Multi-template multi-channel image matching using tensor products of low-dimensional Clifford algebras

Abstract

The rapid expansion of data generated from various sensors, devices, and applications offers the opportunity to exploit complex data relationships in new ways. Multi-dimensional data often appears as multiple, separate data channels, for which multi-channel data representations and analysis techniques have been developed. Alternately, for image matching, multiple-template image matching techniques have been developed. Multi-template approaches use multiple, often singlechannel, templates, exhibiting intra-class variations. Same-class test image exemplars must match all reference templates. In this paper, we combine multiple-template matching techniques with multi-channel data representations to provide multitemplate, multi-channel image matching. We represent image data with pixels taking values in tensor products of lowdimensional Clifford Algebras, for which Fourier transforms exist. Fourier domain matching provides a computational processing improvement over spatial correlation-based matchers. The tensor product approach provides a decomposition of higher-dimensional algebras into combinations of lower dimensional algebras for which Fourier transforms apply. The tensor product approach produces a performance advantage, on data with the appropriate inter-channel correlation characteristics, through exploitation of additional data channel correlations. When these add constructively, a matching performance benefit occurs. We define an anti-involution mapping on a tensor product space, which leads to a definition of image correlation over these spaces. We prove that the correlation satisfies an extended inner product definition. We prove a Cauchy-Schwartz inequality, which validates use of the image correlation as a matcher. We present an example, using synthetic image data, where the approach provides superior matching performance over classical score sum fusion.