Scene -> Optical Image; under relativistic conditions ?

Boris Borcic

In [1] Berthold K.P. Horn argues against linear methods for stereoscopic image
registration in machine vision. Horn explains that

- "Methods based on projective geometry yield a transformation matrix T that in
general does not correspond to a physical imaging situation -- [the latter
amounting to the product of] a rotation, a translation and a perspective
projection".

- "The [general] transformations of projective geometry correspond [instead] to
taking a perspective image of a perspective image [of an scene, what doesn't
occur in an eye or a camera]".

Now of course the physical imaging situations Horn talks about are
non-relativistic. My question is : does special relativity by any chance repair
the situation, in the sense that -relativistic- imaging situations would in
general relate the geometry of a scene to that of its image as would (while
neglecting relativistic effects) taking a perspective image of a perspective
image of the same ?

The motivation of that question is that I believe presenting SR as an "extended
theory of perspective" would help people understand what it is about. And the
above fact - if true - would help justify such as name for it.

TIA, Boris Borcic
--
[1] B. K. P. Horn. Projective Geometry Considered Harmful.
http://www.google.com/search?q=Projective+Harmful , 1999

Igor Khavkine

Boris Borcic wrote:
> In [1] Berthold K.P. Horn argues against linear methods for
> stereoscopic image registration in machine vision. Horn explains that
>
> - "Methods based on projective geometry yield a transformation matrix
> T that in general does not correspond to a physical imaging situation
> -- [the latter amounting to the product of] a rotation, a translation
> and a perspective projection".
>
> - "The [general] transformations of projective geometry correspond
> [instead] to taking a perspective image of a perspective image [of an
> scene, what doesn't occur in an eye or a camera]".
>
> Now of course the physical imaging situations Horn talks about are
> non-relativistic. My question is : does special relativity by any
> chance repair the situation, in the sense that -relativistic- imaging
> situations would in general relate the geometry of a scene to that of
> its image as would (while neglecting relativistic effects) taking a
> perspective image of a perspective image of the same ?

Your question seems to presuppose a certain familiarity with
terminology used in the field of "steroscopic image registration". I
find it hard to decipher what precisely terms like "physical imaging
situation" refer to. I can presume that images are captured by
registring light rays emanating from an object and passing through a
lense or a pinhole on, say, a photographic plate or a film. This is a
non-linear mapping from the points of the object to the points on the
photographic plate, whose details depend on the geometry of the camera.
Based on this simple consideration, it is doubtful that such a mapping
would suddenly become linear if special relativity is taken into
account. In fact, for small object and camera velocities, should should
be only negligible differences the relativistic and non-relativistic
versions of this mapping, the latter of which is already known to be
non-linear.

For more detailed arguments, you have to tell us more precisely what
you mean by "physical imaging situation" and "perspective" in your
question.

> The motivation of that question is that I believe presenting SR as an
> "extended theory of perspective" would help people understand what it
> is about. And the above fact - if true - would help justify such as
> name for it.

Even if that were true (which appears doubtful), the number of people
who are well familiar with the "regular" theory of perspective before
attepmting to study special relativity is not likely to be very large.
However, that's just a guess. You may be better informed on this point.

Igor

tessel@um.bot

On Fri, 29 Sep 2006, Boris Borcic wrote:

> My question is : does special relativity by any chance repair the
> situation, in the sense that -relativistic- imaging situations would in
> general relate the geometry of a scene to that of its image as would
> (while neglecting relativistic effects) taking a perspective image of a
> perspective image of the same ? about. And the above fact - if true -
> would help justify such as name for it.

I don't think I understand your robot vision terminology, but you might
or might not find it useful to know that the effect of Lorentz
transformations of the celestial sphere for an arbitrarily moving
observer in Minkowski vacuum is given by the action of the Moebius group
on the Riemann sphere:

http://math.ucr.edu/home/baez/physic...R/penrose.html

http://math.ucr.edu/home/baez/physic...spaceship.html

So, there's certainly a link with -conformal- geometry on the sphere.
On the surface, you are talking about something different (relavistic
view camera?), but the above should help you get started.

"T. Essel"

Boris Borcic

Igor Khavkine wrote:
> Boris Borcic wrote:
>> In [1] Berthold K.P. Horn argues against linear methods for
>> stereoscopic image registration in machine vision. Horn explains that
>>
>> - "Methods based on projective geometry yield a transformation matrix
>> T that in general does not correspond to a physical imaging situation
>> -- [the latter amounting to the product of] a rotation, a translation
>> and a perspective projection".
>>
>> - "The [general] transformations of projective geometry correspond
>> [instead] to taking a perspective image of a perspective image [of an
>> scene, what doesn't occur in an eye or a camera]".
>>
>> Now of course the physical imaging situations Horn talks about are
>> non-relativistic. My question is : does special relativity by any
>> chance repair the situation, in the sense that -relativistic- imaging
>> situations would in general relate the geometry of a scene to that of
>> its image as would (while neglecting relativistic effects) taking a
>> perspective image of a perspective image of the same ?
>
> Your question seems to presuppose a certain familiarity with
> terminology used in the field of "steroscopic image registration". I
> find it hard to decipher what precisely terms like "physical imaging
> situation" refer to. I can presume that images are captured by
> registring light rays emanating from an object and passing through a
> lense or a pinhole on, say, a photographic plate or a film.

I am no expert on machine vision, just informing myself; the first sentence on
the topic of Horn's article I offered as background, but as it wasn't my concern
maybe I went too fast. Allow me to be more precise by citing a paragraph of
Horn's introduction :

<quote>
We revisit this topic here in the context of a simpler problem, that of exterior
orientation with respect to a planar object. We examine the difference between
the mapping from the object plane to the image plane defined by true perspective
projection and that defined by projective geometry. We show that virtually none
of the transformations allowed by projective geometry correspond to real camera
image-taking situations. We then compare the algorithms and study the
sensitivity to noise using Monte Carlo methods and show that the error
sensitivity of projective geometry based methods is much higher.
<unquote>

Note that he does *not* say that transforms that *do* correspond to "real camera
image-taking situation" *fail* to be allowed by projective geometry.

> This is a
> non-linear mapping from the points of the object to the points on the
> photographic plate,

This is not what Horn says. What he says/shows is that under an adequate choice
of representation/coordinate system :

<quote>
We can always find a matrix T corresponding to a real perspective projection (if
we assume that the principal point is known), using the equations above, but in
general it is not possible to go in the other direction, that is, to find a
perspective projection that corresponds to an arbitrary matrix T . That is,
almost all homogeneous transformations T have the property that they do not
allow a physical interpretation in terms of rigid body motion and perspective
projection.
</unquote>

Should I reproduce the derivations ?

> whose details depend on the geometry of the camera. Based on this simple
> consideration, it is doubtful that such a mapping would suddenly become linear
> if special relativity is taken into account.

Again, what Horn says isn't that the transform fails to be linear, it is that
the whole linear group of transforms under consideration is too large. What I am
asking is whether it stays too large when special relativity is admitted to play
a role.

And of course this question involves unknowns, since stepping up to SR implies
relative motions and getting more precise about shutters and how the image forms
in space-time. So I guess my question should be qualified with "assuming the
most natural hypotheses about shutters".

>> The motivation of that question is that I believe presenting SR as an
>> "extended theory of perspective" would help people understand what it
>> is about. And the above fact - if true - would help justify such as
>> name for it.
>
> Even if that were true (which appears doubtful), the number of people
> who are well familiar with the "regular" theory of perspective before
> attepmting to study special relativity is not likely to be very large.

My point is that "all is perspective" would be much superior to "all is
relative" to speak of what SR says. I can still remember the "aha !" when I
realised at age 4 that distance from any object conditioned its optical
apparence, and I find it hard to imagine anybody endowed with vision who'd fail
to possess that degree of intimate familiarity with the rules of perspective,
what's largely sufficient for the purpose.

Regards,

Boris Borcic
--
"La toute-puissance appartient-elle au pair d'yeux ?"