Greetings,
my question concerns A/D conversion and sinc interpolation. I am
interested
in whether the standard(?) mathematical model for "sampling" applies
to a CCD chip camera, or, if not, whether it can be modified to describe
it.
Sorry for the verbosity, but I think a explanation is needed to make my
point clear.
Given a continous 2d function (e.g, intensity at the CCD surface), the
usual way of preparing this function for digital storage I believe to be
(see e.g. Foley, Van Damme):
1. Band-limit the function by lowpass-filtering according to the
sampling
frequency to be used later on (to filter out frequencies above the
Nyquist
frequency and avoid aliasing),
2. "sample" the lowpass-filtered, continous function on a square grid.
The
samples can then (after digitization) be stored as a finite number of
digital values. "Sampling" means evaluating the continous function
ideally
at a single point; or failing that, integrating the function over a
region
centered about the grid point, where the integral intervall length is
small
compared with the grid size.
If "sampling" is implemented in this manner, the sampling process can
be described (or approximated) by multiplying the continous
lowpass-filtered function with a comb-function (reducing the function
support to a set of discrete points), or, in Fourier space,
convoluting
with the Fourier transform of the comb function (another comb
function).
This replicates the spectrum of the lowpass-filtered function on each
point
of a square grid in Fourier space. Due to the band-limiting in 1.,
the
spectrum of the lowpass-filtered function has finite support and
convolution does not "mix" spectral components.
To reconstruct the lowpass-filtered function, the copies of the spectrum
created by "sampling" (in 2.) have to be removed. This can be done by
multiplying the spectrum of the sampled function with a box function
centered
about (0,0) which passes the "original" spectrum unattenuated and throws
out all the "copies" created by sampling. This corresponds to convolving
the
sampled function with the Fourier transform of the box function, i.e., a
sinc function.
If I am only interested in certain points of the reconstructed function
(i.e.,
I am "resampling" the function), I can evaluate a given point by summing
over
the grid points in its neighbourhood (with known values), giving each
grid
point a weight sinc(kd), where d is the distance of the grid point to
the
point to be evaluated. This performs the above-mentionned sinc
convolution
only for the neighbourhood considered and only at the point to be
evaluated.
Since the sinc function has no finite support, using a finite
neighbourhood
will introduce an error. This problem can be addressed by multiplying
the sinc
with appropriate, non-discontinuous functions with finite support.
Now for the problem:
My "sampling device" is a CCD chip camera. Photons incident to the chip
surface create (several) electron-hole pairs in the semiconductor. These
carriers are separated and the electrons generated in a pixel are
counted.
In terms of the continuous function model, the function (intensity) is
integrated locally over square regions (the pixels), then each point in
the
region is assigned the value of the integration. So: the CCD chip does
not
"sample" the intensity at each pixel center, but integrates it over the
whole
pixel area; the integral intervall length is NOT small compared with the
grid
size (they are equal).
My question: can this CCD-sampling be modelled mathematically? It's not
a linear
transform and not invariant under translation. Does sinc interpolation
(which inverts
the "sampling" described in 2., not the CCD-sampling) make any sense?
Even as
an approximation?
Thanks for reading this. I appreciate all comments, Dennis