Denoising Flow Trajectories

• Denoising Flow Trajectories is the process of inferring plausible vehicle routes by fusing sparse, detailed trajectories with aggregate flow measurements.
• The method employs geometric flow decomposition heuristics using strong Fréchet distance to balance flow accuracy and behavioral realism.
• Comparative analyses show trade-offs between minimizing flow deviation and maintaining interpretable, realistic routes for effective urban planning.

Denoising flow trajectories refers to the process of clarifying or reconstructing the underlying routes taken by vehicles (or agents) in a transportation network using heterogeneous, partial, and noisy data. This problem arises most acutely in traffic analysis and urban planning, where detailed trajectory data is available only for a subset of vehicles (e.g., via map-matched GPS tracks), while aggregate flow data (from, for example, loop detectors) cover the entire network but contain ambiguities on the realized paths. Integrating these partial data sources requires mathematically principled techniques that decompose aggregated flow into realistic, representative routes, while filtering out artifacts and uncertainty resulting from data sparsity or aggregation. Recent approaches formalize this as a combinatorial optimization problem over network flows with realism constraints defined by geometric similarity to representative trajectories, leading to a spectrum of algorithmic solutions with varying trade-offs between fidelity to measured flow and the plausibility of reconstructed routes (Custers et al., 2020).

1. Problem Formulation and Motivation

The core objective in denoising flow trajectories is to infer a set of plausible, physically realistic vehicle routes that collectively "explain" the observed aggregate flow on a transportation network. The problem is motivated by the complementary nature of two data sources:

• Loop-detector measurements: Sensors provide aggregate, time-dependent vehicle counts at fixed network locations, yielding a network-wide, but anonymous and spatially sparse, flow field.
• Representative trajectories: Map-matched GPS trajectories contain rich geometric and behavioral details for a representative—but typically sparse—subset of trips.

The challenge is to construct a comprehensive and realistic set of routes that matches the aggregate counts (thus "denoising" the flow), but that also embodies the spatial and behavioral realism of the available trajectories. This enables robust inference of human mobility patterns, and provides valuable insight for urban planning, infrastructure investment, and emergency response scenarios.

2. Data Fusion and Computational Challenges

Key challenges stem from the partial and conflicting nature of input data:

• Heterogeneity: Loop detectors supply complete, but disaggregated, time-dependent flow data, while trajectories are detailed but sparse and privacy-constrained.
• NP-hardness: The underlying route decomposition—finding a minimum-size set of routes with bounded geometric deviation from representatives and exact flow coverage—is proven NP-hard in multiple formulations (Custers et al., 2020).
• Data sparsity/noise: Sparse trajectory coverage and the ambiguity of aggregated flow data introduce significant noise into feasible reconstructions.

The denoising process must thus account for both overfitting (producing too many short, "nonsensical" routes) and underfitting (missing true patterns due to lack of data), motivating the development of geometric flow decomposition heuristics.

3. Algorithmic Approaches

Three principal heuristic methods are proposed to balance flow explanation and route realism:

A. Fréchet Routes (FR) Heuristic

• Strong Fréchet distance is used to quantify realism: For each basis route $P$, a candidate is accepted if $d_F(P, Q) \leq \epsilon$ for some representative trajectory $Q$.
• Variants:
  • Edge-Inclusion Fréchet Routes (EFR): Candidate generation is biased toward including edges with high "residual" (unexplained) flow using an adapted free-space diagram.
  • Weighted Fréchet Routes (WFR): Candidate generation is further weighted toward edges with high residual flow by assigning interval weights in the free-space diagram.

The reconstructed flow is given by:

$$f(P, c)(e) = \sum_{P \in \text{basis}} M(P, e) \cdot c(P)$$

where $M(P, e)$ counts the number of visits to edge $e$ in route $P$. The residual flow error is

$$\Delta(P, c, \varphi) = \sum_{e} (\varphi(e) - f(P, c)(e))^2$$

and the basis is iteratively expanded by adding routes that minimize $\Delta$.

B. Multi-Commodity Min-Cost Flow (MCMCF)

• Subgraph Decomposition: For each trajectory $T$, a subgraph containing all edges within $\epsilon$ of $T$ is constructed.
• Min-cost flow is solved on each subgraph, with costs derived from the deviation from observed flow. Path-cycle decomposition is used to recombine sub-solutions.

C. Global Min-Cost Flow (GMCF)

• A global min-cost flow is computed directly from the observed aggregate $\varphi$, indifferent to representative trajectories, yielding solutions that minimize overall deviation at the expense of route interpretability and often result in unrealistic, short, or "spurious" paths.

4. Mathematical and Geometric Framework

Key geometric and algebraic constructs underpin these algorithms:

• Network Flow Model: Represents the reconstructed flow as a sum over candidate routes with associated coefficients.
• Strong Fréchet Distance: For curves $P,Q$ parameterized on $[0,1]$,

$$d_F(P, Q) = \inf_{\alpha, \beta} \sup_{t \in [0,1]} \| P(\alpha(t)) - Q(\beta(t)) \|$$

where $\alpha, \beta$ are strictly increasing reparameterizations; implemented via free-space diagram algorithms.

• Optimization Objective: Minimize $\Delta(P, c, \varphi)$ over candidate route collections under geometric realism constraints.

The strong Fréchet criterion ensures that reconstructed routes preserve both the spatial proximity and ordering of path segments, which is crucial for behavioral realism.

5. Empirical Results

Systematic evaluation across synthetic and real-world datasets reveals notable trade-offs:

Method	Flow Deviation Δ	Output Realism	Basis Size (Complexity)	Running Time
FR	Slightly higher	High (few, realistic)	Small	Higher
MCMCF	Moderate	Mostly realistic	Large	Moderate
GMCF	Lowest	Low (many, spurious)	Very large	Lowest

• GMCF achieves the lowest flow deviation (best fit to observed data) but produces many unrealistic or nonsensical routes.
• MCMCF produces realistic routes (guided by representatives), at the expense of a large, redundant route set.
• FR achieves a compact, highly realistic basis of routes that still explains aggregate flow well, trading modestly higher $\Delta$ for much greater interpretability.

Evaluation metrics include flow deviation, mean Fréchet distance to representatives, route coverage of ground-truth, and basis size.

6. Implications, Limitations, and Future Directions

The denoising of flow trajectories through route decomposition guided by geometric constraints enables robust inference of mobility patterns from incomplete data. Key implications include:

• Behavioral Consistency: Using Fréchet-bounded routes enforces compliance with plausible user mobility patterns, necessary for realistic scenario planning and operational traffic management.
• Trade-offs: Minimizing aggregate flow error must be balanced against route realism and interpretability; purely flow-optimal decompositions (GMCF) can yield unrealistic or heavily fragmented travel patterns.
• Scalability and Complexity: The NP-hardness of the underlying combinatorial problem mandates continued development of efficient heuristics and scalable implementations.

Potential extensions involve:

• Incorporating time-dependent flow for real-time traffic analysis.
• Clustering representative trajectories to further reduce basis complexity.
• Leveraging additional sensor modalities (e.g., mobile phone location data) for greater coverage and realism.

In summary, integrating representative route geometry with aggregate flow measurements via geometric flow decomposition establishes an effective, mathematically principled framework for denoising flow trajectories. By grounding candidate route selection in the strong Fréchet distance and optimizing flow allocation to minimize aggregate deviation, this approach balances the competing goals of flow fidelity and behavioral plausibility, advancing the interpretability and utility of reconstructed mobility networks (Custers et al., 2020).