To Örebro University

Motion capture (MoCap) systems are indispensable tools across fields such as biomechanics, computer animation, human-robot interaction, and clinical gait analysis, owing to their ability to accurately record and analyze human movement in 3D space. Marker-based systems use reflective markers attached to subjects and video recordings to track human movement. The tracking requires markers to be detected in the video, which is not always possible due to occlusions, sensor failures, and limited camera coverage. These issues create gaps in recorded trajectories, compromising data integrity and making the motion difficult to utilize in practical applications. Therefore, a wide range of MoCap data completion techniques has been proposed to reconstruct missing trajectories while preserv ing the realism and dynamics of human movement. Human motion data exhibits a low-rank property due to the inherent repetitive nature of human movement as well as the correlations between joints and markers, enforced by the skeletal structure and biomechanical constraints. Low-rank completion techniques exploit this property to reconstruct missing marker positions. This paper reviews state-of-the-art low-rank completion methods for MoCap data completion, focusing specifically on optimization-based low-rank methods. These optimization approaches directly address the missing data completion problem through optimization formulations. We examine two main aspects: kinematic priors, which embed anatomical constraints, joint dependencies, and motion smoothness, and low-rank priors, which exploit inter-marker correlations through matrix and tensor formulations. We further eval uate optimization algorithms for solving these completion problems, such as alternating minimization, proximal algorithms, ADMM, and hybrid schemes, as well as the datasets and tools commonly used in the literature.

The complete eigenstructure, i.e. eigenvalues with multiplicities and minimal indices, of a skew-symmetric matrix pencil may change drastically if the matrix coefficients of the pencil are subjected to (even small) perturbations. These changes can be investigated qualitatively by constructing the stratification (closure hierarchy) graphs of the congruence orbits of the pencils. The results of this paper facilitate the construction of such graphs by providing all closest neighbours for a given node in the graph. More precisely, we prove a necessary and sufficient condition for one congruence orbit of a skew-symmetric matrix pencil, A, to belong to the closure of the congruence orbit of another pencil, B, such that there is no pencil, C, whose orbit contains the closure of the orbit of A and is contained in the closure of the orbit of B.

The gap-filling problem in motion capture (MoCap) data poses a significant challenge in marker-based MoCap systems. These gaps occur due to missing markers during motion recording. MoCap sequence, a time series data characterized by high dimensionality and temporal dependencies, requires accurate recovery of missing markers to ensure smooth motion representation. Tensor decomposition is an effective solution that leverages the multi-way structure of MoCap data. This paper proposes two gap-filling algorithms based on Tucker decomposition, namely Tucker and TuckerTNN. Given the high-dimensional nature of MoCap data, traditional smoothness regularization methods, such as gradient-based techniques, are computationally expensive. Therefore, we introduce the temporal nuclear norm in TuckerTNN as an alternative regularization technique, providing a more efficient solution for large-scale datasets. Both models are minimized using the proximal block coordinate descent (Prox-BCD) method. We evaluated the proposed algorithms using motion capture sequences from the publicly available HDM05 dataset. Our results show that Tucker and TuckerTNN consistently outperform existing approaches, such as CP and SparseCP, in accuracy and efficiency, with TuckerTNN offering the best trade-off between the two.

Motion capture (MoCap) technology is an essential tool for recording and analyzing movements of objects or humans. However, MoCap systems frequently encounter the challenge of missing data, stemming from mismatched markers, occlusion, or equipment limitations. Recovery of these missing data is imperative to maintain the reliability and integrity of MoCap recordings. This paper introduces a novel application of the tensor framework for MoCap data completion. We propose three completion algorithms based on the canonical polyadic (CP) decomposition of tensors. The first algorithm utilizes CP decomposition to capture the low-rank structure of the tensor. However, relying only on low-rank assumptions may be insufficient to deal with complex motion data. Thus, we propose two modified CP decompositions that incorporate additional information, SmoothCP and SparseCP decompositions. SmoothCP integrates piecewise smoothness prior, while SparseCP incorporates sparsity prior, each aiming to improve the accuracy and robustness of MoCap data recovery. To compare and evaluate the merit of the proposed algorithms over other tensor completion methods in terms of several evaluation metrics, we conduct numerical experiments with different MoCap sequences from the CMU motion capture dataset.

We show that the set of m x m complex skew-symmetric matrix polynomials of even grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m x m complex skew-symmetric matrix polynomials of even grade d and rank at most 2r. The analogous problem for the case of skew-symmetric matrix polynomials of odd grade is solved in [24].

We obtain the generic complete eigenstructures of complex Hermitian n x n matrix pencils with rank at most r (with r <= n). To do this, we prove that the set of such pencils is the union of a finite number of bundle closures, where each bundle is the set of complex Hermitian n x n pencils with the same complete eigenstructure (up to the specific values of the distinct finite eigenvalues). We also obtain the explicit number of such bundles and their codimension. The cases r = n, corresponding to general Hermitian pencils, and r < n exhibit surprising differences, since for r < n the generic complete eigenstructures can contain only real eigenvalues, while for r = n they can contain real and nonreal eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures.

In this paper, we consider the solvability conditions of some Sylvester-type quaternion matrix equations. We establish some practical necessary and sufficient conditions for the existence of solutions of a Sylvester-type quaternion matrix equation with five unknowns through the corresponding equivalence relations of the block matrices. Moreover, we present some solvability conditions to some Sylvester-type quaternion matrix equations, including those involving Hermicity. The findings of this article extend related known results.

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it.

The Gaussian phase-space representation can be used to implement quantum dynamics for fermionic particles numerically. To improve numerical results, we explore the use of dynamical diffusion gauges in such implementations. This is achieved by benchmarking quantum dynamics of few-body systems against independent exact solutions. A diffusion gauge is implemented here as a so-called noise-matrix, which satisfies a matrix equation defined by the corresponding Fokker-Planck equation of the phase-space representation. For the physical systems with fermionic particles considered here, the numerical evaluation of the new diffusion gauges allows us to double the practical simulation time, compared with hitherto known analytic noise-matrices. This development may have far reaching consequences for future quantum dynamical simulations of many-body systems. 

A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization to equivalent perturbations in the corresponding matrix polynomial, is needed. In this paper we develop an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations.