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 gains prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photograph technological know-how and electronic snapshot processing, electromagnetic wave propagation, electron microscopy, and the computing tools utilized in most of these domain names.
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Additional resources for Advances in Imaging and Electron Physics, Vol. 111
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, .
2. If ps D 32, then 2ps D 1 mod F4 D 2ps mod 32 . 3. If ps > 32, then ps D ps mod 32 C 32t t D integer and 2ps D 2ps mod 32 Ð 232t D 2ps mod 32 . Equation (19) is then written in the form 31 Gm p,q D 2jpsj ˛hqsi sD0 31 xm s, r 2jrqj mod F 20 rD0 p, q D 0, 1, . . , 31 where j Ð j stands for mod 32. 1. The Twiddle Factor ˛qs The twiddle factor arises from the transformation of the one-dimensional process into a two-dimensional process and acts on the result of the ﬁrst one-dimensional FNT only. It also creates two problems, one of storage and one of time.
The number of nonzero elements in the 2-D FNT is small, especially for the data with Tr ð Tc − M ð N, which is typically the case with a photomask. 48 S. BOUSSAKTA AND A. G. J. HOLT (a) (b) FIGURE 13. (a) Defective photomask with just one pixel error. (b) 2-D FNT of part (a) showing a completely different FNT pattern of the defective mask. Therefore, the detection process may require comparison of only a few 2-D FNT elements. Any differences between the 2-D FNT values being compared automatically imply a defective mask.