Re: Help relating 2-D Fourier Transforms to optics, diffraction, photography for math class project

It was a dark and stormy night in sci.optics, that 10-02-06, but I
could still make out the words murrayatuptowngall...@xxxxxxxxx engraved

in the willow's bark:

The above is particularly disturbing if it's from a prose-bot...

Here are a couple of the physical questions I am trying to get
comfortable with...maybe it's more accurate to say 'lay understanding'
of what seems to be abstract mathematical explanations. You may tell
me to go away and read books before asking questions, but understanding
the significance of the countless articles, papers and book excerpts I
have read don't make it meaningful to me. It doesn't matter if I can do
the integrations by hand or with Maple...I want to have a less abstract
understanding.

Image formation, whether by lens or diffraction by an aperture of any
variety of shapes is apparently all explained or modeled by 2D Fourier
Transforms. For a reality check, I think this is the mathematical
explanation of it. It has been occurring for far longer than the math
was understood, so Fourier Transforms do not create the image, but
explain it...but a lens can be called a Fourier Transformer....hmmm....

I understand the relationship between time and frequency domains and
how a 1D Fourier Transform can provide the information contained in
each.

I don't understand what is being transformed in image formation. They
appear to be one and the same but for inversion. Actually, I see that
forming an image is a distraction...the Fourier transform of an
aperture function is what is relevant to the mathematics, yes? The
aperture function doesn't care what passes thru it...coherent light on
axis or an incoherent light distribution representing a landscape for
example. But a slide projected onto a screen (with a lens) is described
as one example of something on one plane transformed to another, I
guess from some rear reference point of the lens to a front refernce
point, be they back & front focal lengths, principal points or
whatever...I'm limited in my comprehension of how these terms relate
here.

I prefer to focus (sorry) on the incoherent case of image formation in
my pondering for a moment here...is the simple fact that an image is
produced simply as an inversion, and scaled in accordance with image
and object distances, related to profoundly simple summary of a Fourier
Transform and its inverse? One gives the other? (My head is still
swimming with information not tied to envisionable concepts). Yet, a
time-frequency transform appears to produce very different input-output
information....for example, a square wave is Fourier analyzed into an
infinite series of sine and cosine terms. Both the graphical and
mathematical inputs and outputs appear very different, at least very
different perspectives of the same thing.

I expected a physical input (like an eye's view of a lanscape scene) to
be rendered by a transform into something very different.

Am I intersecting with the other side of Fourier Optics, MTF,spatial
frequency, etc? Is this information the 'breakdown' of an image into
it's components, analogous to the sin/cos depiction of other physically
different waveforms?

I hope someone will humor me and help me understand...these are
certainly worthwhile pursuits. It's funny how in some Usenet groups
there is more dialogue about how one should search a group and
essentially know what one is talking about before posting a
question...but I am trying to formulate the questions, define the
mystery.

Thanks for any time invested in replying.

.

Relevant Pages

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