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A fast and gridless ORKA algorithm for tracking moving and deforming objects

Florian Bossmann, Jianwei Ma, Wenze wu

TL;DR

This work tackles the problem of tracking moving and deforming objects across multiple measurements by extending ORKA with a fast gridless, multiresolution framework (fg-ORKA). It replaces the grid-bound, exponential-search steps with an iterative down/upsampling strategy that refines the movement path across resolutions using resampling operators (Wavelet, Fourier, or optimal) and a small per-iteration search radius, achieving complexity roughly $O(3^K)$ per iteration and decoupling from the original data resolution. The authors provide detailed complexity and error analyses, introduce an optimal resampling method to minimize projection error, and demonstrate substantial runtime gains along with improved sub-grid movement estimation on synthetic and real data, including seismic and video sequences. The approach enables gridless, scalable tracking in diverse applications, while also outlining limitations (e.g., periodic shifts) and directions for future enhancement. The combination of iterative multiresolution processing, three resampling strategies, and rigorous analysis constitutes a practical, faster alternative to the original grid-dependent ORKA for moving/deforming object tracking.

Abstract

Identifying objects in given data is a task frequently encountered in many applications. Finding vehicles or persons in video data, tracking seismic waves in geophysical exploration data, or predicting a storm front movement from meteorological measurements are only some of the possible applications. In many cases, the object of interest changes its form or position from one measurement to another. For example, vehicles in a video may change its position or angle to the camera in each frame. Seismic waves can change its arrival time, frequency, or intensity depending on the sensor position. Storm fronts can change its form and position over time. This complicates the identification and tracking as the algorithm needs to deal with the changing object over the given measurements. In a previous work, the authors presented a new algorithm to solve this problem - Object reconstruction using K-approximation (ORKA). The algorithm can solve the problem at hand but suffers from two disadvantages. On the one hand, the reconstructed object movement is bound to a grid that depends on the data resolution. On the other hand, the complexity of the algorithm increases exponentially with the resolution. We overcome both disadvantages by introducing an iterative strategy that uses a resampling method to create multiple resolutions of the data. In each iteration the resolution is increased to reconstruct more details of the object of interest. This way, we can even go beyond the original resolution by artificially upsampling the data. We give error bounds and a complexity analysis of the new method. Furthermore, we analyze its performance in several numerical experiments as well as on real data. We also give a brief introduction on the original ORKA algorithm. Knowledge of the previous work is thus not required.

A fast and gridless ORKA algorithm for tracking moving and deforming objects

TL;DR

This work tackles the problem of tracking moving and deforming objects across multiple measurements by extending ORKA with a fast gridless, multiresolution framework (fg-ORKA). It replaces the grid-bound, exponential-search steps with an iterative down/upsampling strategy that refines the movement path across resolutions using resampling operators (Wavelet, Fourier, or optimal) and a small per-iteration search radius, achieving complexity roughly per iteration and decoupling from the original data resolution. The authors provide detailed complexity and error analyses, introduce an optimal resampling method to minimize projection error, and demonstrate substantial runtime gains along with improved sub-grid movement estimation on synthetic and real data, including seismic and video sequences. The approach enables gridless, scalable tracking in diverse applications, while also outlining limitations (e.g., periodic shifts) and directions for future enhancement. The combination of iterative multiresolution processing, three resampling strategies, and rigorous analysis constitutes a practical, faster alternative to the original grid-dependent ORKA for moving/deforming object tracking.

Abstract

Identifying objects in given data is a task frequently encountered in many applications. Finding vehicles or persons in video data, tracking seismic waves in geophysical exploration data, or predicting a storm front movement from meteorological measurements are only some of the possible applications. In many cases, the object of interest changes its form or position from one measurement to another. For example, vehicles in a video may change its position or angle to the camera in each frame. Seismic waves can change its arrival time, frequency, or intensity depending on the sensor position. Storm fronts can change its form and position over time. This complicates the identification and tracking as the algorithm needs to deal with the changing object over the given measurements. In a previous work, the authors presented a new algorithm to solve this problem - Object reconstruction using K-approximation (ORKA). The algorithm can solve the problem at hand but suffers from two disadvantages. On the one hand, the reconstructed object movement is bound to a grid that depends on the data resolution. On the other hand, the complexity of the algorithm increases exponentially with the resolution. We overcome both disadvantages by introducing an iterative strategy that uses a resampling method to create multiple resolutions of the data. In each iteration the resolution is increased to reconstruct more details of the object of interest. This way, we can even go beyond the original resolution by artificially upsampling the data. We give error bounds and a complexity analysis of the new method. Furthermore, we analyze its performance in several numerical experiments as well as on real data. We also give a brief introduction on the original ORKA algorithm. Knowledge of the previous work is thus not required.
Paper Structure (14 sections, 1 theorem, 48 equations, 9 figures, 1 algorithm)

This paper contains 14 sections, 1 theorem, 48 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. ORKA algorithm
  3. Resolution dependency of ORKA
  4. fast gridless ORKA algorithm
  5. Algorithm concept
  6. Complexity analysis
  7. Error analysis
  8. Optimal resampling
  9. Numerics
  10. Runtime
  11. Non-integer shift
  12. Noisy and high frequency data
  13. Application data
  14. Conclusion

Key Result

Corollary 3

Let $r$, $M$, $R_{\uparrow}$, and $R_{\downarrow}$ be as in Definition def:resamplingPair. Then there exists an $\rho\in\mathbb{R}^M$ such that where $\mathrm{I}_{\frac{M}{r}}$ is the identity matrix of size $\frac{M}{r}\times\frac{M}{r}$.

Figures (9)

  • Figure 1: Runtime of fg-ORKA in different settings: a) $C=5$ for increasing $K$ compared to the original ORKA approach; b) $K=5$ for increasing $C$; c) $K=15$ for increasing $C$.
  • Figure 2: Approximation error for non-integer shifts and different levels of upsampling.
  • Figure 3: Mean approximation error for $K=5$ and noisy data with a maximum shift difference of $C=5$ (left) or $C=27$ (right). In (c) and (d) only high frequency data was used, where the $160$ lowest frequencies are $0$.
  • Figure 4: Best downsampling methods on average for $K=5$, different noise levels and data with low frequencies removed. The maximum shift difference is $C=5$ (a) and $C=27$.
  • Figure 5: Mean approximation error for $K=15$ and noisy data with a maximum shift difference of $C=5$ (left) or $C=27$ (right). In (c) and (d) only high frequency data was used, where the $160$ lowest frequencies are $0$.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Corollary 3
  • proof
  • Example 4
  • Example 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9