arXiv subjects

Jan Krejčí

Publications and source records attributed to Jan Krejčí.

9 recordsLinked to original sources

arXiv2026

Operators on Orlicz sequence spaces and $Δ_2$-fundamentality

A classical result states that the Hardy--Littlewood maximal operator is bounded on an Orlicz space $L^A(\mathbb{R}^n)$ if and only if its conjugate Young function $\tilde{A}$ satisfies the $Δ_2$-condition. The same condition also characterizes the boundedness on $L^A(0,\infty)$ of the Hardy averaging operator. We consider a discrete analogue of the problem, extended to a general interpolation framework. We offer several characterizing conditions for the boundedness of discrete maximal and average operators on Orlicz spaces. Although the principal result is as expected, for its proof some new techniques have to be developed. To this end, we introduce a new notion of the so-called $Δ_2$-fundamental sequence, and give its interesting characterization by a simple condition involving only a limes superior of the ratio of two subsequent terms. We also prove a dual statement concerning operators of Copson type.

math.FA↗

arXiv2026

GOSPA and T-GOSPA quasi-metrics for evaluation of multi-object tracking algorithms

This paper introduces two quasi-metrics for performance assessment of multi-object tracking (MOT) algorithms. One quasi-metric is an extension of the generalised optimal subpattern assignment (GOSPA) metric and measures the discrepancy between sets of objects. The other quasi-metric is an extension of the trajectory GOSPA (T-GOSPA) metric and measures the discrepancy between sets of trajectories. Similar to the GOSPA-based metrics, these quasi-metrics include costs for localisation error for properly detected objects, the number of false objects and the number of missed objects. The T-GOSPA quasi-metric also includes a track switching cost. Differently from the GOSPA and T-GOSPA metrics, the proposed quasi-metrics have the flexibility of penalising missed and false objects with different costs, and the localisation costs are not required to be symmetric. We also explain how to obtain similarity score functions based on these quasi-metrics. The performance of several Bayesian MOT algorithms is assessed with the T-GOSPA quasi-metric via simulations.

cs.CV↗

arXiv2026

Occlusion-Aware Multi-Object Tracking via Expected Probability of Detection

This paper addresses multi-object systems, where objects may occlude one another relative to the sensor. The standard point-object model for detection-based sensors is enhanced so that the probability of detection considers the presence of all objects. A principled tracking method is derived, assigning each object an expected probability of detection, where the expectation is taken over the reduced Palm density, which means conditionally on the object's existence. The assigned probability thus considers the object's visibility relative to the sensor, under the presence of other objects. Unlike existing methods, the proposed method systematically accounts for uncertainties related to all objects in a clear and manageable way. The method is demonstrated through a visual tracking application using the multi-Bernoulli mixture (MBM) filter with marks.

eess.SY↗

arXiv2026

A Quantum Algorithm for the Diffusion Step of Grid-based Filter

We propose a simple quantum algorithm for implementing the diffusion step of grid-based Bayesian filters. The method encodes the advected state density and the process noise density into quantum registers and realizes diffusion using a quantum Fourier transform--based adder. This avoids the explicit convolution required in classical implementations and the repeated coin-flip operations used in quantum random walk approaches. Numerical simulations using a gate-based quantum computing simulator confirm that the proposed approach reproduces the desired probability densities while requiring significantly fewer quantum gates and much shallower circuit depth.

eess.SP↗

arXiv2026

Lagrangian Grid-based Estimation of Nonlinear Systems with Invertible Dynamics

This paper deals with the state estimation of non-linear and non-Gaussian systems with an emphasis on the numerical solution to the Bayesian recursive relations. In particular, this paper builds upon the Lagrangian grid-based filter (GbF) recently-developed for linear systems and extends it for systems with nonlinear dynamics that are invertible. The proposed nonlinear Lagrangian GbF reduces the computational complexity of the standard GbFs from quadratic to log-linear, while preserving all the strengths of the original GbF such as robustness, accuracy, and deterministic behaviour. The proposed filter is compared with the particle filter in several numerical studies using the publicly available MATLAB\textregistered\ implementation\footnote{https://github.com/pesslovany/Matlab-LagrangianPMF}.

eess.SP↗

arXiv2025

Revisiting Functional Derivatives in Multi-object Tracking

Probability generating functionals (PGFLs) are efficient and powerful tools for tracking independent objects in clutter. It was shown that PGFLs could be used for the elegant derivation of practical multi-object tracking algorithms, e.g., the probability hypothesis density (PHD) filter. However, derivations using PGFLs use the so-called functional derivatives whose definitions usually appear too complicated or heuristic, involving Dirac delta ``functions''. This paper begins by comparing different definitions of functional derivatives and exploring their relationships and implications for practical applications. It then proposes a rigorous definition of the functional derivative, utilizing straightforward yet precise mathematics for clarity. Key properties of the functional derivative are revealed and discussed.

eess.SY↗

arXiv2025

TGOSPA Metric Parameters Selection and Evaluation for Visual Multi-object Tracking

Multi-object tracking algorithms are deployed in various applications, each with different performance requirements. For example, track switches pose significant challenges for offline scene understanding, as they hinder the accuracy of data interpretation. Conversely, in online surveillance applications, their impact is often minimal. This disparity underscores the need for application-specific performance evaluations that are both simple and mathematically sound. The trajectory generalized optimal sub-pattern assignment (TGOSPA) metric offers a principled approach to evaluate multi-object tracking performance. It accounts for localization errors, the number of missed and false objects, and the number of track switches, providing a comprehensive assessment framework. This paper illustrates the effective use of the TGOSPA metric in computer vision tasks, addressing challenges posed by the need for application-specific scoring methodologies. By exploring the TGOSPA parameter selection, we enable users to compare, comprehend, and optimize the performance of algorithms tailored for specific tasks, such as target tracking and training of detector or re-ID modules.

eess.SY↗

arXiv2025

Model-based Multi-object Visual Tracking: Identification and Standard Model Limitations

This paper uses multi-object tracking methods known from the radar tracking community to address the problem of pedestrian tracking using 2D bounding box detections. The standard point-object (SPO) model is adopted, and the posterior density is computed using the Poisson multi-Bernoulli mixture (PMBM) filter. The selection of the model parameters rooted in continuous time is discussed, including the birth and survival probabilities. Some parameters are selected from the first principles, while others are identified from the data, which is, in this case, the publicly available MOT-17 dataset. Although the resulting PMBM algorithm yields promising results, a mismatch between the SPO model and the data is revealed. The model-based approach assumes that modifying the problematic components causing the SPO model-data mismatch will lead to better model-based algorithms in future developments.

eess.SY↗

arXiv2024

Pedestrian Tracking with Monocular Camera using Unconstrained 3D Motion Model

A first-principle single-object model is proposed for pedestrian tracking. It is assumed that the extent of the moving object can be described via known statistics in 3D, such as pedestrian height. The proposed model thus need not constrain the object motion in 3D to a common ground plane, which is usual in 3D visual tracking applications. A nonlinear filter for this model is implemented using the unscented Kalman filter (UKF) and tested using the publicly available MOT-17 dataset. The proposed solution yields promising results in 3D while maintaining perfect results when projected into the 2D image. Moreover, the estimation error covariance matches the true one. Unlike conventional methods, the introduced model parameters have convenient meaning and can readily be adjusted for a problem.

cs.CV↗