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research [2018/01/26 00:24]
research [2018/10/17 12:17]
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 ===== Variational Image Analysis on Manifolds and Metric Measure Spaces ===== ===== Variational Image Analysis on Manifolds and Metric Measure Spaces =====
  
-We exploit basic statistical manifolds to devise variational models of low-level image analysis that exhibit favourable properties in comparison to established convex and non-convex models: smoothness, probabilistic interpretation,​ efficiently converging parallel and sparse Riemannian numerical updates that scale up to large problem sizes. The current focus is on the assignment manifold and image labeling, and on learning from image assignments in large-scale unsupervised scenarios, within the mathematical frameworks of information geometry and regularised optimal transport.+**Scope.** ​We exploit basic statistical manifolds to devise variational models of low-level image analysis that exhibit favourable properties in comparison to established convex and non-convex models: smoothness, probabilistic interpretation,​ efficiently converging parallel and sparse Riemannian numerical updates that scale up to large problem sizes. ​
  
-We applied our approach to solve in a novel way the MAP labeling ​problem based on a given graphical model by smoothly combining a geometric reformulation of the local polytope relaxation with rounding to an integral solution. A key ingredient are local `Wasserstein messages'​ that couple local assignment ​measures along edges.+The current focus is on the **//​assignment manifold//​** and image labeling, and on learning from image assignments in large-scale unsupervised scenarios, within ​the mathematical frameworks of information geometry and regularised optimal transport. A novel smooth dynamical system evolving on a statistical manifold, called **//assignment ​flow//**, forms the basis of our work.
  
-  ​* [[https://arxiv.org/abs/1710.01493|preprint, arXiv:1710.014932017]]+**Current work.** We conduct a comprehensive study of //geometric integration//​ techniques, including automatic step size adaption, for numerically computing the assignment flow in a stable, efficient and parameter-free way.  
 +  ​* [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/​Zeilmann2018aa.pdf|Geometric Numerical Integration of the Assignment Flowpreprint: ​arXiv:1810.06970]] 
 +Based on thiswe study how weights for geometric diffusion can be learned from data, by applying optimal control to the assignment flow. This enables to attach a semantic meaning to such weights, a property that is missing in current models of artificial neural networks. 
 + 
 +**Recent work.** We extended the assignment flow to //​unsupervised//​ scenarios, where label evolution on a feature manifold is simultaneously performed together with label assignment to given data - see the  
 +  * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​gcpr2018.pdf|Unsupervised Label Learning on Manifolds by Spatially Regularized Geometric Assignment, preliminary announcement:​ GCPR 2018]]. 
 +This paper sketches a special instance of a more general framework, the //​unsupervised assignment flow//, to be introduced in a forthcoming report. 
 + 
 +We applied our approach to solve in a novel way the //MAP labeling problem// based on a given graphical model by smoothly combining a geometric reformulation of the local polytope relaxation with rounding to an integral solution. A key ingredient are local `//​Wasserstein messages//'​ that couple local assignment measures along edges. 
 + 
 +  * [[https://​epubs.siam.org/​doi/​abs/​10.1137/​17M1150669|Image Labeling Based on Graphical Models Using Wasserstein Messages and Geometric Assignment, SIAM J. on Imaging Science, 11/2 (2018) 1317--1362]]
  
 Kick-off paper that introduces the basic approach: Kick-off paper that introduces the basic approach:
  
-  * [[https://link.springer.com/article/10.1007/​s10851-016-0702-4|J. Math. Imag. Vision2017]] +  * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/​Astroem2017.pdf|Image Labeling by Assignment., ​J. Math. Imag. Vision ​58/2 (2017) 211--238]]
-  * [[https://​www.readcube.com/​articles/​10.1007/​s10851-016-0702-4?​author_access_token=qTJknl5fUiTP-FjpTKUBO_e4RwlQNchNByi7wbcMAY6Xsf53Ss0CTbPqiHjWrFr9KxurTkJxDnblRwd66rV9vVhzVeITjqSsDSC8NWZFxg9y-pWgHhjix00mggjora7T-qHFcXzGInobFGxuIfcnEA%3D%3D|Link to online PDF]]+
   * [[http://​www-rech.telecom-lille.fr/​diff-cv2016/​|Proceedings DIFF-CVML'​16;​ Grenander best paper award]]   * [[http://​www-rech.telecom-lille.fr/​diff-cv2016/​|Proceedings DIFF-CVML'​16;​ Grenander best paper award]]
   * [[https://​ipa.iwr.uni-heidelberg.de/​dokuwiki/​Papers/​Astroem2016d.pdf|Proceedings ECCV'​16]]   * [[https://​ipa.iwr.uni-heidelberg.de/​dokuwiki/​Papers/​Astroem2016d.pdf|Proceedings ECCV'​16]]