Distortion (optics) - Software Correction

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.

$x_\mathrm{u} = (x_\mathrm{d} - x_\mathrm{c})(1 + K_1r^2 + K_2r^4 + \cdots) + (P_1(r^2 + 2(x_\mathrm{d} - x_\mathrm{c})^2) + 2P_2(x_\mathrm{d} - x_\mathrm{c})(y_\mathrm{d} - y_\mathrm{c}))(1 + P_3r^2 + \cdots)$

$y_\mathrm{u} = (y_\mathrm{d} - y_\mathrm{c})(1 + K_1r^2 + K_2r^4 + \cdots) + (P_2(r^2 + 2(y_\mathrm{d} - y_\mathrm{c})^2) + 2P_1(x_\mathrm{d} - x_\mathrm{c})(y_\mathrm{d} - y_\mathrm{c}))(1 + P_3r^2 + \cdots)$

where:

= distorted image point as projected on image plane using specified lens,

= undistorted image point as projected by an ideal pin-hole camera,

= distortion center (assumed to be the principal point),

= radial distortion coefficient,

= tangential distortion coefficient,

=, and

= an infinite series.

Barrel distortion typically will have a positive term for whereas 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 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.

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