Group Theoretical Invariants in Color Image Processing

Many image formation processes are complex interactions of several sub-processes and the analysis of the resulting images requires often to separate the influence of these sub-processes. An example is the formation of a color image which depends on the illumination, the properties of the camera and the objects in the scene, the imaging geometry and many other factors. Color constancy algorithms try to separate the influence of the illumination and the remaining factors and are thus typical examples of the general strategy. An important tool used by these methods are invariants ie. features that do not depend on the state of one (or several) of the sub-processes involved. Illumination invariants are thus features that are independent of illumination changes and depend only on the remaining factors such as material and camera properties. We introduce transformation groups as the descriptors of the sub-processes mentioned above. We then show how they can be used to calculate the number of independent invariants for a given class of transformations. We also show that the theory is constructive in the sense that there are symbolic mathematics packages that can find the invariants as solutions to systems of partial differential equations. We illustrate the general theory with applications from color computer vision. We will describe the construction of invariants from the dichromatic and the Kubelka-Munk reflection models in detail. Space does not permit us to describe the detailed derivation of illumination invariants from PCA models of illumination spectra but it can be shown that the construction of the invariants follows the same mathematical procedure.