Optical transfer function: Difference between revisions

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Although one typically thinks of an image as planar, or two-dimensional, the imaging system will produce a three-dimensional intensity distribution in image space that in principle can be measured. e.g. a two-dimensional sensor could be translated to capture a three-dimensional intensity distribution. The image of a point source is also a three dimensional (3D) intensity distribution which can be represented by a 3D point-spread function. As an example, the figure on the right shows the 3D point-spread function in object space of a wide-field microscope (a) alongside that of a confocal microscope (c). Although the same microscope objective with a numerical aperture of 1.49 is used, it is clear that the confocal point spread function is more compact both in the lateral dimensions (x,y) and the axial dimension (z). One could rightly conclude that the resolution of a confocal microscope is superior to that of a wide-field microscope in all three dimensions.

A three-dimensional optical transfer function can be calculated as the three-dimensional Fourier transform of the 3D point-spread function. Its color-coded magnitude is plotted in panels (b) and (d), corresponding to the point-spread functions shown in panels (a) and (c), respectively. The transfer function of the wide-field microscope has a [[support (mathematics)|support]] that is half of that of the confocal microscope in all three-dimensions, confirming the previously noted lower resolution of the wide-field microscope. Note that along the z-axis, for x=y=0, the transfer function is zero everywhere except at the origin. This ''missing cone'' is a well-known problem that prevents optical sectioning using a wide-field microscope.<ref name=MaciasGarza88>{{cite journalbook |last1= Macias-Garza |first1= F. |last2= Bovik |first2= A. |last3= Diller |first3= K. |last4= Aggarwal |first4= S. |last5= Aggarwal |first5= J. |title= ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing |chapter= The missing cone problem and low-pass distortion in optical serial sectioning microscopy |conference= ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing, Acoustics, Speech, and Signal Processing, 1988. ICASSP-88., 1988 International Conference on, Institute of Electrical & Electronics Engineers (IEEE) |pages= 890–893 |volume= 2 |year= 1988 |urldoi= https:10.1109//ieeexploreICASSP.ieee1988.org/document/196731 }}</ref>

The two-dimensional optical transfer function at the focal plane can be calculated by integration of the 3D optical transfer function along the z-axis. Although the 3D transfer function of the wide-field microscope (b) is zero on the z-axis for z≠0; its integral, the 2D optical transfer, reaching a maximum at x=y=0. This is only possible because the 3D optical transfer function diverges at the origin x=y=z=0. The function values along the z-axis of the 3D optical transfer function correspond to the [[Dirac delta function]].