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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 [[
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, [[
==
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
[[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
|-
! 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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The impulse response of a well-focused optical system is a three-dimensional intensity distribution with a maximum at the focal plane, and could thus be measured by recording a stack of images while displacing the detector axially. By consequence, the three-dimensional optical transfer function can be defined as the three-dimensional Fourier transform of the impulse response. Although typically only a one-dimensional, or sometimes a two-dimensional section is used, the three-dimensional optical transfer function can improve the understanding of microscopes such as the structured illumination microscope.
True to the definition of [[
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|
==
===
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, [[
{{multiple image
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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
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.
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.▼
▲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 [[
=== OTF of an imperfect lens system ===▼
An imperfect, [[Optical Aberrations|aberrated]] imaging system could possess the optical transfer function depicted in the following figure.▼
▲An imperfect, [[
{{multiple image|align=left|total_width=560
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While it could be argued that the resolution of both the ideal and the imperfect system is 2 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 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.
===
[[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).]]
Optical systems, and in particular [[optical aberrations]] are not always rotationally symmetric. Periodic patterns that have a different orientation can thus be imaged with different contrast even if their periodicity is the same. Optical transfer function or modulation transfer functions are thus generally two-dimensional functions. The following figures shows the two-dimensional equivalent of the ideal and the imperfect system discussed earlier, for an optical system with [[
Optical transfer functions are not always real-valued. Period patterns can be shifted by any amount, depending on the aberration in the system. This is generally the case with non-rotational-symmetric aberrations. The hue of the colors of the surface plots in the above figure indicate phase. It can be seen that, while for the rotational symmetric aberrations the phase is either 0 or π and thus the transfer function is real valued, for the non-rotational symmetric aberration the transfer function has an imaginary component and the phase varies continuously.
===
While [[optical resolution]], as commonly used with reference to camera systems, describes only the number of pixels in an image, and hence the potential to show fine detail, the transfer function describes the ability of adjacent pixels to change from black to white in response to patterns of varying spatial frequency, and hence the actual capability to show fine detail, whether with full or reduced contrast. An image reproduced with an optical transfer function that 'rolls off' at high spatial frequencies will appear 'blurred' in everyday language.
Taking the example of a current high definition (HD) video system, with 1920 by 1080 pixels, the [[Nyquist–Shannon sampling theorem|Nyquist theorem]] states that it should be possible, in a perfect system, to resolve fully (with true black to white transitions) a total of 1920 black and white alternating lines combined, otherwise referred to as a spatial frequency of 1920/2=960 line pairs per picture width, or 960 cycles per picture width, (definitions in terms of cycles per unit angle or per mm are also possible but generally less clear when dealing with cameras and more appropriate to telescopes etc.). In practice, this is far from the case, and spatial frequencies that approach the [[Nyquist rate]] will generally be reproduced with decreasing amplitude, so that fine detail, though it can be seen, is greatly reduced in contrast. This gives rise to the interesting observation that, for example, a standard definition television picture derived from a film scanner that uses [[oversampling]], as described later, may appear sharper than a high definition picture shot on a camera with a poor modulation transfer function.
==
[[File:
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 [[
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
==
Most [[
# as the Fourier transform of the incoherent [[point spread function]], or
# as the auto-correlation of the [[pupil function]] of the optical system
Mathematically both approaches are equivalent. Numeric calculations are typically most efficiently done via the Fourier transform; however, analytic calculation may be more tractable using the auto-correlation approach.
===
====
=====
Since the optical transfer function is the [[Fourier transform]] of the [[point spread function]], and the
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
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>
: <math>\
A more detailed discussion can be found in <ref name=Goodman2005
===
The one-dimensional optical transfer function can be calculated as the [[discrete Fourier transform]] of the line spread function. This data is graphed against the [[spatial frequency]] data. In this case, a sixth order polynomial is fitted to the '''MTF vs. spatial frequency''' curve to show the trend. The 50% cutoff frequency is determined to yield the corresponding spatial frequency. Thus, the approximate position of best focus of the '''unit under test''' is determined from this data.
[[
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>\
▲:<math>\text{MTF} = \mathcal{F} \left[ \text{LSF}\right] \qquad \qquad \text{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>\
▲:<math>\text{MTF} = \mathcal{DFT}[\text{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>
:
:
The MTF is then plotted against spatial frequency and all relevant data concerning this test can be determined from that graph.
===
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>{{
==
The optical transfer function is not only useful for the design of optical system, it is also valuable to characterize manufactured systems.
===
The optical transfer function is defined as the [[Fourier transform]] of the [[
===
When the aberrations can be assumed to be spatially invariant, alternative patterns can be used to determine the optical transfer function such as lines and edges. The corresponding transfer functions are referred to as the line-spread function and the edge-spread function, respectively. Such extended objects illuminate more pixels in the image, and can improve the measurement accuracy due to the larger signal-to-noise ratio. The optical transfer function is in this case calculated as the two-dimensional [[
====
The two-dimensional Fourier transform of a line through the origin, is a line orthogonal to it and through the origin. The divisor is thus zero for all but a single dimension, by consequence, the optical transfer function can only be determined for a single dimension using a single '''line-spread function''' (LSF). If necessary, the two-dimensional optical transfer function can be determined by repeating the measurement with lines at various angles.
The line spread function can be found using two different methods. It can be found directly from an ideal line approximation provided by a slit test target or it can be derived from the edge spread function, discussed in the next sub section.
====
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
[[
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 [[
:
▲: <math>\text{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''<sup>th</sup> 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
▲: <math>\text{LSF} = \frac{d}{dx} \text{ESF}(x)</math>
Typically the ESF is only known at discrete points, so the LSF is numerically approximated using the [[finite difference]]:
:<math> \
:<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>\
====
Although 'sharpness' is often judged on grid patterns of alternate black and white lines, it should strictly be measured using a sine-wave variation from black to white (a blurred version of the usual pattern). Where a square wave pattern is used (simple black and white lines) not only is there more risk of aliasing, but account must be taken of the fact that the fundamental component of a square wave is higher than the amplitude of the square wave itself (the harmonic components reduce the peak amplitude). A square wave test chart will therefore show optimistic results (better resolution of high spatial frequencies than is actually achieved). The square wave result is sometimes referred to as the 'contrast transfer function' (CTF).
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In practice, many factors result in considerable blurring of a reproduced image, such that patterns with spatial frequency just below the [[Nyquist rate]] may not even be visible, and the finest patterns that can appear 'washed out' as shades of grey, not black and white. A major factor is usually the impossibility of making the perfect 'brick wall' optical filter (often realized as a '[[phase plate]]' or a lens with specific blurring properties in digital cameras and video camcorders). Such a filter is necessary to reduce [[aliasing]] by eliminating spatial frequencies above the [[Nyquist rate]] of the display.
===
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 [[
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 '
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 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.
===
There has recently been a shift towards the use of large image format [[digital single
==
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.
==
In general, the [[point spread function]], the image of a point source also depends on factors such as the [[wavelength]] ([[
==See also==
* [[Transfer function]]▼
▲* [[Signal transfer function]]
* [[Signal to noise ratio (image processing)]]▼
* [[Strehl ratio]]▼
* [[Wavefront coding]]▼
* [[Bokeh]]
* [[Gamma correction]]
* [[Minimum resolvable contrast]]
* [[Minimum resolvable temperature difference]]
* [[
* [[Signal transfer function]]
▲* [[Strehl ratio]]
▲* [[Transfer function]]
▲* [[Wavefront coding]]
==References==
{{Reflist}}
==
* [
* [
[[Category:Optics|Transfer function]]
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