ping Kennedy

[false_dmitrii]

Kennedy, a very belated thank you for your extensive post on Fourier
Transformation. I had to put off reading it until I could be sure of
absorbing it all.

http://groups.google.com/groups?hl=en&lr=&[email protected]&rnum=14

I never mastered the wave portion of trigonometry to the point where
it was intuitive, so it was a bit tricky to follow your post without
graphical examples. And my physics & electrical background is very
nearly nonexistent. But I think I understood the core of what you
wrote, even if parts were beyond me (hmm...existentialist physics?).
I'm sure I misunderstood some of the details, but rather than
confuse things further and waste your time by asking lots of
questions, I'll use the post as a reference point for further research
if I find myself in more direct need of FT understanding.

I did have two basic points of confusion, at least partly because I
don't recognize many of your technical terms and usages. First, you
wrote that most real-world images are asymmetric, which seemed to
imply that their waveform would be complex. Much later, after you
described the two sampling spectra, you wrote that the original image
was real. Why am I seeing a contradiction where I'm sure none exists?

Second, you refer to multiplying the spatial frequency spectrum with
the FT of the "pixel's response". Can I interpret this "response" to
be the electrical output of the CCD, the light wavelength from the
image, or anything similar? Or does it have an entirely different
meaning here? My impression was that each pixel would have an
infinite, uniform FT based on whatever waveform, electrical or
otherwise, described the color "content" of the pixel. Am I on the
right track?

Sorry for the delay. Whether you wrote all that for the post or dug
it up from storage, it was appreciated and I didn't want you to think
otherwise.

false_dmitrii

(Hmm, Britannica describes FT as an "integral transform". I *thought*
it sounded like calculus! On the topic of old-as-new math, I caught
a special last year in which someone apparently pulled some integral
calculus notes from Archimedes off an old prayer book using UV light
and pushed the discipline's history back a few hundred years. Does
that hold water with you?)

 

[Kennedy McEwen]

Kennedy, a very belated thank you for your extensive post on Fourier
Transformation. I had to put off reading it until I could be sure of
absorbing it all.

Thanks - this is what makes it worthwhile.

I did have two basic points of confusion, at least partly because I
don't recognize many of your technical terms and usages. First, you
wrote that most real-world images are asymmetric, which seemed to
imply that their waveform would be complex.

Yes, it means that the fourier transform of real world images is complex
- unless you happen to have an image with mirror reflections right down
the vertical and horizontal axes, which is unusual to say the least.

Much later, after you
described the two sampling spectra, you wrote that the original image
was real.

The original image is real - the scanner doesn't output complex
variables for the colour of each pixel. However, the fourier transform
of the original image is complex, because the original image is
(usually) asymmetric.

Why am I seeing a contradiction where I'm sure none exists?

You are certainly seeing a contradiction, and it could be either in how
I described it or your interpretation of my description. Either way, no
contradiction exists in practice.

Second, you refer to multiplying the spatial frequency spectrum with
the FT of the "pixel's response". Can I interpret this "response" to
be the electrical output of the CCD, the light wavelength from the
image, or anything similar? Or does it have an entirely different
meaning here? My impression was that each pixel would have an
infinite, uniform FT based on whatever waveform, electrical or
otherwise, described the color "content" of the pixel. Am I on the
right track?

The response I referred to is the electrical output of the individual
pixel to a small point of light as they scan past each other. When the
point of light is nowhere near the pixel, the response is zero, when it
is right in the middle of the pixel, the response is (usually, but not
always) a maximum. As the point passes across the pixel's edges, the
response changes from maximum to minimum, and this may be a sharp
transition or a soft transition depending on the pixel design. The
technical term for this pixel response is the pixel "Point Spread
Function", or PSF.

If you wade through the detailed mathematics you can show that the FT of
the PSF is the pixel response to spatial frequency, and the modulus of
this FT is just the modulation transfer function (MTF) of the pixel.

To help understand this, think of a single pixel being scanned across
the image in infinitely fine steps. If you have no spatial frequency
then the image is simply a single shade of colour and the output of the
pixel is just a single uniform value at every position. As the spatial
frequency increases, the image contains a range of colours from peak
white to peak black and the output of the pixel ranges from maximum to
minimum as the position changes. In these cases, the pixel has a 100%
response.

However, as the spatial frequency in the image is increased, eventually
it will reach a point where the black to white peaks in the image are
exactly half the width as the pixel. In this case, half of the area of
the pixel is black and half of it is white, so the output of the pixel
can only be a value for mid grey. Furthermore, as the pixel scans the
image, the amount of colour that leaves from one edge of the pixel is
exactly replaced by the same colour at the other edge. So the pixel
output stays at this mid grey value irrespective of its position on the
image. Consequently, the pixel produces zero response to that
particular spatial frequency. Clearly, if the pixel has 100% response
to some spatial frequencies and 0% response to others then there must be
some curve which describes the transition between these two cases. That
curve, a plot of the pixel response versus spatial frequency, is the
response that I was referring to.

As it turns out, the response of the pixel is the fourier transform of
the pixels shape. So a square pixel has a response

(Hmm, Britannica describes FT as an "integral transform". I *thought*
it sounded like calculus!

If you look at the mathematics of the fourier transform, then it is
based on an infinite integral. However I often find that those who get
bogged down pursuing the detailed mathematics miss the overall concept
and its general usefulness. Unfortunately, it is usually taught as a
dry mathematical process (by those who understand the equation) rather
than an extremely useful tool (because they can't visualise this). This
lack of vision is one of the reasons that so many people get put off the
FT and never come to appreciate its strength and uses.

On the topic of old-as-new math, I caught
a special last year in which someone apparently pulled some integral
calculus notes from Archimedes off an old prayer book using UV light
and pushed the discipline's history back a few hundred years. Does
that hold water with you?)

Hard to believe, but nothing is impossible. It would certainly add a
new dimension to the old Newton-Leibnitz argument - did they both steal
the idea from an old Archimedes text rather than each other.

 

[false_dmitrii]

false_dmitrii <[email protected]> writes

The original image is real - the scanner doesn't output complex
variables for the colour of each pixel. However, the fourier transform
of the original image is complex, because the original image is
(usually) asymmetric.

Okay. That made the most sense when I was reading the earlier post,
but I couldn't confirm it.

The response I referred to is the electrical output of the individual
pixel to a small point of light as they scan past each other. When the
point of light is nowhere near the pixel, the response is zero, when it
is right in the middle of the pixel, the response is (usually, but not
always) a maximum. As the point passes across the pixel's edges, the
response changes from maximum to minimum, and this may be a sharp
transition or a soft transition depending on the pixel design. The
technical term for this pixel response is the pixel "Point Spread
Function", or PSF.

Got it. It made sense that this was the point at which the image
"content" would enter the process, but some of the terminology was
beyond me.

If you look at the mathematics of the fourier transform, then it is
based on an infinite integral. However I often find that those who get
bogged down pursuing the detailed mathematics miss the overall concept
and its general usefulness. Unfortunately, it is usually taught as a
dry mathematical process (by those who understand the equation) rather
than an extremely useful tool (because they can't visualise this). This
lack of vision is one of the reasons that so many people get put off the
FT and never come to appreciate its strength and uses.

Isn't this true for all mathematics? I never fully took to rigorous
theory and usually tried for an intuitive understanding to
compensate...took a while to "get" limits, but after that basic
calculus isn't too confusing. After a point, though, you have to pull
out a pencil and start doing the hard arithmetic to keep up with it
all.

Hard to believe, but nothing is impossible. It would certainly add a
new dimension to the old Newton-Leibnitz argument - did they both steal
the idea from an old Archimedes text rather than each other.

Depends...did they hang out at Palestine monestaries and pore over the
manuscripts with UV light? It looks like I remembered the story
correctly...if you type "Archimedes palimpsest" into Google a wealth
of trustworthy material comes up. You of all people probably knew
about the document before I'd ever heard of it; the
rectangles-under-a-curve notes and other details were more recent
discoveries, courtesy of digital imaging.

Always interesting to consider whether such rediscovered discoveries
are just historical curiosities that had no impact on their field
after vanishing, or whether instead their lines of thought persisted
quietly within the discipline and subtly influenced future
developments despite the loss of the "breakthrough" treatise. Makes
one wonder what rediscoveries await people in the distant future.

Regards,
false_dmitrii

 

[Kennedy McEwen]

Isn't this true for all mathematics?

Quite possibly. I certainly consider myself to have been very fortunate
in the teachers I had throughout my academic development. My maths
schoolteacher, "Big Joe" as we knew him, not only filled the blackboard
with formulae in record time, but explained everything as he went and
how it related to real life. He used all of his immense skills in model
making and photography to endorse and underpin everything he taught.

Depends...did they hang out at Palestine monestaries and pore over the
manuscripts with UV light?

Remember Newton's treatise entitled "Optiks", where he explained the
nature of colour and light - he could easily have noticed things
fluorescing in ultraviolet and decided to keep it secret.

It looks like I remembered the story
correctly...if you type "Archimedes palimpsest" into Google a wealth
of trustworthy material comes up. You of all people probably knew
about the document before I'd ever heard of it; the
rectangles-under-a-curve notes and other details were more recent
discoveries, courtesy of digital imaging.

On the contrary, I hadn't heard of it at all - and thanks for bringing
it to my notice!!

Having said that, the technique of rectangles under/over curves was a
well known and common approximation prior to Newton and Leibnitz in any
case. A similar approach was used by Pythagoras to approximate pi -
drawing around a circle internal and external polygons with increasing
number of sides. So it doesn't surprise me that the general approach
was known to the Greeks.

What Newton and Leibnitz managed to do was work out the rules governing
what those summations converged to when the width of the rectangles
became zero and the summation infinite, and could thus determine the
area under any curve exactly and directly from its equation, without all
the laborious geometry. The evidence hints that Archimedes might have
managed that step but doesn't really prove it. A few special cases are
there, but the general principle to extend it to any curve and shape
doesn't appear to be - but who knows what other manuscripts contain.

Always interesting to consider whether such rediscovered discoveries
are just historical curiosities that had no impact on their field
after vanishing, or whether instead their lines of thought persisted
quietly within the discipline and subtly influenced future
developments despite the loss of the "breakthrough" treatise.

I think this is actually quite likely in the case you have raised.
Newton wasn't the squeaky clean English gentleman that myths and legend
tech us - he was arrogant, conceited, devious and certainly not above
stealing anyone else's ideas - or rediscovering old ones to claim as his
own. In addition, he spent as much of his life, perhaps more, pursuing
the promises of alchemy to turn lead into gold as he did pursuing
physics. In his alchemaic pursuits he is known to have spent a small
fortune procuring ancient and elicit texts and could, in the process,
quite easily have stumbled on interesting treatise in the process. He
clearly believed in an ancient knowledge that has been lost and
requiring "re-search". Being unique manuscripts and he the only person
in history since their author to have been able to understand them, who
could argue against any claim he made for originality?

Nevertheless, it continually amazes me how much the ancient Greeks
really knew and understood about the world around us. In my own
professional field, thermal imaging, I have developed optical modulators
and over-sampling systems based on designs first described by Archimedes
as well as ferroelectrics, first discovered by Theophrastus, one of
Archimedes' pupils! Nothing new under the sun really. ;-)

Makes
one wonder what rediscoveries await people in the distant future.

Very little from this generation, I fear. We now seem to value
imagination more than achievement. 30 years ago man walked on the moon
- now we make "documentaries" of how we imagine man explored the
planets. :-(