For example, in radiography, we would require the gray values and relative distances in the projection image to be constant, if we move the object or body with respect to the center of the x-ray beam.

If these two conditions are met, the imaging system can be described using the theory of linear, shift invariant systems (LSI), where the input-output-relation is given by a convolution with a single function h, the PSF, which characterizes the system

$g\left(x,y\right)=f\left(x,y\right)*h\left(x,y\right)={\displaystyle \int {\displaystyle \int dx\prime dy\prime f\left(x\prime ,y\prime \right)h\left(x-x\prime ,y-y\prime \right).}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(2.47\right)$

Next, we sketch how equation (2.47) can be obtained, following the derivation presented in [13]. For the sake of simplicity, we consider a 1d input signal f(x). Generalization to higher dimensions ...

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