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 good points prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photo technology and electronic picture processing, electromagnetic wave propagation, electron microscopy, and the computing equipment utilized in some of these domain names.
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Extra info for Advances in Imaging and Electron Physics, Vol. 111
No twiddling occurs afterwards, and hence no time has been wasted. So far, it has been shown how the rows of a large input array can be transformed using a small-length FNT that needs no multiplications but only additions and word shifts (multiplication by powers of 2). The columns of the input array follow the same development, but because it was assumed that the data are available only in rows, the whole matrix must be transposed. If 32 rows of the input array can be accommodated in fast RAM, the whole twodimensional FNT of the 1024 ð 1024 image array can be computed using a 32-length FNT without transposing the array.
34 S. BOUSSAKTA AND A. G. J. HOLT (a) (b) (c) (d) FIGURE 7. Square images of sides 15 and 16 and their 2-D FNTs. (a) Square image of side 15. (b) 2-D FNT of part (a). (c) Square image of side 16. (d) 2-D FNT of part (c). presence of the triangle can be detected by counting the few zero values. If the data are in diminished one representation, the detection of zero values in the 2-D FNT domain is easily done by testing only the most signiﬁcant bit. This may be used, for example, in pattern recognition.
7 otherwise 0 mod F3 92 Thus, period Tr ð Tc D 8 ð 8 leads to a maximum of 64 nonzero elements for an image of 16 ð 16. Figure 10(b) illustrates this. For other periods and cases when the images have different periodicities along rows and columns, the results are tabulated in Tables 3 and 4. 2. Generalization for 2-D FNT Let x m, n be any periodic image with period Tr ð Tc and Ft be any modulus. Then the 2-D FNT of the data is given by M 1N 1 X k, l D x m, n 2mk 2nl mod Ft mD0 nD0 M 1 D N 1 2mk mD0 x m, n 2nl mod Ft 93 nD0 TABLE 3 CASE 1 Tr D Tc .
Advances in Imaging and Electron Physics, Vol. 111 by Peter W. Hawkes