Optical transfer function: Difference between revisions

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Formally, the optical transfer function is defined as the [[Fourier transform]] of the [[point spread function]] (PSF, that is, the [[impulse response]] of the optics, the image of a point source). As a Fourier transform, the OTF is generally complex-valued; however, it is real-valued in the common case of a PSF that is symmetric about its center. In practice, the imaging contrast, as given by the [[Absolute value|magnitude or modulus]] of the optical-transfer function, is of primary importance. This derived function is commonly referred to as the '''modulation transfer function''' ('''MTF''').

The image on the right shows the optical transfer functions for two different optical systems in panels (a) and (d). The former corresponds to the ideal, [[diffraction-limited system|diffraction-limited]], imaging system with a circular [[pupil function|pupil]]. Its transfer function decreases approximately gradually with spatial frequency until it reaches the diffraction-limit, in this case at 500 cycles per millimeter or a period of 2 μm. Since periodic features as small as this period are captured by this imaging system, it could be said that its resolution is 2 μm.<ref>The exact definition of resolution may vary and is often taken to be 1.22 times larger as defined by the [[angular resolution|Rayleigh criterion]].</ref> Panel (d) shows an optical system that is out of focus. This leads to a sharp reduction in contrast compared to the diffraction-limited imaging system. It can be seen that the contrast is zero around 250 cycles/mm, or periods of 4 μm. This explains why the images for the out-of-focus system (e,f) are more blurry than those of the diffraction-limited system (b,c). Note that although the out-of-focus system has very low contrast at spatial frequencies around 250 cycles/mm, the contrast at spatial frequencies nearjust below the diffraction limit of 500 cycles/mm is diffraction-limitedcomparable to that of the ideal system. Close observation of the image in panel (f) shows that the image of the large spoke densities near the center of the [[spoke target]] is relatively sharp.

==Definition and related concepts==

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It can be read from the plot that the contrast gradually reduces and reaches zero at the spatial frequency of 500 cycles per millimeter,. inIn other words the optical resolution of the image projection is 1/500{{sup|th}} of a millimeter, orwhich corresponds to a feature size of 2 micrometer. Beyond 500 cycles per millimeter, the contrast of this imaging system, and therefore its modulation transfer function, is zero. Correspondingly, for this particular imaging device, the spokes become more and more blurred towards the center until they merge into a gray, unresolved, disc.

Note that sometimes the optical transfer function is given in units of the object or sample space, observation angle, film width, or normalized to the theoretical maximum. Conversion between units is typically a matter of a multiplication or division. E.g. a microscope typically magnifies everything 10 to 100-fold, and a reflex camera will generally demagnify objects at a distance of 5 meter by a factor of 100 to 200.

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The intersecting area can be calculated as the sum of the areas of two identical [[circular segment]]s: <math> \theta/2 - \sin(\theta)/2</math>, where <math>\theta</math> is the circle segment angle. By substituting <math>|\nu| = \cos(\theta/2)</math>, and using the equalities <math>\sin(\theta)/2 = \sin(\theta /2)\cos(\theta /2)</math> and <math>1 = \nu^2 + \sin(\arccos(|\nu|))^2</math>, the equation for the area can be rewritten as <math>\arccos(|\nu|) - |\nu|\sqrt{1 - \nu^2}</math>. Hence the normalized optical transfer function is given by:

: <math>\operatorname{OTF}(\nu) = \frac{2}{\pi} \left([\arccoscos^{-1}(|\nu|)-|\nu|\sqrt{1-\nu^2}\right).]</math> for <math>|\nu| < 1</math> and 0 otherwise.

A more detailed discussion can be found in <ref name=Goodman2005/> and.<ref name=Williams2002/>{{rp|152–153}}