By Peter W. Hawkes

**Advances in Imaging & Electron Physics merges long-running serials-- Advances in Electronics & Electron Physics and Advances in Optical & Electron Microscopy . The sequence positive factors prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photo technological know-how and electronic picture processing, electromagnetic wave propagation, electron microscopy, and the computing equipment utilized in some of these domain names.
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3; j D 0, 1, . . , 7 0 otherwise 3 1 x m, n 2mk 2nl 32 mD0 nD0 2ð4 mod F3 if k D 8i; X k, l D l D 4j; 8 i D 0, 1; j D 0, 1, . . , 3 0 otherwise Summing along rows gives Tc M 1 X k, l D 2Tc 1 1 x m, n 2 C mk nl 2 mD0 x m, n 2nl nDTc nD0 N 1 CÐÐÐ C x m, n 2nl mod Ft 94 nDN Tc because x m C Tr , n C Tc D x m, n . Equation (94) can be written as M 1 X k, l D 2mk 1 C 2Tc l C 22Tc l mD0 Tc 1 CÐÐÐ C2 N Tc l x m, n 2nl nD0 mod Ft 95 NUMBER THEORETIC TRANSFORMS AND THEIR APPLICATIONS 43 Summing along columns with respect to the period Tr , Equation (95) can be written as X k, l D 1 C 2Tr k C 22Tr k C Ð Ð Ð C 2 M Tr k ð 1 C 2Tc l C 22Tc l Tr 1 Tc 1 C ÐÐÐ C 2 N Tc l x m, n 2mk 2nl mod Ft 96 mD0 nD0 Factoring Equation (96) gives Tr 1 Tc 1 X k, l D A k Ð A l x m, n 2mk 2nl mod Ft 97 mD0 nD0 where log2 M/Tr Ak D 1 C 2 M/E k ; E D 2q mod Ft 98 1 C 2 N/G l ; G D 2p mod Ft 99 qD1 log2 N/Tc Al D pD1 From Equation (98) and Equation (99): Ak D Al D M/Tr if k D Mi/Tr , 0 otherwise N/Tc if l D Nj/Tc , 0 otherwise i D 0, 1, 2, .

F) 2-D FNT transform of (e). NUMBER THEORETIC TRANSFORMS AND THEIR APPLICATIONS 27 In ﬁnite ﬁelds Equation (60) has two solutions: k D M/2 and l D N/2 61 Thus, the NTT of this square contains one row and one column of zero values in the middle, which divide it into four parts, as shown in Figure 3(b). Note no particular modulus or transform length is speciﬁed, although in the following we use F3 with the transform dimensions M D N D 16 and ˛ D 2. Unless otherwise stated, both the image data and the 2-D FNT are displayed using 32 grey levels, ranging from black for zero intensity to white for the maximum intensity.

The method described here uses only ﬁve I/Os: four to compute the two-dimensional NTT and one to permute the rows. b. Example of 2-D Autoconvolution Obtained from Simulations. Modulus D 28 C 1 D 257, ˛ D 35, array size D 4 ð 4 (padded with zeros to form 64 ð 64), magnitude D 2. NUMBER THEORETIC TRANSFORMS AND THEIR APPLICATIONS 21 Input array: 2 2 2 2 0 0 x m, n D 2 2 2 2 0 0 2 2 2 2 0 0 0 0 0 2 2 2 2 0 0 0 0 0 0 0 0 .. 0 0 0 0 0 0 0 0 0 ... ... ... 0 0 0 0 64 0 0 35 ... 0 64 In this case the two-dimensional FNT is given by 63 63 x k, l D x m, n 35hmki 35hnli mod 257 36 mod 257 37 x n, m 35hnli mod 257 38 xm n 35hnli mod 257 39 mod 257 40 mod 257 41 mD0 nD0 63 D 35hmki mD0 63 G m, l D 63 x m, n 35hnli nD0 nD0 or 63 Gm 1 D nD0 Here h i stands for mod 64.