[argyllcms] Re: Version 0.53 - Gamut mapping warning message.

Graeme Gill schrieb:

One of the things that makes multidimensional function non-monotonicity difficult to identify, is that while a local minima or maxima in an individual PCS channel is a necessary condition for non-monotonicity, it is not sufficient. The color is the combination of PCS values, so actual non-monotonicity is when the same PCS combination value can be produced by more than one device value (a many-to-one mapping). I'm not sure if it's possible to arrive at a deterministic mathematical test for this, given an arbitrary, continuous device function.

Yes, you're right, I was too naive, it's obviously not so easy. Looking at Wikipedia, http://en.wikipedia.org/wiki/Monotonic, the mathematical definition of a monotonic function seems to be:

   Let

   f: P ? Q

   be a function between two sets P and Q, where each set carries a
   partial order (both of which we denote by ?). In calculus one
   focuses on functions between subsets of the reals and the order ? is
   just the usual ordering on real numbers, but this is not essential
   for this definition.

   The function f is monotone if, whenever x ? y, then f(x) ? f(y).
   Stated differently, a monotone function is one that preserves the order.

IMO this implies, monotonicity of a function is only defined with respect to particular ordering relations for the sets P and Q, i.e. a function may be monotonic wrt. one ordering relation, but non-monotonic wrt. another relation, for the same sets P and Q.

For R^N we basically could define various "?" relations, e.g. x1[0]==x2[0] ? x1[1]<=x2[1] : x1[0]<=x2[0] for two 2D numbers x1 and x2, and check for monotonicity wrt. to these orders.

But I doubt that monotonicity with regard to such an "artificial" ordering of multi-dimensional numbers is useful for our desired purpose. As you said above, it's rather more interesting, whether the device -> PCS mapping is bijective or not. So we should probably not use the misleading term "monotonic" (which depends on the chosen ordering relation), but simply the term "bijective" to describe this property, at least for multi-dimensional spaces.

(For 1D real functions and the "usual" meaning of the "?" relation, the term "monotonic" also implies a bijective mapping, so here we can use both terms equally.)

Regards,
Gerhard

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