1. ECE 638: Principles of Digital Color Imaging Systems
Lecture 7: Discrete Wavelength Models
Sensors

2. Synopsis Discretization of model Sensor subspace
Decomposition of stimulus into fundamental and nullspace components
Graphical interpretation

3. Why discretize the model?
So far we have assumed that is continuous
Upside of this assumption
matches real world
Downside of this assumption
for computation, we must discretize the model
we can gain further intuition if we descretize the model

4. Discretization of wavelength
It is common practice to make the following choices:
This implies that we need samples to characterize the visible spectrum.

5. Notation
This material is taken from the paper by Mark Wolski: “A review of linear color descriptor spaces and their applications,” which is available for download at the course website.
Wolski’s notation differs somewhat from that used thus far to describe the continuous-wavelength trichromatic model.
For ease of cross-reference between the lecture notes and Wolski’s paper, we will follow his notation here.

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

7. Characterization of span(S)
Can write any vector in span(S) as
for some coefficient

8. Projection of stimulus onto HVS
Suppose
Task: Find that vector which is closest to
in the sense of the Euclidean norm, i.e.
is the projection of onto

9. Solution for projection
To solve problem, write
for some vector (3x1)
Solution (see HW): satisfies
We can do two things with (1):

10. 1. Obtain expression for Invert to obtain provided is nonsingular.
We can assume that has rank is nonsingular.

11. 2. Identify nullspace component of
We can write (1) as
since
Now we rearrange (2) as
Let
Then we have

12. Application to trichromatic model for color
Consider an arbitrary stimulus
Can write
where
This implies that

13. Restatement of color matching conditions

14. Graphical interpretation
Let number of channels = 1 (achromatic vision)
Let number of samples in wavelength = 2
Assume
Denote stimulus as
Sensor response is a scalar

15. Metamerism
Stimuli and are metameric, i.e. they look the same to the observer with response

16. Non-equivalence of sensors
Stimuli and elicit same response to sensor but not to sensor , i.e.

17. Nullspace component For any stimulus,
where is the fundamental component
and the response to the nullspace component is zero
Question 1: Are there real stimuli that are invisible?
Clarification
By stimulus, we mean signals in the visible wavelength region
By real stimulus, we mean , and there exists some for which

18. Response to monochromatic stimulus
Consider monochromatic stimulus
What is response to ?
i.e. transpose of the i-th row of sensor response matrix

19. What does this tell us about Question 1?
Are there real stimuli that are invisible?
For a stimulus to be invisible, we must have
The human visual system doesn’t have this property
So the answer to Question 1 is “no”.
Question 2: Are there real stimuli for which ?
i.e. there is no blackspace component
Such stimuli might be regarded as being visually efficient
We will answer this question later