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By Samuel Y. Liao

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38) g(k) = g'(k)h(k), where h(z) and h(k) are the resulting PSF and OTF, respectively, and g'(z) can be considered as the gray value image obtained with a perfect sensor, i. , an optical system (including the sensor) whose OTF is identically 1 and whose PSF is a <5-function. Generally, the image formation process results in a blurring of the image; fine details are lost. In Fourier space this leads to an attenuation of high wave numbers. The resulting gray value image is called bandlimited. Sampling Now we perform the sampling.

The white background, however, shows the same feature at the questionable position, where the object might either be adjacent or not. Thus the inconsistency arises, that we may have either two crossing adjacent objects in the 8-neigborhood or two separated objects in the 4-neighborhood. This difficulty can be overcome if we declare the objects as 4-neighboring and the background as 8-neighboring, or vice versa. These complications are a special feature of the reetangular grid. 12e). On a hexagonal grid, we can only define a 6-neighborhood, since pixels which have a joint corner, but no joint edge, do not exist.

The orthogonality condition leaves three matrix elements independent out of nine. Unfortunately, the relationship between the matrix elements and sets of three such parameters turns out to be quite complex and nonlinear. A widely used set of parameters are the three Eulerian rotation angles. Any rotation can be decomposed into three consecutive rotations about the axes of the coordinate system with these angles. A more detailed discussion can be found in textbooks of classical mechanics such as Goldstein [1980].

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