Two-dimensional wreath product group-based image processing

Harmonic analysis is at the heart of much of signal and image processing. This is mainly the harmonic analysis of abelian groups, and ultimately, after sampling, and quantizing, finite abelian groups. Nevertheless, this general group theoretic viewpoint has proved fruitful, yielding a natural group-based multiresolution framework as well as some new and potentially useful nonabelian examples. The papers (Foote et al., 2000; Mirchandani et al., 2000) lay out a general finite groupbased approach to signal and image processing and pay special attention to the use of certain wreath product groups for image processing. The theory explicated there is onedimensional in the sense that the input signal f (respectively its Fourier transform) is represented as a column vector, and all analysis and synthesis is effected as a matrixvector multiply: F · f where F is the Fourier matrix, (respectively its inverse) for some specified finite group. This paper is intended to complement and extend this earlier work in the direction of a two-dimensional (2-D) finite group-based theory, again with an intent of applying this work to image processing. Historically, the notion of a 2-D transform almost always entails a representation of the signal f as a two-dimensional array so that the transform is given by a matrix multiplication of the form AfB for suitable matrices A and B. The most familiar example is that in which A = B is the Fourier matrix and the associated 2-D transform is the usual two-dimensional discrete Fourier transform. The new contributions of this paper are twofold. On the one hand we give a theoretical foundation to the notion of 2-D transform and 2-D signal processing which extends naturally to higher dimensions. We also provide a theory of 2-D group transforms, of which the 2-D Haar transform and 2-D Fourier transform are particular instances. These well-known 2-D transforms are also particular instances of a 2-D wreath product transform (WPT), whose explication is the second major new contribution. This extends the earlier work (ibid.,) which focused exclusively on the 1-D WPT. We include here examples and numerical experiments and indicate its potential use and application for image processing. The presence of an underlying (non-abelian) group afforded by this approach provides