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

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

5. Review: Trichromatic sensor model l are spectral response functions that characterize the sensor

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

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

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

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

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

11. Effect of shape of senor response functions on sensor performance l Consider again the ideal block sensor l 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 l 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 l The resulting curve in the chromaticity diagram is called the spectral locus l Consider case

13. Spectral locus l 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 l The resulting curve in the chromaticity diagram is called the spectral locus l Consider case

14. Spectral locus for block sensor l Is this a good sensor? Sensor cannot distinguish between different stimuli between 0.4 and 0.5  m, or between 0.5 and 0.6  m, or between 0.6 and 0.7  m l How do we fix this problem?

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

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

17. Block sensor response to mixture stimuli l 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 l 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

18. Chromaticity gamut of block sensor l 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 l From the foregoing, we conclude that the chromaticity gamut of the block sensor fills the entire chromaticity diagram

19. Interpretation of regions of chromaticity diagram l 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 0.7  m, we switch from the blue to the green to the red vertices

20. Interpretation (cont.) l 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 l 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

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

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

23. Spectral locus for sensor with response overlap in two channels l Each wavelength between 0.5 and 0.7 m corresponds to a unique chromaticity coordinate l What is the gamut for this sensor?

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

25. Sensor with three channel overlap Response at 0.45  m Response at 0.55  m

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

27. Chromacity gamut for 3 channel overlap sensor l The set of all realizable chromaticities is indicated by the shaded region l Because it is impossible to excite only the green sensor by itself, we cannot get to the vertex l 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

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

29. Prediction of one sensor output based on another l Consider again the block sensor and the two channel overlap sensor l 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

30. Prediction (cont.) l Given two trichomatic sensors with responses l 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.

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