, whereas uniqueness prevents false recognitions. A representation that is invariant and preserves uniqueness, and hence encodes the transformational state, is said to be “invariant in the strong sense,” whereas a representation that is invariant but not unique is called “invariant in the weak sense.”The chapter also present results concerning invariant coding, which are obtained using the theory of Lie transformation groups. It is assumed that all groups considered are one-parameter Lie transformation groups; this is not too restrictive because most groups of interest are one-parameter (Lie) transformation groups.