1153:
426:
397:
367:
1148:{\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}}}
405:
331:
72:
208:(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.
2271:
86:
266:
2574:(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
58:
351:
1921:
1950:
290:
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.
1646:
217:
2266:{\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}}}
1940:
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
416:
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
2581:
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
289:
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
1925:
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.
366:
1916:{\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}}}
2457:
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
2590:
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.
2407:
1573:
238:
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
188:, 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
1944:
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
308:" – "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
1325:
1266:
1211:
1955:
1651:
2425:
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.
2582:
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.
152:, 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.
103:
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
2453:
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
1447:
1386:
2308:
1474:
1937:
of the distortion equation. Lateral chromatic aberration (purple/green fringing) can be significantly reduced by applying such warping for red, green and blue separately.
372:
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
330:
2303:
1608:
1415:
1354:
2737:
1636:
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1933:
the image with a reverse distortion. This involves determining which distorted pixel corresponds to each undistorted pixel, which is non-trivial due to the
2547:: 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
1641:
To model radial distortion, the division model typically provides a more accurate approximation than Brown-Conrady's even-order polynomial model,
148:, 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
2474:
can correct several distortions; for example the fisheye distortion of the popular GoPro Hero3+ Silver camera can be corrected by the command
431:
164:. 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
2575:
2446:
Defishr includes an Unwarp-tool and a
Calibrator-tool. Due to the distortion of a checkerboard pattern, the necessary unwrap is calculated.
2511:
has interactive functions for general pincushion adjustment, and for fringe (adjusting the size of the red, green and blue image parts).
2434:
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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2650:
1277:
1218:
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396:
3005:
228:, particularly large-range zooms, but may also be found in prime lenses, and depends on focal distance – for example, the
41:
305:
2707:
3103:
2752:
2499:
includes a manual Lens
Distortion effect for simple (barrel, fisheye, fisheye spherical and pincushion) distortion.
2496:
2578:; the image itself is not distorted, but is perceived as distorted when viewed from a normal viewing distance.
2556:
2450:
2680:
2521:
Besides these systems that address images, there are some that also adjust distortion parameters for videos:
2610:
2832:
Proceedings of the 2001 IEEE Computer
Society Conference on Computer Vision and Pattern Recognition (CVPR)
2830:
Fitzgibbon, A. W. (2001). "Simultaneous linear estimation of multiple view geometry and lens distortion".
37:
2811:
2402:{\displaystyle {\sqrt {(x_{\mathrm {u} }-x_{\mathrm {c} })^{2}+(y_{\mathrm {u} }-y_{\mathrm {c} })^{2}}}}
1568:{\displaystyle {\sqrt {(x_{\mathrm {d} }-x_{\mathrm {c} })^{2}+(y_{\mathrm {d} }-y_{\mathrm {c} })^{2}}}}
3082:
2859:
2534:
by using the node editor to insert a "Distort/Lens
Distortion" node between the input and output nodes.
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1359:
2782:
2567:
299:
244:
20:
2409:, the Euclidean distance between the undistorted image point and the undistortion/distortion center.
269:
Radial distortions can be understood by their effect on concentric circles, as in an archery target.
3088:
2888:
Wang, J.; Shi, F.; Zhang, J.; Liu, Y. (2008). "A new calibration model of camera lens distortion".
2560:
387:
278:
2493:(from version 5) include a manual Lens Correction filter for simple (pincushion/barrel) distortion
421:. See Zhang for additional discussion of radial distortion. The Brown-Conrady distortion model is
2531:
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1576:
373:
312:(longitudinal) chromatic aberration, which causes aberrations throughout the field, particularly
229:
197:
189:
3045:
254:
A certain amount of pincushion distortion is often found with visual optical instruments, e.g.,
1610:
whereas pincushion distortion will have a positive value. Moustache distortion will have a non-
404:
40:; a projection in which straight lines in a scene remain straight in an image. It is a form of
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104:
2893:
2870:
2835:
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2544:
2514:
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29:
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convert distorted_image.jpg -distort barrel "0.06335 -0.18432 -0.13009" corrected_image.jpg
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2551:: 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.
240:
243:. Mustache distortion is observed particularly on the wide end of zooms, with certain
71:
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1934:
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physical elements in a lens not being perfectly aligned. The latter is also known as
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software can correct complex distortion, and takes into account the focus distance.
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161:
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In pincushion distortion, image magnification increases with the distance from the
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133:
85:
2897:
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In order to understand these distortions, it should be remembered that these are
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2471:
57:
3009:
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2630:
2615:
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can be used to correct distortion, though that is not its primary application.
1213:
is the distorted image point as projected on image plane using specified lens;
286:
255:
165:
2839:
2795:
2770:
2428:
Lensfun is a free to use database and library for correcting lens distortion.
220:
Simulated animation of globe effect (right) compared with a simple pan (left)
2548:
2486:
1611:
356:
In pincushion distortion, corners of squares form elongated points, as in a
225:
149:
2706:
de
Villiers, J. P.; Leuschner, F.W.; Geldenhuys, R. (17–19 November 2008).
2422:
Adobe
Photoshop Lightroom and Photoshop CS5 can correct complex distortion.
216:
3031:
282:
136:. The apparent effect is that of an image which has been mapped around a
23:, a loss of image sharpness that can result from spherical lens surfaces.
2418:
Calibrated systems work from a table of lens/camera transfer functions:
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357:
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2525:
341:
141:
137:
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are wide-angle lenses with heavy barrel distortion and thus exhibit
168:. Convex (plus) spherical lenses tend to have pincushion distortion.
3089:
3D modeling Lens distortion and camera field of view in CCTV design
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395:
264:
248:
215:
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2467:
Manual systems allow manual adjustment of distortion parameters:
2502:
2708:"Centi-pixel accurate real-time inverse distortion correction"
2505:
includes manual lens distortion correction (from version 2.4).
2932:
2860:"Automatic Radial Distortion Estimation from a Single Image"
2715:
2008 International
Symposium on Optomechatronic Technologies
1579:
between the distorted image point and the distortion center.
247:
lenses, and more recently on large-range zooms such as the
224:
In photography, distortion is particularly associated with
1583:
Barrel distortion typically will have a negative term for
3046:"Hugin tutorial – Simulating an architectural projection"
1268:
is the undistorted image point as projected by an ideal
304:
Radial distortion that depends on wavelength is called "
204:
of distance from the center. In mustache distortion the
408:
Barrel distortion corrected with software (this is the
2431:
2819:(Technical report). Microsoft Research. MSR-TR-98-71.
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1320:{\displaystyle (x_{\mathrm {c} },\ y_{\mathrm {c} })}
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1261:{\displaystyle (x_{\mathrm {u} },\ y_{\mathrm {u} })}
1221:
1206:{\displaystyle (x_{\mathrm {d} },\ y_{\mathrm {d} })}
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Mathematically, barrel and pincushion distortion are
1941:
required and the computational resources available.
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176:A mixture of both types, sometimes referred to as
2775:Monthly Notices of the Royal Astronomical Society
2813:A Flexible New Technique for Camera Calibration
2543:Radial distortion is a failure of a lens to be
277:defects; the optical systems in question have
16:Deviation from rectilinear projection (optics)
3032:"ImageMagick v6 Examples -- Lens Corrections"
8:
2701:
2699:
2697:
2555:). This effect can be controlled by using a
400:With uncorrected barrel distortion (at 26mm)
336:In barrel distortion, straight lines bulge
2867:Journal of mathematical imaging and vision
1929:Software can correct those distortions by
3085:with source code and online demonstration
3083:Lens distortion estimation and correction
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2681:"Tamron 18-270mm f/3.5-6.3 Di II VC PZD"
2528:using the "lenscorrection" video filter.
2641:
326:
3006:"Articles: Digital Photography Review"
1449:tangential distortion coefficient; and
281:(omitting non-radial defects), so the
285:correct test image would be a set of
7:
392:Image stitching § straightening
258:, where it serves to counteract the
121:
111:can usually be classified as either
2954:"lensfun – Rev 246 – /trunk/README"
2858:Bukhari, F.; Dailey, M. N. (2013).
2570:, cameras image a cube as a square
2738:"Decentering distortion of lenses"
2381:
2366:
2338:
2323:
2167:
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2016:
2001:
1983:
1964:
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14:
1442:{\displaystyle n^{\mathrm {th} }}
1381:{\displaystyle n^{\mathrm {th} }}
132:decreases with distance from the
365:
349:
329:
84:
70:
56:
1638:the sequence will change sign.
200:, meaning they increase as the
3067:"FFmpeg Filters Documentation"
3008:. Dpreview.com. Archived from
2388:
2357:
2345:
2314:
2253:
2206:
2102:
2055:
1554:
1523:
1511:
1480:
1388:radial distortion coefficient;
1314:
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1080:
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559:
553:
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498:
468:
97:Examples of radial distortions
1:
2898:10.1016/j.patcog.2007.06.012
2751:(3): 444–462. Archived from
2736:Brown, Duane C. (May 1966).
306:lateral chromatic aberration
128:In barrel distortion, image
2745:Photogrammetric Engineering
3120:
2497:Corel Paint Shop Pro Photo
385:
297:
18:
2875:10.1007/s10851-012-0342-2
1327:is the distortion center;
2840:10.1109/CVPR.2001.990465
2810:Zhang, Zhengyou (1998).
2771:"Decentred Lens-Systems"
2557:perspective control lens
2451:Micro Four Thirds system
19:Not to be confused with
2769:Conrady, A. E. (1919).
2611:Cylindrical perspective
340:at the center, as in a
2796:10.1093/mnras/79.5.384
2576:perspective distortion
2403:
2299:
2267:
1917:
1632:
1604:
1569:
1465:
1443:
1411:
1382:
1350:
1321:
1262:
1207:
1149:
419:decentering distortion
413:
401:
270:
221:
38:rectilinear projection
2404:
2300:
2298:{\displaystyle r_{u}}
2268:
1918:
1633:
1605:
1603:{\displaystyle K_{1}}
1570:
1466:
1444:
1412:
1410:{\displaystyle P_{n}}
1383:
1351:
1349:{\displaystyle K_{n}}
1322:
1263:
1208:
1150:
407:
399:
298:Further information:
268:
219:
156:Pincushion distortion
2563:in post-processing.
2309:
2282:
1951:
1647:
1622:
1587:
1475:
1455:
1421:
1394:
1360:
1333:
1278:
1219:
1164:
427:
300:Chromatic aberration
294:Chromatic aberration
182:moustache distortion
36:is a deviation from
21:spherical aberration
2890:Pattern Recognition
2787:1919MNRAS..79..384C
2655:Photographic optics
2250:
2202:
2099:
2051:
388:Image rectification
382:Software correction
374:handlebar mustaches
279:rotational symmetry
178:mustache distortion
172:Mustache distortion
3104:Geometrical optics
2960:on 13 October 2013
2661:on 29 January 2009
2491:Photoshop Elements
2399:
2295:
2263:
2261:
2236:
2188:
2085:
2037:
1913:
1911:
1628:
1600:
1577:Euclidean distance
1565:
1461:
1439:
1407:
1378:
1346:
1317:
1258:
1203:
1145:
1143:
414:
402:
271:
222:
190:handlebar mustache
186:complex distortion
109:radial distortions
42:optical aberration
3004:Wiley, Carlisle.
2758:on 12 March 2018.
2723:10.1117/12.804771
2649:Paul van Walree.
2626:Underwater vision
2539:Related phenomena
2397:
2251:
2204:
2100:
2053:
1904:
1776:
1631:{\displaystyle r}
1563:
1464:{\displaystyle r}
1301:
1242:
1187:
124:Barrel distortion
105:photographic lens
48:Radial distortion
3111:
3071:
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3017:
3001:
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2976:
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2956:. Archived from
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2666:
2657:. Archived from
2646:
2621:Texture gradient
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2021:
2020:
2019:
2006:
2005:
2004:
1993:
1988:
1987:
1986:
1969:
1968:
1967:
1922:
1920:
1919:
1914:
1912:
1905:
1903:
1896:
1895:
1886:
1885:
1873:
1872:
1863:
1862:
1846:
1845:
1844:
1843:
1830:
1829:
1828:
1817:
1812:
1811:
1810:
1793:
1792:
1791:
1777:
1775:
1768:
1767:
1758:
1757:
1745:
1744:
1735:
1734:
1718:
1717:
1716:
1715:
1702:
1701:
1700:
1689:
1684:
1683:
1682:
1665:
1664:
1663:
1637:
1635:
1634:
1629:
1616:geometric series
1609:
1607:
1606:
1601:
1599:
1598:
1574:
1572:
1571:
1566:
1564:
1562:
1561:
1552:
1551:
1550:
1537:
1536:
1535:
1519:
1518:
1509:
1508:
1507:
1494:
1493:
1492:
1479:
1470:
1468:
1467:
1462:
1448:
1446:
1445:
1440:
1438:
1437:
1436:
1416:
1414:
1413:
1408:
1406:
1405:
1387:
1385:
1384:
1379:
1377:
1376:
1375:
1355:
1353:
1352:
1347:
1345:
1344:
1326:
1324:
1323:
1318:
1313:
1312:
1311:
1299:
1295:
1294:
1293:
1267:
1265:
1264:
1259:
1254:
1253:
1252:
1240:
1236:
1235:
1234:
1212:
1210:
1209:
1204:
1199:
1198:
1197:
1185:
1181:
1180:
1179:
1154:
1152:
1151:
1146:
1144:
1131:
1130:
1121:
1120:
1108:
1107:
1098:
1097:
1073:
1072:
1063:
1062:
1061:
1048:
1047:
1046:
1027:
1026:
1014:
1013:
998:
991:
990:
989:
976:
975:
974:
958:
957:
956:
943:
942:
941:
928:
927:
900:
899:
890:
889:
877:
876:
867:
866:
851:
850:
849:
836:
835:
834:
814:
813:
812:
799:
798:
797:
777:
776:
767:
766:
754:
753:
744:
743:
719:
718:
717:
704:
703:
702:
686:
685:
684:
671:
670:
669:
656:
655:
637:
630:
629:
620:
619:
618:
605:
604:
603:
584:
583:
571:
570:
546:
545:
536:
535:
523:
522:
513:
512:
497:
496:
495:
482:
481:
480:
460:
459:
458:
445:
444:
443:
369:
353:
333:
251:18–200 mm.
241:telephoto lenses
237:
235:
88:
74:
60:
30:geometric optics
3119:
3118:
3114:
3113:
3112:
3110:
3109:
3108:
3094:
3093:
3079:
3074:
3065:
3064:
3060:
3050:
3048:
3044:
3043:
3039:
3030:
3029:
3025:
3015:
3013:
3003:
3002:
2998:
2988:
2986:
2978:
2977:
2973:
2963:
2961:
2952:
2951:
2947:
2937:
2935:
2931:
2930:
2926:
2916:
2914:
2910:
2909:
2905:
2887:
2886:
2882:
2862:
2857:
2856:
2847:
2829:
2828:
2824:
2816:
2809:
2808:
2804:
2768:
2767:
2763:
2755:
2740:
2735:
2734:
2730:
2710:
2705:
2704:
2695:
2685:
2683:
2679:
2678:
2674:
2664:
2662:
2648:
2647:
2643:
2639:
2597:
2541:
2479:
2465:
2416:
2387:
2375:
2360:
2344:
2332:
2317:
2307:
2306:
2285:
2280:
2279:
2260:
2259:
2226:
2178:
2174:
2161:
2146:
2145:
2128:
2121:
2109:
2106:
2105:
2075:
2027:
2023:
2010:
1995:
1994:
1977:
1970:
1958:
1949:
1948:
1910:
1909:
1887:
1877:
1864:
1854:
1847:
1834:
1819:
1818:
1801:
1794:
1782:
1779:
1778:
1759:
1749:
1736:
1726:
1719:
1706:
1691:
1690:
1673:
1666:
1654:
1645:
1644:
1620:
1619:
1618:where for some
1590:
1585:
1584:
1553:
1541:
1526:
1510:
1498:
1483:
1473:
1472:
1453:
1452:
1424:
1419:
1418:
1397:
1392:
1391:
1363:
1358:
1357:
1336:
1331:
1330:
1302:
1284:
1276:
1275:
1243:
1225:
1217:
1216:
1188:
1170:
1162:
1161:
1142:
1141:
1122:
1112:
1099:
1089:
1064:
1052:
1037:
1018:
1005:
996:
995:
980:
965:
947:
932:
919:
891:
881:
868:
858:
840:
825:
815:
803:
788:
785:
784:
768:
758:
745:
735:
708:
693:
675:
660:
647:
635:
634:
621:
609:
594:
575:
562:
537:
527:
514:
504:
486:
471:
461:
449:
434:
425:
424:
394:
384:
377:
370:
361:
354:
345:
334:
322:
320:Origin of terms
314:purple fringing
302:
296:
233:
232:
214:
173:
157:
125:
115:distortions or
101:
100:
99:
98:
94:
93:
92:
89:
80:
79:
78:
75:
66:
65:
64:
61:
50:
24:
17:
12:
11:
5:
3117:
3115:
3107:
3106:
3096:
3095:
3092:
3091:
3086:
3078:
3077:External links
3075:
3073:
3072:
3058:
3037:
3023:
3012:on 7 July 2012
2996:
2971:
2945:
2924:
2903:
2880:
2845:
2822:
2802:
2781:(5): 384–390.
2761:
2728:
2693:
2672:
2640:
2638:
2635:
2634:
2633:
2628:
2623:
2618:
2613:
2608:
2603:
2596:
2593:
2584:Fisheye lenses
2540:
2537:
2536:
2535:
2529:
2519:
2518:
2512:
2506:
2500:
2494:
2483:
2482:
2476:
2475:
2464:
2461:
2460:
2459:
2447:
2441:
2438:DxO's PhotoLab
2435:
2429:
2426:
2423:
2415:
2412:
2411:
2410:
2394:
2390:
2383:
2378:
2374:
2368:
2363:
2359:
2356:
2351:
2347:
2340:
2335:
2331:
2325:
2320:
2316:
2292:
2288:
2258:
2255:
2248:
2243:
2239:
2233:
2229:
2225:
2222:
2219:
2214:
2211:
2208:
2200:
2195:
2191:
2185:
2181:
2177:
2169:
2164:
2160:
2154:
2149:
2142:
2136:
2131:
2127:
2124:
2122:
2117:
2112:
2108:
2107:
2104:
2097:
2092:
2088:
2082:
2078:
2074:
2071:
2068:
2063:
2060:
2057:
2049:
2044:
2040:
2034:
2030:
2026:
2018:
2013:
2009:
2003:
1998:
1991:
1985:
1980:
1976:
1973:
1971:
1966:
1961:
1957:
1956:
1908:
1902:
1899:
1894:
1890:
1884:
1880:
1876:
1871:
1867:
1861:
1857:
1853:
1850:
1842:
1837:
1833:
1827:
1822:
1815:
1809:
1804:
1800:
1797:
1795:
1790:
1785:
1781:
1780:
1774:
1771:
1766:
1762:
1756:
1752:
1748:
1743:
1739:
1733:
1729:
1725:
1722:
1714:
1709:
1705:
1699:
1694:
1687:
1681:
1676:
1672:
1669:
1667:
1662:
1657:
1653:
1652:
1627:
1597:
1593:
1581:
1580:
1560:
1556:
1549:
1544:
1540:
1534:
1529:
1525:
1522:
1517:
1513:
1506:
1501:
1497:
1491:
1486:
1482:
1460:
1450:
1435:
1432:
1427:
1404:
1400:
1389:
1374:
1371:
1366:
1343:
1339:
1328:
1316:
1310:
1305:
1298:
1292:
1287:
1283:
1273:
1270:pinhole camera
1257:
1251:
1246:
1239:
1233:
1228:
1224:
1214:
1202:
1196:
1191:
1184:
1178:
1173:
1169:
1140:
1137:
1134:
1129:
1125:
1119:
1115:
1111:
1106:
1102:
1096:
1092:
1088:
1085:
1082:
1079:
1076:
1071:
1067:
1060:
1055:
1051:
1045:
1040:
1036:
1033:
1030:
1025:
1021:
1017:
1012:
1008:
1004:
1001:
999:
997:
994:
988:
983:
979:
973:
968:
964:
961:
955:
950:
946:
940:
935:
931:
926:
922:
918:
915:
912:
909:
906:
903:
898:
894:
888:
884:
880:
875:
871:
865:
861:
857:
854:
848:
843:
839:
833:
828:
824:
821:
818:
816:
811:
806:
802:
796:
791:
787:
786:
783:
780:
775:
771:
765:
761:
757:
752:
748:
742:
738:
734:
731:
728:
725:
722:
716:
711:
707:
701:
696:
692:
689:
683:
678:
674:
668:
663:
659:
654:
650:
646:
643:
640:
638:
636:
633:
628:
624:
617:
612:
608:
602:
597:
593:
590:
587:
582:
578:
574:
569:
565:
561:
558:
555:
552:
549:
544:
540:
534:
530:
526:
521:
517:
511:
507:
503:
500:
494:
489:
485:
479:
474:
470:
467:
464:
462:
457:
452:
448:
442:
437:
433:
432:
383:
380:
379:
378:
371:
364:
362:
355:
348:
346:
335:
328:
321:
318:
295:
292:
213:
210:
194:
193:
174:
171:
169:
158:
155:
153:
146:Fisheye lenses
126:
123:
96:
95:
90:
83:
82:
81:
76:
69:
68:
67:
62:
55:
54:
53:
52:
51:
49:
46:
15:
13:
10:
9:
6:
4:
3:
2:
3116:
3105:
3102:
3101:
3099:
3090:
3087:
3084:
3081:
3080:
3076:
3068:
3062:
3059:
3047:
3041:
3038:
3033:
3027:
3024:
3011:
3007:
3000:
2997:
2985:
2981:
2975:
2972:
2959:
2955:
2949:
2946:
2934:
2928:
2925:
2913:
2907:
2904:
2899:
2895:
2891:
2884:
2881:
2876:
2872:
2868:
2861:
2854:
2852:
2850:
2846:
2841:
2837:
2833:
2826:
2823:
2815:
2814:
2806:
2803:
2797:
2792:
2788:
2784:
2780:
2776:
2772:
2765:
2762:
2754:
2750:
2746:
2739:
2732:
2729:
2724:
2720:
2716:
2709:
2702:
2700:
2698:
2694:
2682:
2676:
2673:
2660:
2656:
2652:
2645:
2642:
2636:
2632:
2629:
2627:
2624:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2606:Angle of view
2604:
2602:
2599:
2598:
2594:
2592:
2589:
2585:
2579:
2577:
2573:
2569:
2564:
2562:
2558:
2554:
2553:conformal map
2550:
2546:
2538:
2533:
2530:
2527:
2524:
2523:
2522:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2488:
2485:
2484:
2478:
2477:
2473:
2470:
2469:
2468:
2462:
2456:
2452:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2420:
2419:
2413:
2392:
2376:
2372:
2361:
2354:
2349:
2333:
2329:
2318:
2290:
2286:
2278:
2277:
2276:
2273:
2256:
2246:
2241:
2237:
2231:
2227:
2223:
2220:
2217:
2212:
2209:
2198:
2193:
2189:
2183:
2179:
2175:
2162:
2158:
2147:
2140:
2129:
2125:
2123:
2110:
2095:
2090:
2086:
2080:
2076:
2072:
2069:
2066:
2061:
2058:
2047:
2042:
2038:
2032:
2028:
2024:
2011:
2007:
1996:
1989:
1978:
1974:
1972:
1959:
1946:
1942:
1938:
1936:
1935:non-linearity
1932:
1927:
1923:
1906:
1900:
1897:
1892:
1888:
1882:
1878:
1874:
1869:
1865:
1859:
1855:
1851:
1848:
1835:
1831:
1820:
1813:
1802:
1798:
1796:
1783:
1772:
1769:
1764:
1760:
1754:
1750:
1746:
1741:
1737:
1731:
1727:
1723:
1720:
1707:
1703:
1692:
1685:
1674:
1670:
1668:
1655:
1642:
1639:
1625:
1617:
1613:
1595:
1591:
1578:
1558:
1542:
1538:
1527:
1520:
1515:
1499:
1495:
1484:
1458:
1451:
1425:
1402:
1398:
1390:
1364:
1341:
1337:
1329:
1303:
1296:
1285:
1274:
1271:
1244:
1237:
1226:
1215:
1189:
1182:
1171:
1160:
1159:
1158:
1155:
1138:
1132:
1127:
1123:
1117:
1113:
1109:
1104:
1100:
1094:
1090:
1086:
1083:
1069:
1053:
1049:
1038:
1031:
1028:
1023:
1019:
1010:
1006:
1002:
1000:
981:
977:
966:
948:
944:
933:
924:
920:
916:
910:
904:
901:
896:
892:
886:
882:
878:
873:
869:
863:
859:
841:
837:
826:
819:
817:
804:
800:
789:
778:
773:
769:
763:
759:
755:
750:
746:
740:
736:
732:
729:
709:
705:
694:
676:
672:
661:
652:
648:
644:
641:
639:
626:
610:
606:
595:
588:
585:
580:
576:
567:
563:
556:
550:
547:
542:
538:
532:
528:
524:
519:
515:
509:
505:
487:
483:
472:
465:
463:
450:
446:
435:
422:
420:
411:
406:
398:
393:
389:
381:
375:
368:
363:
359:
352:
347:
343:
339:
332:
327:
325:
319:
317:
315:
311:
307:
301:
293:
291:
288:
284:
280:
276:
267:
263:
261:
257:
252:
250:
246:
242:
231:
230:Canon EF 50mm
227:
218:
211:
209:
207:
203:
199:
191:
187:
183:
179:
175:
170:
167:
163:
159:
154:
151:
147:
143:
139:
135:
131:
130:magnification
127:
122:
120:
119:distortions.
118:
114:
110:
106:
87:
73:
59:
47:
45:
43:
39:
35:
31:
26:
22:
3061:
3049:. Retrieved
3040:
3026:
3014:. Retrieved
3010:the original
2999:
2987:. Retrieved
2983:
2974:
2962:. Retrieved
2958:the original
2948:
2936:. Retrieved
2927:
2915:. Retrieved
2906:
2892:. Elsevier.
2889:
2883:
2869:. Springer.
2866:
2831:
2825:
2812:
2805:
2778:
2774:
2764:
2753:the original
2748:
2744:
2731:
2714:
2684:. Retrieved
2675:
2663:. Retrieved
2659:the original
2654:
2651:"Distortion"
2644:
2601:Anamorphosis
2587:
2580:
2565:
2542:
2520:
2509:PhotoPerfect
2466:
2417:
2274:
1947:
1943:
1939:
1928:
1924:
1643:
1640:
1582:
1156:
423:
418:
415:
337:
323:
309:
303:
283:didactically
274:
272:
260:globe effect
253:
223:
201:
195:
185:
181:
177:
162:optical axis
134:optical axis
116:
112:
108:
102:
33:
27:
25:
3051:9 September
2984:opencv.org/
2568:perspective
2545:rectilinear
2472:ImageMagick
226:zoom lenses
2989:22 January
2964:13 October
2665:2 February
2637:References
2631:Vignetting
2616:Distortion
2549:trapezoids
2414:Calibrated
386:See also:
287:concentric
256:binoculars
245:retrofocus
212:Occurrence
166:pincushion
117:pincushion
77:Pincushion
34:distortion
2933:"Lensfun"
2917:2 January
2561:corrected
2487:Photoshop
2373:−
2330:−
2221:−
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2938:16 April
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