Laws@STRIPE.SRI.COM (Ken Laws) (06/15/87)
Date: 12 Jun 87 15:52:40 GMT From: mind!harnad@princeton.edu (Stevan Harnad) If discrete photons strike discrete photoreceptors, then discontintuity is transforming into discontinuity. Yet the question can still be asked: Is the transformation preserving physical properties such as intensity and spatial relations by transforming them to physical properties that are isomorphic to them (e.g., intensity to frequency, and spatial adjacency to spatial adjacency) as opposed to merely "standing for" them in some binary code? This makes me uncomfortable. Consider a "hash transformation" that maps a set of "intuitively meaningful" numeric symbols to a set of seemingly random binary codes. Suppose that the transformation can be computed by some [horrendous] information-preserving mapping of the reals to the reals. Now, the hash function satisfies my notion of an analog transformation (in the signal-processing sense). When applied to my discrete input set, however, the mapping does not seem to be analog (in the sense of preserving isomorphic relationships between pairs -- or higher orders -- of symbolic codes). Since information has not been lost, however, it should be possible to define "relational functions" that are analogous to "adjacency" and other properties in the original domain. Once this is done, surely the binary codes must be viewed as isomorphic to the original symbols rather than just "standing for them". The "information" in a signal is a function of your methods for extracting and interpreting the information. Likewise the "analog nature" of an information-preserving transformation is a function of your methods for decoding the analog relationships. We should also keep in mind that information theorists have advanced a great deal since the days of Shannon. Perhaps they have too limited (or general!) a view of information, but they have certainly considered your problem of decoding signal shape (as opposed to detecting modulation patterns). I regret that I am not familiar with their results, but I am sure that methods for decoding both discrete and continuous information in continuous signals are well studied. Not that all the answers are in -- vision workers like myself are well aware that there can be [obvious] information in a signal that is impossible to extract without a good model of the generating process. -- Ken -------