Sequence-Consistent Track Assignment

• Sequence-consistent track assignment is a methodological framework that maintains temporal coherence in multi-object tracking by enforcing continuity constraints across sequences.
• The approach leverages mathematical formulations, dynamic programming, and column generation to optimize object association, ensuring detection disjointness and minimizing fragmentation.
• Recent advancements incorporate deep learning, unsupervised clustering, and graph-based techniques, significantly reducing identity switches and enhancing tracking robustness.

Sequence-consistent track assignment refers to the set of methodologies, architectures, and evaluation metrics that aim to ensure temporally coherent identification, association, and segmentation of objects as they move across video sequences or sensor sweeps. Its principal goal is to avoid fragmented trajectories, identity switches, and association errors caused by local framewise ambiguities, noise, finite detection reliability, or model limitations, by explicitly enforcing temporal continuity and leveraging global context over sequences.

1. Mathematical Formulations for Sequence Consistency

Multi-object tracking (MOT) is fundamentally a set assignment problem, typically cast in the combinatorial language of track selection subject to detection disjointness and temporal consistency constraints. The classical integer linear programming (ILP) formulation models tracks as indicator variables $\gamma \in \{0,1\}^{|\mathcal{P}|}$ over a large space $\mathcal{P}$ of candidate tracks:

$$\text{minimize} \quad \Theta^T \gamma \ \text{subject to} \quad \gamma \in \{0,1\}^{|\mathcal{P}|}, \quad X\gamma \leq 1,$$

where $X \in \{0,1\}^{|\mathcal{D}| \times |\mathcal{P}|}$ encodes whether detection $d$ falls within track $p$, and $\Theta \in \mathbb{R}^{|\mathcal{P}|}$ encodes the cost (negative quality) of each track candidate (Wang et al., 2015). The constraint $X\gamma \leq 1$ guarantees that no two selected tracks share a common detection—enforcing the "track disjointness" axiom for sequence assignment.

To encode higher-order, sequence-level interactions, tracks are decomposed into ordered sequences of subtracks $\mathcal{S}$. Sequence consistency is then enforced by stipulating that any subtrack $s$ may only follow its predecessor $\hat{s}$ if the last $K-1$ elements of $\hat{s}$ equal the first $K-1$ elements of $s$:

$$s_{k-1} = \hat{s}_k, \qquad \forall k = 2, \ldots, K.$$

These constraints, together with dynamic programming solutions over subtracks, form the computational backbone of efficient, consistency-enforcing assignment approaches.

2. Column Generation, Dynamic Programming, and Assignment Guarantees

Solving the combinatorial MWSP problem in its full form is intractable due to the exponential number of candidate tracks. Delayed column generation solves only a restricted LP over a small active set $\hat{\mathcal{P}}$; new candidate tracks are dynamically generated via a "pricing problem" using dynamic programming (Wang et al., 2015). The critical dynamic program recursively constructs cost-to-go $\ell_s$ for each subtrack $s$ via:

$$\ell_s \leftarrow \theta_s + \lambda_{s_K} + \min \left\{ \min_{\hat{s} \Rightarrow s} \ell_{\hat{s}},\, \theta_0 + \sum_{k=0}^{K-1} \lambda_{s_k} \right\}$$

where $\lambda$ are dual variables, and $\theta_0$ initializes a new track. Sequence consistency is strictly enforced, since only allowable subtrack transitions are permitted in the DP state graph. Column generation iterates—expanding $\hat{\mathcal{P}}$ whenever new tracks with negative reduced cost are detected—until all violated constraints are resolved.

Inferred sequence consistency and detection disjointness are preserved throughout optimization: solution tracks are guaranteed not to share any detection, and rounding (Alg 2) removes all conflicting tracks when converting fractional LP solutions to integral decisions.

3. Sequence-Consistent Metrics and Evaluation

Assessing the quality of sequence-consistent assignment requires metrics that reward long-term continuity and penalize fragmentation. The OSPAMT metric (Vu et al., 2018) evaluates the distance between two finite sets of tracks over an entire sequence, combining localization error and cardinality error with many-to-one assignment penalties:

• Localization error: $d(x, y) = \min\{ c, \|x - y\| \}$ (capped Euclidean distance)
• Cardinality error: penalizes unassigned or false tracks, plus additional penalty $A$ for track fragmentation.

Formally,

$$d_p^{OSPAMT}(w, w') = \min\{ d_p(w', w), d_p(w, w') \}$$

This metric can distinguish between solutions with similar per-frame errors but different track fragmentation profiles, rewarding continuous track assignment over broken outputs—a key capability for sequence-optimized trackers.

4. Deep Learning Approaches: Sequence Models and End-to-End Optimization

Recent methods leverage deep sequence models (e.g., LSTM, attention, Transformer) to encode appearance or geometric features across entire tracklets, moving beyond framewise or pairwise affinity. End-to-end tracklet search and ranking (Hu et al., 2020) directly optimize the global tracklet score $f_s(T)$ over candidate sequences, utilizing margin and ranking losses:

$$L_t^{margin} = \sum_{\hat{T}_i^t \in \mathcal{T}_t \setminus T_{gt}^t} \max[0, \alpha - \text{Sigmoid}(f_s(T_{gt}^t)) + \text{Sigmoid}(f_s(\hat{T}_i^t))]$$

This approach exposes the model during training to the same search errors as encountered at inference, thus eliminating exposure bias and improving sequence consistency.

Unified architectures suitable for sequence-consistent assignment employ shared CNN backbones to integrate detection, short-term tracking (via Siamese networks), and re-identification (via embedding-based matching and triplet loss) (Shuai et al., 2020). Long-term consistency is further maintained by explicitly re-activating lost tracks using temporal buffers and similarity measures.

5. Unsupervised and Self-supervised Strategies for Consistency

Spatio-temporal clustering approaches treat track assignment as a clustering problem in a learned latent space—for instance, using deep heterogeneous autoencoders to fuse segmentation and detection features, with constraints graphs enforcing "cannot-link" and "must-link" conditions across time (Siddique et al., 2020). Cluster assignment (via constrained k-means) thus yields temporally consistent track identities, and ablation studies show that combining appearance, location, and temporal constraints leads to near-perfect identity preservation.

Unsupervised deep clustering frameworks employ track-based memory and momentum updates (Alfani et al., 2022), enforcing track-wise consistency despite appearance variation due to pose or viewpoint. Track-based Shannon entropy metrics provide quantitative evaluation of consistency, with significantly reduced entropy in track identity assignment compared to baseline clustering.

Self-supervised methods for tracking in autonomous driving use sequence-level objectives that reward consistency across short and long timescales (Lang et al., 2023). The SubCo loss penalizes assignment discrepancy between propagated short-term associations and direct long-term pairwise associations:

$$L^{SubCo} = \frac{1}{\sum_i [d_i = 0]} \sum_{i \, | \, d_i = 0} \left[ -\log \sum_j (\tilde{A}_{ij} \cdot A_{ij}) \right]$$

This approach demonstrably reduces ID switches and matches supervised performance benchmarks in practical scenarios with large dynamics and noisy appearance changes.

6. Graph-based and Spectral Techniques for Sequence Assignment

Spectral clustering and graph optimization frameworks provide robust mechanisms for enforcing space-time consistency in complex scenes. SFTrack++ (Burceanu, 2020) models the video as a graph of pixels (or voxels), using power iteration and fast 3D Gaussian filtering to compute the dominant eigenvector, interpreted as the object's global segmentation. Multi-channel input fusion and intermediate segmentation maps yield improved temporal consistency and accuracy, particularly in ensemble setups spanning multiple base trackers.

For 3D multi-object tracking, multidimensional assignment via lifted graph edges (skip connections) and batch-processing of a sliding temporal window enables ambiguity resolution over occlusions and missed detections (Papais et al., 27 Feb 2024). The assignment is formulated as a binary integer program, with log-likelihood scoring for each hypothesis, and real-time optimization enabled through graph sparsification and constraint matrix unimodularity.

7. Extensions: Sequence-Aware Training, Diffusion Models, and Reinforcement Learning

Recent extensions emphasize sequence-level training objectives. Approaches based on reinforcement learning roll out episodes across entire videos (Kim et al., 2022), directly optimizing sequence metrics such as average overlap via policies parameterized by the tracking network. REINFORCE and self-critical sequence training further improve robustness to error accumulation.

Diffusion-based denoising frameworks, such as ConsistencyTrack (Jiang et al., 28 Aug 2024), treat joint detection and tracking as a generative process that progressively refines noisy box proposals over pairs of frames. The single/few-step denoising and joint association head minimize identity switches and maintain real-time efficiency, with explicit loss terms for classification, regression, and 3D GIoU.

For radar sensor management, sequence-capable architectures utilizing bidirectional recurrent units and multi-headed self-attention (MHSA) (Ewers et al., 19 Feb 2025) are shown to optimally encode track lists and scan histories, maintaining assignment consistency under rapid dynamics. The use of reinforcement learning rewards that combine search efficiency and tracking covariance improvements naturally promote consistent multi-target assignment, especially when pre-training via behavior cloning and autoencoders is employed.

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In conclusion, contemporary approaches to sequence-consistent track assignment deploy a range of mathematical, algorithmic, and learning-based strategies that jointly address the challenges of data association, error accumulation, fragmented tracking, and temporal hallucination. The field has advanced from initial combinatorial formulations to dynamic programming, deep sequence models, unsupervised clustering, spectral graph methods, and joint generative optimization, with rigorous evaluation via sequence-aware metrics. These developments underpin the reliability, accuracy, and robustness of tracking algorithms in video, robotics, autonomous driving, biological imaging, and radar systems.