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From: alanr@rd.bbc.co.uk (Alan Roberts)
Subject: Re: FAQ purpose/colorspace-faq
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Date: Thu, 11 May 1995 09:43:14 GMT
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Alain Fontaine (fontaine@sri.ucl.ac.be) wrote:

: What triggered my original question was the fact that I encountered the
: words 'perceptually linear'... Charles Poynton suggested, in private
: mail, to use "perceptual uniformity" instead. And so section 1.0 should
: explicitly define both 'linear' (linear-light) and "perceptually uniform"
: (where a change of one unit gives the same perceptual change in any region).
: This is a FAQ, and you cannot expect all readers to know this without being
: told explicitly. 

Ok, I can accept that.
   
: I noticed the careful distinction made between 'conversion equations'
: and 'coding equations', so this is not the problem. To state it otherwise:
: I am questioning the use of the word 'linear' to speak about the 
: transformation itself, not the quantities on which the transformation is
: applied (as in your answer). One can apply mathematically linear 
: transformations on linear (=representing linear physical measurements)
: quantities, non (mathematically) linear transformations on linear quantities
: (giving, of course, non-linear results), mathematically linear transformations
: on non-linear qunatities (as in the coding equations), and even non-linear
: transformation on non-linear quantities (changing the gamma correction on
: an already gamma corrected signal, for example).

: One example of something I have doubts about:

: > [4.2] CIE YUV (1960)
: > 
: > This is a linear transformation of Yxy, in an attempt...
:             ------
: >     u = 2x / (6y - x + 1.5)
: >     v = 3y / (6y - x + 1.5)

: Is it correct to say that it is a 'linear transformation' given the
: type of equations that define it ? I seemed to remember from a distant
: math background that 'linear' transformations (while linear applies to
: the transformation itself, not to the relation that exists between the
: numbers being transformed and some physical quantity) are transformations
: that can be represented by a multiplication by a matrix of constant
: coefficients. Or are all those math courses really too far away ???
:                                                         /AF

We have to be very careful with choice of words here.

A linear transformation is any set of equations that converts values
from one measuring system to another. So your example of xy to uv is
indeed a linear transformation. Since both xy and uv are chromaticity
coordinates derived linearly from tristimulus values, then they are
_as_a_set_ related directly to linear light and all is well.

It is also possible to make linear transformations on non-linear
signals. An example is the coding equations for Rec.709, which generates
three signals (Y Pr Pb) from three other signals (R' G' B') which are
themselves non-linearly related to light through the gamma-correction.
This transformation is clearly reversible, as are all coding equations.

In colour science as I understand it, linearity always means a linear
relationship to measureable light. Non-linearity always applies to
signals or quantitities that are derived from linear signals or
quantities through a non-linear equation such as L* or gamma-correction.

So Yxy, Yuv, Yu'v', RGB, are all linear because they all relate directly
to linear XYZ. But R'G'B', Lab, Luv, Y Pr Pb etc are all non-linear
because they are related via non-linear equation sets.

This is all getting rather deep, and I had hoped that Charles' FAQs
should have covered this. If you feel there is still some room for
confusion then maybe Charles should tackle a new section?

--
************* Alan Roberts **************
* BBC Research & Development Department *
* My views, not necessarily Auntie's    *
*    but they might be, you never know. *
*****************************************
