doug@eris (Doug Merritt) (04/05/88)
In article <8310@agate.BERKELEY.EDU> doug@eris.berkeley.edu (Doug Merritt) writes: >The standard conceptual model for doing this with *any* bitmap images, >including but not limited to fonts, is to do a spatial lowpass filtering I forgot to mention that this also tends to *add* serifs to sans serif fonts, which has some interesting implications about why serif fonts look pleasing to the eye, considering that the visual system does something akin to an FFT during processing. (Actually a Gabor transform, according to Dr. Karl Pribram [NeuroPsychology chairman at Stanford], which is a finite rather than infinite equivalent of the Fourier transform with implications of a sort of Uncertainty Principle of resolvable details. Not to digress or anything...) The reason for the serif embellishment is that, to create a perfectly straight line requires infinite high spatial frequencies (to the resolution of the display, anyway). The fourier transform of a square wave (which is analogous to a straight line/rectangle) is composed of an infinite sequence of spatial sinusoids. This means that to draw a perfect square wave, or perfectly straight line/rectangle, you need all the high frequency components you can display. If you filter out any of the high frequency components (for instance in order to accomplish the smoothing I describe), then sharp square edges will tend to get more rounded, which in this application means that, at a sufficiently large magnification level, the fonts will be more curvaceous, with serifs. If you wanted to end up with a sans serif font with horizontal and vertical straight lines preserved, modify the lowpass filter a bit to preserve the purely vertical and purely horizontal high spatial frequencies, but filter out the ones with both a vertical and horizontal component. Do this by averaging only over diagonal neighbors, not including the horizontal and vertical neighbors. If you want your font to have straight diagonal lines along with horizontal and vertical lines, but still to have no curves, then the filter gets even more complex. In general you can draw an image composed of all of the types of features you are concerned with, and take an FFT of it. The result can be used directly as a filter to delete those features, or its complement can be used to preserve only those features. For further details see any text on one dimensional signal processing, or on Fourier optics for two dimensional signal processing, such as "Introduction to Fourier Optics" by Joseph Goodman (rigorous), or "Optical Information Processing" by Francis Yu (more accessible, and with photos, still mathematical), or "Array Signal Processing" by Justice/Owsley/Yen/Kak (more general, e.g. includes phased array radar and CT techniques). The fact that you have a choice like this as to the appearance of the magnified font is a consequence of the fact that there are several different ways to introduce new high resolution (high spatial frequency) detail where there was none to begin with. Interestingly enough, it is possible to model most different styles of fonts as appropriately filtered versions of vector fonts, keeping in mind that there are several *entirely different* ways of visually symbolizing the same letter (e.g. look at "A" versus "a") in the vector font. Doug Merritt doug@mica.berkeley.edu (ucbvax!mica!doug) or ucbvax!unisoft!certes!doug