Color Is Not a Metric Space

Using a metric feature space for pattern recognition, data mining, and machine learning greatly simplifies the mathematics because distances are preserved under rotation and translation in feature space. A metric space also provides a “ruler”, or absolute measure of how different two feature vectors are. In the computer vision community color can easily be misstreated as a metric distance. This paper serves as an introduction to why using a non-metric space is a challenge, and provides details of why color is not a valid Euclidean distance metric. Keywords—metric dimension; pattern recognition; feature space; distance learning; color space; CIELAB I. EXAMPLE NON-METRIC MEASUREMENT SPACE Color is often used as an appearance cue [19] for image or object clustering.[20] As a result it is easy to think of color as a metric space, which is not the case. Here we demonstrate the problem of using a non-metric space, and the extension to color measures. A. Non-commensurate axes Suppose that there are two surveillance suspects: Smith and Thomas. To tell them apart, we measure their features. Smith is 6 feet tall, with 1 FPSI. Thomas is 5.5 feet tall, with 3 FPSI. When plotted in feature space, (Figure 1.) we can ask, “What is the difference between Smith and Thomas?” It would be very easy to compute a “Euclidean” distance: D = (6! 5.5) + (3!1) = 2.06 But, what are the units of this distance? The units here are intentionally vague. Whatever units FPSI are, they are not feet. Consequently, D cannot be in feet. D is not a metric distance. Furthermore, there is no understanding of the relative importance between the height measurement in feet, versus the skin complexion, in freckles per square inch [FPSI]. The only way for D to be a valid metric distance would be if both axes had the same units. Consider the consequence of changing units. If the units of height were changed to centimeters, the value of D would change to 15.4. Height would be more important than FPSI. If the height was in kilometers, the difference in heights becomes nearly zero, and the D would only reflect the difference in FPSI, or 2.0. The bias between any two dissimilar axes is inherent in the measurement units. B. Application to color Consider when one the units of the axes are changed to the amounts of green and red. The distance, D, still cannot have units associated with it. We can call it a color difference, but the distance is neither only a difference in red, nor only a difference in green. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 4.5 5 5.5 6 6.5 7