Re: Digicam

[Kennedy McEwen <rkm@xxxxxxxxxxxxxxxxxxxx>]

In article <FAEBJHPJNNGCAGDNGLNPMEIOEIAA.gsellani@accesscom.com>, gary 
<gsellani@accesscom.com> writes
>I have to disagree with you on the first count (sampling). Think about this
>a bit more before you reply. Input aliasing should not be part of the
>discussion. Period. I am focusing (heh heh) on the sample "width", which
>leads to the pixel width.
>
I didn't mention input aliasing in my response because I agree it is not 
relevant to the analogy you made, however the finite pixel size (or 
width as you now term it) certainly *IS* equivalent to the pre-filter in 
the analogue domain of the audio analogy - as is the MTF of the optical 
system and, in the case of scanners, the MTF of the film and the 
original camera optics.

This is easiest to visualise when you realise that the pixels perform 
more than just the sampling function.  The output of an array of 
identical pixels is physically and mathematically exactly the same as 
the output produced by first scanning (convolving) the image with a 
single pixel of the same shape and size (point spread function) as those 
in the array and then multiplying the resulting analogue signal map with 
an array of delta functions (ideal sampling).

In spatial frequency terms this is the same thing as multiplying the 
spatial spectrum of the image with the spatial frequency response of the 
pixel (ie. its MTF, the fourier transform of the point spread function 
of the pixel) and then convolving the result with an array of delta 
functions at a pitch equal to the spatial sampling frequency.

In other words the array of identical finite extent pixels is exactly 
the same as scanning a single pixel and an array of infinitely fine 
samples.  Visualised in this way it is *obvious* that the pixel MTF is 
prefilter of an ideal 2 dimensional sampling function.

This is EXACTLY the same as the one dimensional audio case, where the 
signal is first filtered and the sample is (to all intents and purposes) 
instant.  The audio filter applied to the audio spectrum is 
mathematically EXACTLY the same as a time convolution of the audio 
waveform with a function of a certain shape and time extent - just the 
FT of the frequency response of the filter.

Consequently, the output of each pixel in the array CAN be considered to 
be an infinitely fine sample - it is the output produced by placing the 
pixel in EXACTLY that spatial position.  A nanometer left or right 
produces a different output, just the same as introducing sampling lead 
or lag on the audio waveform would.

Before anyone asks the obvious question of where, within the extended 
pixel area, these infinitely fine samples occur - it is irrelevant. 
Since the pre-filter of the imaging system is intimately tied to the 
sampling process, changing the exact theoretical point of sampling is 
balanced by an equal and opposing change in the phase of the filter. 
This is just the same as changing the sampling time on an audio 
waveform, while simultaneously changing the phase of the pre-filter to 
perfectly compensate.  Thus an array of extended pixels with the actual 
sample points being considered in the top left corner of each pixel is 
mathematically exactly equivalent to an array of identical pixels with 
the sampling points in their centres - the phase change in the delta 
functions is matched by an equal and opposite change in the phase of the 
spatial filter.

Incidentally, should anyone question the value of this apparently 
academic exercise in imaging analysis, it is by separating the 
visualisation of the sampling and filtering function of the detector 
array that leads to the conclusion, so aptly exploited by Fuji (and 
others in different fields), that a pixel shape other than the 
conventional rectangle yields better image resolution, and that this can 
be sampled more accurately by matching the sampling density (pixel 
positions) to the information content of the resulting filtered image 
spectrum.  That is why a SuperCCD camera from Fuji actually needs much 
more than the same number of pixels on a conventional display to 
reproduce all of the information that the camera resolves.

> Now it is true that imaging does not have a
>pre-filter, but so what. Dogs have tails, but this is not relevant either.

Not at all, imaging DOES have a prefilter - usually at least two (and 
several more if the image reconstruction from the sampled data is 
considered):

1. The optical system (which I have ignored in the discussion above, but 
which IS band limited, although not in a brick wall form), and

2. The pixel size and shape in terms of its electro-optic response (few 
devices have a flat response across the active pixel area, although they 
can be approximated as such), both of which alter the MTF of the sensor 
itself.

The most significant differences between the audio analogy and the 
imaging case are:

1.  In audio the analogue prefilter is single sided since the analogue 
circuits can only respond to their input after it has been received, 
whilst in the imaging case all of the samples occur simultaneously and 
the filter is nominally symmetric about the spatial frequency axis in 
all orientations.  Of course, if the optical system includes certain 
distortions, such as coma, then the implied filter becomes slightly 
asymmetric also.  In modern high performance audio systems the 
asymmetric filter is replaced by oversampling and application of a 
symmetric FIR digital filter, making it even more similar to the imaging 
case in this respect.

2. Being completely separate functions in the audio analogy, the filter 
can be tailored to meet the constraints of the sampling system, with a 
brick wall filter.  In imaging this is not so easy to implement, 
although it is possible, and the filter is normally determined by the 
constraints of the optical MTF and the shape and size of the pixel 
itself.

-- 
Kennedy
Yes, Socrates himself is particularly missed;
A lovely little thinker, but a bugger when he's pissed.
Python Philosophers
-
Turn off HTML mail features. Keep quoted material short. Use accurate
subject lines. http://www.leben.com/lists for list instructions.