1. ECE 638: Principles of Digital Color Imaging Systems
Lecture 4: Chromaticity Diagram

2. Goal is to understand the origins and meaning of everyday chromaticity diagrams, such as the CIE xy diagram

3. Synopsis Brief review of trichromatic theory
Develop chromaticity diagram
Examine conditions for prediction of response of one sensor based on another
Limitations of trichromatic theory

4. Review: Development of trichromatic theory
Sensor model
Reinterpretation of conditions for color match
Metamerism
Linearity of sensor model
Response to monochromatic stimuli
Chromaticity diagram

5. Review: Trichromatic sensor model
Show three sensor response curves, and then the equations for the sensor output 3-tuple.
are spectral response functions that characterize the sensor

6. Review: Metamerism
Stimuli and are said to be metameric with respect to the sensor if they elicit identical responses
Metamerism is both a blessing and a curse for color imaging systems designers

7. Review: Plane where R+G+B=1
Represents points of roughly equal lightness
All colors that differ by a scalar multiple will intersect this plane at the same point

8. Review: Chromaticity diagram
For a particular set of nonorthognal axes, the lengths of the perpendiculars illustrated below can be shown to satisfy

9. Development of chromaticity diagram
Shape of spectral response functions for sensor
Consider three example sensors
Spectral locus
Mixture stimuli
Polar coordinate interpretation of chromaticity coordinates
Chromaticity gamut
Spectral response functions for the HVS

10. Notation Use upper case symbols for sensor response
Use lower case symbols for corresponding chromaticity coordinates

11. Effect of shape of senor response functions on sensor performance
Consider again the ideal block sensor
Intuitively, we might expect that this would be a good sensor
full coverage of spectral band at uniform sensitivity
no overlap of response from different channels minimize cross-talk or “confusion”

12. Spectral locus
Recall that the response of a sensor to an arbitrary stimulus can be expressed in terms of the response to monochromatic stimuli at all wavelengths
The resulting curve in the chromaticity diagram is called the spectral locus
Consider case

13. Review: Chromaticity diagram
For a particular set of nonorthognal axes, the lengths of the perpendiculars illustrated below can be shown to satisfy

14. Spectral locus
Recall that the response of a sensor to an arbitrary stimulus can be expressed in terms of the response to monochromatic stimuli at all wavelengths
The resulting curve in the chromaticity diagram is called the spectral locus
Consider case

15. Spectral locus for block sensor
Is this a good sensor?
Sensor cannot distinguish between different stimuli between 0.4 and m, or between 0.5 and m, or between 0.6 and m
How do we fix this problem?

16. Mixture of two stimuli
Consider two stimuli and the responses of a trichromatic sensor to these stimuli
Now suppose we create a third stimulus as a mixture of these two stimuli with mixture parameter
Then by linearity of the sensor, we have that

17. Geometric interpretation of mixture
As ranges from 0 to 1, traces a line in the sensor space connecting with
We observe analogous behavior in the chromaticity diagram

18. Block sensor response to mixture stimuli
Any three stimuli possibly, but not necessarily monochromatic, with support confined to the ranges , respectively, will correspond to chromaticity coordinates at the vertices of the chromaticity diagram
The chromaticity coordinate of any stimulus that is a mixture of two or three of the stimuli will lie within the convex hull formed by the chromaticity coordinates

19. Chromaticity gamut of block sensor
We define the chromaticity gamut for a sensor to be the set of all points in the chromaticity space that correspond to the response of that sensor to some real stimulus
From the foregoing, we conclude that the chromaticity gamut of the block sensor fills the entire chromaticity diagram

20. Interpretation of regions of chromaticity diagram
We have seen that stimuli confined to a narrow spectral band will excite only one channel, and will yield a chromaticity point at one of the three vertices of the chromaticity diagram
As we increase wavelength from 0.4 to m, we switch from the blue to the green to the red vertices

21. Interpretation (cont.)
A stimulus which equally excites all three channels will correspond to a chromaticity coordinate at the center of the diagram. Visually, this color should look achromatic
As we move from this point outward, the stimulus response is concentrated in one or two channels the color should appear more spectrally pure or more saturated

22. Polar coordinate representation of color
This suggests a polar coordinate interpretation of color
Origin of system is at center of chromaticity diagram – corresponding to intersection of neutral axis in RGB sensor space with chromaticity plane
Angle of chromaticity coordinate with respect to horizontal axis is a correlate of hue
Distance from origin is a correlate of saturation

23. Sensor with response overlap in two channels
Response for is same as before
Consider response at
Similarly,

24. Spectral locus for sensor with response overlap in two channels
Each wavelength between 0.5 and m corresponds to a unique chromaticity coordinate
What is the gamut for this sensor?

25. Chromaticity gamut for sensor with two channel overlap
Chromaticity gamut is same as that for block sensor

26. Sensor with three channel overlap
Response at m
Response at m

27. Spectral locus for sensor with 3 channel overlap
Now every wavelength corresponds to a unique chromaticity coordinate
What is the downside of the using this sensor?

28. Chromacity gamut for 3 channel overlap sensor
The set of all realizable chromaticities is indicated by the shaded region
Because it is impossible to excite only the green sensor by itself, we cannot get to the vertex
Thus we see that while overlap of sensor responses is desirable from the standpoint of yielding a unique chromaticity coordinate for each distinct wavelength, it also limits the sensor gamut

29. Cone sensitivity functions for the HVS
reproduced from Boynton, 1992, p. 153.
data is due to Smith and Pokorny, 1975.

30. Prediction of one sensor output based on another
Consider again the block sensor and the two channel overlap sensor
Given the response of Sensor 1 to an arbitrary stimulus, can we determine how Sensor 2 would response to that same input?
Sensor 1: Block
Sensor 2: 2 channel overlap

31. Prediction (cont.) Given two trichomatic sensors with responses
A necessary and sufficient condition for us to be able to predict the response of Sensor 2 to any stimulus from the response of Sensor 1 to that same stimulus is that we be able to express the Sensor 2 functions as a linear combination of those for Sensor 1, i.e.

32. Limitations of trichromatic theory
Does not yield a uniform color space
Fails to account for color opponency
Does not predict color appearance