1. ECE 638: Principles of Digital Color Imaging Systems
Lecture 8: Discrete Wavelength Models
Primaries

2. Synopsis Review of discrete wavelength model
Definition of primary set for discrete wavelength model
Color matching condition and graphical interpretation
Fundamental component of primary set
Computation of primary match amount
Transformation between two different primary sets
Color matching matrix

3. Discrete-wavelength trichomatic model
Stimulus
Sensor response
Response of the i-th channel
Define
Span(S) defines HVS subspace
Stack sensor outputs

4. Primaries Discrete-wavelength representation of a single primary
Primary set
Stimuli generated using primary set consist of linear combinations of columns of

5. Color matching condition
Stimulus is matched by primary combination if
Graphical illustration of color matching condition for two wavelength samples, a monochromatic sensor, and a single primary
Since ,
All possible primaries lie on this line

6. Fundamental component of primary set
Recall that the fundamental component of a stimulus is determined by projecting onto the span of
So we have where
Considering the primary set as a set of three stimuli (columns of ), we can write its fundamental component as
Then for any arbitrary primary combination , we have

7. Span of P*
Can match any stimulus seen by sensor with some combination of the primary set , i.e , and
provided
In this case, we say that (or ) is a set of visually independent primaries.
Any stimulus in this 2-D space can be matched by some primary weight which will lie on the line shown.

8. Determining the primary mixture amount
Color matching condition is given by
If is non-singular, then primary mixture amount is given by
Is nonsingular?
If is visually independent, can write that for some invertible 3x3 matrix
But is invertible, which implies that is invertible, since it is a product of two invertible matrices

9. Transforming between primaries
Consider a fixed stimulus and two visually independent primary sets and .
Suppose we know the mixture amount of primary that yields a match to the stimulus .
Can we find the mixture amount of primary that also matches , if we just know , , and ; and we do not know directly?
Since both primaries match the stimulus, we have that
Also, since the primary set is visually independent, we have that

10. Color matching matrix Recall that (i-th monochromatic stimulus)
Then (31x31 identity matrix)
Define according to
Concatenating these equations side-by-side for yields
or , where (3x31)
31x3 matrix is color matching matrix whose i-th row is amount of primary needed to match i-th wavelength
i-th column

11. Equivalence between human visual subspace and span of color matching matrix
Theorem:
Proof:
This means that we can freely transform among color matching functions and the HVS sensitivity matrix

12. Example transformations between color matching functions
We earlier introduced the 1931 CIE RGB color space with color matching matrix , and discussed the fact that the CIE also specified an alternative space – the CIE 1931 XYZ space with the properties that the the color matching functions are all non-negative and the color matching function is the relative luminous efficiency function.
Since the color matching functions and represent primary amounts needed to match monochromatic stimuli with the two respective primary sets and , the matrix that transforms between these color matching functions also transforms between arbitrary coordinates (mixture amounts) in these two color spaces.

13. Primaries and color matching functions for 1931 CIE RGB and XYZ color spaces
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