1164:
437:
408:
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1159:{\displaystyle {\begin{alignedat}{3}x_{\mathrm {u} }=x_{\mathrm {d} }&+(x_{\mathrm {d} }-x_{\mathrm {c} })(K_{1}r^{2}+K_{2}r^{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_{3}r^{2}+P_{4}r^{4}\cdots )\\y_{\mathrm {u} }=y_{\mathrm {d} }&+(y_{\mathrm {d} }-y_{\mathrm {c} })(K_{1}r^{2}+K_{2}r^{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_{3}r^{2}+P_{4}r^{4}\cdots ),\end{alignedat}}}
416:
342:
83:
219:(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.
2282:
97:
277:
2585:(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
69:
362:
1932:
1961:
301:
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.
1657:
228:
2277:{\displaystyle {\begin{aligned}x_{\mathrm {d} }&=x_{\mathrm {c} }+{\frac {x_{\mathrm {u} }-x_{\mathrm {c} }}{2K_{1}r_{u}^{2}}}(1-{\sqrt {1-4K_{1}r_{u}^{2}}})\\y_{\mathrm {d} }&=y_{\mathrm {c} }+{\frac {y_{\mathrm {u} }-y_{\mathrm {c} }}{2K_{1}r_{u}^{2}}}(1-{\sqrt {1-4K_{1}r_{u}^{2}}}),\end{aligned}}}
1951:
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
427:
Radial distortion, whilst primarily dominated by low-order radial components, can be corrected using Brown's distortion model, also known as the Brown–Conrady model based on earlier work by
Conrady. The Brown–Conrady model corrects both for radial distortion and for tangential distortion caused by
2592:
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
300:
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
1936:
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. Using this model, a single term is usually sufficient to model most cameras.
377:
1927:{\displaystyle {\begin{aligned}x_{\mathrm {u} }&=x_{\mathrm {c} }+{\frac {x_{\mathrm {d} }-x_{\mathrm {c} }}{1+K_{1}r^{2}+K_{2}r^{4}+\cdots }}\\y_{\mathrm {u} }&=y_{\mathrm {c} }+{\frac {y_{\mathrm {d} }-y_{\mathrm {c} }}{1+K_{1}r^{2}+K_{2}r^{4}+\cdots }},\end{aligned}}}
2468:
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
2601:
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.
2418:
1584:
249:
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
199:, 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
1955:
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. In this case, the distorted pixels are given by
319:" – "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
1336:
1277:
1222:
1966:
1662:
2436:
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.
2593:
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.
163:, barrel distortion appears in the middle of the lens's focal length range and is worst at the wide-angle end of the range. Concave (minus) spherical lenses tend to have barrel distortion.
114:
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
2464:
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
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2319:
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1948:
of the distortion equation. Lateral chromatic aberration (purple/green fringing) can be significantly reduced by applying such warping for red, green and blue separately.
383:
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
341:
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1619:
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the image with a reverse distortion. This involves determining which distorted pixel corresponds to each undistorted pixel, which is non-trivial due to the
2558:: 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
1652:
To model radial distortion, the division model typically provides a more accurate approximation than Brown-Conrady's even-order polynomial model,
159:, 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
2485:
can correct several distortions; for example the fisheye distortion of the popular GoPro Hero3+ Silver camera can be corrected by the command
442:
175:. 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
2586:
2457:
Defishr includes an Unwarp-tool and a
Calibrator-tool. Due to the distortion of a checkerboard pattern, the necessary unwrap is calculated.
2522:
has interactive functions for general pincushion adjustment, and for fringe (adjusting the size of the red, green and blue image parts).
2445:
is an open-source BSD-licensed library for computer vision (multi-language, multi-OS). It features a module for camera calibration.
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1229:
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407:
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239:, particularly large-range zooms, but may also be found in prime lenses, and depends on focal distance – for example, the
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316:
2718:
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2763:
2510:
includes a manual Lens
Distortion effect for simple (barrel, fisheye, fisheye spherical and pincushion) distortion.
2507:
2589:; the image itself is not distorted, but is perceived as distorted when viewed from a normal viewing distance.
2567:
2461:
2691:
2532:
Besides these systems that address images, there are some that also adjust distortion parameters for videos:
2621:
2843:
Proceedings of the 2001 IEEE Computer
Society Conference on Computer Vision and Pattern Recognition (CVPR)
2841:
Fitzgibbon, A. W. (2001). "Simultaneous linear estimation of multiple view geometry and lens distortion".
48:
2822:
2413:{\displaystyle {\sqrt {(x_{\mathrm {u} }-x_{\mathrm {c} })^{2}+(y_{\mathrm {u} }-y_{\mathrm {c} })^{2}}}}
1579:{\displaystyle {\sqrt {(x_{\mathrm {d} }-x_{\mathrm {c} })^{2}+(y_{\mathrm {d} }-y_{\mathrm {c} })^{2}}}}
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2870:
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by using the node editor to insert a "Distort/Lens
Distortion" node between the input and output nodes.
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2793:
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310:
255:
31:
2420:, the Euclidean distance between the undistorted image point and the undistortion/distortion center.
280:
Radial distortions can be understood by their effect on concentric circles, as in an archery target.
3099:
2899:
Wang, J.; Shi, F.; Zhang, J.; Liu, Y. (2008). "A new calibration model of camera lens distortion".
2571:
398:
289:
2504:(from version 5) include a manual Lens Correction filter for simple (pincushion/barrel) distortion
432:. See Zhang for additional discussion of radial distortion. The Brown-Conrady distortion model is
2542:
2501:
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384:
323:(longitudinal) chromatic aberration, which causes aberrations throughout the field, particularly
240:
208:
200:
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265:
A certain amount of pincushion distortion is often found with visual optical instruments, e.g.,
1621:
whereas pincushion distortion will have a positive value. Moustache distortion will have a non-
415:
51:; a projection in which straight lines in a scene remain straight in an image. It is a form of
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115:
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convert distorted_image.jpg -distort barrel "0.06335 -0.18432 -0.13009" corrected_image.jpg
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2562:: lines are imaged as lines, but the angles between them are not preserved (tilt is not a
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The names for these distortions come from familiar objects which are visually similar.
251:
254:. Mustache distortion is observed particularly on the wide end of zooms, with certain
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1945:
1941:
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physical elements in a lens not being perfectly aligned. The latter is also known as
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17:
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software can correct complex distortion, and takes into account the focus distance.
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172:
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In pincushion distortion, image magnification increases with the distance from the
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In order to understand these distortions, it should be remembered that these are
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68:
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can be used to correct distortion, though that is not its primary application.
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is the distorted image point as projected on image plane using specified lens;
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176:
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2806:
2781:
2439:
Lensfun is a free to use database and library for correcting lens distortion.
231:
Simulated animation of globe effect (right) compared with a simple pan (left)
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1622:
367:
In pincushion distortion, corners of squares form elongated points, as in a
236:
160:
2717:
de
Villiers, J. P.; Leuschner, F.W.; Geldenhuys, R. (17–19 November 2008).
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Adobe
Photoshop Lightroom and Photoshop CS5 can correct complex distortion.
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293:
147:. The apparent effect is that of an image which has been mapped around a
34:, a loss of image sharpness that can result from spherical lens surfaces.
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Calibrated systems work from a table of lens/camera transfer functions:
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368:
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352:
152:
148:
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are wide-angle lenses with heavy barrel distortion and thus exhibit
179:. Convex (plus) spherical lenses tend to have pincushion distortion.
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3D modeling Lens distortion and camera field of view in CCTV design
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259:
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Manual systems allow manual adjustment of distortion parameters:
2513:
2719:"Centi-pixel accurate real-time inverse distortion correction"
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includes manual lens distortion correction (from version 2.4).
2943:
2871:"Automatic Radial Distortion Estimation from a Single Image"
2726:
2008 International
Symposium on Optomechatronic Technologies
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between the distorted image point and the distortion center.
258:
lenses, and more recently on large-range zooms such as the
235:
In photography, distortion is particularly associated with
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Barrel distortion typically will have a negative term for
3057:"Hugin tutorial – Simulating an architectural projection"
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is the undistorted image point as projected by an ideal
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Radial distortion that depends on wavelength is called "
215:
of distance from the center. In mustache distortion the
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Barrel distortion corrected with software (this is the
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2830:(Technical report). Microsoft Research. MSR-TR-98-71.
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1331:{\displaystyle (x_{\mathrm {c} },\ y_{\mathrm {c} })}
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1272:{\displaystyle (x_{\mathrm {u} },\ y_{\mathrm {u} })}
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1217:{\displaystyle (x_{\mathrm {d} },\ y_{\mathrm {d} })}
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Mathematically, barrel and pincushion distortion are
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required and the computational resources available.
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187:A mixture of both types, sometimes referred to as
2786:Monthly Notices of the Royal Astronomical Society
2824:A Flexible New Technique for Camera Calibration
2554:Radial distortion is a failure of a lens to be
288:defects; the optical systems in question have
27:Deviation from rectilinear projection (optics)
3043:"ImageMagick v6 Examples -- Lens Corrections"
8:
2712:
2710:
2708:
2566:). This effect can be controlled by using a
411:With uncorrected barrel distortion (at 26mm)
347:In barrel distortion, straight lines bulge
2878:Journal of mathematical imaging and vision
1940:Software can correct those distortions by
3096:with source code and online demonstration
3094:Lens distortion estimation and correction
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2692:"Tamron 18-270mm f/3.5-6.3 Di II VC PZD"
2539:using the "lenscorrection" video filter.
2652:
337:
3017:"Articles: Digital Photography Review"
1460:tangential distortion coefficient; and
292:(omitting non-radial defects), so the
296:correct test image would be a set of
7:
403:Image stitching § straightening
269:, where it serves to counteract the
132:
122:can usually be classified as either
2965:"lensfun – Rev 246 – /trunk/README"
2869:Bukhari, F.; Dailey, M. N. (2013).
2581:, cameras image a cube as a square
2749:"Decentering distortion of lenses"
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1453:{\displaystyle n^{\mathrm {th} }}
1392:{\displaystyle n^{\mathrm {th} }}
143:decreases with distance from the
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360:
340:
95:
81:
67:
1649:the sequence will change sign.
211:, meaning they increase as the
3078:"FFmpeg Filters Documentation"
3019:. Dpreview.com. Archived from
2399:
2368:
2356:
2325:
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2217:
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2066:
1565:
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1399:radial distortion coefficient;
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108:Examples of radial distortions
1:
2909:10.1016/j.patcog.2007.06.012
2762:(3): 444–462. Archived from
2747:Brown, Duane C. (May 1966).
317:lateral chromatic aberration
139:In barrel distortion, image
2756:Photogrammetric Engineering
3131:
2508:Corel Paint Shop Pro Photo
396:
308:
29:
2886:10.1007/s10851-012-0342-2
1338:is the distortion center;
2851:10.1109/CVPR.2001.990465
2821:Zhang, Zhengyou (1998).
2782:"Decentred Lens-Systems"
2568:perspective control lens
2462:Micro Four Thirds system
30:Not to be confused with
2780:Conrady, A. E. (1919).
2622:Cylindrical perspective
351:at the center, as in a
2807:10.1093/mnras/79.5.384
2587:perspective distortion
2414:
2310:
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1928:
1643:
1615:
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430:decentering distortion
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281:
232:
49:rectilinear projection
2415:
2311:
2309:{\displaystyle r_{u}}
2279:
1929:
1644:
1616:
1614:{\displaystyle K_{1}}
1581:
1477:
1455:
1423:
1421:{\displaystyle P_{n}}
1394:
1362:
1360:{\displaystyle K_{n}}
1333:
1274:
1219:
1161:
418:
410:
309:Further information:
279:
230:
167:Pincushion distortion
18:Pincushion distortion
2574:in post-processing.
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2293:
1962:
1658:
1633:
1598:
1486:
1466:
1432:
1405:
1371:
1344:
1289:
1230:
1175:
438:
311:Chromatic aberration
305:Chromatic aberration
193:moustache distortion
47:is a deviation from
32:spherical aberration
2901:Pattern Recognition
2798:1919MNRAS..79..384C
2666:Photographic optics
2261:
2213:
2110:
2062:
399:Image rectification
393:Software correction
385:handlebar mustaches
290:rotational symmetry
189:mustache distortion
183:Mustache distortion
3115:Geometrical optics
2971:on 13 October 2013
2672:on 29 January 2009
2502:Photoshop Elements
2410:
2306:
2274:
2272:
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2199:
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1924:
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1588:Euclidean distance
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1357:
1328:
1269:
1214:
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1154:
425:
413:
282:
233:
201:handlebar mustache
197:complex distortion
120:radial distortions
53:optical aberration
3015:Wiley, Carlisle.
2769:on 12 March 2018.
2734:10.1117/12.804771
2660:Paul van Walree.
2637:Underwater vision
2550:Related phenomena
2408:
2262:
2215:
2111:
2064:
1915:
1787:
1642:{\displaystyle r}
1574:
1475:{\displaystyle r}
1312:
1253:
1198:
135:Barrel distortion
116:photographic lens
59:Radial distortion
16:(Redirected from
3122:
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2967:. Archived from
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2668:. Archived from
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2632:Texture gradient
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2047:
2046:
2033:
2032:
2031:
2030:
2017:
2016:
2015:
2004:
1999:
1998:
1997:
1980:
1979:
1978:
1933:
1931:
1930:
1925:
1923:
1916:
1914:
1907:
1906:
1897:
1896:
1884:
1883:
1874:
1873:
1857:
1856:
1855:
1854:
1841:
1840:
1839:
1828:
1823:
1822:
1821:
1804:
1803:
1802:
1788:
1786:
1779:
1778:
1769:
1768:
1756:
1755:
1746:
1745:
1729:
1728:
1727:
1726:
1713:
1712:
1711:
1700:
1695:
1694:
1693:
1676:
1675:
1674:
1648:
1646:
1645:
1640:
1627:geometric series
1620:
1618:
1617:
1612:
1610:
1609:
1585:
1583:
1582:
1577:
1575:
1573:
1572:
1563:
1562:
1561:
1548:
1547:
1546:
1530:
1529:
1520:
1519:
1518:
1505:
1504:
1503:
1490:
1481:
1479:
1478:
1473:
1459:
1457:
1456:
1451:
1449:
1448:
1447:
1427:
1425:
1424:
1419:
1417:
1416:
1398:
1396:
1395:
1390:
1388:
1387:
1386:
1366:
1364:
1363:
1358:
1356:
1355:
1337:
1335:
1334:
1329:
1324:
1323:
1322:
1310:
1306:
1305:
1304:
1278:
1276:
1275:
1270:
1265:
1264:
1263:
1251:
1247:
1246:
1245:
1223:
1221:
1220:
1215:
1210:
1209:
1208:
1196:
1192:
1191:
1190:
1165:
1163:
1162:
1157:
1155:
1142:
1141:
1132:
1131:
1119:
1118:
1109:
1108:
1084:
1083:
1074:
1073:
1072:
1059:
1058:
1057:
1038:
1037:
1025:
1024:
1009:
1002:
1001:
1000:
987:
986:
985:
969:
968:
967:
954:
953:
952:
939:
938:
911:
910:
901:
900:
888:
887:
878:
877:
862:
861:
860:
847:
846:
845:
825:
824:
823:
810:
809:
808:
788:
787:
778:
777:
765:
764:
755:
754:
730:
729:
728:
715:
714:
713:
697:
696:
695:
682:
681:
680:
667:
666:
648:
641:
640:
631:
630:
629:
616:
615:
614:
595:
594:
582:
581:
557:
556:
547:
546:
534:
533:
524:
523:
508:
507:
506:
493:
492:
491:
471:
470:
469:
456:
455:
454:
380:
364:
344:
262:18–200 mm.
252:telephoto lenses
248:
246:
99:
85:
71:
41:geometric optics
21:
3130:
3129:
3125:
3124:
3123:
3121:
3120:
3119:
3105:
3104:
3090:
3085:
3076:
3075:
3071:
3061:
3059:
3055:
3054:
3050:
3041:
3040:
3036:
3026:
3024:
3014:
3013:
3009:
2999:
2997:
2989:
2988:
2984:
2974:
2972:
2963:
2962:
2958:
2948:
2946:
2942:
2941:
2937:
2927:
2925:
2921:
2920:
2916:
2898:
2897:
2893:
2873:
2868:
2867:
2858:
2840:
2839:
2835:
2827:
2820:
2819:
2815:
2779:
2778:
2774:
2766:
2751:
2746:
2745:
2741:
2721:
2716:
2715:
2706:
2696:
2694:
2690:
2689:
2685:
2675:
2673:
2659:
2658:
2654:
2650:
2608:
2552:
2490:
2476:
2427:
2398:
2386:
2371:
2355:
2343:
2328:
2318:
2317:
2296:
2291:
2290:
2271:
2270:
2237:
2189:
2185:
2172:
2157:
2156:
2139:
2132:
2120:
2117:
2116:
2086:
2038:
2034:
2021:
2006:
2005:
1988:
1981:
1969:
1960:
1959:
1921:
1920:
1898:
1888:
1875:
1865:
1858:
1845:
1830:
1829:
1812:
1805:
1793:
1790:
1789:
1770:
1760:
1747:
1737:
1730:
1717:
1702:
1701:
1684:
1677:
1665:
1656:
1655:
1631:
1630:
1629:where for some
1601:
1596:
1595:
1564:
1552:
1537:
1521:
1509:
1494:
1484:
1483:
1464:
1463:
1435:
1430:
1429:
1408:
1403:
1402:
1374:
1369:
1368:
1347:
1342:
1341:
1313:
1295:
1287:
1286:
1254:
1236:
1228:
1227:
1199:
1181:
1173:
1172:
1153:
1152:
1133:
1123:
1110:
1100:
1075:
1063:
1048:
1029:
1016:
1007:
1006:
991:
976:
958:
943:
930:
902:
892:
879:
869:
851:
836:
826:
814:
799:
796:
795:
779:
769:
756:
746:
719:
704:
686:
671:
658:
646:
645:
632:
620:
605:
586:
573:
548:
538:
525:
515:
497:
482:
472:
460:
445:
436:
435:
405:
395:
388:
381:
372:
365:
356:
345:
333:
331:Origin of terms
325:purple fringing
313:
307:
244:
243:
225:
184:
168:
136:
126:distortions or
112:
111:
110:
109:
105:
104:
103:
100:
91:
90:
89:
86:
77:
76:
75:
72:
61:
35:
28:
23:
22:
15:
12:
11:
5:
3128:
3126:
3118:
3117:
3107:
3106:
3103:
3102:
3097:
3089:
3088:External links
3086:
3084:
3083:
3069:
3048:
3034:
3023:on 7 July 2012
3007:
2982:
2956:
2935:
2914:
2891:
2856:
2833:
2813:
2792:(5): 384–390.
2772:
2739:
2704:
2683:
2651:
2649:
2646:
2645:
2644:
2639:
2634:
2629:
2624:
2619:
2614:
2607:
2604:
2595:Fisheye lenses
2551:
2548:
2547:
2546:
2540:
2530:
2529:
2523:
2517:
2511:
2505:
2494:
2493:
2487:
2486:
2475:
2472:
2471:
2470:
2458:
2452:
2449:DxO's PhotoLab
2446:
2440:
2437:
2434:
2426:
2423:
2422:
2421:
2405:
2401:
2394:
2389:
2385:
2379:
2374:
2370:
2367:
2362:
2358:
2351:
2346:
2342:
2336:
2331:
2327:
2303:
2299:
2269:
2266:
2259:
2254:
2250:
2244:
2240:
2236:
2233:
2230:
2225:
2222:
2219:
2211:
2206:
2202:
2196:
2192:
2188:
2180:
2175:
2171:
2165:
2160:
2153:
2147:
2142:
2138:
2135:
2133:
2128:
2123:
2119:
2118:
2115:
2108:
2103:
2099:
2093:
2089:
2085:
2082:
2079:
2074:
2071:
2068:
2060:
2055:
2051:
2045:
2041:
2037:
2029:
2024:
2020:
2014:
2009:
2002:
1996:
1991:
1987:
1984:
1982:
1977:
1972:
1968:
1967:
1919:
1913:
1910:
1905:
1901:
1895:
1891:
1887:
1882:
1878:
1872:
1868:
1864:
1861:
1853:
1848:
1844:
1838:
1833:
1826:
1820:
1815:
1811:
1808:
1806:
1801:
1796:
1792:
1791:
1785:
1782:
1777:
1773:
1767:
1763:
1759:
1754:
1750:
1744:
1740:
1736:
1733:
1725:
1720:
1716:
1710:
1705:
1698:
1692:
1687:
1683:
1680:
1678:
1673:
1668:
1664:
1663:
1638:
1608:
1604:
1592:
1591:
1571:
1567:
1560:
1555:
1551:
1545:
1540:
1536:
1533:
1528:
1524:
1517:
1512:
1508:
1502:
1497:
1493:
1471:
1461:
1446:
1443:
1438:
1415:
1411:
1400:
1385:
1382:
1377:
1354:
1350:
1339:
1327:
1321:
1316:
1309:
1303:
1298:
1294:
1284:
1281:pinhole camera
1268:
1262:
1257:
1250:
1244:
1239:
1235:
1225:
1213:
1207:
1202:
1195:
1189:
1184:
1180:
1151:
1148:
1145:
1140:
1136:
1130:
1126:
1122:
1117:
1113:
1107:
1103:
1099:
1096:
1093:
1090:
1087:
1082:
1078:
1071:
1066:
1062:
1056:
1051:
1047:
1044:
1041:
1036:
1032:
1028:
1023:
1019:
1015:
1012:
1010:
1008:
1005:
999:
994:
990:
984:
979:
975:
972:
966:
961:
957:
951:
946:
942:
937:
933:
929:
926:
923:
920:
917:
914:
909:
905:
899:
895:
891:
886:
882:
876:
872:
868:
865:
859:
854:
850:
844:
839:
835:
832:
829:
827:
822:
817:
813:
807:
802:
798:
797:
794:
791:
786:
782:
776:
772:
768:
763:
759:
753:
749:
745:
742:
739:
736:
733:
727:
722:
718:
712:
707:
703:
700:
694:
689:
685:
679:
674:
670:
665:
661:
657:
654:
651:
649:
647:
644:
639:
635:
628:
623:
619:
613:
608:
604:
601:
598:
593:
589:
585:
580:
576:
572:
569:
566:
563:
560:
555:
551:
545:
541:
537:
532:
528:
522:
518:
514:
511:
505:
500:
496:
490:
485:
481:
478:
475:
473:
468:
463:
459:
453:
448:
444:
443:
394:
391:
390:
389:
382:
375:
373:
366:
359:
357:
346:
339:
332:
329:
306:
303:
224:
221:
205:
204:
185:
182:
180:
169:
166:
164:
157:Fisheye lenses
137:
134:
107:
106:
101:
94:
93:
92:
87:
80:
79:
78:
73:
66:
65:
64:
63:
62:
60:
57:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3127:
3116:
3113:
3112:
3110:
3101:
3098:
3095:
3092:
3091:
3087:
3079:
3073:
3070:
3058:
3052:
3049:
3044:
3038:
3035:
3022:
3018:
3011:
3008:
2996:
2992:
2986:
2983:
2970:
2966:
2960:
2957:
2945:
2939:
2936:
2924:
2918:
2915:
2910:
2906:
2902:
2895:
2892:
2887:
2883:
2879:
2872:
2865:
2863:
2861:
2857:
2852:
2848:
2844:
2837:
2834:
2826:
2825:
2817:
2814:
2808:
2803:
2799:
2795:
2791:
2787:
2783:
2776:
2773:
2765:
2761:
2757:
2750:
2743:
2740:
2735:
2731:
2727:
2720:
2713:
2711:
2709:
2705:
2693:
2687:
2684:
2671:
2667:
2663:
2656:
2653:
2647:
2643:
2640:
2638:
2635:
2633:
2630:
2628:
2625:
2623:
2620:
2618:
2617:Angle of view
2615:
2613:
2610:
2609:
2605:
2603:
2600:
2596:
2590:
2588:
2584:
2580:
2575:
2573:
2569:
2565:
2564:conformal map
2561:
2557:
2549:
2544:
2541:
2538:
2535:
2534:
2533:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2499:
2496:
2495:
2489:
2488:
2484:
2481:
2480:
2479:
2473:
2467:
2463:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2431:
2430:
2424:
2403:
2387:
2383:
2372:
2365:
2360:
2344:
2340:
2329:
2301:
2297:
2289:
2288:
2287:
2284:
2267:
2257:
2252:
2248:
2242:
2238:
2234:
2231:
2228:
2223:
2220:
2209:
2204:
2200:
2194:
2190:
2186:
2173:
2169:
2158:
2151:
2140:
2136:
2134:
2121:
2106:
2101:
2097:
2091:
2087:
2083:
2080:
2077:
2072:
2069:
2058:
2053:
2049:
2043:
2039:
2035:
2022:
2018:
2007:
2000:
1989:
1985:
1983:
1970:
1957:
1953:
1949:
1947:
1946:non-linearity
1943:
1938:
1934:
1917:
1911:
1908:
1903:
1899:
1893:
1889:
1885:
1880:
1876:
1870:
1866:
1862:
1859:
1846:
1842:
1831:
1824:
1813:
1809:
1807:
1794:
1783:
1780:
1775:
1771:
1765:
1761:
1757:
1752:
1748:
1742:
1738:
1734:
1731:
1718:
1714:
1703:
1696:
1685:
1681:
1679:
1666:
1653:
1650:
1636:
1628:
1624:
1606:
1602:
1589:
1569:
1553:
1549:
1538:
1531:
1526:
1510:
1506:
1495:
1469:
1462:
1436:
1413:
1409:
1401:
1375:
1352:
1348:
1340:
1314:
1307:
1296:
1285:
1282:
1255:
1248:
1237:
1226:
1200:
1193:
1182:
1171:
1170:
1169:
1166:
1149:
1143:
1138:
1134:
1128:
1124:
1120:
1115:
1111:
1105:
1101:
1097:
1094:
1080:
1064:
1060:
1049:
1042:
1039:
1034:
1030:
1021:
1017:
1013:
1011:
992:
988:
977:
959:
955:
944:
935:
931:
927:
921:
915:
912:
907:
903:
897:
893:
889:
884:
880:
874:
870:
852:
848:
837:
830:
828:
815:
811:
800:
789:
784:
780:
774:
770:
766:
761:
757:
751:
747:
743:
740:
720:
716:
705:
687:
683:
672:
663:
659:
655:
652:
650:
637:
621:
617:
606:
599:
596:
591:
587:
578:
574:
567:
561:
558:
553:
549:
543:
539:
535:
530:
526:
520:
516:
498:
494:
483:
476:
474:
461:
457:
446:
433:
431:
422:
417:
409:
404:
400:
392:
386:
379:
374:
370:
363:
358:
354:
350:
343:
338:
336:
330:
328:
326:
322:
318:
312:
304:
302:
299:
295:
291:
287:
278:
274:
272:
268:
263:
261:
257:
253:
242:
241:Canon EF 50mm
238:
229:
222:
220:
218:
214:
210:
202:
198:
194:
190:
186:
181:
178:
174:
170:
165:
162:
158:
154:
150:
146:
142:
141:magnification
138:
133:
131:
130:distortions.
129:
125:
121:
117:
98:
84:
70:
58:
56:
54:
50:
46:
42:
37:
33:
19:
3072:
3060:. Retrieved
3051:
3037:
3025:. Retrieved
3021:the original
3010:
2998:. Retrieved
2994:
2985:
2973:. Retrieved
2969:the original
2959:
2947:. Retrieved
2938:
2926:. Retrieved
2917:
2903:. Elsevier.
2900:
2894:
2880:. Springer.
2877:
2842:
2836:
2823:
2816:
2789:
2785:
2775:
2764:the original
2759:
2755:
2742:
2725:
2695:. Retrieved
2686:
2674:. Retrieved
2670:the original
2665:
2662:"Distortion"
2655:
2612:Anamorphosis
2598:
2591:
2576:
2553:
2531:
2520:PhotoPerfect
2477:
2428:
2285:
1958:
1954:
1950:
1939:
1935:
1654:
1651:
1593:
1167:
434:
429:
426:
348:
334:
320:
314:
294:didactically
285:
283:
271:globe effect
264:
234:
212:
206:
196:
192:
188:
173:optical axis
145:optical axis
127:
123:
119:
113:
44:
38:
36:
3062:9 September
2995:opencv.org/
2579:perspective
2556:rectilinear
2483:ImageMagick
237:zoom lenses
3000:22 January
2975:13 October
2676:2 February
2648:References
2642:Vignetting
2627:Distortion
2560:trapezoids
2425:Calibrated
397:See also:
298:concentric
267:binoculars
256:retrofocus
223:Occurrence
177:pincushion
128:pincushion
88:Pincushion
45:distortion
2944:"Lensfun"
2928:2 January
2572:corrected
2498:Photoshop
2384:−
2341:−
2232:−
2224:−
2170:−
2081:−
2073:−
2019:−
1912:⋯
1843:−
1784:⋯
1715:−
1623:monotonic
1550:−
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1144:⋯
1061:−
989:−
956:−
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790:⋯
717:−
684:−
618:−
562:⋯
495:−
423:computer)
209:quadratic
161:zoom lens
118:. These
3109:Category
2991:"OpenCV"
2949:16 April
2923:"PTlens"
2845:. IEEE.
2728:. SPIE.
2697:20 March
2606:See also
2500:CS2 and
2469:designs.
349:outwards
102:Mustache
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2577:Due to
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1942:warping
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1428:is the
1367:is the
369:cushion
217:quartic
3027:3 July
2537:FFMPEG
2474:Manual
2455:proDAD
2443:OpenCV
2286:where
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286:radial
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2722:(PDF)
2570:, or
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421:ENIAC
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260:Nikon
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3064:2009
3029:2013
3002:2018
2977:2013
2951:2022
2930:2012
2699:2013
2678:2009
2599:both
2514:GIMP
2460:The
401:and
151:(or
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2730:doi
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