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# Research

## Geometric Low-Level Variational Image Analysis on 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.

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.

Kick-off paper that introduces the basic approach:

## Estimating Vehicle Ego-Motion and Piecewise Planar Scene Structure from Optical Flow in a Continuous Framework

We propose a variational approach for estimating egomotion and structure of a static scene from a pair of images recorded by a single moving camera. In 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 domain. The continuously parametrized planes are determined along with the extrinsic camera parameters by jointly minimizing a non-convex smooth objective function, that comprises a data term based on the pre-calculated optical flow between the input images and suitable priors on the scene variables.

**Researchers**: Andreas Neufeld, Johannes Berger, Florian Becker, Frank Lenzen, Christoph Schnörr

## 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 property) underlying 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.

**Researchers**: Andreea Denitiu, Stefania Petra, Christoph Schnörr

## Segmentation of Thin Fiber Structures in 3D Tomographical Data

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 covering, Gibbs distributions and marked point processes.

**Researchers**:
Peter Markowsky, Tabea Zuber, Gabriele Steidl (TU Kaiserslautern), Christoph Schnörr

## 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 space, we 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 Lenzen, Christoph Schnörr

## Partial Optimality in MAP-MRF

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 general. We propose a novel polynomial time algorithm to obtain a part of its optimal non-relaxed integral solution. For 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 variables. The method is very general, as 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 Swoboda, Bogdan Savchynskyy, Alexander Shekhovtsov, Jörg Hendrik Kappes, Christoph Schnörr

Details: 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 (CSIs) appear (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.

These CSI models are more expressive than graphical models but they lack the compactness and the theoretical framework of the latter. By 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.

A long-term objective is to better understand the properties of deep learning architectures, in particular those with a probabilistic interpretation.

**Researchers**: Mattia Desana, Christoph Schnörr