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From: turk@apple.UUCP (Ken "Turk" Turkowski)
Newsgroups: comp.graphics
Subject: Re: effective resolution and anti-aliasing
Message-ID: <1014@apple.UUCP>
Date: Thu, 11-Jun-87 22:00:55 EDT
Article-I.D.: apple.1014
Posted: Thu Jun 11 22:00:55 1987
Date-Received: Sat, 20-Jun-87 13:47:14 EDT
References: <1713@ames.UUCP>
Reply-To: turk@apple.UUCP (Ken "Turk" Turkowski)
Organization: Apple Computer Inc., Cupertino, USA
Lines: 44
Keywords: resolution, anti-aliasing

In article <1713@ames.UUCP> lamaster@pioneer.arc.nasa.gov (Hugh LaMaster) writes:
>I have heard that it is possible to represent lines of less than 1 pixel in
>width with greyscale or color images using data filtered from a higher
>resolution image or object.  I have never seen either the algorithms or the
>theoretical basis for an algorithm.  [An example would be a representation,
>using a greyscale picture, of a line less than one pixel or dot width in size].
>I have looked in standard graphics textbooks and find no mention of this [but
>I could have missed something or not understood what I was looking at].  Does
>anyone know of any references for this?  Are there any devices out there which
>make use of it [e.g. a monitor with a much higher resolution frame buffer than
>the tube and a filter to reduce the resolution before output to the tube]?

The theory is really quite simple.  Imagine an infinite resolution
image over which a low-pass filter is passed, with cutoff frequency
consistent with the Sampling Theorem.  Sample the sucker, and what do
you get?  A line of one pixel in width whose amplitude is attenuated
linearly as a function of the line width.  Note that this model also
accommodates sub-pixel positioning.

Unfortunately, such lines do not look quite as good as 1 pixel wide
lines because there are less quantized intensity values available to
interpolate with.  If the line gets real thin, you end up with a
Bresenham.

To do the computations in image space one could certainly use a
Bresenham line-drawer at a higher resolution and then decimate
(down-sample) the image, but I am am a proponent of object space
algorithms.  For references, theory, and algorithms, see:

Turkowski, Keneth
Anti-Aliasing through the use of Coordinate Transformations
ACM Transactions on Graphics, Vol. 1, No. 3, July 1982 pp. 215-234.

For thinner lines, you would just scale the line spread function.
Another reference is:

Gupta, S and Sproull, R
Filtering Edges for Gray-scale Displays
SIGGRAPH 1981 proceedings, vol. 15, no. 3, pp. 1-5.
-- 
Ken Turkowski @ Apple Computer, Inc., Cupertino, CA
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