# Research

## Variational Image Analysis on Manifolds and Metric Measure Spaces

**Scope.** We exploit basic statistical manifolds to devise variational models of low-level image analysis that exhibit favorable 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. A novel smooth dynamical system evolving on a statistical manifold, called

**, forms the basis of our work.**

*assignment flow*
**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.

A more classical *additive* variational reformulation provides a smooth geometric version of the continuous cut approach.

**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 discretization and optimization operations. Results reveal the steerability of the assignment flow and its potential for pattern *formation*.

**Unsupervised label learning.** Our recent work concerns the emergence of labels in a completely unsupervised way by data *self*-assignment. The resulting *self-assignment flow* has connections to low-rank matrix factorization and discrete optimal mass transport that are explored in our current work.

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*.

**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.

**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.

**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

## 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

Details: Paper

## 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 a 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 labeled variables. The method is very general, as it is applicable to models with arbitrary factors of 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