arXiv subjects

Jan Lellmann

Publications and source records attributed to Jan Lellmann.

At least 19 recordsLinked to original sources

arXiv2026

Diffeomorphism-Equivariant Neural Networks

Incorporating group symmetries via equivariance into neural networks has emerged as a robust approach for overcoming the efficiency and data demands of modern deep learning. While most existing approaches, such as group convolutions and averaging-based methods, focus on compact, finite, or low-dimensional groups with linear actions, this work explores how equivariance can be extended to infinite-dimensional groups. We propose a strategy designed to induce diffeomorphism equivariance in pre-trained neural networks via energy-based canonicalisation. Formulating equivariance as an optimisation problem allows us to access the rich toolbox of already established differentiable image registration methods. Empirical results on segmentation and classification tasks confirm that our approach achieves approximate equivariance and generalises to unseen transformations without relying on extensive data augmentation or retraining.

cs.LG↗

arXiv2024

Stationary Velocity Fields on Matrix Groups for Deformable Image Registration

The stationary velocity field (SVF) approach allows to build parametrizations of invertible deformation fields, which is often a desirable property in image registration. Its expressiveness is particularly attractive when used as a block following a machine learning-inspired network. However, it can struggle with large deformations. We extend the SVF approach to matrix groups, in particular $\SE(3)$. This moves Euclidean transformations into the low-frequency part, towards which network architectures are often naturally biased, so that larger motions can be recovered more easily. This requires an extension of the flow equation, for which we provide sufficient conditions for existence. We further prove a decomposition condition that allows us to apply a scaling-and-squaring approach for efficient numerical integration of the flow equation. We numerically validate the approach on inter-patient registration of 3D MRI images of the human brain.

cs.CV↗

arXiv2023

Riemannian geometry for efficient analysis of protein dynamics data

An increasingly common viewpoint is that protein dynamics data sets reside in a non-linear subspace of low conformational energy. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich structure to account for a wide range of geometries that can be modelled after an energy landscape. Second, many standard data analysis tools initially developed for data in Euclidean space can also be generalised to data on a Riemannian manifold. In the context of protein dynamics, a conceptual challenge comes from the lack of a suitable smooth manifold and the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a novel local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold of point clouds modulo rigid body group actions and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. It performs exceptionally well on coarse-grained molecular dynamics simulated data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop.

q-bio.BM↗

arXiv2023

A Universal Quantum Algorithm for Weighted Maximum Cut and Ising Problems

We propose a hybrid quantum-classical algorithm to compute approximate solutions of binary combinatorial problems. We employ a shallow-depth quantum circuit to implement a unitary and Hermitian operator that block-encodes the weighted maximum cut or the Ising Hamiltonian. Measuring the expectation of this operator on a variational quantum state yields the variational energy of the quantum system. The system is enforced to evolve towards the ground state of the problem Hamiltonian by optimizing a set of angles using normalized gradient descent. Experimentally, our algorithm outperforms the state-of-the-art quantum approximate optimization algorithm on random fully connected graphs and challenges D-Wave quantum annealers by producing good approximate solutions. Source code and data files are publicly available.

quant-ph↗

arXiv2022

Regularising orientation estimation in Cryo-EM 3D map refinement through measure-based lifting over Riemannian manifolds

Motivated by the trade-off between noise-robustness and data-consistency for joint 3D map reconstruction and rotation estimation in single particle cryogenic-electron microscopy (Cryo-EM), we propose ellipsoidal support lifting (ESL), a measure-based lifting scheme for regularising and approximating the global minimiser of a smooth function over a Riemannian manifold. Under a uniqueness assumption on the minimiser we show several theoretical results, in particular well-posedness of the method and an error bound due to the induced bias with respect to the global minimiser. Additionally, we use the developed theory to integrate the measure-based lifting scheme into an alternating update method for joint homogeneous 3D map reconstruction and rotation estimation, where typically tens of thousands of manifold-valued minimisation problems have to be solved and where regularisation is necessary because of the high noise levels in the data. The joint recovery method is used to test both the theoretical predictions and algorithmic performance through numerical experiments with Cryo-EM data. In particular, the induced bias due to the regularising effect of ESL empirically estimates better rotations, i.e., rotations closer to the ground truth, than global optimisation would.

math.OC↗

arXiv2022

Inverse Scale Space Iterations for Non-Convex Variational Problems: The Continuous and Discrete Case

Non-linear filtering approaches allow to obtain decompositions of images with respect to a non-classical notion of scale, induced by the choice of a convex, absolutely one-homogeneous regularizer. The associated inverse scale space flow can be obtained using the classical Bregman iteration with quadratic data term. We apply the Bregman iteration to lifted, i.e. higher-dimensional and convex, functionals in order to extend the scope of these approaches to functionals with arbitrary data term. We provide conditions for the subgradients of the regularizer -- in the continuous and discrete setting -- under which this lifted iteration reduces to the standard Bregman iteration. We show experimental results for the convex and non-convex case.

math.NA↗

arXiv2021

Inverse Scale Space Iterations for Non-Convex Variational Problems Using Functional Lifting

Non-linear filtering approaches allow to obtain decompositions of images with respect to a non-classical notion of scale. The associated inverse scale space flow can be obtained using the classical Bregman iteration applied to a convex, absolutely one-homogeneous regularizer. In order to extend these approaches to general energies with non-convex data term, we apply the Bregman iteration to a lifted version of the functional with sublabel-accurate discretization. We provide a condition for the subgradients of the regularizer under which this lifted iteration reduces to the standard Bregman iteration. We show experimental results for the convex and non-convex case.

math.NA↗

arXiv2021

An Inexact Semi-smooth Newton Method on Riemannian Manifolds with Application to Duality-based Total Variation Denoising

We propose a higher-order method for solving non-smooth optimization problems on manifolds. In order to obtain superlinear convergence, we apply a Riemannian Semi-smooth Newton method to a non-smooth non-linear primal-dual optimality system based on a recent extension of Fenchel duality theory to Riemannian manifolds. We also propose an inexact version of the Riemannian Semi-smooth Newton method and prove conditions for local linear and superlinear convergence that hold independent of the sign of the curvature. Numerical experiments on l2-TV-like problems with dual regularization confirm superlinear convergence on manifolds with positive and negative curvature.

math.OC↗

arXiv2020

On the Connection between Dynamical Optimal Transport and Functional Lifting

Functional lifting methods provide a tool for approximating solutions of difficult non-convex problems by embedding them into a larger space. In this work, we investigate a mathematically rigorous formulation based on embedding into the space of pointwise probability measures over a fixed range $\Gamma$. Interestingly, this approach can be derived as a generalization of the theory of dynamical optimal transport. Imposing the established continuity equation as a constraint corresponds to variational models with first-order regularization. By modifying the continuity equation, the approach can also be extended to models with higher-order regularization.

math.OC↗

arXiv2020

Deformable Groupwise Image Registration using Low-Rank and Sparse Decomposition

Low-rank and sparse decompositions and robust PCA (RPCA) are highly successful techniques in image processing and have recently found use in groupwise image registration. In this paper, we investigate the drawbacks of the most common RPCA-dissimi\-larity metric in image registration and derive an improved version. In particular, this new metric models low-rank requirements through explicit constraints instead of penalties and thus avoids the pitfalls of the established metric. Equipped with total variation regularization, we present a theoretically justified multilevel scheme based on first-order primal-dual optimization to solve the resulting non-parametric registration problem. As confirmed by numerical experiments, our metric especially lends itself to data involving recurring changes in object appearance and potential sparse perturbations. We numerically compare its peformance to a number of related approaches.

cs.CV↗

arXiv2019

Lifting methods for manifold-valued variational problems

Lifting methods allow to transform hard variational problems such as segmentation and optical flow estimation into convex problems in a suitable higher-dimensional space. The lifted models can then be efficiently solved to a global optimum, which allows to find approximate global minimizers of the original problem. Recently, these techniques have also been applied to problems with values in a manifold. We provide a review of such methods in a refined framework based on a finite element discretization of the range, which extends the concept of sublabel-accurate lifting to manifolds. We also generalize existing methods for total variation regularization to support general convex regularization.

math.NA↗

arXiv2019

Functional Liftings of Vectorial Variational Problems with Laplacian Regularization

We propose a functional lifting-based convex relaxation of variational problems with Laplacian-based second-order regularization. The approach rests on ideas from the calibration method as well as from sublabel-accurate continuous multilabeling approaches, and makes these approaches amenable for variational problems with vectorial data and higher-order regularization, as is common in image processing applications. We motivate the approach in the function space setting and prove that, in the special case of absolute Laplacian regularization, it encompasses the discretization-first sublabel-accurate continuous multilabeling approach as a special case. We present a mathematical connection between the lifted and original functional and discuss possible interpretations of minimizers in the lifted function space. Finally, we exemplarily apply the proposed approach to 2D image registration problems.

math.NA↗

arXiv2018

Fully-deformable 3D image registration in two seconds

We present a highly parallel method for accurate and efficient variational deformable 3D image registration on a consumer-grade graphics processing unit (GPU). We build on recent matrix-free variational approaches and specialize the concepts to the massively-parallel manycore architecture provided by the GPU. Compared to a parallel and optimized CPU implementation, this allows us to achieve an average speedup of 32.53 on 986 real-world CT thorax-abdomen follow-up scans. At a resolution of approximately $256^3$ voxels, the average runtime is 1.99 seconds for the full registration. On the publicly available DIR-lab benchmark, our method ranks third with respect to average landmark error at an average runtime of 0.32 seconds.

cs.CV↗

arXiv2018

Higher-Order Total Directional Variation: Imaging Applications

We introduce a class of higher-order anisotropic total variation regularisers, which are defined for possibly inhomogeneous, smooth elliptic anisotropies, that extends the Total Generalized Variation (TGV) regulariser and its variants. We propose a primal-dual hybrid gradient approach to approximate numerically the associated gradient flow. This choice of regularisers allows to preserve and enhance intrinsic anisotropic features in images. This is illustrated on various examples from different imaging applications: image denoising, wavelet-based image zooming, and reconstruction of surfaces from scattered height measurements.

math.NA↗

arXiv2018

A multi-contrast MRI approach to thalamus segmentation

Thalamic alterations are relevant to many neurological disorders including Alzheimer's disease, Parkinson's disease and multiple sclerosis. Routine interventions to improve symptom severity in movement disorders, for example, often consist of surgery or deep brain stimulation to diencephalic nuclei. Therefore, accurate delineation of grey matter thalamic subregions is of the upmost clinical importance. MRI is highly appropriate for structural segmentation as it provides different views of the anatomy from a single scanning session. Though with several contrasts potentially available, it is also of increasing importance to develop new image segmentation techniques that can operate multi-spectrally. We hereby propose a new segmentation method for use with multi-modality data, which we evaluated for automated segmentation of major thalamic subnuclear groups using T1-, T2*-weighted and quantitative susceptibility mapping (QSM) information. The proposed method consists of four steps: highly iterative image co-registration, manual segmentation on the average training-data template, supervised learning for pattern recognition, and a final convex optimisation step imposing further spatial constraints to refine the solution. This led to solutions in greater agreement with manual segmentation than the standard Morel atlas based approach. Furthermore, we show that the multi-contrast approach boosts segmentation performances. We then investigated whether prior knowledge using the training-template contours could further improve convex segmentation accuracy and robustness, which led to highly precise multi-contrast segmentations in single subjects. This approach can be extended to most 3D imaging data types and any region of interest discernible in single scans or multi-subject templates.

cs.CV↗

arXiv2018

A matrix-free approach to parallel and memory-efficient deformable image registration

We present a novel computational approach to fast and memory-efficient deformable image registration. In the variational registration model, the computation of the objective function derivatives is the computationally most expensive operation, both in terms of runtime and memory requirements. In order to target this bottleneck, we analyze the matrix structure of gradient and Hessian computations for the case of the normalized gradient fields distance measure and curvature regularization. Based on this analysis, we derive equivalent matrix-free closed-form expressions for derivative computations, eliminating the need for storing intermediate results and the costs of sparse matrix arithmetic. This has further benefits: (1) matrix computations can be fully parallelized, (2) memory complexity for derivative computation is reduced from linear to constant, and (3) overall computation times are substantially reduced. In comparison with an optimized matrix-based reference implementation, the CPU implementation achieves speedup factors between 3.1 and 9.7, and we are able to handle substantially higher resolutions. Using a GPU implementation, we achieve an additional speedup factor of up to 9.2. Furthermore, we evaluated the approach on real-world medical datasets. On ten publicly available lung CT images from the DIR-Lab 4DCT dataset, we achieve the best mean landmark error of 0.93 mm compared to other submissions on the DIR-Lab website, with an average runtime of only 9.23 s. Complete non-rigid registration of full-size 3D thorax-abdomen CT volumes from oncological follow-up is achieved in 12.6 s. The experimental results show that the proposed matrix-free algorithm enables the use of variational registration models also in applications which were previously impractical due to memory or runtime restrictions.

cs.CV↗

arXiv2017

Measure-Valued Variational Models with Applications to Diffusion-Weighted Imaging

We develop a general mathematical framework for variational problems where the unknown function assumes values in the space of probability measures on some metric space. We study weak and strong topologies and define a total variation seminorm for functions taking values in a Banach space. The seminorm penalizes jumps and is rotationally invariant under certain conditions. We prove existence of a minimizer for a class of variational problems based on this formulation of total variation, and provide an example where uniqueness fails to hold. Employing the Kan\-torovich-Rubinstein transport norm from the theory of optimal transport, we propose a variational approach for the restoration of orientation distribution function (ODF)-valued images, as commonly used in Diffusion MRI. We demonstrate that the approach is numerically feasible on several data sets.

math.NA↗

arXiv2017

Image reconstruction with imperfect forward models and applications in deblurring

We present and analyse an approach to image reconstruction problems with imperfect forward models based on partially ordered spaces - Banach lattices. In this approach, errors in the data and in the forward models are described using order intervals. The method can be characterised as the lattice analogue of the residual method, where the feasible set is defined by linear inequality constraints. The study of this feasible set is the main contribution of this paper. Convexity of this feasible set is examined in several settings and modifications for introducing additional information about the forward operator are considered. Numerical examples demonstrate the performance of the method in deblurring with errors in the blurring kernel.

math.NA↗