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, picture technology and electronic photo processing, electromagnetic wave propagation, electron microscopy, and the computing tools utilized in some of these domain names.
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Additional info for Advances in Imaging and Electron Physics, Vol. 111
A possible application could be in the process for integrated or ﬁlm circuits, where photomasks are used for fabrication steps. Typically, these photomasks possess a repeated or periodic pattern. Before being used the masks must be carefully inspected for any possible defects, which, if not detected, would result in the manufacture of faulty circuits. In this case, it may be possible to apply the 2-D FNT method to readily show the existence of a mask defect. The inspection process begins by taking a digitized image (of dimension M ð N) of the photomask followed by the computation of its 2-D FNT.
Thus, the regular zero values of B k, l are the same as the zeros of: 1 C ˛3k 1 C ˛3l 1 2 D0 mod F 71 which gives: k D M 2i C 1 /6 and l D N 2j C 1 /6 72 where i, j are integers such that k and l are also integers and less than M and N, respectively. For M D N D 16, there are two solutions, corresponding to k D 8 and l D 8 73 Thus, a square image of side L D 6 is characterized by one row and one column of zeros in the middle of the NTT image dividing it into four parts, which is the same as a square image of size 2 ð 2.
HOLT Theorem If an image x m, n is periodic with period Tr ð Tc , the number of nonzero pixels in its 2-D FNT is less than or equal to the product Tr ð Tc . These nonzero pixels are determined by the following equation: Tr 1 Tc 1 X k, l D M/Tr N/Tc nl x m, n ˛mk 1 ˛2 mod Ft mD0 nD0 with k D Mi/Tr ; l D Nj/Tc ; i D 0, 1, 2, . . , Tr 1; j D 0, 1, 2, . . , Tc 1. 103 The result in Equation (103) is derived for Fermat number transforms. It can be extended to other number theoretic transforms. 3. The Effect of Data Errors on 2-D FNT We have considered cases where the data are periodic.