What about distortion correction?

Since Telecentric Lenses are a real world object, they show some residual distortion which can affect measurement accuracy. Distortion is calculated as the percent difference between the real and expected image height and can be approximated by a second order polynomial.

If we define the radial distances from the image centre as follows

Ra = actual radius

Re = expected radius

the distortion is computed as a function of Ra:

dist (Ra) = (Ra - Re)/Ra = c*Raˆ2 + b*Ra + a

where a, b and c are constant values that define the distortion curve behavior; note that "a" is usually zero as the distortion is usually zero at the image centre. In some cases, a third order polynomial could be required to get a perfect fit of the curve.

In addition to radial distortion, also trapezoidal distortion must be taken into account. This effect can be thought of as the perspective error due to the misalignment between optical and mechanical components, whose consequence is to transform parallel lines in object space into convergent (or divergent) lines in image space. Such effect, also known as "keystone" or "thin prism", can be easily fixed by means of pretty common algorithms which compute the point where convergent bundles of lines cross each other.

An interesting aspect is that radial and trapezoidal distortion are two completely different physical phenomena, hence they can be mathematically corrected by means of two independent space transform functions which can also be applied subsequently.

An alternative (or additional) approach is to correct both distortions locally and at once: the image of a grid pattern is used to define the distortion error amount and its orientation zone by zone. The final result is a vector field where each vector associated to a specific image zone defines what correction has to be applied to the x,y coordinate measurements within the image range.