Re: Nonlinear Mapping - Object to Image Space



Jim Klein wrote:
> mpate@xxxxxxxxxxx wrote:
>
> >I am familiar with collinear mapping from object to image space where
> >we can map point to point, lines to lines, and planes to planes through
> >an optical system.
> >
> >I would like to find the nonlinear? mapping where we can map from say
> >an elliptical object to the corresponding image space shape, or a
> >cylinder object surface shape to the corresponding image space shape.
> >
> >Have googled around a bit looking for answers but didnt find much.
> >What is this field of math/mapping called?
> >Any online or book references would be appreciated.
>
> Hi,
>
> Don't be alarmed. I will try to produce a helpful answer. Really, I
> have thought about this problem a lot.
>
> Given that you have an optical design program or optical raytrace
> program (or you wrote your own) that can specify an image point in
> terms of the to be determined starting object location, this problem
> is in theoretically easy to solve.
>
> The software takes the final image location (like on a focal plane)
> and iteratively uses the targeted relative aperture stop surface
> location and iteratively finds the object space position that the ray
> must start from so that the ray hits the location on the image plane
> desired. Sounds difficult but it is no more complicated than HS
> Algebra. It is a double nested loop.
>
> As far as I know, no one has commercially solved the problem ( except
> paraxially which is easy and non-iterative) except yours truely. FRED
> solves it but FRED is not an optical design program.
>
> I'd love to send you the software to do this, but if I did, I would
> get into a lot of lawyer trouble and loose my pension from the last
> Aerospace company I worked for.
>
> Think about it. It needs to be a double iteration so that the ray hits
> the correct position at the image plane and at the aperture stop
> surface.
>
> The cheap solution is to reverse the optical system and trace from the
> image plane throght the desired stop position and out to the object
> (not the image) location. You can do that with any commercial program.
>
> Jim Klein
>
> If this is confusing, send questions to jameseklein@xxxxxxxxxxxxx

If I understand you correctly the method you describe is only part way
to the correct answer. What is required is to optimize for the best
image quality for each object point. This gives correct mapping that
includes the effect of optical aberrations. This is the method I used
back in the early 1980s to map distortion for holographic HUDs.
(Although in this case only in 1 plane, not through focus.)

What you describe is similar to determining the holographic fringe
geometry during construction of a hologram. In this case one
iteratively traces from the object and image beam sources to a specific
point on the hologram.

.

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