Barrel distortion
Encyclopedia
In geometric optics and cathode ray tube
Cathode ray tube
The cathode ray tube is a vacuum tube containing an electron gun and a fluorescent screen used to view images. It has a means to accelerate and deflect the electron beam onto the fluorescent screen to create the images. The image may represent electrical waveforms , pictures , radar targets and...

(CRT) displays, 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 optical aberration
Aberration in optical systems
Aberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after transmission through the system does not converge...

.

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
Photographic lens
A camera lens is an optical lens or assembly of lenses used in conjunction with a camera body and mechanism to make images of objects either on photographic film or on other media capable of storing an image chemically or electronically.While in principle a simple convex lens will suffice, in...

. The radial distortion can usually be classified as one of two main types:

Barrel distortion: In "barrel distortion", image magnification
Magnification
Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called "magnification"...

decreases with distance from the optical axis
Optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens or microscope.The optical axis is an imaginary line that defines the path along which light propagates through the system...

. The apparent effect is that of an image which has been mapped around a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

(or barrel
Barrel
A barrel or cask is a hollow cylindrical container, traditionally made of vertical wooden staves and bound by wooden or metal hoops. Traditionally, the barrel was a standard size of measure referring to a set capacity or weight of a given commodity. A small barrel is called a keg.For example, a...

). Fisheye lens
Fisheye lens
In photography, a fisheye lens is a wide-angle lens that takes in a broad, panoramic and hemispherical image. Originally developed for use in meteorology to study cloud formation and called "whole-sky lenses", fisheye lenses quickly became popular in general photography for their unique, distorted...

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.

Pincushion distortion: In "pincushion distortion", image magnification increases with the distance from the optical axis
Optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens or microscope.The optical axis is an imaginary line that defines the path along which light propagates through the system...

. 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
Pincushion
A pincushion is a small cushion, typically 3–5 cm across, which is used in sewing to store pins or needles with their heads protruding so as to take hold of them easily, collect them, and keep them organized....

. A certain amount of pincushion distortion is often found with visual optical instruments, e.g. binoculars
Binoculars
Binoculars, field glasses or binocular telescopes are a pair of identical or mirror-symmetrical telescopes mounted side-by-side and aligned to point accurately in the same direction, allowing the viewer to use both eyes when viewing distant objects...

, where it serves to eliminate the globe effect
Globe effect
The globe effect, sometimes called the rolling ball effect or the spinning globe effect, is an optical phenomenon that occurs with visual optical instruments, in particular binoculars and telescopes, that are designed to be free of distortion. When these instruments are panned, the moving image...

.

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.

Mathematically, barrel and pincushion distortion are quadratic
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms...

, meaning they increase as the square of distance from the center. In mustache distortion the quartic
Quartic
In mathematics, the term quartic describes something that pertains to the "fourth order", such as the function x^4. It may refer to one of the following:* Quartic function, a polynomial function of degree 4* Quartic curve, an algebraic curve of degree 4...

(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.

Occurrence
In photography, distortion is particularly associated with zoom lens
Zoom lens
A zoom lens is a mechanical assembly of lens elements for which the focal length can be varied, as opposed to a fixed focal length 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
Canon EF 50mm lens
The EF 50mm lenses are a group of normal prime lenses made by Canon that share the same focal length. These lenses are based on the classic double-Gauss lens, with the f/1.8 being a standard six-element double-Gauss with an air gap and powers between element 2 and 3 and its faster cousins adding...

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 telephoto distortion is often seen in older or low-end telephoto lens
Telephoto lens
In photography and cinematography, a telephoto lens is a specific type of a long-focus lens in which the physical length of the lens is shorter than the focal length. This is achieved by incorporating a special lens group known as a telephoto group that extends the light path to create a long-focus...

es. Mustache distortion is observed particularly on the wide end of some zooms, with certain retrofocus
Angenieux retrofocus
The Angénieux retrofocus photographic lens is a wide-angle lens design that uses an inverted telephoto configuration. The popularity of this lens design made the name retrofocus synonymous with this type of lens...

lenses, and more recently on large-range zooms such as the Nikon
Nikon
, also known as just Nikon, is a multinational corporation headquartered in Tokyo, Japan, specializing in optics and imaging. Its products include cameras, binoculars, microscopes, measurement instruments, and the steppers used in the photolithography steps of semiconductor fabrication, of which...

18–200 mm.

In order to understand these distortions, it should be remembered that these are radial defects; the optical systems in question have rotational symmetry
Rotational symmetry
Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...

(omitting non-radial defects), so the didactically correct test image would be a set of concentric
Concentric
Concentric objects share the same center, axis or origin with one inside the other. Circles, tubes, cylindrical shafts, disks, and spheres may be concentric to one another...

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

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
Purple fringing
In photography, and particularly in digital photography, purple fringing is the term for an out-of-focus purple or magenta "ghost" image on a photograph...

.

### Gallery

The names for these distortions come from familiar objects which are visually similar.

## Software correction

Radial distortion, whilst primarily dominated by low order radial components, can be corrected using Brown's distortion model. Brown's model caters for both radial distortion and for tangential distortion caused by physical elements in a lens not being perfectly aligned. The latter is thus also known as decentering distortion.

where: = undistorted image point, = distorted image point, = distortion center (assumed to be the principal point
Principal Point
Principal Point is a prominent ice-covered headland lying 4 nautical miles east of Cape Errera and forming the southeast end of Wiencke Island, in the Palmer Archipelago. First charted by the French Antarctic Expedition under Charcot, 1903-05. The name, applied by the Argentine Antarctic...

), = radial distortion coefficient, = tangential distortion coefficient, = , and = an infinite series.

Barrel distortion typically will have a positive term for where as pincushion distortion will have a negative value. Moustache distortion will have a non-monotonic radial geometric series where for some the sequence will change sign.

Software can correct those distortions by warping
Image warping
Image warping is the process of digitally manipulating an image such that any shapes portrayed in the image have been significantly distorted. Warping may be used for correcting image distortion as well as for creative purposes...

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. Lateral chromatic aberration (purple/green fringing) can be significantly reduced by applying such warping for red, green and blue separately.

An alternative method iteratively computes the undistorted pixel position.

### Calibrated

Calibrated systems work from a table of lens/camera transfer functions:
• 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.
• DxO Labs
DxO Labs
DxO Labs is a privately held image processing software company based in Boulogne-Billancourt, France.DxO Optics Pro is a personal computer package which corrects for various optical aberrations, notably image distortion, with corrections tuned to particular lenses and cameras. It also adjusts...

' Optics Pro can correct complex distortion, and takes into account the focus distance.
• The Micro Four Thirds system
Micro Four Thirds system
The Micro Four Thirds system is a standard created by Olympus and Panasonic, and announced on August 5, 2008, for mirrorless interchangeable lens digital cameras and camcorders design and development...

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 converter software. The optics of most of these lenses feature substantially more distortion than their counterparts in systems that don't offer such automatic corrections, but the software-corrected final images show noticeably less distortion than competing designs.

### Manual

Manual systems allow manual adjustment of distortion parameters:
• Photoshop CS2 and Photoshop Elements (from version 5) include a manual Lens Correction filter for simple (pincushion/barrel) distortion
• Corel Paint Shop Pro Photo include a manual Lens Distortion effect for simple (barrel, fisheye, fisheye spherical and pincushion) distortion.
• The GIMP
GIMP
GIMP is a free software raster graphics editor. It is primarily employed as an image retouching and editing tool and is freely available in versions tailored for most popular operating systems including Microsoft Windows, Apple Mac OS X, and Linux.In addition to detailed image retouching and...

includes manual lens distortion correction (from version 2.4).
• PhotoPerfect
PhotoPerfect
PhotoPerfect is a proprietary commercial photo editing software program for Microsoft Windows, published by Arcadia Software AG. The first version was introduced in 2002....

has interactive functions for general pincushion adjustment, and for fringe (adjusting the size of the red, green and blue image parts).
• Hugin
Hugin (software)
Hugin is a cross-platform open source panorama photo stitching and HDR merging program developed by Pablo d'Angelo and others. It is a GUI front-end for Helmut Dersch's Panorama Tools and Andrew Mihal's Enblend and Enfuse...

can be used to correct distortion, though that is not its primary application.

## Related phenomena

Radial distortion is a failure of a lens to be rectilinear
Rectilinear lens
In photography, a rectilinear lens is a photographic lens that yields images where straight features, such as the walls of buildings, appear with straight lines, as opposed to being curved. In other words, it is a lens with little or no barrel or pincushion distortion...

: 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
Trapezoid
In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...

s: lines are imaged as lines, but the angles between them are not preserved (tilt is not a conformal map
Conformal map
In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

). This effect can be controlled by using a perspective control lens, or corrected in post-processing.

Due to perspective
Perspective (visual)
Perspective, in context of vision and visual perception, is the way in which objects appear to the eye based on their spatial attributes; or their dimensions and the position of the eye relative to the objects...

, cameras image a cube as a square frustum
Frustum
In geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....

(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
Perspective distortion (photography)
In photography and cinematography, perspective distortion is a warping or transformation of an object and its surrounding area that differs significantly from what the object would look like with a normal focal length, due to the relative scale of nearby and distant features...

; 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
Fisheye lens
In photography, a fisheye lens is a wide-angle lens that takes in a broad, panoramic and hemispherical image. Originally developed for use in meteorology to study cloud formation and called "whole-sky lenses", fisheye lenses quickly became popular in general photography for their unique, distorted...

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.

• Anamorphosis
Anamorphosis
Anamorphosis or anamorphism may refer to any of the following:*Anamorphosis, in art, the representation of an object as seen, for instance, altered by reflection in a mirror...

• Angle of view
Angle of view
In photography, angle of view describes the angular extent of a given scene that is imaged by a camera. It is used interchangeably with the more general term field of view....

• Cylindrical perspective
Cylindrical perspective
Cylindrical perspective is a form of distortion caused by fisheye and panoramic lenses which reproduce straight horizontal lines above and below the lens axis level as curved while reproducing straight horizontal lines on lens axis level as straight...

• Distortion
Distortion
A distortion is the alteration of the original shape of an object, image, sound, waveform or other form of information or representation. Distortion is usually unwanted, and often many methods are employed to minimize it in practice...

Texture gradient is the distortion in size which closer objects have compared to objects farther away. It also involves groups of objects appearing denser as they move farther away...

• Underwater vision
Underwater vision
Light rays bend when they travel from one medium to another; the amount of bending is determined by the refractive indices of the two media. If one medium has a particular curved shape, it functions as a lens. The cornea, humours, and crystalline lens of the eye together form a lens that focuses...

• Vignetting
Vignetting
In photography and optics, vignetting  is a reduction of an image's brightness or saturation at the periphery compared to the image center. The word vignette, from the same root as vine, originally referred to a decorative border in a book. Later, the word came to be used for a photographic...