Two-Stage Sensing Framework

• The paper introduces a two-stage sensing framework that first reduces uncertainty with a coarse, fast estimation process before applying computationally intensive refinement.
• It employs a robust initial stage to narrow the search space and a secondary stage that leverages model-adaptive techniques for fine parameter estimation.
• This approach is extensively used in communications, quantum estimation, and anomaly detection, enhancing both detection reliability and resource efficiency.

A two-stage sensing framework is an architectural or algorithmic paradigm in which information acquisition, estimation, or detection proceeds through two explicitly separated and functionally distinct stages. The critical rationale for bifurcation into two stages is the exploitation of complementary strengths: typically, an initial coarse, rapid, or robust process reduces uncertainty or search space, followed by a second, computationally or statistically intensive refinement restricted to a manageable region, dataset, or hypothesis set. The two-stage strategy is pervasive across disciplines, including communications, quantum estimation, wireless sensing, anomaly detection, compressive imaging, and spatiotemporal data reconstruction.

1. Foundational Principles of Two-Stage Sensing

The two-stage sensing paradigm is grounded in the principle of hierarchical information extraction:

• Stage 1 ("coarse" or "preliminary" stage): Employs simple, robust, or efficient sensing, detection, or estimation techniques to extract initial coarse information or to narrow the search space. This stage is often model-agnostic or uses fixed, parameter-independent methods, aiming for broad coverage or high recall at low resource cost.
• Stage 2 ("refined" or "secondary" stage): Utilizes specialized, often computationally intensive, model-adaptive or parameter-optimized techniques that operate within the narrower uncertainty or search region produced by Stage 1. Refinement may involve high-resolution parameter estimation, denoising, or context-aware hypothesis discrimination.

These two stages may employ heterogeneous sensor modalities, distinct signal models, disparate learning paradigms (e.g., unsupervised autoencoders followed by recurrent nets, variational Bayesian inference after subspace estimation), or cascade decision strategies (e.g., entropy-based test with confirmatory secondary sampling).

Formally:

• In compressed sensing or estimation, two-stage methods may alternate between analytic procedures and model-driven neural inference (Zheng et al., 2022, Jeong et al., 2022).
• In spectrum or cognitive radio, two-stage protocols trade off detection reliability (via conservative guard zones) against minimal false alarms, subject to system throughput and collision constraints (Gabran et al., 2010, Zhao, 2011).
• In radar and Wi-Fi sensing, an initial coarse estimator addresses ambiguity or high-dimensional search, with the secondary stage focusing on fine parameter tuning via convex or non-convex estimation, denoising, or spatial clustering (Liu et al., 2020, Bacchielli et al., 2024, Hu et al., 2022, Wu et al., 18 Nov 2025).

2. Formal Architectures and Algorithmic Realizations

Representative 2-Stage Architectures

Application Domain	Stage 1 Function	Stage 2 Function
Quantum Sensing (Gong et al., 2024)	θ-independent measurement, pre-estimator for parameter	θ̂₁-dependent SLD measurement, MLE refinement
OFDM Radar (Liu et al., 2020)	Delay (range) estimation via compressed sensing	Angle estimation via array processing (MUSIC/Capon)
WiFi Sensing (Hu et al., 2022)	Weighted root-MUSIC for coarse delays	Stochastic particle-based VBI for refinement
Anomaly Detection (Jeong et al., 2022)	Signal-type-specific DAE (recall-optimized)	LSTM-DAE on sensor data (precision-optimized)
Spatiotemporal Reconstruction (Sun et al., 2024)	Diffusion model (spatial, sparse input)	Diffusion model (temporal patterning, fine-scale)
Spectrum Sensing (Gabran et al., 2010)	Fast, low-resolution energy detection	Conservative, high-resolution energy test

Algorithmic Structure

1. Coarse Estimation/Detection
   • Channel subspace estimation, entropy-based decision, or spatial/temporal feature extraction under robust but non-adaptive or generic assumptions.
   • Often accompanied by uncertainty quantification or confidence assessment to determine whether escalation to Stage 2 is necessary (Zhao, 2011).
2. Refined Inference or Disambiguation
   • Exploits the preliminary outcome to select a focused region for high-accuracy parameter estimation (MUSIC, ML, Bayesian, RL-based policies, denoising neural networks).
   • Adaptive measurement protocols or iterative optimization (e.g., Newton-Raphson, variational Bayes, unwrapped phase fitting, federated reinforcement learning) (Guo et al., 16 Jun 2025, Sun et al., 2024, Wu et al., 18 Nov 2025).

3. Performance Analysis and Asymptotic Properties

Two-stage frameworks are motivated by critical trade-offs in sensing theory, including resource allocation, statistical efficiency, computational complexity, and operational constraints.

• Cramér–Rao/Quantum Cramér–Rao Bound Saturation: In quantum metrology, two-stage estimators achieve the quantum Fisher information limit, with initial θ-independent measurements followed by SLD measurements adapted via preliminary estimates, ensuring consistency and asymptotic normality under mild conditions (Gong et al., 2024).
• Complexity versus Accuracy: Stage 1 reduces the problem size or search range, enabling Stage 2 to apply computationally intensive or high-resolution estimation within feasible regions—critical for ultra-wideband or high-dimensional sensor architectures (Bacchielli et al., 2024, Hu et al., 2022).
• False Alarm and Detection Probability: Adaptive two-stage protocols outperform single-stage strategies, especially under strict error constraints, as in spectrum sensing (energy detection plus confirmatory check) (Gabran et al., 2010, Zhao, 2011). The fusion of multi-bit decisions in cooperative settings further improves detection rates without significant added complexity.
• Robustness to Model Misspecification and Sensor Constraints: Fresh mask adaptation, scaling to large spatial domains, accommodating modality heterogeneity (e.g., sparse/irregular sensors), and variable masking (e.g., real-time battery modeling, multiband Wi-Fi) are systematically enabled by the two-stage split (Wei, 2023, Zheng et al., 2022).

4. Domain-Specific Instantiations

Quantum Sensing and Estimation

Two-stage quantum estimation, as formalized in (Gong et al., 2024), bypasses the measurement paradox by deploying an initial approximation step via θ-independent measurements (e.g., homodyne detection), followed by SLD-optimized classical inference over an exponentially dominant sample count. The key regularity relaxation broadens admissible estimators, admits nuisance parameter extensions, and delivers quantum-enhanced mean-squared error scaling ($\sim1/N$).

Integrated Sensing and Communication (ISAC), Radar, and WiFi

ISAC two-stage frameworks unify communication pilot structures (e.g., 5G NR) with custom sparse pilot insertions for UAV and target localization. A cascade of angular estimation (MUSIC), beamformed spatial filtering, and multi-resolution FFT is followed by fine-grained off-grid parameter extraction using unwrapped-phase and iterative relaxation (RELAX), culminating in spatial clustering (DBSCAN) for hypothesis grouping and error minimization (Wu et al., 18 Nov 2025).

Radar signal processing architectures deploy channel sparsity exploitation (compressed sensing/group LASSO), with range–angle coupling resolved through eigenvalue decomposition and beam-space models, allowing the decoupling of delay/range and angle/DoA estimation. Multiband WiFi sensing employs low-complexity root-MUSIC for global search reduction, with SPVBI yielding sharp, near-CRB parameter identification (Liu et al., 2020, Hu et al., 2022, Bacchielli et al., 2024).

Deep Learning and Anomaly Detection

Heterogeneous time series or spatiotemporal field estimation mandates model-agnostic, signal-type-specific anomaly detectors (e.g., DAE for event logs, LSTM-DAE for temporally coherent data). The separation ensures that each stage is tuned to the statistical structure of its data type. Joint or concatenated treatments are outperformed due to the statistical mismatch between features (Jeong et al., 2022). Similar cascades—diffusion-based spatial completion followed by temporal upsampling—are used for missing value imputation in complex spatiotemporal datasets (Sun et al., 2024).

Adaptive Sensing, Field Fusion, and Resource-Constrained Sensing

Meta-learning-based two-stage adaptive sensing in WSN settings deploys a base-learner (proximal SGD parameter update) and a meta-learner (policy-gradient-driven sensor query selector). The policy jointly optimizes error reduction and communication cost, yielding sample complexity and mean-square error advantages versus uniform or static importance sampling (Wu et al., 2019).

5. Experimental and Theoretical Performance Benchmarks

Two-stage approaches universally exhibit empirical and analytical superiority over single-stage methods under operational metrics tailored to application:

Reference (arXiv ID)	Application	Metric(s)	Two-Stage Gain over Baseline
(Zheng et al., 2022)	Video CS	PSNR	+1.7 dB (1→2 stages), +2.3 dB and >17× speed-up (color VCS)
(Jeong et al., 2022)	Anomaly detection	F₁	+7–12 points over best single-stage, best F₁=0.77
(Gong et al., 2024)	Quantum estimation	MSE, QCRB	Achieves 1/(N QFI) scaling, QFI continuity with relaxed conditions
(Liu et al., 2020, Bacchielli et al., 2024)	Radar/THz ISAC	RMSE, PEB	Achieves sub-cm accuracy, mm-level tracking at SNR > –10dB
(Hu et al., 2022)	WiFi sensing	RMSE	Near-CRB performance, >2× faster convergence than PVBI
(Wu et al., 18 Nov 2025)	UAV ISAC	Range RMSE	Matches or surpasses 2D-FFT/MUSIC, scales with multi-targets

6. Limitations, Practical Considerations, and Extensions

Despite their advantages, two-stage frameworks may entail:

• Additional per-stage latency in uncertain regimes if both stages are required (e.g., spectrum sensing under low-SNR (Zhao, 2011)).
• Occasional minor loss in asymptotic optimality relative to fully joint, computationally infeasible approaches (e.g., quantum two-stage estimators).
• Inference speed bottlenecks in models based on deep diffusion or generative architectures (Sun et al., 2024).
• Need for careful per-stage thresholding or uncertainty calibration, as in entropy detection or Bayesian classification (Gabran et al., 2010).

Potential extensions include multi-stage generalizations (for extremely high-dimensional parameter spaces), hybrid reinforcement learning for resource-constrained adaptive sensing, and real-time updating for streaming or online deployments (Wu et al., 2019, Sun et al., 2024).

7. Connections, Variants, and Generalizations

Two-stage frameworks can be viewed as instances of:

• Sequential detection, confirmation, or estimation (e.g., Wald SPRT, Neyman–Pearson cascades).
• Deep unfolding/rolling in neural network reconstruction, with the number of stages determined by the trade-off between capacity and generalization (Zheng et al., 2022).
• Coarse-to-fine search and resolution adaptive approaches in signal processing, with analogues in numerical optimization and machine learning (e.g., meta-learning, curriculum learning).
• Variational Bayesian and block-wise optimization schemes that exploit dimensionality reduction or subspace alignment.

Overall, the two-stage sensing framework emerges as a unifying meta-architecture for balancing statistical efficiency, computational expense, and adaptability, as demonstrated across a wide spectrum of state-of-the-art applications in quantum metrology, communications, radar, imaging, and data-centric industrial sensing.