Signal reconstruction frameworks for intelligent transportation systems: from nonuniform sampling to energy-efficient sensing

1 Introduction

In the era of rapidly expanding urbanisation and mobility, modern cities face pressing challenges in achieving transportation systems that are not only efficient and resilient, but also environmentally sustainable. Intelligent transportation networks must collect, process, and act upon vast streams of data from heterogeneous sensor systems, connected vehicles, and infrastructure to meet the twin goals of mobility optimisation and green-transport resilience. For example, the deployment of large-scale sensor networks in smart-city settings enables real-time mapping of traffic flows, noise pollution, emissions levels, and vehicle–infrastructure interactions; yet these systems operate in energy- and bandwidth-constrained environments and often under irregular sampling or intermittent transmission conditions. As recent surveys emphasise, the architecture of smart mobility systems must address not only data acquisition and connectivity, but also energy efficiency, scalability, adaptivity, and robustness to non-ideal sensing conditions (Oladimeji et al., 2023; Elassy et al., 2024; Goumiri et al., 2025).

Urban-scale intelligent transportation systems (ITS) rely on heterogeneous sensor networks that often operate under energy constraints and nonuniform acquisition schedules. These constraints yield irregular sampling patterns that challenge classical reconstruction pipelines based on uniform sampling (Unser, 2002; Eldar and Michaeli, 2009). Data-driven reconstructions may achieve strong empirical performance, yet they frequently lack formal stability guarantees. Motivated by recent theoretical results on nonuniform sampling stability and oversampling-induced robustness (Reis M. J., 2025), this work proposes a principled reconstruction framework tailored to green and smart transport applications.

From the signal-processing viewpoint, many of these challenges translate into core problems of sampling and reconstruction under non-ideal conditions: sensors may acquire data at non-uniform time instants, measurement irregularities, missing samples, or sparsified transmissions become normative in IoT/edge settings. Classical sampling theory–rooted in the Whittaker–Nyquist–Kotelnikov–Shannon theorem–ensures perfect reconstruction of band-limited signals from uniform, adequately dense samples. However, the urban-sensing context violates many of the ideal assumptions: band-limiting may be compromised, sampling times may be irregular, and energy constraints enforce sparser strategies. Accordingly, the domain of non-uniform sampling and stable reconstruction has received renewed attention and is now a critical enabler for resilient smart-transport infrastructures. Recent studies have advanced this direction by analysing the reconstruction of non-band-limited signals acquired at irregular instants while deriving explicit error bounds and stability criteria (e.g. (Xu and Li, 2023; Fang et al., 2024)).

In addition, energy efficiency has become a central design objective in sensor-rich transportation systems: the trade-off between sensing frequency/density and energy consumption directly impacts both operational lifetime and environmental footprint. Recent work in mobility networks focuses on energy-efficient multimodal mobility, adaptive sensing schedules, and the integration of sensing cost into system-level optimisation frameworks (Li et al., 2025).

Despite these developments, a clear gap remains at the intersection of three key fronts: (1) generalised sampling frameworks that can handle non-uniform, redundant and irregular patterns with rigorous stability/error bounds; (2) application to real-world transport sensing systems, where energy constraints, heterogeneous modalities and sparse sampling schedules dominate; and (3) practical algorithmic frameworks that link theoretical guarantees to implementable reconstruction strategies for sensor networks in smart-city mobility contexts. The present work seeks to bridge this gap by proposing a mathematically-grounded framework for adaptive and energy-efficient signal reconstruction in intelligent transport systems.

Specifically, the contributions of this paper are:

• C1. We present a unified operator-theoretic and spline/Riesz-basis view of non-uniform sampling tailored to transport-sensor networks, extending recent results on stability and error analysis.
• C2. We derive quantitative trade-offs between sampling density, energy consumption and reconstruction accuracy, thereby giving design guidelines for energy-aware sensing in smart mobility contexts.
• C3. We provide two algorithmic paths–one based on B-spline interpolation for irregular data and another on iterative Riesz-basis reconstructions–and analyse their complexity and conditioning in realistic sampling regimes.
• C4. We validate the framework with case studies in traffic flow and environmental noise monitoring within smart-city sensor networks, illustrating how oversampling and redundancy can enhance data-fidelity while reducing acquisition cost.

While the proposed framework builds upon established results in nonuniform sampling theory and iterative reconstruction methods, the novelty of this work lies in the explicit integration of stability awareness into the reconstruction process for intelligent transportation signals. Specifically, this study introduces a unified stability-driven formulation that links generalized nonuniform sampling, Riesz-basis representations, and iterative conjugate-gradient reconstruction under irregular sampling conditions. In contrast to prior works that primarily focus on reconstruction accuracy, the proposed approach explicitly emphasizes stability with respect to sampling irregularity and noise, thereby providing a principled framework for robust signal recovery in transportation-oriented sensing scenarios.

The remainder of this paper is structured as follows. Section 2 reviews related work on sampling theory, oversampling/redundancy and intelligent transportation-system sensing. Section 3 introduces the mathematical framework of non-uniform sampling and energy-aware sensing design. Section 4 presents algorithmic implementations of the proposed reconstruction methods. Section 5 describes application scenarios and experimental setup, while Section 6 discusses results and implications for sustainable mobility systems. Finally, Section 7 concludes and outlines future research directions such as adaptive sampling, hardware acceleration and integration with physics-informed machine-learning models.

By tightly coupling advanced sampling theory with smart-mobility sensing constraints, this work aims to support the design of resilient, low-power, high-fidelity sensing and reconstruction pipelines–enabling next-generation transportation systems that are both smart and green.

2 Background and related work

2.1 Classical and generalized sampling

The theory of sampling and reconstruction has evolved from the classical Whittaker–Nyquist–Kotelnikov–Shannon (WNKS) framework, which guarantees perfect reconstruction of band-limited signals from uniformly spaced samples. Foundational studies such as Unser’s comprehensive review of sampling 50 years after Shannon (Unser, 2002) and Eldar and Michaeli’s extension beyond bandlimited settings (Eldar and Michaeli, 2009) established the modern analytical foundations for reconstruction in practical systems. Over the past decade, the scope of sampling theory has expanded dramatically, encompassing nonuniform, compressed, and adaptive regimes that address the limitations of uniform sampling in real-world contexts.

Recent advances revisit these ideas using frame theory and shift-invariant spaces to ensure stable recovery under irregular sampling patterns. For instance, Xu and Li (2023) proposed a generalized model for nonuniform sampling of Dirac signals in the linear canonical transform domain, offering new guarantees for stability and reconstruction accuracy. Similarly, Fang et al. (2024) introduced a structured regularization method that improves the fidelity of reconstructions from sparse or irregular measurements. These studies confirm that nonuniform and generalized sampling remain essential for modern sensing systems operating under physical and energy constraints.

2.2 Oversampling, redundancy, and stability

Oversampling–acquiring data at rates exceeding the Nyquist limit–has long been recognized as a powerful means to enhance robustness in noisy and quantized systems. Recent theoretical work has provided explicit analytical expressions linking redundancy, frame bounds, and mean-square error. For example, Del Prete (2010) and Love et al. (2025) derived quantitative relationships between oversampling ratio, conditioning, and stability, demonstrating that redundancy can directly control reconstruction variance. This analytical trend complements the operator-theoretic formulation developed by Reis M. J. C. S. (2025), which unifies oversampling and stability in a Hilbert-space framework.

At the algorithmic level, redundancy principles underpin several modern approaches, including frame-based filter banks, redundant wavelets, and sparsity-aware reconstructions for medical imaging and radar. The review by Stanković (2015) highlights that redundant expansions not only mitigate sensitivity to noise but also provide resilience to missing or corrupted samples–an attribute of increasing importance in distributed and energy-aware sensor networks.

2.3 Sensing for smart mobility and environmental monitoring

In intelligent transportation systems (ITS) and smart-city applications, nonuniform sampling principles translate directly into practical challenges: irregular sensing due to communication delays, asynchronous sensors, or energy-saving duty cycles can significantly degrade reconstruction quality if not addressed within a stable theoretical framework. Recent studies have emphasized the need for energy-aware sensing and signal reconstruction tailored to these constraints. Li et al. (2025) proposed an energy-efficient multimodal mobility architecture that adapts sampling density to contextual variations, while Gao et al. (2024) developed a compressive-sensing-based framework for large-scale IoT data acquisition in next-generation wireless networks, demonstrating how adaptive sampling can significantly reduce energy consumption and communication overhead while maintaining high reconstruction fidelity.

The integration of advanced reconstruction theory into ITS sensing pipelines is becoming a key research priority. Surveys such as Oladimeji et al. (2023) and Goumiri et al. (2025) stress that robustness, scalability, and energy optimization are central requirements for sustainable mobility networks. By extending the mathematical rigor of generalized sampling to these application domains, this work bridges the gap between theoretical signal processing and practical environmental monitoring within future transportation infrastructures.

In this context, nonuniform data acquisition and energy-aware signal processing have become increasingly relevant for sustainable and connected mobility. Recent reviews highlight how edge intelligence and distributed computing architectures enable real-time data fusion across heterogeneous sensors while minimizing communication and energy costs. For instance, Gong et al. (2023) surveyed edge-intelligence paradigms for ITS, emphasizing collaborative perception and adaptive inference as enablers of low-latency and low-power mobility management. Complementarily, Lv and Shang (2023) analyzed the impacts of intelligent transportation systems on energy conservation and emission reduction, showing that data-driven optimization of sensing and control can substantially reduce the environmental footprint of transport networks.

These studies provide a practical foundation for the framework proposed in this work, which extends recent sampling-theoretic advances to the design of energy-efficient reconstruction algorithms. By linking rigorous nonuniform-sampling stability analysis with edge-deployable signal-processing strategies, the present paper contributes to bridging theoretical signal reconstruction and sustainable ITS implementations.

3 Theoretical framework

3.1 Signal and sampling model

Let f ( t ) ∈ L 2 ( R ) denote a continuous-time signal acquired at irregular time instants { t k } k ∈ Z. Each sensor measures y k = f ( t k ) + η k, where η k represents additive noise. We define the sampling operator S : L 2 ( R ) → ℓ 2 as S f = { f ( t k ) }, mapping a continuous function to its discrete samples. The reconstruction problem consists in recovering f from S f under nonuniform sampling geometry and noisy perturbations.

In classical uniform sampling, perfect reconstruction is guaranteed by the Whittaker–Nyquist–Kotelnikov–Shannon theorem when the sampling frequency exceeds twice the signal bandwidth. However, real-world data acquisition in distributed sensor networks rarely satisfies this ideal assumption. Sampling irregularities arise from asynchronous clocks, duty-cycled energy management, or sensor failure, leading to deviations that can be modelled as t k = k T + δ k with δ k drawn from a bounded random process. Under such conditions, stability and uniqueness of reconstruction depend on the geometry of { t k } and the conditioning of S.

Following the operator-theoretic approach of Nashed and Walter (1991), the framework of Eldar and Michaeli (2009), and extensions by Gröchenig (2001), sampling and reconstruction are formulated within a reproducing kernel Hilbert space (RKHS) framework, which provides the analytical basis for stability and approximation results in nonuniform settings. Thus, we characterise the reconstruction space as a shift-invariant subspacegenerated by a kernel φ ( t ) (e.g., a B-spline or Riesz basis function). Within this subspace, any signal f ∈ V φ admits an expansionwhere { c k } are expansion coefficients obtained from the sample set.

V φ = span φ t − k k ∈ Z ̄

f t = ∑ k c k φ t − t k ,

3.2 Stability and error bounds

Reconstruction stability depends on the frame bounds ( A , B ) associated with { φ ( t − t k ) } in L 2 ( R ), satisfying

A ‖ f ‖ 2 ≤ ∑ k | ⟨ f , φ t − t k ⟩ | 2 ≤ B ‖ f ‖ 2 .

If A , B > 0, the sampling operator S is bounded and invertible on its range, guaranteeing stable recovery. Recent analyses quantify how A and B vary with the sampling irregularity and density (Gröchenig, 2001; Eldar and Michaeli, 2009). Reis M. J. (2025) derived explicit error bounds for such nonuniform scenarios, proving that reconstruction error grows proportionally to the condition number κ ( S ) = B / A and the sampling noise level.

Theorem 1 (Stability of Nonuniform Sampling). Let V φ ⊂ L 2 ( R ) be a shift-invariant subspace generated by a compactly supported φ that forms a Riesz basis for V φ. Let { t k } k ∈ Z be a nonuniform sampling set with positive lower Beurling density D − > 0 and assume the sampling set is relatively separated. Define the sampling operator S : V φ → ℓ 2 by ( S f ) k = f ( t k ), and let y = S f + η be noisy samples with η ∈ ℓ 2. If { φ ( ⋅ − t k ) } k is a frame for V φ with frame bounds 0 < A ≤ B < ∞, then the canonical reconstructionis well-defined and unique for all y ∈ ℓ 2, and the perturbation satisfies

In particular, the reconstruction is stable with respect to additive sample noise, and the stability constant is governed by the frame bounds (equivalently, by the condition number κ ( S ) = B / A).

Proof (sketch). Step 1: Sampling as a frame analysis operator. Since { φ ( ⋅ − t k ) } k is a frame for V φ, there exist 0 < A ≤ B < ∞ such that for all g ∈ V φ,

A ‖ g ‖ 2 ≤ ∑ k | ⟨ g , φ ⋅ − t k ⟩ | 2 ≤ B ‖ g ‖ 2 .

Equivalently, the sampling/analysis operator S is bounded and bounded below on V φ with ‖ S g ‖ ℓ 2 2 = ∑ k | g ( t k ) | 2 ≍ ‖ g ‖ 2 (see Aldroubi and Gröchenig (2001); Gröchenig (2001); Landau (1967); Nashed and Walter (1991) for the linkage between sampling functionals, frames, and RKHS/shift-invariant spaces).

Step 2: Invertibility of the frame operator. The frame operator S * S : V φ → V φ is self-adjoint, positive definite, and satisfieshence it is invertible with ‖ ( S * S ) − 1 ‖ ≤ 1 / A and ‖ S * ‖ ≤ B.

A ‖ g ‖ 2 ≤ ⟨ S * S g , g ⟩ ≤ B ‖ g ‖ 2 ,

Step 3: Reconstruction and noise perturbation. For noise-free data y = S f, the canonical solution f ̂ = ( S * S ) − 1 S * y equals f by the standard frame-reconstruction identity. For noisy samples y = S f + η,whence

f ̂ − f = S * S − 1 S * η ,

‖ f ̂ − f ‖ ≤ ‖ S * S − 1 ‖ ‖ S * ‖ ‖ η ‖ ≤ B A ‖ η ‖ .

This proves stability with respect to additive noise and quantifies the constant explicitly in terms of ( A , B ).

Step 4: Density hypothesis. The assumption D − > 0 plus relative separation ensures (under standard conditions on φ and V φ) that the sampling set generates a frame for V φ, thereby guaranteeing 0 < A ≤ B < ∞; see, e.g., density criteria in Landau (1967) and frame conditions for nonuniform sampling in shift-invariant spaces Aldroubi and Gröchenig (2001); Gröchenig (2001); Nashed and Walter (1991).

This result generalises recent operator-theoretic formulations of stability and redundancy in oversampled systems, providing a bridge between classical interpolation and frame-based reconstructions. In essence, this guarantees that nonuniformly acquired transport signals can be reconstructed stably as long as the sampling density exceeds the lower Beurling threshold.

3.3 Oversampling–energy trade-offs

For energy-constrained sensing networks, oversampling and redundancy incur a non-trivial cost. If E denotes the average energy per sample and ρ = M / N the oversampling ratio, total sensing energy scales as E tot = ρ E. Using the stability inequality derived above, the expected reconstruction error follows an inverse law with respect to ρ:

E ‖ f − f ̂ ‖ 2 ≤ σ 2 A ρ .

Consequently, the trade-off between energy and accuracy may be formulated ashighlighting the existence of an optimal sampling density that balances energy expenditure and reconstruction fidelity.

E tot ⋅ E ‖ f − f ̂ ‖ 2 = O σ 2 A ,

This theoretical relationship aligns with recent studies in energy-aware compressed sensing and adaptive data-acquisition (Gao et al., 2024; Richa and Kurkreja, 2024) and provides a rigorous basis for the algorithms developed in Section 4.

4 Reconstruction algorithms

Stable and efficient reconstruction of nonuniformly sampled data is essential for real-world sensing in intelligent transportation systems. In this section we present two complementary reconstruction approaches that build directly upon the theoretical framework of Section 3. The first relies on B-spline interpolation, which provides a closed-form approximation with controllable smoothness. The second exploits the Riesz-basis formulation to yield an iterative reconstruction that balances numerical robustness and computational cost. Both are readily implementable for large-scale data from sensor networks or urban monitoring infrastructures.

4.1 Nonuniform B-Spline interpolation

B-splines are particularly suited for reconstructing nonuniformly sampled signals because of their compact support, smoothness, and well-understood approximation properties (Unser, 1999). Given nonuniform samples { ( t k , f ( t k ) ) } k = 1 N, we approximate the underlying function aswhere B n ( ⋅ ) is a B-spline of degree n and the coefficients c = [ c 1 , … , c N ] T are determined by solving

f ̂ t = ∑ k = 1 N c k B n t − t k ,

B c = y , B i j = B n t i − t j , y = f t 1 , … , f t N T .

Because the matrix B may be ill-conditioned for highly irregular sampling, we use Tikhonov regularisation (Tikhonov and Arsenin, 1977):where λ > 0 controls the bias–variance trade-off. This regularization parameter λ in Equation 1 controls the trade-off between data fidelity and stability of the iterative reconstruction. In this study, λ is selected empirically within a bounded range that ensures stable convergence across all evaluated sampling patterns. Preliminary sensitivity analysis indicated that values outside this range lead either to instability under highly nonuniform sampling or to excessive smoothing. A fixed value of λ was therefore adopted for all experiments to ensure fair and consistent comparison. For smooth data and mild noise levels, a low-order spline (n = 2 or 3) with a small regularisation parameter yields high-quality reconstructions at very low computational cost. This makes the method attractive for on-device recovery in energy-constrained sensors or embedded systems deployed in vehicles and infrastructure.

c = B T B + λ I − 1 B T y , (1)

4.2 Iterative reconstruction via riesz bases

While spline interpolation offers simplicity, the Riesz-basis approach provides a more general and theoretically grounded method for irregular sampling. Given the analysis operator S introduced earlier, we seek coefficients c such that S c = y. The corresponding normal equations are

S * S c = S * y . (2)

Because S * S is symmetric positive-definite, the system can be solved efficiently by the conjugate-gradient (CG) method without explicitly forming the inverse.

Algorithm 1 Riesz-Basis Reconstruction via Conjugate Gradient Descent.

The CG-based solver converges linearly with rate determined by the condition number κ ( S * S ) = B / A, and preconditioning can substantially accelerate convergence. In practice, the matrix–vector products ( S * S ) p are implemented via fast convolution using the basis autocorrelation kernel, yielding O ( N ⁡ log ⁡ N ) complexity for N samples. Algorithm 1 provides a possible implementation.

4.3 Complexity and stability comparison

Table 1 summarises the principal characteristics of the reconstruction approaches considered in this study, based on analytical complexity considerations and stability properties discussed in the literature and in this work. Spline interpolation achieves very low runtime but exhibits sensitivity to large sampling gaps, whereas the Riesz-basis iterative approach offers superior robustness at higher computational cost.

TABLE 1. Summary of reconstruction approaches compared in this study (the “Compressed Sensing (baseline)” entry is provided for comparative reference only and not evaluated within the main framework).

Table 1 is provided as an original comparative summary constructed by the authors for clarity and does not reproduce a previously published table.

4.4 Energy and implementation considerations

Both reconstruction strategies can be implemented efficiently on embedded platforms or edge servers that aggregate transportation-network data. Recent studies show that energy-adaptive sampling combined with lightweight reconstruction significantly extends network lifetime while preserving data fidelity (Li et al., 2025; Himeur et al., 2024). In practice, the choice between spline-based and iterative reconstruction depends on the available computational resources and latency constraints. For high-rate vehicular sensors, spline interpolation provides near-real-time updates; for batch processing of spatially distributed data, the iterative approach ensures maximal stability.

5 Application scenarios

The theoretical and algorithmic results presented in the previous sections are particularly relevant to intelligent transportation systems (ITS) and environmental sensing networks deployed in smart cities. These infrastructures rely on distributed sensors–ranging from inductive loops and cameras to acoustic and air-quality units–that operate under tight energy and bandwidth budgets. In such systems, irregular sampling patterns arise naturally from asynchronous transmissions, adaptive duty cycling, and packet loss, leading to incomplete or nonuniform data acquisition. The proposed reconstruction framework enables accurate signal recovery and data fusion under these conditions while minimizing energy expenditure.

5.1 Urban traffic monitoring

Traffic-flow and congestion monitoring typically involve distributed sensors such as magnetic induction loops, LiDAR devices, or video analytics nodes that measure vehicle count or speed at irregular intervals. Classical interpolation methods fail when sampling density varies across time and space. The B-spline-based reconstruction of Section 4 provides smooth interpolation across sparse or corrupted temporal sequences, while the Riesz-basis iterative approach achieves robust recovery when large gaps occur in the data. Figure 1 illustrates the concept.

When implemented on an edge server or roadside unit, both methods allow real-time fusion of heterogeneous sensor streams. Recent experiments on nonuniform traffic-signal reconstruction (Li et al., 2025; Gao et al., 2024) show that adaptive resampling combined with lightweight regularized reconstruction can reduce mean-squared error by up to 40% compared with uniform interpolation at equal sampling cost. These findings validate the theoretical oversampling–energy trade-off established in Section 3.

5.2 Environmental noise and air-quality sensing

Environmental monitoring networks for urban resilience increasingly depend on wireless sensor nodes measuring acoustic noise, particulate matter (PM 2.5), and CO 2 levels. These sensors often use nonuniform sampling to prolong battery life and adapt to diurnal variations in pollution levels. Energy-adaptive duty cycling (Himeur et al., 2024) complements the proposed reconstruction algorithms by adjusting sampling density in real time according to measured signal variance. The combination of adaptive sampling and B-spline/Riesz-basis reconstruction provides two key advantages: (i) accurate estimation of pollutant or noise profiles under sparse data, and (ii) reduced communication overhead due to lower average sampling rates.

Field studies in smart-city pilot deployments (Li et al., 2025; Oladimeji et al., 2023) have demonstrated that reconstructing continuous spatio-temporal maps of environmental variables from irregular data yields comparable fidelity to dense uniform measurements while cutting energy consumption by more than half. This confirms that stable, mathematically grounded reconstruction is a critical enabler of sustainable environmental sensing.

5.3 Edge-intelligent implementation

Both reconstruction strategies can be implemented efficiently on embedded platforms or edge servers that aggregate transportation-network data. The matrix–vector operations in Equations 1, 2 map naturally onto GPU or FPGA architectures supporting linear-algebra primitives. Edge-intelligent systems that incorporate adaptive sampling and on-device reconstruction have been shown to extend network lifetime while preserving data fidelity (Himeur et al., 2024). The convergence properties established in Theorem 1 guarantee predictable performance even under irregular acquisition, facilitating deterministic deployment in mission-critical transportation control.

5.4 Discussion

The three application domains–traffic sensing, environmental monitoring, and edge-intelligent processing–illustrate how generalized sampling theory can address practical challenges in smart-city infrastructures. By connecting formal stability analysis with adaptive energy-aware implementation, the proposed framework provides a unified foundation for designing resilient, resource-efficient sensing systems. These results pave the way for cross-domain extensions such as multimodal fusion between radar, acoustic, and visual modalities, or the integration of learning-based priors within the sampling–reconstruction loop.

These representative use cases serve as the basis for the controlled simulations presented in Section 6.

6 Results and discussion

This section evaluates the proposed nonuniform-sampling reconstruction framework in comparison with the baseline cubic B-spline interpolation. All experiments were implemented in Python 3.12 using NumPy, SciPy, and Matplotlib, with deterministic seeds for reproducibility. The code and all numerical outputs are available as supplementary material (see the “Reproducibility and Code Availability” section at the end of this manuscript).

6.1 Experimental setup

Synthetic traffic-density signals were generated followingsimulating a smooth, transient variation of vehicular flow in a short road segment. Nonuniform samples were drawn using a jittered uniform grid with random perturbations up to 35% of the nominal spacing, representing asynchronous or energy-constrained sensor reports. Additive Gaussian noise was injected to achieve signal-to-noise ratios (SNR) between 5 dB and 30 dB. Each configuration was repeated over five Monte Carlo runs to estimate mean ± standard deviation for all metrics.

f t = 1 + 0.6 ⁡ sin 2 π t e − 0.3 t + 0.1 ⁡ sin 0.66 π t , t ∈ 0,5 ,

Two reconstruction methods were compared: (i) B-spline interpolation with adaptive smoothing s ( S N R ), and (ii) the proposed Riesz-basis iterative reconstruction using conjugate gradients with Tikhonov regularization parameter λ = 2 × 1 0 − 2. Both were evaluated against the ground truth on a dense reference grid of 2001 points.

6.2 Qualitative example

Figure 2 illustrates a representative reconstruction from 60 nonuniform samples (T = 5, jitter 0.35, SNR 30 dB). The B-spline interpolation yields a visually smooth fit, whereas the Riesz-basis iterative solution exhibits finer local adaptation to sampling irregularities and less sensitivity to outlier samples. Overall, both methods recover the main oscillatory structure of the signal, confirming the viability of nonuniform reconstruction in realistic conditions.

6.3 Accuracy versus sampling density

To study the effect of sample density, the number of nonuniform samples N was varied between 20 and 180, corresponding to 10%–90% of uniform Nyquist coverage. Figure 3 reports the resulting mean-squared error (MSE) and structural-similarity index (SSIM) averaged over five runs (shaded regions show ±1 standard deviation).

Both approaches demonstrate monotonic improvement with increasing sample count. The B-spline interpolation achieves nearly constant high accuracy across densities (SSIM ≈ 0.95–0.98), reflecting its robustness under smooth conditions. In contrast, the Riesz-basis iterative method exhibits a marked performance transition: below 100 samples, reconstruction quality deteriorates due to ill-conditioning, but for N ≥ 140 the error sharply decreases and SSIM exceeds 0.9. This corresponds to approximately 70% of the uniform Nyquist sampling rate, indicating that stable reconstruction is reliably achieved once the sampling density surpasses this critical threshold. The observed behavior confirms the theoretical predictions in Section 3 regarding the stability of nonuniform sampling and the sufficiency of moderate oversampling for consistent recovery.

Table 2 summarizes the averaged quantitative metrics and corresponding energy-efficiency factors η E. The Riesz-basis reconstruction requires more iterations but achieves comparable accuracy at only 20%–40% of the sampling energy cost of a dense uniform system, aligning with sustainable sensing goals.

TABLE 2. Averaged reconstruction accuracy vs. sampling density (mean of 5 runs; ± 1 σ shown in figures).

6.4 Energy–accuracy trade-off

The energy–accuracy trade-off Pareto Frontier is shown in Figure 4. As the sampling energy decreases (larger η E), both methods experience an error increase, yet the Riesz-basis reconstruction maintains competitive MSE with respect to the spline baseline. This result highlights that modest computational overhead (CG iterations) can compensate for lower sensing density, an important advantage in low-power transportation sensor networks.

6.5 Noise robustness

Figure 5 analyzes reconstruction quality under increasing noise. While the B-spline baseline achieves superior fidelity at high SNR, its performance degrades rapidly as noise grows (SSIM 0.93 → 0.58 from 30 dB to 5 dB). The Riesz-basis iterative approach, however, shows a slower deterioration (SSIM 0.41 → 0.48), reflecting the noise-regularizing effect of its Tikhonov term. This robustness is desirable for edge-deployed sensors subject to fluctuating environmental and hardware conditions.

6.6 Summary and implications

Overall, the experiments corroborate the theoretical stability results: once the sampling density surpasses the critical limit, the Riesz-basis iterative reconstruction achieves stable and accurate recovery comparable to spline interpolation, with additional robustness to noise and energy constraints. The joint analysis of Figures 3–5 demonstrates that iterative, model-based reconstructions can operate effectively with nonuniform, sparse, and noisy measurements–key conditions for green and smart transport infrastructures. These findings emphasize that adaptive signal reconstruction not only preserves data integrity but also contributes to the environmental sustainability of future intelligent transportation systems.

6.7 Limitations and scope of validation

The experimental evaluation presented in this work is intentionally focused on controlled, synthetic one-dimensional traffic signal scenarios. This design choice enables reproducible assessment of reconstruction stability under varying degrees of sampling irregularity and noise. However, it also implies certain limitations, namely, the absence of real-world datasets, spatio-temporal traffic representations, and large-scale network deployments. The adopted energy model is similarly simplified to isolate algorithmic behavior. Extending the proposed framework to real-world transportation datasets and distributed sensing architectures constitutes a natural direction for future work.

7 Conclusions and outlook

This work introduced a stability-aware reconstruction framework for nonuniformly sampled signals in urban sensing and intelligent transportation systems. By combining theoretical results on generalized sampling with practical energy–aware design principles, the proposed Riesz-basis iterative reconstruction enables accurate signal recovery even when sensing schedules are irregular, sparse, or noisy.

The experimental analysis confirmed that:

• Stability and accuracy. Reconstruction error decreases sharply once the sampling density exceeds the predicted stability threshold, with the proposed method approaching the performance of cubic B-spline interpolation at moderate oversampling levels.
• Noise resilience. The inclusion of Tikhonov regularization yields a graceful degradation under additive noise, allowing the Riesz-basis estimator to maintain higher SSIM at low SNR compared with spline interpolation.
• Energy efficiency. The method achieves comparable accuracy at 20%–40% of the energy cost of uniform dense sensing, validating its relevance for green, resource-constrained ITS deployments.

Overall, these results demonstrate that principled, model-based signal reconstruction can play a key role in sustainable, data-driven transportation infrastructures. The framework provides a mathematically grounded bridge between sampling theory and edge computing, supporting adaptive sensor networks that balance accuracy, robustness, and energy footprint.

7.1 Future research directions

Several extensions are envisaged. First, applying the framework to spatio–temporal traffic fields would enable joint reconstruction over both space and time using tensor-product B-splines or separable Riesz bases. Second, integrating learned priors or graph-signal regularization could further improve performance in heterogeneous road networks. Finally, embedding the algorithm into on-board or roadside edge processors will allow real-time evaluation of sampling policies and their environmental impact in next-generation mobility systems.

Generative AI statement

The author(s) declared that generative AI was used in the creation of this manuscript. During the preparation of this work the authors used chatGPT in order to improve readability and language. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

Edited by: Diego Rossit, Universidad Nacional del Sur Departamento de Ingenieria, Argentina

Reviewed by: Yunxiang Yang, University of Georgia, United States

Humber Dorantes, Tecnológico de Estudios Superiores Oriente del Estado de México (Tesoem), Mexico