Nonlinear Mapping - Object to Image Space

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mpate@oscintl.com - 23 Jul 2005 01:30 GMT

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.

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Helpful person - 23 Jul 2005 18:01 GMT

All you need to know (for a well corrected imaging system) is the
system distortion. Then it is a simple manner to geometrically map
from one object point to another.

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Paul Hichinel - 25 Jul 2005 18:50 GMT

> 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.

There is only one purely optical solution:

1 to 1doubly telecentric (image space and object space) imaging.

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Steve Eckhardt - 26 Jul 2005 17:34 GMT

>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
[quoted text clipped - 7 lines]
>What is this field of math/mapping called?
>Any online or book references would be appreciated.

Hi, Michael

I'm having a little difficulty understanding the question, which is why my
answer is entirely different from the other two. It seems that you are asking
about the mapping of three dimensional objects, if I can presume that you mean
ellipsoid when you say "elliptical object".

If I do understand your question correctly, the collinear mapping will still
apply. In the case of the ellipsoid, just take the equation for the object
and apply the equations of the collinear transformation. You will end up with
a quartic equation, if I recollect properly. The reason you get a quartic
rather than a quadratic is that the appropriate coordinate system for image
space is no longer rectilinear. (It's still linear, but no longer are the
coordinates orthogonal.)

If your cylindrical object is on-axis (the axis of the cylinder is coincident
with the axis of symmetry of the optical system), then you can just map one
point from each end of the cylinder in the meridional plane of the optical
system to the image space, then generate the resulting truncated cone by
revolving the points about the axis.

When I apply such theory to the imaging of an arc by a reflector, I generally
find that the aberrations of the system are large enough to strongly affect
the result. Unfortunately, I have not had the leisure to develop a higher
order theory of transformation that would give more insight and predictability
to this calculation.

If you are interested in pursuing this, you might begin by learning more about
nonimaging optics. Welford and Winston are the gurus in this field.

HTH.

Signature

Best regards,
Steve Eckhardt
skeckhardt at mmm dot com

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mpate@oscintl.com - 27 Jul 2005 06:27 GMT

Steve:

> I'm having a little difficulty understanding the question, which is why my
> answer is entirely different from the other two. It seems that you are asking
> about the mapping of three dimensional objects, if I can presume that you mean
> ellipsoid when you say "elliptical object".

Sorry for the confusion, now I understand why I didnt get many replys
on target. I did mean the mapping of an ellipsoid 3D object (concave
or convex) to the corresponding 3D shape in image space.

> If I do understand your question correctly, the collinear mapping will still
> apply. In the case of the ellipsoid, just take the equation for the object
[quoted text clipped - 3 lines]
> space is no longer rectilinear. (It's still linear, but no longer are the
> coordinates orthogonal.)

In reading my Shack notes I dont think it is collinear mapping because:

"For there to be a mapping of any kind between the object and the image
space, there must be a one-to-one correspondence between the points in
the two spaces. One and only one image point corresponds to each
object point and vice versa. The mapping will be collinear if, for any
set of collinear object points, the corresponding set of image points
is also collinear. Thus every straight line in object space has a
straight line in image space correspoindgin to it. As a consequence of
the point and straight line correspondence, there is also a one-to-one
correspondence between planes in the two spaces. Although we have
three corespoindences (points, stright lines, and planes), satisfying
any two of them insures that the mapping is collinear."

I hope I am not taking the notes out of context as he was really
talking about typical object and image conjugates. When I look at the
last criteria with my 3D ellipsoid object and the corresponding 3D
image I dont satify the requirement for collinear mapping. I only can
map points to points. I dont have a line or a plane in the 3D object
or image.

It seems then that it must be some other type of mapping that governs
this situation. I have looked at spherical and hyperbolic geometry and
differential geometry as mapping possiblities but dont know much about
these fields. I was hoping someone else had solved this problem in 3D
mapping or nonlinear mapping.

> If you are interested in pursuing this, you might begin by learning more about
> nonimaging optics. Welford and Winston are the gurus in this field.

I have also looked in Elmers book, and the new Nonimaging OPtics book
by Winston, Minano, and Benitez with not much luck on this mapping
topic.

Any comments on the above are appreciated

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Phil Hobbs - 27 Jul 2005 15:05 GMT

(snip)

> "For there to be a mapping of any kind between the object and the image
> space, there must be a one-to-one correspondence between the points in
[quoted text clipped - 15 lines]
> or image.
> (snip)

> I have also looked in Elmers book, and the new Nonimaging OPtics book
> by Winston, Minano, and Benitez with not much luck on this mapping
> topic.

We had a discussion of this awhile back--search Google Groups on
"Scheimpflug". In the neighbourhood of an image, if the lateral
magnification is M_lat == di/do, the longitudinal magnification M_long
== d(di)/d(do) = M_lat**2. You can derive the Scheimpflug condition
from that. Depending on whether the optical system is telecentric or
not, M may or may not depend on distance.

There is no linear mapping over all object space, since at do = f, di
goes from +infinity to -infinity, and then continues negative as do ->
0. Linear mappings are all continuous, of course.

Cheers,

Phil Hobbs

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Steve Eckhardt - 27 Jul 2005 19:24 GMT

>I hope I am not taking the notes out of context as he was really
>talking about typical object and image conjugates. When I look at the
>last criteria with my 3D ellipsoid object and the corresponding 3D
>image I dont satify the requirement for collinear mapping. I only can
>map points to points. I dont have a line or a plane in the 3D object
>or image.

Phil Hobbs is correct when he points out the discontinuity that makes a
collinear transformation impossible. But if you can stay away from the
discontinuity, a collinear transformation works well.

I'm afraid you're misunderstanding Shack's notes. He is giving the necessary
conditions for a collinear transformation. Optical systems (when you ignore
aberrations) do map points to points and lines to lines. This means that a
collinear transformation will map object space to image space (again ignoring
the discontinuity Phil mentions.) And if the collinear transformation
for the spaces exists, then one can use it to map any point within those
spaces. Thus, you may map every point on the surface of the ellipsoid or
cylinder in object space to the corresponding point in image space, creating
an image.

If this still isn't clear, email me for a phone #. I'll set up my SPAM filter
to allow your mail.

Signature

Best regards,
Steve Eckhardt
skeckhardt at mmm dot com

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mpate@oscintl.com - 31 Jul 2005 18:06 GMT

> Phil Hobbs is correct when he points out the discontinuity that makes a
> collinear transformation impossible. But if you can stay away from the
[quoted text clipped - 5 lines]
> cylinder in object space to the corresponding point in image space, creating
> an image.

Phil & Steve

Thanks I now believe that the collinear mapping will work for this
case. I just need to perform the transform in Mathematica or another
program so see how the point on my ellipsoid or concave spherical
object plane will map to the image plane.

Michael

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Jim Klein - 28 Jul 2005 04:34 GMT

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@earthlink.net

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Helpful person - 28 Jul 2005 13:10 GMT

> >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
[quoted text clipped - 44 lines]
>
> If this is confusing, send questions to jameseklein@earthlink.net

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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Natural Science Forum / Physics / Optics / July 2005

Nonlinear Mapping - Object to Image Space