Discrete Transforms Produced from Two Natural Numbers and Applications

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 Ri ,Rj , (i,j=1,...,m) of an R-matrix satisfies either Ri ⊙Rj =R max {i,j} or Ri ⊙Rj =(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.