Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
research [2019/03/20 09:59]
ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces]
research [2019/10/24 11:52] (current)
ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces]
Line 7: Line 7:
 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. 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.
  
-**Mathematical aspects.** The assignment flow evolves non-locally for any data given on a graph. ​Variational ​aspects, extensions to continuous domains ​and scale separation are investigated. A preliminary ​step concerns ​a more classical //​additive//​ variational ​formulation that provides a smooth geometric version of the continuous cut approach.+**Mathematical aspects.** The assignment flow evolves non-locally for any data given on a graph. ​Geometric and variational ​aspects, extensions to continuous domainsscale separation ​and models of knowledge representation across the scales ​are investigated. ​ 
 + 
 +A preliminary ​extension from graphs to the continuous domain in the `zero-scale limit' (local interaction only) reveals the interplay between the underlying geometry and variational aspects. 
 +  * [[https://​arxiv.org/​abs/​1910.07287|Continuous-Domain Assignment Flow, preprint arXiv:​1910.07287]]. 
 +a more classical //​additive//​ variational ​reformulation ​provides a smooth geometric version of the continuous cut approach.
   * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​Savarino2019aa.pdf|A Variational Perspective on the Assignment Flow, SSVM 2019]].   * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​Savarino2019aa.pdf|A Variational Perspective on the Assignment Flow, SSVM 2019]].
  
-**Parameter learning.** We study how weights for geometric diffusion that parametrize the adaptivity of the assignment flow can be learned from data. Symplectic integration ensures the commutativity of discretisation and optimisation operations. ​We currently investigate this approach in connection with more general objective functions+**Parameter learning.** We study how weights for geometric diffusion that parametrize the adaptivity of the assignment flow can be learned from data. Symplectic integration ensures the commutativity of discretisation and optimisation operations. ​Results reveal the steerability of the assignment flow and its potential for pattern //​formation//​. 
-  * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​Huhnerbein2019aa.pdf|Learning Adaptive Regularization for Image Labeling Using Geometric Assignment, SSVM 2019]].+  * [[https://​arxiv.org/​abs/​1910.09976|Learning Adaptive Regularization for Image Labeling Using Geometric Assignment, preprint arXiv:​1910.09976]] 
 +  * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​Huhnerbein2019aa.pdf|Conference version, SSVM 2019]].
  
 **Unsupervised label learning.** Our recent work concerns the emergence of labels in a completely unsupervised way by data //​self//​-assignment. The resulting unsupervised assignment flow has connections to low-rank matrix factorisation and discrete optimal mass transport that are explored in our current work. **Unsupervised label learning.** Our recent work concerns the emergence of labels in a completely unsupervised way by data //​self//​-assignment. The resulting unsupervised assignment flow has connections to low-rank matrix factorisation and discrete optimal mass transport that are explored in our current work.
   * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​Zisler2019aa.pdf|Unsupervised Labeling by Geometric and Spatially Regularized Self-Assignment,​ SSVM 2019]].   * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​Zisler2019aa.pdf|Unsupervised Labeling by Geometric and Spatially Regularized Self-Assignment,​ SSVM 2019]].
  
-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. This paper sketches a special instance of a more general framework, ​the //​unsupervised assignment flow//, ​to be introduced in a forthcoming report.+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. The following papers introduce ​the corresponding ​//​unsupervised assignment flow//
 +  * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​Zern2019aa.pdf|Unsupervised Assignment Flow: Label Learning on Feature Manifolds by Spatially Regularized Geometric Assignmentpreprint arXiv:1904.10863]]
   * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​gcpr2018.pdf|Unsupervised Label Learning on Manifolds by Spatially Regularized Geometric Assignment, GCPR 2018]].   * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​gcpr2018.pdf|Unsupervised Label Learning on Manifolds by Spatially Regularized Geometric Assignment, GCPR 2018]].
  
 **Geometric numerical integration.** We conducted 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.  **Geometric numerical integration.** We conducted 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 Flow, preprint: arXiv:​1810.06970]]+  ​* [[https://​iopscience.iop.org/​article/​10.1088/​1361-6420/​ab2772|Geometric Numerical Integration of the Assignment Flow, Inverse Problems, 2019]] 
 +  ​* [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​Zeilmann2018aa.pdf|preprint:​ arXiv:​1810.06970]]
  
 **Evaluation of discrete graphical models.** 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. **Evaluation of discrete graphical models.** 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 Science11/2 (2018) 1317--1362]]+  * [[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: