pepke@loligo.cc.fsu.edu (Eric Pepke) (01/13/90)
I am looking for a word to describe a property of transformations and other mappings. The property is this: Say I have a mapping from R(n) to R(n) where n is greater than one. Call the independent variable in the dimension k where 1 <= k <= n a(k) and the dependent variable b(k). If the mapping has this property for which I need a name, b(i) is not dependent on a(j) when i is not equal to j. Consider transformation of 2-d images. In that case, n is 2. If a transformation had this property, then the transformed Y value would only depend on the original Y value, likewise for X. If the transformation did not have this property, the transformed Y could depend on both the original X and Y. With this property, a mapping (X1, Y1) -> (X2, Y2) could be decomposed into two mappings: X1 -> X2 and Y1 -> Y2. Without this property, no such decomposition would be feasible. Uniform scaling and stretching along one axis, even if nonlinear, would have this property. Skewing and rotation would not. This is a very elementary and important property and is basic to algorithm design, but I can't for the life of me think of the word for it! I looked up all the words in Webster's that begin with "ortho" but to no avail. I could call the individual component mappings independent from each other, or I could say that the composite mapping was decomposable, but there must be a better and more specific term. Does this ring anybody's bell? Eric Pepke INTERNET: pepke@gw.scri.fsu.edu Supercomputer Computations Research Institute MFENET: pepke@fsu Florida State University SPAN: scri::pepke Tallahassee, FL 32306-4052 BITNET: pepke@fsu Disclaimer: My employers seldom even LISTEN to my opinions. Meta-disclaimer: Any society that needs disclaimers has too many lawyers.
markv@gauss.Princeton.EDU (Mark VandeWettering) (01/14/90)
In article <447@fsu.scri.fsu.edu> pepke@scri1.scri.fsu.edu (Eric Pepke) writes: >Consider transformation of 2-d images. In that case, n is 2. If a >transformation had this property, then the transformed Y value would only >depend on the original Y value, likewise for X. If the transformation did not >have this property, the transformed Y could depend on both the original X and >Y. With this property, a mapping (X1, Y1) -> (X2, Y2) could be decomposed >into two mappings: X1 -> X2 and Y1 -> Y2. Without this property, no such >decomposition would be feasible. >Uniform scaling and stretching along one axis, even if nonlinear, would have >this property. Skewing and rotation would not. Hmmm. Very interesting question, and it even brings a few questions into my head. The term originally thought of by myself is "separable", but that has typically been used to say that only one coordinate is modified during a pass, not that it was the only one used to specify the transformation. Without knowing the precise use that you intend, this may be enough to describe the algorithms you intend. Alvy Ray Smith and Edwin Catmull published a paper called "3-D transformation in scanline order" which outlines the basic method of image transformation composed as a series of seperable (1-D) transformations. Their work has been followed up by a number of articles dealing with image warping. Fant had a much less general, but possibly easier to follow paper in IEEE CG&A. If your usage of "decomposeable" is consistent with the terms separable, then I would advise using the already existing terminology. Otherwise, its up to somebody else to make a suggestion. Mark
ksbooth@watcgl.waterloo.edu (Kelly Booth) (01/15/90)
In article <447@fsu.scri.fsu.edu> pepke@scri1.scri.fsu.edu (Eric Pepke) writes: >I am looking for a word to describe a property of transformations and other >mappings. The property is this: Say I have a mapping from R(n) to R(n) >where n is greater than one. Call the independent variable in the dimension >k where 1 <= k <= n a(k) and the dependent variable b(k). If the mapping >has this property for which I need a name, b(i) is not dependent on a(j) >when i is not equal to j. Separable? F(a1,a2,...,an) = [f1(a1),f2(a2),...,fn(an)] Mappings as above are often said to be "defined componentwise".