In this chapter, we present a variation to the HT, which is called a centeredweighted HT, such as the reverse jacket transform (RJT), complex RJT (CRJT), extended CRJT (ECRJT) and extended CRJT over finite fields, and the generalized RJT. Centered-weighted HTs have found several interesting applications in image processing, communication sequencing, and cryptology that have been pointed out. These transforms have a similar simplicity to that of the HT, but offer a better quality of representation over the same region of the image. The reason for developing this theory is motivated by the fact that (1) the human visual system is most sensitive to the special (in general midspatial) fragments and (2) the same part of data sequences or the middle range of frequency components are more important.

First, we present the recursive generation of the real weighted HT matrices. Then we introduce the methods of generating complex weighted Hadamard matrices.

12.1 Introduction to Jacket Matrices

Definition 12.1.1: The square matrix A = (a _i,j) of order n is called a jacket matrix if its entries are nonzero and real, complex, or from a finite field, and satisfy

(12.1)

where I_n is the identity matrix of order n, B = 1/n (a⁻¹_i,j)^T; and T denotes the transpose of the matrix. In other words, the inverse of a jacket matrix is determined by its elementwise or blockwise inverse. The definition above may also be expressed as

(12.2)