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From: perry@netcom.com (Perry West)
Subject: Re: Re : Resolution
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Date: Mon, 26 Jun 1995 20:05:24 GMT
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: 1. First of all, the sampling frequency of an image is limited by the pixels size in the captor. No matter your processing, you will never go further this limit. This means that details above this limit will not be resolved. The condition Fe > 2.Fc is n
ot respected. You rather have Fc = Fe/2, that is : the maximum spatial frequency you can resolve depends on your captor cut-off frequency.

I agree, sampling a signal limits the frequecy response of information in 
the image.  There is a potential loss of information.  The loss depends 
on the scene being digitized.  If a person's goal is to recapture the 
information lost by sampling, too bad.  I know you can use peaking 
filters to enhance the higher frequencied, but they add distortion to the 
image that may or may not be tolerable.

: 2. The captors do not really sample, they are blocker-samplers, they integrate the intensity on a surface. This means the signal spectrum undergoes a non-harmless transformation.

In general, I don't agree.  We have two sampling processes going on.  One 
is spatial sampling, the other is amplitude sampling.  Since most systems 
now use solid-state image sensors, I will relate my comments to them.  
The solid-state image sensors do integrate the incident light energy over 
their sensing areas.  However, for most of the sensors in use today, 
interline transfer CCD and frame transfer CCD, the fill factor (ratio of 
light sensing area to the area computed from the center-to-center spacing 
of pixels is a fraction.  Therefore, the sensor is usually providing some 
additional limitiations to what one expects from traditional sampling 
which implies averaging over the sampled interval.  Also, if you are 
working without a pixel clock between the camera and the frame grabber, a 
very common situation, there is an additional source of complication too 
deep to go into here.  Finally, the A/D convertor used in most frame 
grabbers is a flash A/D.  It does not average, but captures the signal at 
one point in time (space).  So, you see how careful we must be when we 
expect to have the best performance from our camera and frame grabber 
combination.  Generally, the performance is not as good as simple 
sampling theory leads us to believe.

: 3. Even if the Shannon condition is respected (i.e. you take picture of low-frequency objects), the reconstruction of the signal must be undertaken by an ideal filter, which needs an infinity of samples. This leads to aliasing and Gibbs phenomena.

: But :

: Provided your image has not so high frequencies,
: Provided your take into account the integration on your captor,
: Provided you use the best interpolation kernels, and prove the ringing in the image domain is limited to an epsilon strictly inferior to your quantization step,

: You could reconstruct any higher resolution image than your original.

: This seems like getting out information of nowhere, but as information theory has not been disproved, it works well !

This assumes the resolution (frequency response) of our image is well 
below the limitations imposed by our sampling.  Then, the information is 
all captured in the digitized image.  Zooming in does not create new 
information -- the fourier spectrum of the zoomed image will not differ 
from the fourier spectrum of the original when we account for the zoom 
factor.  In fact, because of quantization errors, the interpolated pixel 
values may well introduce minor frequency artifacts which were not 
present in the original scene or the original digitized image.

Resolution and its implications is a favorite topic of mine.  Thank you 
to the person who initiated this thread and to Nicolas for his 
thoughtfull response.

Regards,
Perry

