Generalized analytic signals in image processing: Comparison, theory and their applications

This article is intended as both a mathematical overview of the generalizations of analytic signals to higher dimensional problems as well as their applications and comparisons on artificial and real-world image samples. We first start by reviewing the basic concepts behind analytic signal theory and derive its mathematical background based on boundary value problems of one dimensional analytic functions. Based on that, two generalizations are motivated by means of higher dimensional complex analysis or Clifford analysis. Both approaches are proven to be valid generalizations of the known analytic signal concept. In a last part we experimentally motivate the choice of such higher dimensional analytic or monogenic signal representations in the context of image analysis. We see how one can take advantage of one or another representation depending on the application. Mathematics Subject Classification (2010). Primary 94A12; Secondary 44A12, 30G35.