Iterative Multiplicative Filters for Data Labeling
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Based on an idea in Åström et al. (J Math ImagingVis, doi: 10.1007/s10851-016-0702-4, 2017) we propose a new iterative multiplicative filtering algorithm for label assignment matrices which can be used for the supervised partitioning of data. Starting with a row-normalized matrix containing the averaged distances between prior features and observed ones, the method assigns in a very efficient way labels to the data. We interpret the algorithm as a gradient ascent method with respect to a certain function on the product manifold of positive numbers followed by a reprojection onto a subset of the probability simplex consisting of vectors whose components are bounded away from zero by a small constant. While such boundedness away from zero is necessary to avoid an arithmetic underflow, our convergence results imply that they are also necessary for theoretical reasons. Numerical examples show that the proposed simple and fast algorithm leads to very good results. In particular we apply the method for the partitioning of manifold-valued images.
KeywordsLabeling Supervised partitioning Multiplicative filter Partitioning Manifold-valued images
We are grateful to Ch. Schnörr (University of Heidelberg) for stimulating discussions. Many thanks to R. Hielscher (University of Chemnitz) for supporting the work on EBSD data. We thank the referees for requesting a discussion of condition (PI). Funding by the German Research Foundation (DFG) within the Project STE 571/13-1 & BE 5888/2-1 and within the Research Training Group 1932 “Stochastic Models for Innovations in the Engineering Sciences”, project area P3, is gratefully acknowledged. Furthermore, G. Steidl acknowledges the support by the German Federal Ministry of Education and Research (BMBF) through Grant 05M13UKA (AniS).
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