Optical transfer function: Difference between revisions

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[[File:Illustration of the optical transfer function and its relation to image quality.svg|thumb|right|400px|Illustration of the optical transfer function (OTF) and its relation to image quality. The optical transfer function of a well-focused (a), and an out-of-focus optical imaging system without aberrations (d). As the optical transfer function of these systems is real and non-negative, the optical transfer function is by definition equal to the modulation transfer function (MTF). Images of a point source and a [[spoke target]] with high [[spatial frequency]] are shown in (b,e) and (c,f), respectively. Note that the scale of the point source images (b,e) is four times smaller than the spoke target images.]]

The '''optical transfer function''' ('''OTF''') of an optical system such as a [[camera]], [[microscope]], [[human eye]], or [[image projector|projector]] specifiesis howa differentscale-dependent description of their imaging contrast. Its magnitude is the image contrast of the [[Sine and cosine|harmonic]] intensity pattern, <math>1 + \cos(2\pi \nu \cdot x)</math>, as a function of the spatial frequenciesfrequency, are<math>\nu</math>, capturedwhile orits transmitted[[Argument (complex analysis)|complex argument]] indicates a phase shift in the periodic pattern. ItThe optical transfer function is used by optical engineers to describe how the optics project light from the object or scene onto a photographic film, [[Image sensor|detector array]], [[retina]], screen, or simply the next item in the optical transmission chain. A variant, the '''modulation transfer function''' ('''MTF'''), neglects phase effects, but is equivalent to the OTF in many situations.

Either [[transfer function]] specifiesFormally, the responseoptical to a periodic [[sine-wave]] pattern passing through the lens system, as atransfer function of its spatial frequency or period, and its orientation. Formally, the OTF 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; buthowever, it will beis real-valued in the common case of a PSF that is symmetric about its center. TheIn MTFpractice, isthe formallyimaging definedcontrast, as given by the [[Absolute value|magnitude (absoluteor value)modulus]] of the complexoptical-transfer function, is of OTFprimary 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 near the diffraction limit of 500 cycles/mm is diffraction-limited. 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.

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==Limitations==

In general, the [[point spread function]], the image of a point source also depends on factors such as the [[wavelength]] ([[visible spectrum|color]]), and [[field of view|field]] angle]] (lateral point source position). When such variation is sufficiently gradual, the optical system could be characterized by a set of optical transfer functions. However, when the image of the point source changes abruptly upon lateral translation, the optical transfer function does not describe the optical system accurately. Inaccuracies can often be mitigated by a collection of optical transfer functions at well-chosen wavelengths or field-positions. However, a more complex characterization may be necessary for some imaging systems such as the [[Light field camera]].

==See also==