Abstract
Let m be a natural number and be the usual Hadamard product operation on finite data of length m. In [1] we built a large class of m ×m boolean invertible matrices (called R-matrices) determined by a pair of permutations (ρ,s) of the set {1,...,m}. To do that we required that any pair of rows R i ,R j , (i,j = 1,...,m) of an R-matrix satisfies either R i ⊙ R j = R max {i,j} or R i ⊙ R j = (0,...,0), a property mainly observed in matrices associated with multiscale linear transforms. In this paper we deal with R-transforms, i.e. linear transforms whose corresponding matrices are R-matrices. We prove that the inverse transform \(T_R^{-1}\) has a simple representation depending only on the pair (ρ,s) identifying the matrix R. As a result we obtain a fast encoding/decoding scheme. Finally we demonstrate a method for constructing R-transforms with desired properties from a recursive equation based on dilation operators on permutations and we present applications.
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© 2012 Springer-Verlag Berlin Heidelberg
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Atreas, N., Karanikas, C. (2012). Discrete Transforms Produced from Two Natural Numbers and Applications. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2011. EUROCAST 2011. Lecture Notes in Computer Science, vol 6928. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27579-1_39
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DOI: https://doi.org/10.1007/978-3-642-27579-1_39
Publisher Name: Springer, Berlin, Heidelberg
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