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

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

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

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

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

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

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

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

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

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

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”

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

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

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?

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

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

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

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

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

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

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

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

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?

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

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

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?

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

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

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

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.

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