On Periodic Patterns and their Spectra (*) Claudio Moraga European Centre for Soft Computing 33600 Mieres, Spain Technical University of Dortmund 44149 Dortmund, Germany mvl@claudio-moraga.eu Radomir Stanković Dept. Computer Science Faculty of Electronics University of Niš 18000 Niš, Serbia Radomir.Stankovic@gmail.com Jaakko Astola Tampere Int. Center for Signal Processing Tampere University of Technology 33101 Tampere Finland Jaakko.Astola@tut.fi Abstract * Two-sided spectra of patterns based on classes of orthogonal matrices are introduced and their main properties discussed. In particular, the “mosaicness” of patterns is studied. It is shown that this property of patterns may easily be recognized in the spectral domain, albeit the mosaic structure cannot be unambiguously determined. 2D-Dirichlet kernels based on non-Abelian groups can however be used for a more efficicient analysis of mosaics. 1. Introduction Spectral Techniques have a well established position in the MVL Community. See [5], [9], [6] for a review of 30 years of activity in this area. Most applications are related to multiple-valued logic design and to the analysis of properties of discrete functions. In the areas of Signal Processing and Pattern Recognition, it has been the Fourier transform the one that by far has the leading role [4]. In the present paper, new (and old) families of discrete transforms are studied and applied to generate two-sided spectra of patterns and analyze some of their properties, with emphasis in periodicity. Preliminary contributions in this direction may be found in e.g. [10] and [8]. The paper is structured in the following way. All required formalisms, definitions and lemmata will be introduced first. A later section will study the connection between the spectral analysis of patterns and the Sampling Theorem. Some examples are offered to illustrate relevant aspects of the paper. * Work leading to this paper was partially supported by the Foundation for the Advancement of Soft Computing, Mieres, Asturias, Spain. The work of R.S. Stankovic and J. Astola was supported by the Academy of Finland, Finnish Center of Excellence Programme, Grant No. 213462. 2. Formalisms Capital letters will be used to denote matrices. The dimensions of the matrix will be given as a pair of indices in brackets. Should a matrix be square, just one index will be given. A pattern is a 2 dimensional object represented by a matrix with non-negative real entries. These numbers will be interpreted as coding of the colour of a pixel (or a block) of the pattern. In this sense, a pattern may be interpreted as a digitized picture. Entries resulting from operations between two matrices representing patterns are expected to have a well defined chromatic interpretation. Definition 1: Let T (n) be a complex-valued orthogonal matrix and (T (n) ) a its adjoint (i.e. its transpose and complex conjugate). The product of an orthogonal matrix and its adjoint is commutative and gives an identity matrix scaled by a factor whose value depends on the structure of the matrix. T (n) ·(T (n) ) a = (T (n) ) a ·T (n) = k n ·I (n) (1) Recall that both the matrix product as well as the Kronecker product of two orthogonal matrices is orthogonal. Therefore T (n) may be generated as the product of several, not necessarily equal, orthogonal matrices. Furthermore, the adjoint of the Kronecker product of matrices equals the Kronecker product of the adjoints of the corresponding factor-matrices † , meanwhile the adjoint of a matrix product is the product of the adjoint factor-matrices, but in the reverse order. The resulting scaling factor will be in both cases the (simple) product of the individual scaling factors. Definition 2: † For properties of the Kronecker product, see http://en.wikipedia.org/wiki/Kronecker_product