View source for Distortion (optics)

{{short description|Deviation from rectilinear projection (optics)}}
{{hatnote|Not to be confused with [[spherical aberration]], a loss of image sharpness that can result from spherical lens surfaces.}}
{{Use dmy dates|date=March 2024}}

In [[geometric optics]], '''distortion''' is a deviation from [[rectilinear projection]]; a projection in which straight lines in a scene remain straight in an image. It is a form of [[aberration in optical systems|optical aberration]] that may be distinguished from other aberrations such as [[spherical aberration]], [[Coma (optics)|coma]], [[chromatic aberration]], [[Petzval field curvature|field curvature]], and [[Astigmatism (optical systems)|astigmatism]] in a sense that these impact the image sharpness without changing an object shape or structure in the image (e.g., a straight line in an object is still a straight line in the image although the image sharpness may be degraded by the mentioned aberrations) while distortion can change the object structure in the image (so named as distortion).

==Radial distortion==

{{multiple image
 | direction = vertical
 | width     = 100
 | footer    = Examples of radial distortions
 | image1    = Barrel distortion.svg
 | alt1      = Barrel distortion
 | caption1  = Barrel
 | image2    = Pincushion distortion.svg
 | alt2      = Pincushion distortion
 | caption2  = Pincushion
 | image3    = Mustache distortion.svg
 | alt3      = Mustache distortion
 | caption3  = Mustache
}}

Although distortion can be irregular or follow many patterns, the most commonly encountered distortions are radially symmetric, or approximately so, arising from the symmetry of a [[photographic lens]].  These ''radial distortions'' can usually be classified as either ''barrel'' distortions or ''pincushion'' distortions.<ref name="van Walree">{{cite web |url= http://toothwalker.org/optics/distortion.html |title= Distortion |author= Paul van Walree |work= Photographic optics |access-date= 2 February 2009 |archive-url= https://web.archive.org/web/20090129134205/http://toothwalker.org/optics/distortion.html |archive-date= 29 January 2009 |url-status= dead }}</ref>

{{glossary}}
{{term|Barrel distortion}}
{{defn|1=
In barrel distortion, image [[magnification]] decreases with distance from the [[optical axis]]. The apparent effect is an image that seems to be mapped around a [[sphere]] (or [[barrel]]). [[Fisheye lens]]es, which take hemispherical views, utilize this type of distortion as a way to map an infinitely wide object plane into a finite image area. In a [[zoom lens]], barrel distortion appears in the middle of the lens's focal length range and is worst at the wide-angle end of the range.<ref>{{cite web |url=http://www.cnet.com.au/tamron-18-270mm-f3-5-6-3-di-ii-vc-pzd-339323047.htm |title=Tamron 18-270mm f/3.5-6.3 Di II VC PZD |access-date=20 March 2013}}</ref>

Concave (minus) spherical lenses tend to have barrel distortion.
}}

{{term|Pincushion distortion}}
{{defn|1=
In pincushion distortion, image magnification increases with the distance from the [[optical axis]]. The visible effect is that lines that do not go through the centre of the image are bowed inwards, towards the centre of the image, like a [[pincushion]].

Convex (plus) spherical lenses tend to have pincushion distortion.
}}

{{term|Mustache distortion}}
{{defn|1=
A mixture of both types, sometimes referred to as ''mustache distortion'' (''moustache distortion'') or ''complex distortion'', is less common but not rare. It starts out as barrel distortion close to the image center and gradually turns into pincushion distortion towards the image periphery, making horizontal lines in the top half of the frame look like a [[handlebar mustache]].
}}
{{glossary end}}

Mathematically, barrel and pincushion distortion are [[quadratic function|quadratic]], meaning they increase as the ''square'' of distance from the center. In mustache distortion the [[quartic function|quartic]] (degree 4) term is significant: in the center, the degree 2 barrel distortion is dominant, while at the edge the degree 4 distortion in the pincushion direction dominates. Other distortions are in principle possible – pincushion in center and barrel at the edge, or higher order distortions (degree 6, degree 8) – but do not generally occur in practical lenses, and higher order distortions are small relative to the main barrel and pincushion effects.

===Origin of terms===
The names for these distortions come from familiar objects which are visually similar.
<gallery widths="150" heights="180">
File:Barrel (PSF).png|In barrel distortion, straight lines bulge ''outwards'' at the center, as in a [[barrel]].
File:Cushion.jpg|In pincushion distortion, corners of squares form elongated points, as in a [[cushion]].
File:Handlebar-moustache.svg|In mustache distortion, horizontal lines bulge up in the center, then bend the other way as they approach the edge of the frame (if in the top of the frame), as in curly [[handlebar mustache]]s.
</gallery>

===Occurrence===
[[File:globe effect.gif|thumb|Simulated animation of globe effect (right) compared with a simple pan (left)]]
In photography, distortion is particularly associated with [[zoom lens]]es, particularly large-range zooms, but may also be found in prime lenses, and depends on focal distance – for example, the [[Canon EF 50mm lens|Canon EF 50mm]] {{f/}}1.4 exhibits barrel distortion at extremely short focal distances. Barrel distortion may be found in wide-angle lenses, and is often seen at the wide-angle end of zoom lenses, while pincushion distortion is often seen in older or low-end [[telephoto lens]]es. Mustache distortion is observed particularly on the wide end of zooms, with certain [[Angenieux retrofocus|retrofocus]] lenses, and more recently on large-range zooms such as the [[Nikon]] 18–200&nbsp;mm.

A certain amount of pincushion distortion is often found with visual optical instruments, e.g., [[binoculars]], where it serves to counteract the [[globe effect]].

[[File:WA 80 cm archery target.svg|thumb|Radial distortions can be understood by their effect on concentric circles, as in an archery target.]]
In order to understand these distortions, it should be remembered that these are ''radial'' defects; the optical systems in question have [[rotational symmetry]] (omitting non-radial defects), so the [[didactic]]ally correct test image would be a set of [[concentric]] circles having even separation – like a shooter's target.  It will then be observed that these common distortions actually imply a nonlinear radius mapping from the object to the image: What is seemingly pincushion distortion, is actually simply an exaggerated radius mapping for large radii in comparison with small radii.  A graph showing radius transformations (from object to image) will be steeper in the upper (rightmost) end.  Conversely, barrel distortion is actually a diminished radius mapping for large radii in comparison with small radii.  A graph showing radius transformations (from object to image) will be less steep in the upper (rightmost) end.

===Chromatic aberration===
{{Further|Chromatic aberration}}
Radial distortion that depends on wavelength is called "[[lateral chromatic aberration]]" – "lateral" because radial, "chromatic" because dependent on color (wavelength). This can cause colored fringes in high-contrast areas in the outer parts of the image. This should not be confused with ''axial'' (longitudinal) chromatic aberration, which causes aberrations throughout the field, particularly [[purple fringing]].

==Software correction==
{{See also|Image rectification|Image stitching#straightening}}
[[file:ENIAC at Ft. Sill, OK, US, with barrel distortion.jpg|right|thumb|With uncorrected barrel distortion (at 26mm)]]
[[file:ENIAC, Ft. Sill, OK, US (78).jpg|right|thumb|Barrel distortion corrected with software (this is the [[ENIAC]] computer)]]

Radial distortion, whilst primarily dominated by low-order radial components,<ref name="devilliers">{{cite conference |last1 = de Villiers | first1 = J. P. | last2 = Leuschner | first2 = F.W. | last3 = Geldenhuys| first3 = R. | title = Centi-pixel accurate real-time inverse distortion correction| book-title = 2008 International Symposium on Optomechatronic Technologies | publisher = SPIE | date = 17–19 November 2008 | url = http://researchspace.csir.co.za/dspace/bitstream/10204/3168/1/De%20Villiers_2008.pdf |doi=10.1117/12.804771}}</ref> can be corrected using Brown's distortion model,<ref name="brown">{{Cite journal| last=Brown |first=Duane C. |title=Decentering distortion of lenses |journal=Photogrammetric Engineering |volume=32 |issue=3 |pages= 444–462 |date=May 1966 | url=https://www.asprs.org/wp-content/uploads/pers/1966journal/may/1966_may_444-462.pdf | archive-url=https://web.archive.org/web/20180312205006/https://www.asprs.org/wp-content/uploads/pers/1966journal/may/1966_may_444-462.pdf | url-status=dead | archive-date=2018-03-12 }}</ref> also known as the Brown–Conrady model based on earlier work by Conrady.<ref>{{cite journal|bibcode=1919MNRAS..79..384C|title=Decentred Lens-Systems|last1=Conrady|first1=A. E.|journal=Monthly Notices of the Royal Astronomical Society|volume=79|issue=5|pages=384–390|year=1919|doi=10.1093/mnras/79.5.384|doi-access=free}}</ref> The Brown–Conrady model corrects both for radial distortion and for tangential distortion caused by physical elements in a lens not being perfectly aligned. The latter is also known as ''decentering distortion''. See Zhang<ref name="zhang">{{cite tech report |first=Zhengyou |last=Zhang |title=A Flexible New Technique for Camera Calibration |number=MSR-TR-98-71 |institution=Microsoft Research |year=1998 |url=http://research.microsoft.com/en-us/um/people/zhang/Papers/TR98-71.pdf }}</ref> for additional discussion of radial distortion. The Brown-Conrady distortion model is

<math display=block>\begin{alignat}{3}
x_\mathrm{u} = x_\mathrm{d} & + (x_\mathrm{d} - x_\mathrm{c})(K_1r^2 + K_2r^4 + \cdots) + 
(P_1(r^2 + 2(x_\mathrm{d} - x_\mathrm{c})^2) \\
& + 2P_2(x_\mathrm{d} - x_\mathrm{c})(y_\mathrm{d} - y_\mathrm{c}))(1 + P_3r^2 + P_4r^4 \cdots) \\
y_\mathrm{u} = y_\mathrm{d} & + (y_\mathrm{d} - y_\mathrm{c})(K_1r^2 + K_2r^4 + \cdots) + 
(2P_1(x_\mathrm{d} - x_\mathrm{c})(y_\mathrm{d} - y_\mathrm{c}) \\
& + P_2(r^2 + 2(y_\mathrm{d} - y_\mathrm{c})^2))(1 + P_3r^2 + P_4r^4 \cdots),
\end{alignat}
</math>

where

*<math>(x_\mathrm{d},\ y_\mathrm{d})</math> is the distorted image point as projected on image plane using specified lens;
*<math>(x_\mathrm{u},\ y_\mathrm{u})</math> is the undistorted image point as projected by an ideal [[pinhole camera]];
*<math>(x_\mathrm{c},\ y_\mathrm{c})</math> is the distortion center;
*<math>K_n</math> is the <math>n^{\mathrm{th}}</math> radial distortion coefficient;
*<math>P_n</math> is the <math>n^{\mathrm{th}}</math> tangential distortion coefficient; and
*<math>r</math> = <math>\sqrt{(x_\mathrm{d}-x_\mathrm{c})^2 + (y_\mathrm{d}-y_\mathrm{c})^2}</math>, the [[Euclidean distance]] between the distorted image point and the distortion center.<ref name="devilliers"/>

Barrel distortion typically will have a negative term for <math>K_1</math> whereas pincushion distortion will have a positive value. Moustache distortion will have a non-[[monotonic]] radial [[geometric series]] where for some <math>r</math> the sequence will change sign.

To model radial distortion, the division model<ref name="fitzgibbon">{{cite conference |last1 = Fitzgibbon | first1 = A. W. | title = Simultaneous linear estimation of multiple view geometry and lens distortion| book-title = Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR) | publisher = IEEE | date = 2001|doi=10.1109/CVPR.2001.990465}}</ref> typically provides a more accurate approximation than Brown-Conrady's even-order polynomial model,<ref name="bukhari">{{cite conference |last1 = Bukhari | first1 = F. | last2 = Dailey | first2 = M. N. | title = Automatic Radial Distortion Estimation from a Single Image| book-title = Journal of mathematical imaging and vision | publisher = Springer | date = 2013 | url = http://www.cs.ait.ac.th/~mdailey/papers/Bukhari-RadialDistortion.pdf |doi=10.1007/s10851-012-0342-2}}</ref>

<math display=block>\begin{align}
x_\mathrm{u} & = x_\mathrm{c} + \frac{x_\mathrm{d} - x_\mathrm{c}}{1 + K_1r^2 + K_2r^4 + \cdots} \\
y_\mathrm{u} & = y_\mathrm{c} + \frac{y_\mathrm{d} - y_\mathrm{c}}{1 + K_1r^2 + K_2r^4 + \cdots},
\end{align}
</math>

using the same parameters previously defined. For radial distortion, this division model is often preferred over the Brown–Conrady model, as it requires fewer terms to more accurately describe severe distortion.<ref name="bukhari"/> Using this model, a single term is usually sufficient to model most cameras.<ref name="wang">{{cite conference |last1 = Wang | first1 = J. | last2 = Shi | first2 = F. | last3 = Zhang | first3 = J. | last4 = Liu | first4 = Y. | title = A new calibration model of camera lens distortion| book-title = Pattern Recognition | publisher = Elsevier | date = 2008 |doi=10.1016/j.patcog.2007.06.012}}</ref>

Software can correct those distortions by [[image warping|warping]] the image with a reverse distortion. This involves determining which distorted pixel corresponds to each undistorted pixel, which is non-trivial due to the [[non-linearity]] of the distortion equation.<ref name="devilliers"/> Lateral chromatic aberration (purple/green fringing) can be significantly reduced by applying such warping for red, green and blue separately.

Distorting or undistorting requires either both sets of coefficients or inverting the non-linear problem which, in general, lacks an analytical solution. Standard approaches such as approximating, locally linearizing and iterative solvers all apply. Which solver is preferable depends on the accuracy required and the computational resources available.

In addition to usually being sufficient to model most cameras, as mentioned, the single-term division model has an analytical solution to the reverse-distortion problem.<ref name="bukhari"/> In this case, the distorted pixels are given by

<math display=block>\begin{align}
x_\mathrm{d} & = x_\mathrm{c} + \frac{x_\mathrm{u} - x_\mathrm{c}}{2K_1r_u^2} (1 - \sqrt{1 - 4K_1r_u^2}) \\
y_\mathrm{d} & = y_\mathrm{c} + \frac{y_\mathrm{u} - y_\mathrm{c}}{2K_1r_u^2} (1 - \sqrt{1 - 4K_1r_u^2}),
\end{align}
</math>

where

*<math>r_u</math> = <math>\sqrt{(x_\mathrm{u}-x_\mathrm{c})^2 + (y_\mathrm{u}-y_\mathrm{c})^2}</math>, the Euclidean distance between the undistorted image point and the undistortion/distortion center.

===Calibrated===
Calibrated software works from a table of lens/camera transfer functions:
* Adobe Photoshop Lightroom and Photoshop CS5 can correct complex distortion.
* PTlens is a Photoshop plugin or standalone application which corrects complex distortion.  It not only corrects for linear distortion, but also second degree and higher nonlinear components.<ref>{{cite web |url=http://epaperpress.com/ptlens/ |title=PTlens |access-date=2 January 2012}}</ref>
* Lensfun is a free to use database and library for correcting lens distortion.<ref>{{cite web|url=https://lensfun.github.io/|title=Lensfun|access-date=16 April 2022}}</ref><ref>{{cite web |url=http://svn.berlios.de/wsvn/lensfun/trunk/README?rev=246 |archive-url=https://archive.today/20131013142525/http://svn.berlios.de/wsvn/lensfun/trunk/README?rev=246 |url-status=dead |archive-date=13 October 2013 |title=lensfun – Rev 246 – /trunk/README |access-date=13 October 2013 }}</ref>
* [https://opencv.org/ OpenCV] is an open-source BSD-licensed library for computer vision (multi-language, multi-OS). It features a module for camera calibration.<ref>{{cite web|title=OpenCV|url=https://docs.opencv.org/2.4/doc/tutorials/calib3d/camera_calibration/camera_calibration.html|website=opencv.org/|access-date=22 January 2018}}</ref>
* [[DxO PhotoLab|DxO's PhotoLab]] software can correct complex distortion, and takes into account the focus distance.
* [[proDAD]] Defishr includes an Unwarp-tool and a Calibrator-tool. Due to the distortion of a checkerboard pattern, the necessary unwrap is calculated.
* The [[Micro Four Thirds system]] cameras and lenses perform automatic distortion correction using correction parameters that are stored in each lens's firmware, and are applied automatically by the camera and [[Raw image format|raw]] converter software.  The optics of most of these lenses feature substantially more distortion than their counterparts in systems that do not offer such automatic corrections, but the software-corrected final images show noticeably less distortion than competing designs.<ref>{{cite web |last=Wiley |first=Carlisle |url=http://www.dpreview.com/articles/distortion/ |title=Articles: Digital Photography Review |publisher=Dpreview.com |access-date=2013-07-03 |url-status=dead |archive-url=https://web.archive.org/web/20120707100436/http://www.dpreview.com/articles/distortion/ |archive-date=7 July 2012 }}</ref>

===Manual===
Manual software allows manual adjustment of distortion parameters:
* [[ImageMagick]] can correct several distortions; for example the fisheye distortion of the popular GoPro Hero3+ Silver camera can be corrected by the command<ref>{{cite web|url=http://www.imagemagick.org/Usage/lens/|title=ImageMagick v6 Examples -- Lens Corrections}}</ref>
:<code>convert distorted_image.jpg -distort barrel "0.06335 -0.18432 -0.13009" corrected_image.jpg</code>
* [[Photoshop]] CS2 and [[Photoshop Elements]] (from version 5) include a manual Lens Correction filter for simple (pincushion/barrel) distortion
* [[Corel Paint Shop Pro Photo]] includes a manual Lens Distortion effect for simple (barrel, fisheye, fisheye spherical and pincushion) distortion.
* [[GIMP]] includes manual lens distortion correction (from version 2.4).
* [[PhotoPerfect]] has interactive functions for general pincushion adjustment, and for fringe (adjusting the size of the red, green and blue image parts).
* [[Hugin (software)|Hugin]] can be used to correct distortion, though that is not its primary application.<ref>{{cite web|title=Hugin tutorial – Simulating an architectural projection |url=http://hugin.sourceforge.net/tutorials/architectural/en.shtml |access-date=9 September 2009}}</ref>
Besides these systems that address images, there are some that also adjust distortion parameters for videos:
* [[FFMPEG]] using the "lenscorrection" video filter.<ref>{{cite web|url=http://www.ffmpeg.org/ffmpeg-filters.html#lenscorrection|title=FFmpeg Filters Documentation}}</ref>
* [[Blender (software)|Blender]] by using the node editor to insert a "Distort/Lens Distortion" node between the input and output nodes.

==Perceived distortion (perspective distortion)==
Radial distortion is a failure of a lens to be [[rectilinear lens|rectilinear]]: a failure to image lines into lines. If a photograph is not taken straight-on then, even with a perfect rectilinear lens, rectangles will appear as [[trapezoid]]s: lines are imaged as lines, but the angles between them are not preserved (tilt is not a [[conformal map]]). This effect can be controlled by using a [[perspective control lens]], or [[Perspective control|corrected]] in post-processing.

Due to [[Perspective (visual)|perspective]], cameras image a cube as a square [[frustum]] (a truncated pyramid, with trapezoidal sides) – the far end is smaller than the near end. This creates perspective, and the rate at which this scaling happens (how quickly more distant objects shrink) creates a sense of a scene being deep or shallow. This cannot be changed or corrected by a simple transform of the resulting image, because it requires 3D information, namely the depth of objects in the scene. This effect is known as [[Perspective distortion (photography)|perspective distortion]]; the image itself is not distorted but is perceived as distorted when viewed from a normal viewing distance.

Note that if the center of the image is closer than the edges (for example, a straight-on shot of a face), then barrel distortion and wide-angle distortion (taking the shot from close) both increase the size of the center, while pincushion distortion and telephoto distortion (taking the shot from far) both decrease the size of the center. However, radial distortion bends straight lines (out or in), while perspective distortion does not bend lines, and these are distinct phenomena. [[Fisheye lens]]es are wide-angle lenses with heavy barrel distortion and thus exhibit ''both'' these phenomena, so objects in the center of the image (if shot from a short distance) are particularly enlarged: even if the barrel distortion is corrected, the resulting image is still from a wide-angle lens, and will still have a wide-angle perspective.

==See also==
* [[Anamorphosis]]
* [[Angle of view]]
* [[Cylindrical perspective]]
* [[Distortion]]
* [[Texture gradient]]
* [[Underwater vision]]
* [[Vignetting]]

==References==
<references/>

==External links==
{{Commons category}}
*[http://www.ipol.im/pub/algo/ags_algebraic_lens_distortion_estimation/ Lens distortion estimation and correction] with source code and online demonstration
*[http://cctvcad.com/videocad_help/index.html?about_distortion.htm 3D modeling Lens distortion and camera field of view in CCTV design]

{{DEFAULTSORT:Distortion (Optics)}}
[[Category:Geometrical optics]]