# Distortion

#### Notice: As of Imatest 4.5, the Checkerboard Module performs distortion and SFR calculations, and is resistant to significant amounts of fisheye distortion. We recommend users to switch to this module for testing distortion and sharpness using checkerboard charts.

Imatest™ Distortion

• measures radial lens distortion, an aberration that causes straight lines to curve,
• calculates coefficients for removing it, and
• provides additional information on geometric distortions in digital images.

Distortion models assume circular symmetry about a central point, where undistorted and distorted radii ru and rd (distances from the center) are related by equations of the form ru = fr), where f is one of several functions.

In the simplest distortion (third order) model,

ru = rd + k1 rd3.

Because this third-order equation is not adequate for all lenses, Distortion also calculates the coefficients for the fifth-order model,

ru = rd + h1 rd3 + h2 rd5.

Distortion also calculates the coefficients for Picture Window Pro, which uses a tangent/arctangent distortion model,

ru = tan(10 p1 rd )/ (10 p1 ) ;      h1 > 0   (barrel distortion)
ru = tan-1(10 p1 rd )/ (10 p1 ) ;    h1 < 0   (pincushion distortion)

Distortion has two forms, barrel (k1 > 0) or pincushion (k1 < 0), as illustrated below.

 None Barrel Pincushion

Distortion tends to be most serious in extreme wide angle, telephoto, and zoom lenses. It is most objectionable in architectural photography and photogrammetry — photography used for measurement (metrology). It can be highly visible on tangential lines near the boundaries of the image, but it isn’t visible on radial lines. Distortion by Paul van Walree is excellent background reading.

To measure distortion, you’ll need a rectangular or (preferably) square grid pattern, which you can create using Test Charts. Print the chart, photograph it (taking care to avoid glare; matte surface recommended), and enter the image into the Imatest Distortion module.

An example of Distortion output is shown below for the Sigma 18-125 mm f/3.5-5.6 DC lens (designed for APS-C-sized sensors, such as the Canon EOS-30D, Digital Rebel, Nikon D100, D70, etc. The Sigma is an excellent lens — a bargain — except for its autofocus. Mine doesn’t autofocus as reliably as Canon lenses, but it works beautifully on manual. The autofocus problem is plainly visible when working with the distortion chart.

The Sigma has modest amounts of pincushion distortion at 125 mm and barrel distortion at 18mm, its widest angle setting. Results displayed below the image, in the column on the left, include,

• TV Distortion from the SMIA specification, §5.20. Referring to the image on the right,

SMIA TV Distortion = 100( A-B )/B ;
A = ( A1+A2 )/2

The box on the right is described in the SMIA spec as “nearly filling” the image. Since the test chart grid may not do this, Distortion uses a simulated box whose height is 98% that of the image. Note that the sign is opposite of k1.

Although any number in this list can be used as a measure of distortion, SMIA TV distortion may be the best choice because it’s the easiest to visualize.

• Coefficient k1 from the third-order equation, ru = rd + k1 rd3  where r is normalized to the center-corner distance. k1 = 0 for no distortion; k1 < 0 for pincushion distortion; k1 > 0 for barrel distortion.
• Coefficients h1 and h1 from the fifth-order equation, ru = rd + h1 rd3 + h2 rd5.
• The Lens Distortion correction coefficient and scale factor for Picture Window Pro. The sign is the same as k1.

Distortion also calculates decentering: the deviation of the center of distortion symmetry from the geometric center of the image. Decentering can result from poor manufacturing quality or mechanical shock (whoops!). Details are in the Distortion instructions.

The plot includes arrows that illustrate the change in radius when distortion is corrected. Distortion was too low on the above plot to make the arrows visible. They are illustrated in the plot below for a large amount of simulated barrel distortion.

Distortion can also be used with a single line if its curvature is visible. Long tangential lines near the edge of the image are preferred. The ISO 12233 chart contains two such lines, which are adequate but not ideal: they would be better if they were thinner and further from the image center. Here is an example, illustrating a modest amount of pincushion distortion.

In Imatest Master, Distortion  can also produce an intersection points figure, showing points where lines cross, and a radius correction figure, showing fine details of the distortion profile.

 Full instructions for using Distortion are in Distortion Instructions.