QUOTE (djellison @ Jul 7 2006, 01:31 PM)
The reason that the simple "multiply 0.273 by 1024" equation works for Pancam is simply because there is *no* geometric distortion in the optical system. We tried hard to measure it so we could characterize and correct for it, if needed, but as we wrote in our 2003 JGR paper we couldn't detect *any* distortion down to a residual of 0.01% or so across the field--even in the corners.
Doug, thanks for directing me to the more appropriate thread.
I have to disagree with that figure for distortion of 0.01%. There is a fundamental
intrinsic distortion present in any
optical system, completely independent of the quality of the optics (indeed it exists for a pinhole camera!). It arises because the goal of imaging is always to represent a section of an imaginary sphere surrounding the camera on a flat image plane. In exactly the same way that you cannot represent a section of the earth without distortion on a flat map, any flat image contains fundamentally irreducible distortion (ie even image processing cannot eliminate it).
I suspect the distortion figure that Bell was quoting referred to the amount of pinhole or barrel distortion. These can
be eliminated. Thus if we image a grid of straight lines on a card normal to the line of sight, the lines will be straight on the image. But distortions remain, in particular radial and azimuthal angular image scales will have to differ.
Just as with maps of earth, the irreducible distortions increase with the size of the section of the sphere we're imaging, ie with the field of view. It's straightforward to calculate the size of the irreducible distortion.
Suppose we're imaging a circle, normal to the line of sight, whose radius subtends angle theta(radians) at the camera. Then it's angular circumference will be 2 pi sin(theta). Lets suppose the angular image scale is constant in the radial direction. Then, working from the image, if that image scale gives us theta for the radius of the circle, it must give us 2 pi theta for the circumference, in disagreement with the actual circumference of 2 pi sin(theta).
The relative error is specified by the ratio, R = 2 pi sin(theta) / 2 pi theta = sin(theta)/theta. As expected this ratio approaches unity, ie distortion approaches zero, as fov theta approaches zero. For pancam we have theta = 8 degrees, which gives R = 0.9968. Therefore the radial and azimuthal angular image scales must differ by (1 - R) = 0.32%. (Actually the value will be even larger in the corners.) This irreducible distortion is much larger than the quoted value 0.01%.
This ratio R is important when converting pixel separations to angular separations, since the scale factor used to do this (0.273mrad/pixel) must
vary by at least 0.32% across the frame and for different orientations. Eg, when comparing a line through the centre of the fov (radial) vs. a line across one edge (more or less azimuthal).
Of course in the end the effect is still small. 0.32% corresponds to an error of only 3 pixels for a 1024 pixel line! Indeed according to the Bell et al JGR (2003) paper, the fov's of the 4 pancams vary by a comparable amount! (It's not clear if this is just measurement uncertainty).