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research [2017/11/02 02:53]
research [2024/02/15 19:13] (current)
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 ====== Research ====== ====== Research ======
  
-===== Geometric Low-Level Variational Image Analysis ​on Metric Measure Spaces ​=====+===== Assignment Flows: Dynamical Systems ​on Riemannian Manifolds for Data Analysis ​=====
  
-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 modelssmoothness, probabilistic interpretation,​ efficiently converging parallel and sparse Riemannian numerical updates that scale up to large problem sizesThe 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.+This project is funded by the DFG within the [[https://www.foundationsofdl.de|Priority Programme ​on the Theoretical Foundations ​of Deep Learning]]
  
-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 solutionA key ingredient ​are local `Wasserstein messages'​ that couple local assignment measures ​along edges.+**Scope.** ​We study product spaces of elementary Riemannian manifolds for the context-sensitive analysis of data observed in any metric space. State spaces interact dynamically ​by geometric ​averaging and locally according ​to the adjacency structure of an underlying graphThe corresponding interaction parameters ​are learned from data. Geometric integration of the resulting continuous-time flow generates layers of a neural network. Our approach enables to study dynamical relations of inference and learning in neural networks from a geometric viewpoint, ​along with a probabilistic interpretation of contextual decision making. From the numerical point of view, the approach copes with high dimensions and large problem sizes.
  
-  ​[[https://​arxiv.org/​abs/​1710.01493|preprintarXiv:​1710.014932017]]+**Mathematical aspects.** Information geometrycoupled and regularized information transportgeometric mechanics on manifolds and variational principles, geometric numerical integration,​ statistical performance characterization using PAC-Bayesian analysis.
  
-Kick-off paper that introduces the basic approach: +**Recent work.** [[https://​ipa.math.uni-heidelberg.de/​publications|download]] 
- +  * extension to generative assignment flows for discrete joint distributions (arXiv:2402.07846, 2024) 
-  ​[[https://​link.springer.com/​article/​10.1007/​s10851-016-0702-4|J. Math. Imag. Vision, 2017]] +  * geometric embedding ​approach to multiple games and populations (arXiv::​2401.059182024) 
-  ​[[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]] +  quantum state assignment flows (Entropy2023) 
-  ​[[http://​www-rech.telecom-lille.fr/​diff-cv2016/​|Proceedings DIFF-CVML'​16;​ Grenander best paper award]] +  * geometric mechanics ​of assignment flows (Information Geometry2023) 
-  ​* [[https://​ipa.iwr.uni-heidelberg.de/​dokuwiki/​Papers/​Astroem2016d.pdf|Proceedings ECCV'​16]] +  novel PAC-Bayes bound for structured prediction (NeurIPS2023) 
-===== Estimating Vehicle Ego-Motion and Piecewise Planar Scene Structure from Optical Flow in a Continuous Framework ===== +  * self-certifying classification by linearized deep assignment flows (PAMM, 2023) 
- +  * non-local graph PDE for structured labeling based on assignment flows (SIAM SIIMS, 2023) 
-We propose a variational approach ​for estimating egomotion and structure of a static scene from a pair of images recorded by a single moving cameraIn our approach ​the scene structure is described by a set of 3D planar surfaces, which are linked ​to a SLIC superpixel decomposition of the image domainThe continuously parametrized planes are determined along with the extrinsic camera parameters by jointly minimizing a non-convex smooth objective functionthat comprises a data term based on the pre-calculated optical flow between the input images and suitable priors on the scene variables. +  * learning linearized assignment flows (JMIV, 2023) 
- +  * convergence and stability of assignment flows (Information Geometry2022) 
- +  * continuous-domain assignment flows (EuropJApplMath., 2021) 
-**Researchers**:​ Andreas Neufeld, Johannes Berger, Florian Becker, Frank LenzenChristoph Schnörr +  * order-constrained 3D OCT segmentation using assignment flows (IJCV, 2021) 
- +  * self-assignment flows (SIAM SIIMS, 2020) 
-[[research:​hflow:​start|Details]] +  unsupervised assignment flows (JMIV, 2020) 
- +  ​geometric integration of assignment flows (Inverse Problems2020) 
- +  ​* assignment flows for labeling: introduction (HandbookVarMethNonlGeomData, 2020) 
- +  * assignment flows for metric data labeling (JMIV, 2017)
-===== Minimum Energy Filtering on Lie Groups with Application to Structure and Motion Estimation from Monocular Videos ===== +
- +
-We investigate Minimum Energy Filters on Lie Groups in order to reliably estimate camera motion relative to a static scene from noisy data. In addition to properly taking into account the geometry ​of the state spacewe also deal with nonlinearities of the observation equation. A long-term objective concerns the estimation of accelerated camera motion in connection with scene depth from monocular videos. +
- +
-**Researchers**:​ +
-Johannes Berger, Andreas Neufeld, Florian Becker, Frank LenzenChristoph Schnörr +
- +
-[[https://​hciweb.iwr.uni-heidelberg.de/​node/​2385|Details]] +
-===== Partial Optimality in MAP-MRF ===== +
- +
-We consider the energy minimization problem for undirected graphical modelsalso known as MAP-inference problem for Markov random fields which is NP-hard in generalWe propose a novel polynomial time algorithm to obtain a part of its optimal non-relaxed integral solutionFor this task we devise a novel pruning strategy that utilizes standard MAP-solvers as subroutineWe show that our pruning strategy is in a certain sense theoretically optimalAlso empirically our method outperforms previous approaches in terms of the number of persistently labelled variables. The method is very generalas it is applicable to models with arbitrary factors of an arbitrary ​order and can employ any solver for the considered relaxed problem. Our method’s runtime is determined by the runtime of the convex relaxation solver for the MAP-inference problem. +
- +
-**Researchers**:​ Paul SwobodaBogdan Savchynskyy,​ Alexander Shekhovtsov,​ Jörg Hendrik Kappes, Christoph Schnörr\\ ​ +
-Details: ​ [[https://ipa.iwr.uni-heidelberg.de/​dokuwiki/​Papers/​Swoboda2016.pdf|Paper]]+