Abstract
This article is intended as a mathematical overview of the generalizations of analytic signals to higher-dimensional problems, as well as of their applications to and of their comparison 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. Following 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 the last part we experimentally motivate the choice of such higherdimensional analytic or monogenic signal representations in the context of image analysis. We see how one can take advantage of one or the other representation depending on the application.
Mathematics Subject Classification (2010). Primary 94A12; secondary 44A12, 30G35.
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Bernstein, S., Bouchot, JL., Reinhardt, M., Heise, B. (2013). Generalized Analytic Signals in Image Processing: Comparison, Theory and Applications. In: Hitzer, E., Sangwine, S. (eds) Quaternion and Clifford Fourier Transforms and Wavelets. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0603-9_11
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