2016/03/16 by Freddie Åström, Stefania Petra, Bernhard Schmitzer +1 · 1 citation
Computer Science · Engineering · Mathematics · #3D Shape Modeling and Analysis #Algorithm #Artificial intelligence #Computer science #Flow (mathematics) #Geometric data analysis #Geometric flow #Geometry #Image (mathematics) #Initialization #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Medical Image Segmentation Techniques #Metric (unit) #Multiplicative function #Point (geometry) #Pure mathematics #Riemannian manifold #Topological and Geometric Data Analysis #cs.CV #math.OC #msc:62H35 #msc:62M40 #msc:65K05 #msc:68U10
paper · pdf · doi:10.1007/s10851-016-0702-4
arxiv created 2016/03/16 · openalex publication_date 2017/01/12 · arxiv updated 2017/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We introduce a novel geometric approach to the image labeling problem. Abstracting from specific labeling applications, a general objective function is defined on a manifold of stochastic matrices, whose elements assign prior data that are given in any metric space, to observed image measurements. The corresponding Riemannian gradient flow entails a set of replicator equations, one for each data point, that are spatially coupled by geometric averaging on the manifold. Starting from uniform assignments at the barycenter as natural initialization, the flow terminates at some global maximum, each of which corresponds to an image labeling that uniquely assigns the prior data. Our geometric variational approach constitutes a smooth non-convex inner approximation of the general image labeling problem, implemented with sparse interior-point numerics in terms of parallel multiplicative updates that converge efficiently.