[filmscanners] Re: Film resolution - was: Re: 3 year wait

["Mike Kersenbrock" <michaelk@aracnet.com>]

Austin Franklin wrote:

> But, one thing that is VERY important is there is a difference between
> sampling sine waves and square waves.  These test patterns are square waves.
> Though at Nyquist, you guarantee detecting the frequency, you do not
> guarantee full amplitude.
>
> To sample square waves, and GUARANTEE getting at least ONE "sensor" that
> contains full amplitude, you MUST sample at 4x f (or 2x the line width, NOT
> line pair width)...and in the case of an image, f really doesn't matter, but
> the thickness of the line (which would be 2f).

When sampling a square wave at the nyquist frequency, the square wave can
be perfectly reconstructed from the resulting samples.  But you've
missed my very point.

The nyquist rate (which is a minimum, by the way) is twice the frequency
of the highest component.  If you have black and white "square wave" bars
at some alternating frequency f, the nyquist sampling rate is NOT f, NOT
2f, not even 4f or 8f.  It's actually twice infinity!  The highest frequency
component of a perfect square-wave is at infinity!  A simple fourier
transform will demonstrate this.

This is why the nyquist rate has been so badly maligned -- there's nothing
wrong with it, it just has been badly applied and it's no surprise that
it didn't seem to work.  It's like Rosannadanna in the old Saturday Night
live skits.  :-)

If the light/dark stripes are at frequency f and the light/dark is changing
in a sinusoidal fashion, then the nyquist rate is 2f.  Note also that when
reconstructing the image from the samples, the output needs to be bandwidth
limited to f -- and when done so, the fastest the intensity can change will
be a sinusoid of f (any faster change requires a higher frequency component
and because there isn't any, it can't).  The math really does work out.

Real scans are done at a sub-nyquist rate (like in your examples where
twice infinity is the nyquist rate).  Doing so also provides some error
sources that I'll not go into (I'm not a real expert at this) but I do
know that at least some higher end digital cameras will have a filter
in front of the CCD (a scanner of sorts) that restricts the frequency
of the image coming in to keep things under the nyquist rate -- but
doing so by lowing the frequency of the input image rather than upping
the CCD resolution.  In other words, effectively blurring the incoming
image so the black and white bars are smoothed to a variable gray sinusoid.
Sounds counter productive, but it actually comes out better.  There have
been some articles about this in electronic (design) mags some time ago
that I read.

Mike K.

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