# How to estimate radial distortion from lens characteristics?

Is there a way to estimate the radial distortion coefficients without calibration, given the lens characteristics, such as its diameter and its focal length?

Typically for camera calibration a pinhole camera model is used, and then radial distortion is added on top of that. Radial distortion is a 4th or 6th degree polynomial with 2 or 3 coefficients. During calibration, these coefficients are not derived from anything. They are initially assumed to be 0, and then estimated from data using numeric optimization.

What I am asking is whether or not there is an equation relating lens characteristics that you can read off the manufacturer's spec (focal length, aperture, field of view) to the radial distortion coefficients from the typical polynomial distortion model? In other words, is it possible to get some approximation of the distortion coefficients from the camera spec, before doing the calibration.

• To those who voted to close: how exactly is this too broad? Either there is an equation that answers my question, or there isn't. – Dima Jul 14 '15 at 17:02

## 2 Answers

There is a lot of variability because distortion can be compensated for optically and lens designs differ. Some lenses are marketed as rectilinear, most not. You would be making the distortion worse for some lenses by doing any correction advised by focal length alone. For focal lengths shorter than about 30 mm, the distortion seems dominantly barrel, according to the data of:

Neale, W., Hessel, D., and Terpstra, T., "Photogrammetric Measurement Error Associated with Lens Distortion," SAE Technical Paper 2011-01-0286, 2011, doi:10.4271/2011-01-0286.

They only tested zoom lengths though.

Find the equations (for instance in here). And solve them.

If this is good enough for you, you can make use of the usual quadratic formulae.

In parallel to this I've heard of the actual use of ray tracing as a Monte-Carlo-typed method to get the same information.

However, I guess that even with that, a few effects are not taken into account. But again, it depends on the precision you need.

Edit: AFAIK, diameter and focal length is not enough: the full shape is important. With the diameter and the focal length alone, you need to make extra assumptions.

For instance, if you are in the thin-lens approximation, the answer is easy: there is no distortion whatever its diameter or its focal length.

I guess I understand now, intrinsics such as the lens actual curvature, shape and composition?

It is perfectly possible: that's what's called geometrical optics.

What you want is ray tracing. If the shape is good enough, you can solve everything analytically and therefore derive an equation linking what you want.

However, in contrast with a real calibration, with real images, this technique does not take a few effects into account (like diffraction, interferences, non-linearities or the errors in the specifications compared to the actual lens).

The main equation is the refraction one (used in ray-tracing setups) and there is little more to say about it.

• thanks, but these particular equations do not help me. They treat the distortion separately from the intrinsics. What I need is some idea of what the radial coefficients may be as a function of the physical characteristics of the lens that you can measure or read of a spec sheet. – Dima Jul 14 '15 at 16:36
• What do you mean then? Which intrinsics exactly? – user13706 Jul 14 '15 at 17:29
• I updated my answer. – user13706 Jul 14 '15 at 17:55