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Software correction

Radial distortion, whilst primarily dominated by low order radial components,[3] can be corrected using Brown's distortion model,[4] also known as the Brown–Conrady model based on earlier work by Conrady.

Radial distortion, whilst primarily dominated by low order radial components,[3] can be corrected using Brown's distortion model,[4] also known as the Brown–Conrady model based on earlier work by Conrady.[5] 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[6] for additional discussion of radial distortion.

= distorted image point as projected on image plane using specified lens,
= undistorted image point as projected by an ideal pinhole camera,
= distortion center,
= radial distortion coefficient,
= tangential distortion coefficient,
= , and
= an infinite series.

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

To model radial distortion, the division model[7] typically provides a more accurate approximation than Brown-Conrady's even-order polynomial model:[8]

[7]

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 describe more accurately severe distortion.[8] Using this model, a single term is usually sufficient to model most cameras.[9]

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.[3] 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.

Calibrated

Calibrated systems work from a table of lens/camera transfer functions:

Manual

Manual systems allow manual adjustment of distortion parameters:

convert distorted_image.jpg -distort barrel "0.06335 -0.18432 -0.13009" corrected_image.jpg