International Symposium on Nonlinear Theory and its Applications


Session Number:3-4-2



Invariants: A Group Theoretical Foundation and some Applications in Signal Processing and Pattern Recognition

Reiner Lenz,  Thanh Bui,  


Publication Date:2005/10/18

Online ISSN:2188-5079


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Many signal processing problems can be described by a black-box model. In this paper the input of the black-box is an object, the black-box is a measurement device and the output are the measurements produced. A typical example of an object is a sheet of white paper illuminated by some source. A camera is the measurement device and the produced image is the measurement. The black-box has a number of internal degrees of freedom that are unrelated to the input but that will affect the produced output. In the example we are interested of the properties of the sheet of paper but the image depends on the illumination and the camera used, the geometrical relation between camera, illumination and paper, the time when the measurements are recorded and many other additional factors. In many applications we want to extract properties of the object from the measurements, independent of the state of the black-box. This is the basic motivation behind all invariance frameworks in pattern recognition and signal processing. In this paper we will assume that the variations of the measurement device can be modeled by transformation groups. We will then first describe a class of invariants that can be derived from the theory of group representations. We call these invariants integral invariants. Then we introduce another class of invariants derived from the Lie-theory of differential equations. We call them differential invariants. We will illustrate the general theory with some examples from color constancy, pattern recognition and linear system theory.