Optical transfer function: Difference between revisions

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{{short description|Function that specifies how different spatial frequencies are captured by an optical system}}

[[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>, handledwhile byits [[Argument (complex analysis)|complex argument]] indicates a phase shift in the systemperiodic 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-limited system|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 system|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|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 spokeimage structure is relatively sharp forof the large spoke densities near the center of the [[spoke target]] is relatively sharp.

==Definition and related concepts==

Since the optical transfer function<ref name=Williams2002>{{cite book |first=Charles S.|last=Williams|year=2002|title=Introduction to the Optical Transfer Function|publisher=SPIE -– The International Society for Optical Engineering|isbn=0-8194-4336-0}}</ref> (OTF) is defined as the Fourier transform of the point-spread function (PSF), it is generally speaking a [[complex number|complex-valued]] function of [[spatial frequency]]. The projection of a specific periodic pattern is represented by a complex number with absolute value and [[complex argument]] proportional to the relative contrast and translation of the projected projection, respectively.

[[File:Definitions PSF OTF MTF PhTF.svg|right|thumb|400px|Various closely related characterizations of an optical system exhibiting coma, a typical aberration that occurs off-axis. (a) The point-spread function (PSF) is the image of a point source. (b) The image of a line is referred to as the line-spread function, in this case a vertical line. The line-spread function is directly proportional to the vertical integration of the point-spread image. The optical-transfer function (OTF) is defined as the Fourier transform of the point-spread function and is thus generally a two-dimensional complex function. Typically only a one-dimensional slice is shown (c), corresponding to the Fourier transform of the line-spread function. The thick green line indicates the real part of the function, and the thin red line the imaginary part. Often only the absolute value of the complex function is shown, this allows visualization of the two-dimensional function (d); however, more commonly only the one-dimensional function is shown (e). The latter is typically normalized at the spatial frequency zero and referred to as the modulation transfer function (MTF). For completeness, the complex argument is sometimes provided as the phase transfer function (PhTF), shown in panel (f).]]

{| class="wikitable floatright"

|-

! Dimensions !! Spatial function !! Fourier transform

|-

! 1D

| Line-spread function<br />(derivative of edge-spread function)

| 1D section of 2D optical-transfer function

|-

! 2D

| Point-spread function

| (2D) Optical transfer function

|-

! 3D

| 3D Point-spread function

| 3D Optical-transfer function

|}

Often the contrast reduction is of most interest and the translation of the pattern can be ignored. The relative contrast is given by the absolute value of the optical transfer function, a function commonly referred to as the '''modulation transfer function''' ('''MTF'''). Its values indicate how much of the object's contrast is captured in the image as a function of spatial frequency. The MTF tends to decrease with increasing spatial frequency from 1 to 0 (at the diffraction limit); however, the function is often not [[monotonic]]. On the other hand, when also the pattern translation is important, the [[complex argument]] of the optical transfer function can be depicted as a second real-valued function, commonly referred to as the '''phase transfer function''' ('''PhTF'''). The complex-valued optical transfer function can be seen as a combination of these two real-valued functions:

:<math>\mathrm{OTF}(\nu)=\mathrm{MTF}(\nu)e^{i\,\mathrm{PhTF}(\nu)}</math>

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Generally, the optical transfer function depends on factors such as the spectrum and polarization of the emitted light and the position of the point source. E.g. the image contrast and resolution are typically optimal at the center of the image, and deteriorate toward the edges of the field-of-view. When significant variation occurs, the optical transfer function may be calculated for a set of representative positions or colors.

Sometimes it is more practical to define the transfer functions based on a binary black-white stripe pattern. The transfer function for an equal-width black-white periodic pattern is referred to as the '''contrast transfer function (CTF)'''.<ref name=CTF >{{cite web |title=Contrast Transfer Function|url=http://www.microscopyu.com/articles/optics/mtfintro.html|accessdateaccess-date=16 November 2013}}</ref>

==Examples==

===The OTF of an idealIdeal lens system===

A perfect lens system will provide a high contrast projection without shifting the periodic pattern, hence the optical transfer function is identical to the modulation transfer function. Typically the contrast will reduce gradually towards zero at a point defined by the resolution of the optics. For example, a perfect, [[optical aberration|non-aberrated]], [[F-number|f/4]] optical imaging system used, at the visible wavelength of 500&nbsp;nm, would have the optical transfer function depicted in the right hand figure.

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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 the twounits 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.

The resolution of a digital imaging device is not only limited by the optics, but also by the number of pixels, more in particular by their separation distance. As explained by the [[Nyquist–Shannon sampling theorem]], to match the optical resolution of the given example, the pixels of each color channel should be separated by 1 micrometer, half the period of 500 cycles per millimeter. A higher number of pixels on the same sensor size will not allow the resolution of finer detail. On the other hand, when the pixel spacing is larger than 1 micrometer, the resolution will be limited by the separation between pixels; moreover, [[aliasing]] may lead to a further reduction of the image fidelity.

===OTF of an imperfectImperfect lens system===

An imperfect, [[optical aberration|aberrated]] imaging system could possess the optical transfer function depicted in the following figure.

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While it could be argued that the resolution of both the ideal and the imperfect system is 2&nbsp;μm, or 500 LP/mm, it is clear that the images of the latter example are less sharp. A definition of resolution that is more in line with the perceived quality would instead use the spatial frequency at which the first zero occurs, 10&nbsp;μm, or 100 LP/mm. Definitions of resolution, even for perfect imaging systems, vary widely. A more complete, unambiguous picture is provided by the optical transfer function.

===The OTF of an opticalOptical system with a non-rotational symmetric aberration===

[[File:Trefoil aberration PSF OTF and example image.svg|right|thumb|600px|When viewed through an optical system with trefoil aberration, the image of a point object will look as a three-pointed star (a). As the point-spread function is not rotational symmetric, only a two-dimensional optical transfer function can describe it well (b). The height of the surface plot indicates the absolute value and the hue indicates the complex argument of the function. A spoke target imaged by such an imaging device is shown by the simulation in (c).]]

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==The three-dimensional optical transfer function==

[[File:3DPSF_3DMTF_widefield_confocal3DPSF 3DMTF widefield confocal.png|right|thumb|600px|The three-dimensional point spread functions (a,c) and corresponding modulation transfer functions (b,d) of a wide-field microscope (a,b) and confocal microscope (c,d). In both cases the numerical aperture of the objective is 1.49 and the refractive index of the medium 1.52. The wavelength of the emitted light is assumed to be 600 nm and, in case of the confocal microscope, that of the excitation light 500 nm with circular polarization. A section is cut to visualize the internal intensity distribution. The colors as shown on the logarithmic color scale indicate the irradiance (a,c) and spectral density (b,d) normalized to the maximum value.]]

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''&nbsp;=&nbsp;''y''&nbsp;=&nbsp;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 <!-- unsupported parameter |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= http://ieeexplore10.ieee.org/xpl1109/articleDetailsICASSP.1988.jsp?arnumber=196731 |s2cid= 120191405 }}</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''z''&nbsp;≠&nbsp;0; its integral, the 2D optical transfer, reaching a maximum at ''x''&nbsp;=&nbsp;''y''&nbsp;=&nbsp;0. This is only possible because the 3D optical transfer function diverges at the origin ''x''&nbsp;=&nbsp;''y''&nbsp;=&nbsp;''z''&nbsp;=&nbsp;0. The function values along the ''z''-axis of the 3D optical transfer function correspond to the [[Dirac delta function]].

==Calculation==

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The pupil function of an ideal optical system with a circular aperture is a disk of unit radius. The optical transfer function of such a system can thus be calculated geometrically from the intersecting area between two identical disks at a distance of <math>2\nu</math>, where <math>\nu</math> is the spatial frequency normalized to the highest transmitted frequency.<ref name=Williams2002/> In general the optical transfer function is normalized to a maximum value of one for <math>\nu = 0</math>, so the resulting area should be divided by <math>\pi</math>.

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>\mathitoperatorname{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}}

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[[File:MTF example graph.jpg|thumb|right|310px|The '''MTF''' data versus spatial frequency is normalized by fitting a sixth order polynomial to it, making a smooth curve. The 50% cut-off frequency is determined and the corresponding '''spatial frequency''' is found, yielding the approximate position of '''best focus'''.]]

The Fourier transform of the line spread function (LSF) can not be determined analytically by the following equations{{Citation needed|reason=Many functions have analytical Fourier Transforms|date=August 2024}}:

:<math>\textoperatorname{MTF} = \mathcal{F} \left[ \textoperatorname{LSF}\right] \qquad \qquad \textoperatorname{MTF}= \int f(x) e^{-i 2 \pi\, x s}\, dx</math>

Therefore, the Fourier Transform is numerically approximated using the discrete Fourier transform <math>\mathcal{DFT}</math>.<ref>Chapra, S.C.; Canale, R.P. (2006). ''Numerical Methods for Engineers (5th ed.). New York, New York: McGraw-Hill</ref>

:<math>\textoperatorname{MTF} = \mathcal{DFT}[\textoperatorname{LSF}] = Y_k = \sum_{n=0}^{N-1} y_n e^{-ik \frac{2 \pi}{N} n} \qquad k\in [0, N-1] </math>

where

* <math>Y_k\,</math> = the <math>k^\text{th}</math> value of the MTF

* <math>N\,</math> = number of data points

* <math>n\,</math> = index

* <math>k\,</math> = <math>k^\text{th}</math> term of the LSF data

* <math>y_n\,</math> = <math>n^\text{th}\,</math> pixel position

* <math>i=\sqrt{-1}</math>

:<math> e^{\pm ia} = \cos(a) \, \pm \, i \sin(a) </math>

:<math>\textoperatorname{MTF}= \mathcal{DFT}[\textoperatorname{LSF}] = Y_k = \sum_{n=0}^{N-1} y_n \left[\cos\left(k\frac{2 \pi}{N} n\right) - i\sin\left(k \frac{2 \pi}{N} n\right)\right] \qquad k\in[0,N-1]</math>

The MTF is then plotted against spatial frequency and all relevant data concerning this test can be determined from that graph.

===The vectorial transfer function===

At high numerical apertures such as those found in microscopy, it is important to consider the vectorial nature of the fields that carry light. By decomposing the waves in three independent components corresponding to the Cartesian axes, a point spread function can be calculated for each component and combined into a ''vectorial'' point spread function. Similarly, a ''vectorial'' optical transfer function can be determined as shown in (<ref name=Sheppard1997>{{cite journal |last1= Sheppard |first1= C.J.R. |last2= Larkin |first2= K. |title= Vectorial pupil functions and vectorial transfer functions |journal= Optik-Stuttgart |volume= 107 |pages= 79–87 |year= 1997 |pmid= |pmc= |url= http://www.nontrivialzeros.net/KGL_Papers/28_Vectorial_OTF_Optik_1997.pdf}}</ref>) and (<ref name=Arnison2002>{{cite journal |last1= Arnison |first1= M. R. |last2= Sheppard |first2= C. J. R. |doi= 10.1016/S0030-4018(02)01857-6 |title= A 3D vectorial optical transfer function suitable for arbitrary pupil functions |journal= Optics Communications |volume= 211 |issue= 1–6 |pages= 53–63 |year= 2002 |pmid= |pmc= |url= http://www.purplebark.net/mra/research/votf/|bibcode= 2002OptCo.211...53A|url-access= subscription }}</ref>).

==Measurement==

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====Edge-spread function====

The two-dimensional Fourier transform of an edge is also only non-zero on a single line, orthogonal to the edge. This function is sometimes referred to as the '''edge spread function''' (ESF).<ref>Holst, G.C. (1998). ''Testing and Evaluation of Infrared Imaging Systems'' (2nd ed.). Florida:JCD Publishing, Washington:SPIE.</ref><ref name="ElectroOpticalTestLab">{{cite web|url=http://www.electro-optical.com/html/toplevel/educationref.asp|title=Test and Measurement -– Products -– EOI|website=www.Electro-Optical.com|access-date=2 January 2018|archive-url=https://web.archive.org/web/20080828124035/http://www.electro-optical.com/html/toplevel/educationref.asp|archive-date=28 August 2008|url-status=dead}}</ref> However, the values on this line are inversely proportional to the distance from the origin. Although the measurement images obtained with this technique illuminate a large area of the camera, this mainly benefits the accuracy at low spatial frequencies. As with the line spread function, each measurement only determines a single axes of the optical transfer function, repeated measurements are thus necessary if the optical system cannot be assumed to be rotational symmetric.

[[File:MTF knife-edge target.jpg|thumb|right|215px|In evaluating the '''ESF''', an operator defines a box area equivalent to 10%{{citation needed|date=August 2013}} of the total frame area of a '''knife-edge test target''' back-illuminated by a '''black body'''. The area is defined to encompass the edge of the target image.]]

As shown in the right hand figure, an operator defines a box area encompassing the edge of a '''knife-edge test target''' image back-illuminated by a [[black body]]. The box area is defined to be approximately 10%{{citation needed|date=August 2013}} of the total frame area. The image [[pixel]] data is translated into a two-dimensional array ([[pixel]] intensity and pixel position). The amplitude (pixel intensity) of each [[line (video)|line]] within the array is [[normalization (statistics)|normalized]] and averaged. This yields the edge spread function.

:<math>\textoperatorname{ESF} = \frac{X - \mu}{\sigma} \qquad \qquad \sigma\, = \sqrt{\frac{\sum_{i=0}^{n-1} (x_i-\mu\,)^2}{n}} \qquad \qquad \mu\, = \frac{\sum_{i=0}^{n-1} x_i}{n} </math>

where

* ESF = the output array of normalized pixel intensity data

* <math>X\,</math> = the input array of pixel intensity data

* <math>x_i\,</math> = the ''i''^th element of <math>X\,</math>

* <math>\mu\,</math> = the average value of the pixel intensity data

* <math>\sigma\,</math> = the standard deviation of the pixel intensity data

* <math>n\,</math> = number of pixels used in average

The line spread function is identical to the [[derivative|first derivative]] of the edge spread function,<ref name=Mazzetta2007>Mazzetta, J.A.; Scopatz, S.D. (2007). Automated Testing of Ultraviolet, Visible, and Infrared Sensors Using Shared Optics.'' Infrared Imaging Systems: Design Analysis, Modeling, and Testing XVIII, Vol. 6543'', pp. 654313-1 654313-14</ref> which is differentiated using [[numerical analysis|numerical methods]]. In case it is more practical to measure the edge spread function, one can determine the line spread function as follows:

:<math>\textoperatorname{LSF} = \frac{d}{dx} \textoperatorname{ESF}(x)</math>

Typically the ESF is only known at discrete points, so the LSF is numerically approximated using the [[finite difference]]:

:<math> \textoperatorname{LSF} = \frac{d}{dx}\textoperatorname{ESF}(x) \approx \frac{\Delta \textoperatorname{ESF}}{\Delta x}</math>

:<math> \textoperatorname{LSF} \approx \frac{\textoperatorname{ESF}_{i+1} - \textoperatorname{ESF}_{i-1}}{2(x_{i+1} - x_i)}</math>

where:

* <math>i\,</math> = the index <math>i = 1,2,\dots,n-1</math>

* <math>x_i\,</math> = <math>i^\text{th}\,</math> position of the <math>i^\text{th}\,</math> pixel

* <math>\textoperatorname{ESF}_i\,</math> = ESF of the <math>i^\text{th}\,</math> pixel

====Using a grid of black and white lines====

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===Oversampling and downconversion to maintain the optical transfer function===

The only way in practice to approach the theoretical sharpness possible in a digital imaging system such as a camera is to use more pixels in the camera sensor than [[sampling (signal processing)|samples]] in the final image, and 'downconvert' or 'interpolate' using special digital processing which cuts off high frequencies above the [[Nyquist rate]] to avoid aliasing whilst maintaining a reasonably flat MTF up to that frequency. This approach was first taken in the 1970s when flying spot scanners, and later [[charge-coupled device|CCD]] line scanners were developed, which sampled more pixels than were needed and then downconverted, which is why movies have always looked sharper on television than other material shot with a video camera. The only theoretically correct way to interpolate or downconvert is by use of a steep low-pass spatial filter, realized by [[convolution]] with a two-dimensional sin(''x'')/''x'' [[weighting]] function which requires powerful processing. In practice, various mathematical approximations to this are used to reduce the processing requirement. These approximations are now implemented widely in video editing systems and in image processing programs such as [[Photoshop]].

Just as standard definition video with a high contrast MTF is only possible with oversampling, so HD television with full theoretical sharpness is only possible by starting with a camera that has a significantly higher resolution, followed by digitally filtering. With movies now being shot in [[4K resolution|4k]] and even 8k video for the cinema, we can expect to see the best pictures on HDTV only from movies or material shot at the higher standard. However much we raise the number of pixels used in cameras, this will always remain true in absence of a perfect optical spatial filter. Similarly, a 5-megapixel image obtained from a 5-megapixel still camera can never be sharper than a 5-megapixel image obtained after down-conversion from an equal quality 10-megapixel still camera. Because of the problem of maintaining a high contrast MTF, broadcasters like the [[BBC]] did for a long time consider maintaining standard definition television, but improving its quality by shooting and viewing with many more pixels (though as previously mentioned, such a system, though impressive, does ultimately lack the very fine detail which, though attenuated, enhances the effect of true HD viewing).

Another factor in digital cameras and camcorders is lens resolution. A lens may be said to 'resolve' 1920 horizontal lines, but this does not mean that it does so with full modulation from black to white. The 'Modulationmodulation Transfertransfer Functionfunction' (just a term for the magnitude of the optical transfer function with phase ignored) gives the true measure of lens performance, and is represented by a graph of amplitude against spatial frequency.

Lens aperture diffraction also limits MTF. Whilst reducing the aperture of a lens usually reduces aberrations and hence improves the flatness of the MTF, there is an optimum aperture for any lens and image sensor size beyond which smaller apertures reduce resolution because of diffraction, which spreads light across the image sensor. This was hardly a problem in the days of plate cameras and even 35&nbsp;mm film, but has become an insurmountable limitation with the very small format sensors used in some digital cameras and especially video cameras. First generation HD consumer camcorders used 1/4-inch sensors, for which apertures smaller than about f4 begin to limit resolution. Even professional video cameras mostly use 2/3&nbsp;inch sensors, prohibiting the use of apertures around f16 that would have been considered normal for film formats. Certain cameras (such as the [[Pentax K10D]]) feature an "MTF autoexposure" mode, where the choice of aperture is optimized for maximum sharpness. Typically this means somewhere in the middle of the aperture range.<ref>{{cite web|url=http://www.b2bvideosource.com/mm5/merchant.mvc?Screen=CAMERA_TERMINOLOGY&Store_Code=BVS|title=B2BVideoSource.com: Camera Terminology|website=www.B2BVideoSource.com|access-date=2 January 2018}}</ref>

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There has recently been a shift towards the use of large image format [[digital single-lens reflex camera]]s driven by the need for low-light sensitivity and narrow [[depth of field]] effects. This has led to such cameras becoming preferred by some film and television program makers over even professional HD video cameras, because of their 'filmic' potential. In theory, the use of cameras with 16- and 21-megapixel sensors offers the possibility of almost perfect sharpness by downconversion within the camera, with digital filtering to eliminate aliasing. Such cameras produce very impressive results, and appear to be leading the way in video production towards large-format downconversion with digital filtering becoming the standard approach to the realization of a flat MTF with true freedom from aliasing.

==Digital inversion of the optical transfer functionOTF==

Due to optical effects the contrast may be sub-optimal and approaches zero before the [[Nyquist–Shannon sampling theorem|Nyquist frequency]] of the display is reached. The optical contrast reduction can be partially reversed by digitally amplifying spatial frequencies selectively before display or further processing. Although more advanced digital [[Iterative reconstruction|image restoration]] procedures exist, the [[Wiener deconvolution]] algorithm is often used for its simplicity and efficiency. Since this technique multiplies the spatial spectral components of the image, it also amplifies noise and errors due to e.g. aliasing. It is therefore only effective on good quality recordings with a sufficiently high signal-to-noise ratio.

==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==

* [[Bokeh]]

* [[Gamma correction]]

* [[Minimum resolvable contrast]]

* [[Minimum resolvable temperature difference]]

* [[Optical resolution]]

* [[Signal-to-noise ratio]]

* [[Signal transfer function]]

* [[Strehl ratio]]

* [[Transfer function]]

* [[Wavefront coding]]

==References==

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==External links==

* [https://spie.org/publications/tt52_131_modulation_transfer_function "Modulation transfer function"], by Glenn D. Boreman on SPIE Optipedia.

* [https://www.optikos.com/wp-content/uploads/2013/11/How-to-Measure-MTF.pdf "How to Measure MTF and other Properties of Lenses"], by Optikos Corporation.

[[Category:Optics|Transfer function]]