Optical transfer function: Difference between revisions

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{{short description|Function that specifies how different spatial frequencies are handledcaptured 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-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 spokeimage structure is relatively sharp forof the large spoke densities near the center of the [[spoke target]] is relatively sharp.

==Definition and related concepts==

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[[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).]]

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! Dimensions !! Spatial function !! Fourier transform

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==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 book |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 |doi= 10.1109/ICASSP.1988.196731 |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''&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]].

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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>\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}}

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

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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>

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===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 |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 |url= http://www.purplebark.net/mra/research/votf/|bibcode= 2002OptCo.211...53A|url-access= subscription }}</ref>).

==Measurement==

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

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

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

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

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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]]