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An imaging system can be characterized by its point spread function (PSF), which in most cases is space-variant. The final image is the result of the convolution of the PSF with the object (2d representation). Also, we know that the operation of convolution can achieved by multiplication in the Fourier domain.

So the question is: Can an optical system be described by a single unique matrix operator in the Fourier Domain which we can multiply with the object's Fourier transform to achieve the final image (in Fourier space, of course)?

EDIT: I've been doing some thinking about my topic, and it occurred to me for every PSF there's a corresponding MTF (modulation transfer function), that determines how well can a certain feature in the object plain will be imaged be the optical system. So it follows that to characterize an optical system in Fourier space, each point\matrix element needs to have a MTF curve. So instead of a single unique matrix of constants, we need a unique matrix of functions (MTFs), which are modulation dependent. This suggests to me that this operator cannot be multiplied be the object's Fourier transform (which was the assumption in the original question), but rather, we need a modulation representation of the object, to go with the MTFs elements.

Your thoughts?

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  • $\begingroup$ Evidently, as you've proved, this is possible. Have you got more specific questions about this? $\endgroup$ – Emilio Pisanty Sep 25 '15 at 22:26
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    $\begingroup$ Taking the Fourier transform from the spatial domain to spatial frequency domain gives the optical transfer function , which is certainly used in optical analysis. I think the question may be a bit too broad. $\endgroup$ – paisanco Sep 26 '15 at 0:13
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A single operator cannot be used with a space-variant point spread function. You need to do the calculation for every pixel in this case. Calculating the image with a space variant PSF is therefore slow and the PSF needs to be measured at every position (or guessed at usually by trying to interpolate it from a sparse set of measurements). It is therefore a lot easier and faster to make the approximation that the PSF is space invariant.

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As in Emilio Pisanty's comment:

Evidently, as you've proved, this is possible...

an assumption of shift invariance validates the use of the Fourier transfer function exactly as you describe. Practically, however, the point spread function varies, sometimes drastically, across the whole field of view so that shift invariance is a more often than not a poor assumption, unless your image is nonzero only in a small region compared with the whole field of view.

For example, for brightfield microscopy objectives, the assumption is made that most of the human user's information will be in the center of the image. Therefore, very high correction / low aberration is demanded in the middle fifth of the field of view. But a human operator only uses the edge information for navigation within the sample, so the designer will allow the performance to drop off swiftly with increasing distance from the optical axis. I've seen in commercial objectives the Strehl ratio vary from 0.9 (diffraction limited, for most purposes) to as low as 0.1 at the edge of the field of view, with a steady dropoff with increasing lateral distance from the optical axis, and this does not seem to degrade the objective's utility noticeably. In aberration terms, this corresponds to an optical path difference of roughly 0.05 waves on axis, to of the order of 0.2 waves at the edge of the FOV. Clearly the point spread function varies a great deal across the field of view of such a system.

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