Camera Lens Distortion - Image Engineering (2024)

How to measure distortion

ISO 90391¹ is the standard that defines methods to measure a lens that is separated from a camera. Sometimes, however, you cannot remove the lens from a camera (such as in a mobile phone), and thus the time-consuming methods described in ISO 9039 will not be adequate. As a result, ISO 17850² was introduced to define methods to measure distortion using a camera lens combination.

TV distortion method (ISO 9039)

The oldest method that uses the camera-lens combination is the TV distortion method, created to analyze TV camera systems. This method requires a test chart with a regular grid of geometric structures such as those shown below.

The TV distortion method is essentially a system that shows a steadily increasing distortion from the image center to the corners. The bending of a straight line in the original image is quantified at the image's top edge (see figures below). The ratio of the bending over the height of the image multiplied by 100 is the percentage of picture height distortion. This method is compliant with the process described in EBU Tech 3249 3249³.

$$D=\frac{\Delta H}{H}\cdot 100$$

Line geometric distortion method (ISO 17850)

Specific lens systems (particularly small ones in mobile devices) will correct distortion at the maximum image height. These systems show the highest distortion level at lower distances from the optical center. The resulting type of distortion is often a mixture of barrel and pincushion distortion and is described with the term wave distortion.

For systems with a wave distortion, the measured picture height distortion can be zero or close to zero even if a strong distortion is visible in the image. We can break down the line geometric distortion method into three different methods to measure these systems' distortion as specified in ISO 17850.

1) Horizontal line distortion

This method is applicable when the vertical line Ai is located closer to the vertical line at the center of the image than to Bi, use formula 2: $$Dhi=\frac{\left(Bi-Ai\right)}{2V}\times100%$$

Otherwise, use formula 3: $$Dhi=\frac{\left(Ai-Bi\right)}{2V}\times100%%$$

Where:
i = a suffix representing each picture height;
Ai, Bi, and V shall be represented by the number of pixels from the output image.

2) Vertical line distortion

Use this method when the horizontal line αi is located closer to the horizontal line through the center of the image than to βi, use formula 4: $$Dvi=\frac{\left(\beta i-\alpha i\right)}{2V}\times100%$$

Otherwise, use formula 5: $$Dvi=\frac{\left(\alpha i-\beta i\right)}{2V}\times100%$$

Where:
i is a suffix representing each picture width;
αi, βi, and V shall be represented by the number of pixels in the output image.

3) Total line distortion

The total line distortion is calculated as: $$\left|D_{\mathrm{line}}i\right|=\sqrt{\mathrm{Dh}\mathrm{i}^2+\mathrm{Dv}\ \mathrm{i}^2}%$$

The sign for Dlinei represents the higher absolute value of either the horizontal or the vertical line distortion.

Local geometric distortion method (ISO 17850)

The local geometric distortion method is utilized when a single number for the distortion is not sufficient. In other words, we need a function for the distortion to correct the distortion in image processing. Keep in mind that when you need to address a specific image height, the local geometric distortion is more reliable.

When measuring the local geometric distortion, we assume that the distortion close to the optical center is zero. Then you can calculate a regular grid based on the geometric positions of the nine structures (3x3) in the center of the image. This grid is expanded to the whole image and defines the nominal positions for each of the structures.

The distortion is then measured using the following formula:

$$D=\frac{\Delta H}{H}\cdot 100=\frac{H^{*}-H}{H}\cdot 100$$

Where:

H* is the distance of the dot from the image center

H is the nominal distance of the dot from the image center based on an expanded regular grid.

If deriving a single number from the local geometric distortion, the maximum distortion measured for any of the image's geometric structures is the one that is reported.

ISO 17850 describes this method in detail.