1. ECE 638: Principles of Digital Color Imaging Systems
Lecture 9: Discrete Wavelength Models
Projection Operator

2. Synopsis
Review of discrete wavelength model

3. Discrete-wavelength trichomatic model
Stimulus
Sensor response
Response of the i-th channel
Define
Span(S) defines HVS subspace
Stack sensor outputs
Set of visually independent primaries
Color matching matrix for this set of visually independent primaries

4. Review

5. Definition of projection operator
Projection operator is a linear operator that extracts the fundamental component of the stimulus , i.e.
Since , we can write that
Following the earlier development of the fundamental component , we have that
Substituting (3) into (2), we obtain
Comparing (4) with (1), we get

6. Alternate forms of projection operator
Based on fact that , we obtained
But for visually independent primaries.
Thus
Equations (5a), (5b), and (5c) are all equivalent.

7. Projection of monochromatic stimuli onto the human visual subspace
Recall that a monochromatic stimulus at wavelength can be expressed as
So the identity matrix represents the complete set of monochromatic stimuli.
Consider that
Thus ; so the columns of are the fundamental components of the monochromatic stimuli.

8. Relation between projection operator and fundamental component of primary mixtures for monochromatic stimuli
Recall that
Since is the fundamental component of , we also have that So
Since the responses to and are identical, their fundamental components are also the same. But is the fundamental component of the monochromatic stimuli; so it already lies within the human visual subspace, i.e.
Thus, we have from Eq. (6) that

9. Extraction of fundamental component of stimulus: an imaging systems interpretation
A complete imaging system, can be thought of as a capture device followed by a display device.
The process of extracting the fundamental component of the stimulus, as described by Eq. (7a) from the preceding slide can be viewed in this way
i.e.
We also have
Combining (8) and (9) yields
Sensor
Display

10. Questions about the human visual subspace
Recall two earlier questions:
Question 1: Are there real stimuli that are invisible?
Answer: “No”
Question 2: Are there real stimuli for which ?
Not yet answered
Now introduce a third question:
Question 3: What colors in the fundamental space are physically realizable with the addition of a black space component, i.e. what colors have a physically realizable metamer?

11. Characterization of subset of fundamental space with physically realizable metamers
Theorem: A stimulus has a physically realizable metamer its fundamental component can be written as a non-negative linear combination of the columns of
Proof:
Suppose is physically realizable. (Here is fixed; and is one possible metamer for .)
but since is physically realizable, its components are all non-negative. Therefore, Eq. (11) shows that the fundamental component is a non-negative linear combination of the columns of .

12. Characterization (cont.)
Theorem: A stimulus has a physically realizable metamer its fundamental component can be written as a non-negative linear combination of the columns of
Proof:
Suppose , where This immediately leads to the conclusion that is a physically realizable metamer for .
This completes the proof of the theorem.