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research [2017/11/01 17:33]
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 framework ​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]]
  
-**Researchers**: Freddie AströmRuben GarskeJudit Recknagel, Fabrizio Savarino, Christoph Schnörr, Alexander Zeilmann+**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 graph. The 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 viewpointalong with a probabilistic interpretation of contextual decision making. From the numerical point of viewthe approach copes with high dimensions and 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.+**Mathematical aspects.** Information geometry, coupled and regularized information transport, geometric mechanics ​on manifolds and variational principles, ​geometric ​numerical integration,​ statistical performance characterization using PAC-Bayesian analysis.
  
-  ​[[https://​arxiv.org/​abs/​1710.01493|preprint,​ arXiv:​1710.01493,​ 2017]] +**Recent work.** [[https://​ipa.math.uni-heidelberg.de/​publications|download]] 
- +  * extension to generative assignment flows for discrete joint distributions (arXiv:2402.078462024) 
-Kick-off paper that introduces the basic approach: +  * geometric embedding approach ​to multiple games and populations (arXiv::​2401.059182024) 
- +  quantum state assignment flows (Entropy2023) 
-  ​[[https://​link.springer.com/​article/​10.1007/​s10851-016-0702-4|J. Math. Imag. Vision, 2017]] +  * geometric mechanics ​of assignment flows (Information Geometry, 2023) 
-  ​[[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]] +  * novel PAC-Bayes bound for structured prediction ​(NeurIPS, 2023
-  * [[http://​www-rech.telecom-lille.fr/​diff-cv2016/​|Proceedings DIFF-CVML'​16;​ Grenander best paper award]] +  self-certifying classification by linearized deep assignment flows (PAMM2023) 
-  ​* [[https://​ipa.iwr.uni-heidelberg.de/​dokuwiki/​Papers/​Astroem2016d.pdf|Proceedings ECCV'​16]] +  * non-local graph PDE for structured labeling based on assignment flows (SIAM SIIMS2023) 
-===== Estimating Vehicle Ego-Motion and Piecewise Planar Scene Structure from Optical Flow in a Continuous Framework ===== +  learning linearized assignment flows (JMIV2023
- +  * convergence ​and stability ​of assignment flows (Information Geometry2022) 
-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 surfaceswhich 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. +  continuous-domain assignment flows (EuropJApplMath., 2021) 
- +  * order-constrained 3D OCT segmentation using assignment flows (IJCV, 2021) 
- +  * self-assignment flows (SIAM SIIMS, 2020) 
-**Researchers**:​ Andreas NeufeldJohannes Berger, Florian Becker, Frank Lenzen, Christoph Schnörr +  unsupervised assignment flows (JMIV2020) 
- +  ​* geometric integration ​of assignment flows (Inverse Problems, 2020) 
-[[research:​hflow:​start|Details]] +  * assignment flows for labeling: introduction ​(Handbook: VarMethNonlGeomData2020) 
- +  assignment flows for metric data labeling (JMIV2017)
-===== Phase Transitions and Recovery ​of Cosparse Objects Through Limited Angle Tomography ===== +
- +
-Sampling patterns as used in industrial tomographical set-ups with limited numbers of projections fall short of the common assumptions ​(e.g.~restricted isometry propertyunderlying compressed sensing. In this project, we investigate the relation between the number of sufficient tomographic projections and the co-/​sparsity of volume functions for unique recovery of these functions from given projection data. We also investigate approaches to efficiently solve the corresponding large numerical optimization problem in the 3D case. +
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-**Researchers**:​ Andreea DenitiuStefania Petra, Christoph Schnörr +
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-[[research:​tomography:​start|Details]] +
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-===== Segmentation of Thin Fiber Structures in 3D Tomographical Data ===== +
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-Recognizing the microstructure of fiber reinforced polymers is a necessary prerequisite for inferring macroscopic material properties. This project aims to combine low-level image processing and stochastic models of fiber distributions in order to reliably segment fibers in noisy image data. A major aspect concerns variational methods ​for approximate inference in connection with set coveringGibbs distributions and marked point processes. +
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-**Researchers**:​  +
-Peter MarkowskyTabea Zuber, Gabriele Steidl (TU Kaiserslautern), Christoph Schnörr +
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-===== Minimum Energy Filtering on Lie Groups with Application to Structure ​and Motion Estimation from Monocular Videos ===== +
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-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. +
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-**Researchers**:​ +
-Johannes Berger, Andreas Neufeld, Florian Becker, Frank Lenzen, Christoph Schnörr +
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-[[https://​hciweb.iwr.uni-heidelberg.de/​node/​2385|Details]] +
-===== Partial Optimality in MAP-MRF ===== +
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-We consider the energy minimization problem for undirected graphical models, also 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 subroutine. We show that our pruning strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous approaches in terms of the number of persistently labelled variablesThe 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. +
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-**Researchers**:​ Paul SwobodaBogdan Savchynskyy,​ Alexander Shekhovtsov,​ Jörg Hendrik Kappes, Christoph Schnörr\\ ​ +
-Details: ​ [[https://​ipa.iwr.uni-heidelberg.de/​dokuwiki/​Papers/​Swoboda2016.pdf|Paper]] +
-===== Context Specific Independence and Graphical Models ===== +
- +
-This projects focuses on the connections between the field of probabilistic graphical models and the family of models that are able to represent efficiently distributions in which Context Specific Independences ​(CSIsappear ​(dependences that are not fixed but only happen given a certain "​context"​ state)Such models include And/Or graphs, Arithmetic Circuits, Sum-Product Networks, and others. +
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-These CSI models are more expressive than graphical models but they lack the compactness and the theoretical framework of the latterBy creating new connections between the two fields we aim to translate algorithms and methodologies from one to the other, thereby obtaining a compact representation of CSI enjoying the sound probabilistic framework of graphical models. +
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-A long-term objective is to better understand the properties of  +
-deep learning architecturesin particular those with a probabilistic  +
-interpretation. +
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-**Researchers**:​ Mattia DesanaChristoph Schnörr+