Manifold-valued Graph Neural Networks (MATH+ AA5-3)
Project Head together with G. Steidl

Geometry-aware, data-analytic approaches improve understanding and assessment of pathophysiological processes. We will derive a new theoretical framework for deep neural networks that can cope with geometric data and apply it for classification of musculoskeletal illness from both shape and movement patterns.

A Soft-Correspondence Approach to Shape Analysis (MATH+ EF3-13)
Project Head together with D. Baum, A. Bobenko, and F. Fless

Despite many advances, frameworks for geometric morphometry still rely on point-to-point correspondences between shapes, either explicitly in form of homologous landmarks or implicitly in terms of diffeomorphisms of the ambient space. Point-to-point correspondences, however, have fundamental limitations that prohibit the analysis of shape collections with incomplete or topologically varying objects. The goal of this project is to extend the scope of shape analysis methodology in order to overcome these limitations.

Spline Models for Shape Trajectory Analysis (MATH+ EF2-3)
Project Head together with H.-C. Hege, A. Hennemuth, and F. Fless

The aim of the project is the development of novel methods for the identification of systematic differences in empirically defined distributions of shape trajectories. The forcus will be on the derivation of hierarchical statistical models that account for the correlation inherent to time series of shapes. To this end, we will generalize the mixed-effects framework to manifold-valued data based on Riemannian spline models. The development will be driven by applications from cardiology and archaeology.

Analysis of Empirical Shape Trajectories (ECMath/MATH+ CH15)
Project Head together with H.-C. Hege and T. Sullivian

The focus of this project is to develop new mathematical methods for the analysis, processing and reconstruction of empirically defined shape trajectories. By treating the trajectories as curves in shape space, we plan to exploit the rich geometric structure inherent to these spaces. In consequence, we expect the derived schemes to benefit from a compact encoding of constraints and a superior consistency as compared to their Euclidean counterparts.
 

In-vivo soft tissue kinematics from dynamic MRI

The aim of this project is the development of MRI-based methods and technologies for the non-invasive in-vivo assessment of mechanical loads in soft tissue structures of the knee joint. Given the short acquisition times available in the dynamic imaging setup only partial information (low resolution, 2D slices instead of 3D volume) can be acquired. Therefore, specialized reconstruction methods have to be developed to extract the motion of soft tissues from the incomplete data. The obtained time-dependent configurations of the structures can then be interpreted as shape trajectories allowing morphometric analysis employing manifold-valued statistical tools.

KneeLaxity – Dynamic multi-modal knee joint registration

The goal of the project is to characterize knee joint laxity in vivo as well as their consequences on the onset and progression of osteoarthritis. To this end, patient specific 3D joint structures are to be matched to 2D fluoroscopic images. While MRI provides a non-invasive imaging basis, registering the according images to fluoroscopic data automatically is a challenging task as the acquired information does not map directly to the X-ray absorption properties of the imaged tissue. Based on the developed 3D reconstruction techniques we will furthermore assess and improve skin marker-based reconstruction methods.
 

X-ray based anatomy reconstruction with low radiation exposure

This project aims at improving 3D reconstruction methods from 2D X-ray images. A special focus is on finding a small set of X-ray imaging directions for a sufficiently accurate 3D reconstruction of anatomical structures, thus reducing the radiation dose of X-ray based medical imaging. We will improve the reconstruction accuracy obtained with existing 3D-from-2D techniques and the data acquisition in form of an optimal design of experiments to reduce the radiation dose. Within the project we are aiming at the development a fast design optimization algorithm to optimize individual imaging plans on the fly.

3D reconstruction of anatomical structures from 2D X-ray images

2D X-ray images play a crucial role for the diagnosis and the therapy planning in orthopaedics. This imaging technique is not only widely available but is also, in contrast to more advanced 3D imaging methods like CT or MRI, considered a fast and inexpensive procedure. However, musculoskeletal anatomy is inherently 3D, and accurate assessment of the patients anatomy is therefore largely dependent upon the individual surgeon’s experience and ability to assess the 3D anatomy by analysis of 2D images only. To overcome this limitation, and in order to enable reliable, quantitative planning of surgical interventions that is less dependent upon the experience of the surgeon, the goal of this project is to provide a method aimed at 3D reconstruction of musculoskeletal anatomy from standard 2D projection X-ray images.

MATHEON F4 – Geometric Shape Optimization

Geometric shape optimization is concerned with novel discrete surface energies and computational geometry algorithms for processing and optimizing polyhedral meshes in industrial applications such as computer aided design (CAD) and computer graphics. The main focus of this project addresses the development of efficient mesh processing algorithms and robust modeling tools. The mathematical tools are novel discrete differential and curvature operators.
 

MATHEON F9 – Trajectory Compression

For large, nonlinear, and time dependent PDE constrained optimization problems with 3D spatial domain, reduced methods are a viable algorithmic approach. The computation of reduced gradients by adjoint methods requires the storage of 4D data, which can be quite expensive from both a capacity and bandwidth point of view. This project investigates lossy compression schemes for storing the state trajectory, based on hierarchical interpolation in adaptively refined meshes as a general predictor.

MATHEON F11 – Accelerating Curvature Flows on the GPU

The goal of this project to speed up the computation of curvature flows, previously implemented on the CPU. This project will in particular demonstrate numerics that fully utilize the SIMT (single-instruction, multiple-thread) architecture of modern Graphics cards (GPU), and provide a proof-of-concept implementation of advanced geometry processing on the GPU. In particular, industrial design and CAD companies could use such fast curvature flow or smoothing algorithms for their laser scanners to quickly capture models accurately and cheaply compared to their current setup.