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I had written about a distorted image of an object on the celestial sphere-
"Why must the scale distortion be the same in one direction (largely 
circumferential, about the centre of the image) as it is radial? Far from 
it."

And Frank responded- Well, no, not "far from it". That could easily be the 
case.

No. Frank is wrong. It's a matter of geometry. It's only possible for the 
radial scale factor and the azimuthal scale factor, at any part of the 
image, to remain equal to each other if there's no distortion of the image. 
That happens in a pinhole camera, and is approached with a "rectilinear" 
lens, optimised to reproduce straight lines on an object-plane as 
straight-lines on an image plane.

In any normal camera, there must be complete symmetry about the optic axis. 
A piece of polar graph paper, placed on a plane and centred on the optic 
axis, will create a plane image, as another polar graph, also centred on 
the optic axis. The radial lines will be reproduced exactly. Any distortion 
shows up in the spacing of the imaged circles. If they are equally spaced, 
there's no distortion. With any lens system, there will always be a 
deviation from that perfection. A plot of the "transfer function" of y (the 
radius of an image circle) against x (the radius of the object cirle that 
produces it) tells us all there is to know about the distortions of that 
system. If that plot is a straight line, there's no distortion. If it 
curves upward at increasing radius, this is pincushion distortion. If it 
droops, it's barrel distortion.

At any radius, just as a result of geometry, the radial scale and azimuthal 
scale are equal only if, in that transfer function y = f (x), dy/y = dx/x. 
Which implies that the transfer function is a straight line pointing toward 
the origin. That corresponds to the transfer function of a no-distortion 
system, in which y = k x.

There's a useful wikipedia article, on distortion (optics), at 
http://en.wikipedia.org/wiki/Distortion_(optics) , which shows the effects 
of different types of distortion.

What we have been considering is the imaging of a plane object on a flat 
plane, the sort of job a camera for document copying or aerial survey was 
spefically designed to do precisely, but a job that any other camera copes 
with pretty well.

Next, consider instead portraying an object, such as the Sun, which always 
occupies a known angle, on to a flat image plane. Even in the absence of 
any lens distortion (that is, using a pinhole or a perfect rectilinear 
lens), the transfer function of angle A (measured from the optic axis) to 
radius y will not be a straight line, but y = k Tan A. This in itself gives 
rise to gross distortion near the edges of the photo in question, when the 
angle A becomes 20º. If the camera lens added no distortion of its own 
(that is if it was a rectilinear design) then that tangent factor would 
cause radial dimensions , around those outer Sun images 1, to be expanded 
by a factor of 1.13 to 1 (compared with the centre) and azimuthal 
dimensions to be enhanced by a very different factor of 1.04 to 1. The 
image is then very far from being "orthomorphic"; in other words, it fails 
to preserve the Sun's circular disc, and turns it into an ellipse, pointing 
radally. That was the situation with optics that are nominally "rectlinear" 
with a plane object, and any deviation from that ideal will combine with 
the distortion we have considered, enhancing or reducing it depending on 
the sign.

Frank seems to be postulating that some sort of special lens has been used 
which somehow corrects for that lack of orthomorphism (and yet has failed 
to linearise the radial scale) when he states- "Trouble is, cameras and 
other optical systems are designed with various sorts of distortions built 
in which go around the theoretical argument you've proposed". Does he know 
if such a system exists, then, which might conceivably rectify that 
distortion, or is he simply imagining it? Does Frank think it likely that 
someone taking photos of Arctic sunsets is likely to be emplying a lens 
system which is intended to create distortion, and to deliberately depart 
from the rectilinear norm?

Frank called in aid some comments from another posting, which he ascribed 
to Mike Burkes but were actually from an earlier contribution by Peter 
Fogg. As that posting confused distortion with perspective, it had little 
relevance to the matter in hand.

=========================

This question takes on a bit of importance because postings appear on this 
list, quite often, from proponents of the use of cameras for making 
celestial measurements. The geometrical distortions discussed here, that 
arise from portraying a spherical surface on to a plane array, add serious 
complications to interpreting measurements, and deriving scale factors; 
complications which are often neglected. It would indeed be possible to 
calibrate the transfer function of a lens system and remap images on a 
corrected grid, though this would call for a lot of work if a zoom lens is 
being used.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

   

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