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Wine glasses create non-uniform distortion of their background

In geometric optics and cathode ray tube (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.

Contents

Radial distortion

Barrel distortion simulation
Pincushion distortion simulation

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

Barrel distortion
In "barrel distortion", image magnification decreases with distance from the optical axis. The apparent effect is that of an image which has been mapped around a sphere. Fisheye lenses, 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. 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. In photography, this aberration is often seen in older or low-end telephoto lenses. A certain amount of pincushion distortion is often found with visual optical instruments, e.g. binoculars, where it serves to eliminate the globe effect.

A mixture of both types, sometimes referred to as moustache 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. It is observed with certain retrofocus lenses, also more recently on large-range zooms such as the Nikon 18–200 mm.

In order to understand these distortions, it should be remembered that the optical systems in question have rotational symmetry, so the didactically 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 exaggregated 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.

Purple/green fringing in the outer parts of an image indicate lateral chromatic aberration: The radius distortion varies with the color (wavelength) of the light, and may be non-linear.

Software correction

Radial distortion, whilst primarily dominated by low order radial components,[1] can be corrected using Brown's distortion model.[2] 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.

xu = xd + (xdxc)(K1r2 + K2r4 + ...) + (P1(r2 + 2(xdxc)2) + 2P2(xdxc)(ydyc))(1 + P3r2 + ...)
yu = yd + (ydyc)(K1r2 + K2r4 + ...) + (P2(r2 + 2(ydyc)2) + 2P1(xdxc)(ydyc))(1 + P3r2 + ...)

where:

(x_\mathrm{u},\ y_\mathrm{u}) = undistorted image point,
(x_\mathrm{d},\ y_\mathrm{d}) = distorted image point,
(x_\mathrm{c},\ y_\mathrm{c}) = centre of distortion (ie. the principal point),
Kn = nth radial distortion coefficient,
Pn = nth tangential distortion coefficient,
r = \sqrt{(x_\mathrm{d}-x_\mathrm{c})^2 + (y_\mathrm{d}-y_\mathrm{c})^2}, and
... = an infinite series.

Barrel distortion typically will have a positive term for K1 where as pincushion distortion will have a negative value. Moustache distortion will have a non-monotonic radial geometric series where for some r the sequence will change sign.

Software can correct those distortions by 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.[1] Lateral chromatic aberration (purple/green fringing) can be significantly reduced by applying such warping for red, green and blue separately.

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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' Optics Pro can correct complex distortion, and takes into account the focus distance

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
  • The 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 can be used to correct distortion, though that is not its primary application.[3]

Related phenomena

Radial distortion is a failure of a lens to be 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 trapezoids: 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 corrected in post-processing.

Due to 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.

See also

Optical aberration
Piston

Tilt
Defocus
Spherical
Coma
Astigmatism
Field curvature
Distortion
Chromatic aberration

References

  1. ^ a b de Villiers, J. P.; Leuschner, F.W.; Geldenhuys, R. (17–19 November 2008). "Centi-pixel accurate real-time inverse distortion correction". 2008 International Symposium on Optomechatronic Technologies. SPIE. http://spie.org/x648.html?product_id=804771.  
  2. ^ Brown DC (1966). "Decentering distortion of lenses.". Photogrammetric Engineering. 7: 444–462.  
  3. ^ "Hugin tutorial – Simulating an architectural projection". http://hugin.sourceforge.net/tutorials/architectural/en.shtml. Retrieved 09 Sep. 2009.  

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