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 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] Both sides previous revision Previous revision 2019/10/24 11:52 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2019/10/24 11:37 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2019/04/25 10:44 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2019/03/20 09:59 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/10/17 12:17 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/09/23 15:31 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/05/28 09:58 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/05/28 09:57 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/03/01 21:39 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/02/19 23:35 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/02/19 23:33 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/01/26 00:24 ipa [Geometric Low-Level Variational Image Analysis on Metric Measure Spaces] 2017/11/02 11:49 ipa [Minimum Energy Filtering on Lie Groups with Application to Structure and Motion Estimation from Monocular Videos] 2017/11/02 11:49 ipa [Minimum Energy Filtering on Lie Groups with Application to Structure and Motion Estimation from Monocular Videos] 2017/11/02 02:53 ipa [Context Specific Independence and Graphical Models] 2017/11/02 02:53 ipa [Segmentation of Thin Fiber Structures in 3D Tomographical Data] 2017/11/02 02:53 ipa [Phase Transitions and Recovery of Cosparse Objects Through Limited Angle Tomography] 2017/11/02 02:51 ipa [Geometric Low-Level Variational Image Analysis on Metric Measure Spaces] 2017/11/01 22:15 ipa [Geometric Low-Level Variational Image Analysis on Metric Measure Spaces] 2017/11/01 22:14 ipa [Geometric Low-Level Variational Image Analysis on Metric Measure Spaces] 2017/11/01 17:33 ipa [Geometric Low-Level Variational Image Analysis on Metric Measure Spaces] 2017/11/01 17:32 ipa [Geometric Low-Level Variational Image Analysis] 2017/10/28 15:38 ipa [Geometric Low-Level Variational Image Analysis] 2017/10/28 15:37 ipa [Geometric Low-Level Variational Image Analysis] 2017/05/21 11:18 ipa use https for links2017/01/19 14:11 ipa [Geometric Low-Level Variational Image Analysis] 2016/10/18 16:19 ipa [Geometric Low-Level Variational Image Analysis] 2016/09/06 12:54 ipa [Geometric Low-Level Variational Image Analysis] 2016/07/19 13:38 ipa [Geometric Low-Level Variational Image Analysis] Next revision Previous revision 2019/10/24 11:52 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2019/10/24 11:37 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2019/04/25 10:44 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2019/03/20 09:59 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/10/17 12:17 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/09/23 15:31 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/05/28 09:58 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/05/28 09:57 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/03/01 21:39 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/02/19 23:35 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/02/19 23:33 ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces] 2018/01/26 00:24 ipa [Geometric Low-Level Variational Image Analysis on Metric Measure Spaces] 2017/11/02 11:49 ipa [Minimum Energy Filtering on Lie Groups with Application to Structure and Motion Estimation from Monocular Videos] 2017/11/02 11:49 ipa [Minimum Energy Filtering on Lie Groups with Application to Structure and Motion Estimation from Monocular Videos] 2017/11/02 02:53 ipa [Context Specific Independence and Graphical Models] 2017/11/02 02:53 ipa [Segmentation of Thin Fiber Structures in 3D Tomographical Data] 2017/11/02 02:53 ipa [Phase Transitions and Recovery of Cosparse Objects Through Limited Angle Tomography] 2017/11/02 02:51 ipa [Geometric Low-Level Variational Image Analysis on Metric Measure Spaces] 2017/11/01 22:15 ipa [Geometric Low-Level Variational Image Analysis on Metric Measure Spaces] 2017/11/01 22:14 ipa [Geometric Low-Level Variational Image Analysis on Metric Measure Spaces] 2017/11/01 17:33 ipa [Geometric Low-Level Variational Image Analysis on Metric Measure Spaces] 2017/11/01 17:32 ipa [Geometric Low-Level Variational Image Analysis] 2017/10/28 15:38 ipa [Geometric Low-Level Variational Image Analysis] 2017/10/28 15:37 ipa [Geometric Low-Level Variational Image Analysis] 2017/05/21 11:18 ipa use https for links2017/01/19 14:11 ipa [Geometric Low-Level Variational Image Analysis] 2016/10/18 16:19 ipa [Geometric Low-Level Variational Image Analysis] 2016/09/06 12:54 ipa [Geometric Low-Level Variational Image Analysis] 2016/07/19 13:38 ipa [Geometric Low-Level Variational Image Analysis] 2016/07/19 13:35 ipa [Geometric Low-Level Variational Image Analysis] 2016/03/22 15:12 ipa 2016/03/04 16:29 ipa [Partial Optimality in MAP-MRF] 2016/03/04 16:28 ipa [Partial Optimality in MAP-MRF] 2016/03/04 16:27 ipa [Partial Optimality in MAP-MRF] 2015/09/21 11:16 aneufeld 2015/09/21 11:16 aneufeld [Monocular 3D Reconstruction of Traffic Scenes from Optical Flow] 2015/08/21 10:50 ipa [Minimum Energy Filtering on Lie Groups with Application to Structure and Motion Estimation from Monocular Videos] 2015/08/20 13:03 aneufeld 2015/08/20 13:03 aneufeld 2015/08/13 15:55 aneufeld 2015/08/12 18:41 aneufeld 2015/08/12 18:34 aneufeld [Monocular 3D Reconstruction of Traffic Scenes from Optical Flow] 2015/08/12 18:33 aneufeld [Monocular 3D Reconstruction of Traffic Scenes from Optical Flow] 2015/08/12 18:33 aneufeld 2015/08/12 18:30 aneufeld 2015/08/12 18:30 aneufeld 2015/08/10 17:10 aneufeld 2015/08/10 17:09 aneufeld 2015/08/07 11:44 aneufeld 2015/08/07 11:43 aneufeld 2015/08/07 11:43 aneufeld 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 domains, scale 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 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 Assignment, preprint 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 Science, 11/2 (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: