Abstract
In this paper, we describe the formulation of a novel transform called Multi-Dimensional Rajan Transform, which is an extension of Rajan Transform. Basically, Rajan Transform operates on a number sequence, whose length is a power of two. It transforms any sequence of arbitrary numbers into a sequence of interrelated numbers. As regards 2D Rajan Transform, there are two methods to implement it: (i) Row- Column method and (ii) Column-Row method. The 2D Rajan Transform obtained using the first method need not be the same as that obtained using second method. Similarly, one can implement 3-D Rajan Transform using the following approaches: (i) Row-Column-Depth approach, (ii) Row-Depth- Column approach, (iii) Column-Row-Depth approach, (iv) Column-Depth-Row approach, (v) Depth-Row-Column approach and (vi) Depth-Column-Row approach. This paper explains these approaches to implement two and three dimensional Rajan Transforms.
Highlights
Two Dimensional Rajan Transform and its ImplementationOne can implement twodimensional Rajan Transformin two different ways: (i)
If x(n) is a number sequence of length N=2k; Rajan transform essentially operates on a number sequence, whose length is a power of two
It transforms any sequence of arbitrary numbers k>0, its Rajan Transform(RT) is denoted as X(k)
Summary
One can implement twodimensional Rajan Transformin two different ways: (i). The 2D-RT spectra of H4 obtained using RowColumn method is Xr,c(k1,k2) and it is shown below. 64 08 16 00 stated previously, 2D Rajan Spectrum obtained using first method need not be the same as the spectrum [Xr,c(k1,k2)] =. The 2D-RT spectra of H4 obtained using Column-Row method is Xc,r(k1,k2) and it is shown below. Consider the two-dimensional array A showing aT like pattern. One can verify that [Xc,r(K1,k2)] for both arrays A and B remain the same. One can verify that [Xr,c(k1,k2)] remain the same for both arrays A and B. This amounts to saying that 2D-Rajan Transform is essentially a translation invariant function, which could be effectively used in pattern recognition
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