Metric Zoom Lens.pdf

Metric Zoom Lens Model

Background

Military test ranges use cameras extensively to determine time-space-position information (TSPI) of the weapons and targets involved in test events. Using suitably calibrated video recorded from multiple cameras, one can reconstruct TSPI with software such as TrackEye (Image Systems AB, Linköping, Sweden). Up to now, the convention has been to use lenses at a single focal length. However, in recent years, there has been a growing interest in employing “metric zoom” lenses. A recent example is the US Army funded Advanced Range and Tracking Imaging System (ARTIS). For TrackEye to properly utilize video from zoom lenses will require the lenses to be characterized far more extensively than fixed-focal length ones.

Scope

Certain real-world phenomena are not repeatable and therefore make the creation of a complete zoom lens model problematic. Thermal effects are one example. Rarely, especially in an outdoor environment, is the temperature of a lens completely uniform throughout. This is even though analytical thermal models of a lens design almost always assume this. Therefore, even though temperature may have a significant effect, there may be no benefit in including it impossible to merely provide a temperature to the model. We have seen measurements of boresight errors vs. temperature dependency in excess of 10 pixels, so this is a problem that can’t be ignored. Neither mechanical backlash nor “lens clunk” (which happens when a lens is inverted) are repeatable, either. Since no mount is infinitely stiff, azimuth and elevation and reported in the metadata are subject to errors. With most lenses, changing the focus will change the effective focal length.

Nevertheless, a metric zoom lens model can still be useful, provided the user is aware of the model’s limitations. Benefits could be realized by performing certain lens calibrations immediately before or after a test event. For example, on tests where the lens is pointed by a tracking mount, distortion and boresight calibrations could be conducted in the field by sweeps on a calibration target or a star. Fixed cameras are somewhat different but the same principles would apply. It is assumed that lens focal length would be somehow provided in the video metadata.

The Model

The lens model consists of three parts:

1) The nominal lens design (focal length and detector pixel pitch)

2) Boresight error vs. focal length

3) Optical distortion vs. focal length

In general, focal length is treated as an independent variable.

The nominal lens design is presumably what is implemented in TrackEye currently. There is assumed to be no image roll. In other words, the pixels are aligned perfectly with the camera’s azimuth and elevation axes. If the azimuth and elevation of a point in space are α and β, and the corresponding image on the detector array has pixel coordinates (x, y), the center of the array having coordinates (0, 0), then 𝑥 = 𝑓 ∙ 𝑡𝑎𝑛(𝛼) 𝑝 and 𝑦 = 𝑓 ∙ 𝑡𝑎𝑛(𝛽) 𝑝 with f being the focal length and p being the pixel pitch. It is assumed that the detector has square pixels.

Boresight Error Vs. Focal Length

For convenience, the boresight error is described by a polynomial function of focal length:

𝑥 = 𝑥 + 𝑎 (𝑓 − 𝑓) and 𝑦 = 𝑦 + 𝑏 (𝑓 − 𝑓)

Where fmax is the maximum focal length, a and b are the polynomial coefficients, and x’ and y’ are the pixel coordinates having been subjected to boresight error. The value for n depends on the nature of the measured boresight error vs. focal length curves. The polynomial is expanded about (fmax-f) because boresight would typically be measured relative to the maximum zoom, where it is the most difficult to center a boresight target in the field of view.

Optical Distortion Vs. Focal Length

It is not uncommon for zoom lenses to have, for example, pincushion distortion at one end of the zoom range and barrel distortion at the other. If ru is the distance from the center of the focal plane of the point image with undistorted optics, and rd is the distance, with distortion, then 𝑟 = 𝑎 𝑓 𝑟

This does not include image roll and non-radial symmetric distortion. As before, the number of terms, m and n, depend on the nature of the distortion and the zooming.

A Final Note

To be useful, this model needs to define the reverse process. In other words, it should remove distortion and misalignments from an imperfect image. This may be addressed in a future version of this document.