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Re: Pinhole camera





On Sat, 28 Jun 1997, Donald E. Simanek wrote:

Does the image formed by a single lens even deserve to be called a
one-to-one mapping? After all it loses 3-D depth information. A hologram
does depth, but has limitations of resolution, and other problems that
lens images do not.

**********************

I think a point concerning the mapping may have been missed. It
is a mapping from one space (Not a plane) into another space. A pinhole
because of its infinite depth of field maps all of the object space on to
the plane of its screen, but the screen can be in an infinite number of
positions. Thus any point in the object plane is mapped to an infinite
number of points in the image space and vice versa. (All assuming perfect
geometric optics.)

Thanks
Roger Haar


That depends on how loosely you define 'mapping'. If you define the image
point at the "best" resolved convergence point of light from the object
point, then the 3d object maps to a 3d image (an array of points in three
dimensional space). This image is not geometrically 1:1, because the axial
magnification (along the lens axis) isn't linear, and isn't the same as
the lateral magnification. However, pointwise, the relation between object
points and image points is 1:1, for each object point maps to one image
point. Placing a screen somewhere in this space will result in a 'picture'
on the screen which is mostly made up of circles of confusion. If these
are small enough, they still allow one to recognize the picture as a
representation of the object. Points on the object map to fuzzy
overlapping circles on the image.

In fact, so few of the object points are mapped to points that we can say
that the overwhelming number of objects points map to fuzzy circles.
(Mathematically the ratio of circles to points on the screen is infinite,
but who's counting?) In the case of the pinhole camera, *all* of the
object points map to small, fuzzy circles. So, are these pinhole images
all that different in character from those of the lens? If you take
aberrations into account, *none* of the mapped points on the screen are
really points, even for the lens.

The discussion gets irrelevant and academic very quickly, though
discussing these things helps to expose fine points of the physics
(sweating the details) even if one doesn't reach a definite semantic
conclusion.

And we haven't even gotten to the interesting mathematical aspects of
images. To qualify as an image, the array of image points much have
certain relations to each other. For every point, its neighboring points
must be the same in image and object. There must also be *continuity*. For
every imaged line, the points on the line must be in the same order in the
image as in the object, and this must hold for every line you can
construct. In short, the image points can't be scrambled. Geometric
distortions (stretching, compression, warping) can be present, however,
without compromising the continuity and neighborhood requirements noted.

[I'm sure the mathematical language can be tightened here, but it's been a
long time since I never took topology. :-) ]

This brings up the important difference between a hologram and a lens. In
the hologram, the emergent light maps to a very particular deterministic
scrambling of the object points (a Fourier transform). This process can be
repeated (second Fourier transform) to recover a 3d replica of the
original object, a reasonably faithful (except for resolution limitations
and noise) image of the original object. Personally, I think it's not good
to call the hologram an image, because of the nature of the mapping. But
the reconstructed hologram does qualify as an image of the original
object.

-- Donald

......................................................................
Dr. Donald E. Simanek Office: 717-893-2079
Prof. of Physics Internet: dsimanek@eagle.lhup.edu
Lock Haven University, Lock Haven, PA. 17745 CIS: 73147,2166
Home page: http://www.lhup.edu/~dsimanek FAX: 717-893-2047
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