Two-Stage Signal Reconstruction

• Two-stage signal reconstruction algorithms are methods that decompose recovery into a fast, coarse inference stage followed by precise, fine-tuning optimization.
• They leverage complementary strengths by first performing rapid, rough recovery and then refining the estimate to enforce structural and model-based constraints.
• Applications include compressed sensing, imaging, communications, and deep learning, where these methods enhance both computational efficiency and reconstruction accuracy.

A two-stage signal reconstruction algorithm is a class of methods that decompose the signal recovery process into two distinct and complementary phases, typically to exploit different algorithmic strengths or to enforce multiple structural constraints explicitly. These algorithms appear in diverse domains such as compressed sensing, spectral phase retrieval, block-sparse approximation, spectrogram inversion, communication systems, and deep unfolding networks, among others. The two-stage paradigm allows for separation of coarse (fast or structurally simple) inference from fine (precise or globally consistent) optimization, enabling both computational efficiency and enhanced reconstruction fidelity. Multiple rigorous frameworks, convergence results, and practical implementations have established the significance and versatility of two-stage reconstruction across contemporary signal processing literature.

1. Mathematical Principles and General Framework

Two-stage reconstruction methods are grounded in problem structures where signal information is only partially accessible—due to undersampling, nonlinear acquisition, noise, or lossy preprocessing. The goal is to obtain a feasible or optimal signal estimate that is consistent with the measurements while simultaneously enforcing desirable signal priors, structured sparsity or other model-based regularization.

Let $x \in \mathbb{R}^N$ (or $\mathbb{C}^N$) denote the unknown signal and $y = \mathcal{A}(x) + n$ the data, where $\mathcal{A}$ is a linear or nonlinear measurement operator. Two-stage frameworks typically operate as follows:

1. Coarse Reconstruction: Perform a fast partial recovery—e.g., by projecting $y$ onto a “measurement-consistency” set, erasing zeros, or deconvolving dominant effects—yielding an initial estimate or a greatly reduced problem.
2. Fine Reconstruction: Apply a more sophisticated, computationally intensive, or model-based optimization, often constrained to the feasible set delineated from Stage 1, refining the solution to satisfy stringent signal model constraints or achieve minimum residual error.

The explicit division of algorithmic labor permits careful trade-offs between accuracy, complexity, and robustness under practical limitations (Ma et al., 2014, Ma et al., 2013, Xia et al., 20 Dec 2025, Mukhopadhyay et al., 2020, Guo et al., 2017, Zheng et al., 2022, Perlmutter et al., 2019, Chen et al., 2013, Thao et al., 13 May 2025).

2. Representative Methodologies

A variety of two-stage signal reconstruction methodologies have been formalized and extensively analyzed:

• Alternating Projection (POCS): Defined for reconstructing $x \in \mathcal{U}$ from measurements $w = P_\mathcal{V}x$ (orthogonal projection onto $\mathcal{V}$), the method alternates between projection onto the model space $\mathcal{U}$ and the affine data-consistency set $\mathcal{W}$, i.e.,

$$u^{n+1} = P_\mathcal{U}[u^n + P_\mathcal{V}(w - P_\mathcal{V}u^n)]$$

guaranteeing convergence to the intersection or to a least-squares solution under realistic geometric assumptions (Thao et al., 13 May 2025, Gadde et al., 2015).

• Two-Part (Sudocodes) CS Framework: An initial rapid support pruning is performed using sparse, binary measurements to identify zeros (“zero-identification”), followed by conventional dense compressed sensing recovery on the resulting subproblem, e.g., using Approximate Message Passing (AMP) or Binary Iterative Hard Thresholding (BIHT) (Ma et al., 2014, Ma et al., 2013).
• Guided/Convex Interpolation: Constructs a convex set interpolating between a sample-consistent affine plane and a model-based subspace, e.g., the line segment $$\mathcal{R} = \{\alpha x_c + (1-\alpha)x_g : \alpha \in [0,1]\}$$ between the least out-of-model-energy and purely model-projected points. Efficient Krylov/CGLS solvers are employed (Gadde et al., 2015).
• Two-Stage Block-OMP: For cluster-sparse signals, the algorithm first coarsely identifies possible block locations, then finely selects the actual clusters inside these windows, improving upon block OMP and sliding GBOMP in cases of unknown or non-uniform block boundaries (Mukhopadhyay et al., 2020).
• Projection-Correction Models (PCM): In Fourier imaging, first coefficients are projected onto a discretized frame expansion (e.g., via a Galerkin method), followed by a correction stage minimizing a hybrid variational functional (e.g., TV–TFV) with efficient proximal/Bregman steps (Guo et al., 2017).
• Deep Two-Stage Unfolding Networks: For compressive video sensing, a first stage applies a learned projection (with explicit inversion for measurement operator) and an initial denoising network; the second stage iterates the same architecture for refined regularization, achieving state-of-the-art performance with flexibility to unseen masks and arbitrary scaling factors (Zheng et al., 2022).
• Aliased Wigner Deconvolution + Angular Synchronization: Inverting STFT magnitude data is split into (1) algebraic deconvolution to solve for the diagonals of the rank-one matrix $\hat x \hat x^*$ and (2) eigenvector-based angular synchronization to recover phases, enabling robust phase retrieval from dramatically reduced data (Perlmutter et al., 2019).
• Two-Stage APTBM Reconstruction: In PA-distorted communications, dominant nonlinear distortion is removed via deterministic compensation (coarse), followed by constrained trust-region minimization enforcing modulation-specific amplitude/phase coupling (fine) (Xia et al., 20 Dec 2025).
• Super-resolution and Phase Retrieval: Recovery from low-pass magnitude data is achieved through (1) harmonic retrieval of unlabeled autocorrelation terms (e.g., matrix pencil), and (2) combinatorial disentanglement (sorting, distance geometry) to reconstruct the original parameter set (Chen et al., 2013).

3. Formal Guarantees, Analysis, and Regularization

Rigorous theoretical analyses exist for convergence and optimality:

• POCS: Converges geometrically to the unique solution in the intersection of the trial space and data affine set whenever their intersection is nonempty; alternately, converges to the model-constrained least-squares solution for inconsistent/noisy cases, with semi-convergence properties providing implicit regularization (Thao et al., 13 May 2025).
• TSGBOMP: Recovery guarantees leverage a generalized block-RIP (“pseudoblock-interleaved block RIP”) and provide explicit bounds on the minimum singular value for successful support recovery under dynamically varying cluster structures (Mukhopadhyay et al., 2020).
• Noisy Sudocodes + AMP/BIHT: Asymptotic support error and runtime tradeoffs are characterized precisely via binomial statistics for zero-test failure/false alarm probabilities and AMP state evolution (Ma et al., 2014, Ma et al., 2013).
• Two-stage PCM-TV-TFV: The hybrid variational correction improves on standard TV or ℓ1 schemes; empirically and theoretically, the combination suppresses both oscillatory Gibbs effects and staircasing while balancing edge and smooth region restoration (Guo et al., 2017).
• Spectrogram Inversion: Algebraic recovery of diagonal bands followed by eigenvector synchronization is provably robust to noise, with explicit sample complexity reductions compared to classic phase retrieval (Perlmutter et al., 2019).
• Two-stage APTBM: By decoupling dominant from residual distortion and leveraging geometric symmetry constraints, the algorithm demonstrates up to 4 dB IBO reduction and up to 59.1% PAE gain over prior methods in both simulation and hardware (Xia et al., 20 Dec 2025).

4. Algorithmic Structure and Complexity

Across methodologies, two-stage frameworks separate a cheap, coarse, or high-throughput inference from an expensive, precise, model-constrained refinement. Salient computational characteristics include:

• Dimension Reduction: Fast zero-detection or localization drastically shrinks the active problem size before invoking iterative optimization.
• Closed-form/FFT-friendly Steps: Initial projections, deconvolutions, matrix pencils, or deep-projection layers are often amenable to highly parallel or FFT-based evaluation.
• Iterative Refinement: Constrained iterative solvers (e.g., trust-region/CGLS/AMP/LSQR/Bregman) are deployed on reduced or better-initialized problems, improving convergence rates and solution quality.
• Regularization via Early Stopping: In Landweber or POCS-type methods, stopping iteration short of convergence regularizes ill-posed inverses.
• Discretization for DSP Implementation: Methods such as POCS with discrete basis projections map naturally to O($K^2$) DSP hardware for moderate $K$ (Thao et al., 13 May 2025).

Illustrative complexity results from the literature:

Method	Stage 1 Cost	Stage 2 Cost	Comment
Noisy Sudocodes+AMP	$O(N\log N)$	$O(\tilde N M_2)$	$N$ = original, $\tilde N$ = reduced
PCM-TV-TFV	$O(N)$ (projection)	$O(N)$ per C-iteration	C-stage uses ADMM/sparse solve
TSGBOMP	$O(Kmn)$	$O(m(Kpb)^3)$	$K$ clusters, $b$ = block size
Deep Unfolding (VCS)	$O(N)$ (GAP+1 Net)	$O(N)$ (GAP+1 Net)	Per stage, end-to-end learnable

5. Applications and Empirical Performance

Two-stage algorithms have demonstrated efficacy in a range of tasks:

• Compressed Sensing: Substantially improves runtime and support recovery probability in sparse and block-sparse settings, especially when non-uniform clusters or block boundaries are present (Ma et al., 2014, Ma et al., 2013, Mukhopadhyay et al., 2020).
• Fourier and STFT Phase Retrieval: Achieves exact or robust recovery from severely undersampled magnitude-only data, outperforming convex liftings and Gerchberg–Saxton-type approaches (Chen et al., 2013, Perlmutter et al., 2019).
• Imaging from Nonuniform Data: PCM-TV/TFV achieves higher PSNR/SSIM and reduced fine-scale error compared to standard TV, isotropic, or ℓ1-based recovery, especially notable in MRI, astronomical, and remote sensing images (Guo et al., 2017).
• Communications: In APTBM-based PAs, the two-stage method achieves substantial input back-off (IBO) reduction and power amplifier efficiency gain under both simulation and testbed conditions (Xia et al., 20 Dec 2025).
• Deep Flexible Video CS: A two-stage deep unfolding network matches or exceeds multi-stage deep networks for color/gray video compressive sensing, with greater flexibility to unseen masks and wide scaling (Zheng et al., 2022).

6. Extensions and Generalizations

Numerous extensions have been explored in the literature:

• Multi-stage and Adaptive Frameworks: Generalizations to more than two stages allow for incremental refinement or cascading of structural priors (Ma et al., 2014).
• Matrix and Tensor Extensions: For signals with matrix- or tensor-valued components (e.g., images, spectrograms), recovery algorithms incorporate Frobenius-norm correlations and associated RIP extensions (Mukhopadhyay et al., 2020).
• Nonlinear and Ill-posed Models: Two-stage paradigms are robust to heavy measurement noise, sub-Nyquist sampling, nonlinearities (e.g., in modulator hardware), and can incorporate advanced regularization (early stopping, variable splitting).
• Plug-and-Play Extensions: Deep architectures fuse explicit measurement-projection stages with learned regularizers (Zheng et al., 2022).
• Multidimensional Harmonic Retrieval: Phase retrieval super-resolution methods can be adapted to recover high-dimensional spike trains via block-Hankel pencils and multidimensional distance geometry (Chen et al., 2013).

7. Summary of Theoretical and Practical Impact

The two-stage signal reconstruction paradigm has reshaped the design space for signal recovery under partial, noisy, or nonlinear measurement regimes. By leveraging the synergy of stagewise inference—coarse pruning, constraint satisfaction, or deconvolution followed by precise, constrained or learned refinement—these algorithms rigorously and efficiently address challenges that neither one-stage regularization nor iterative consistent projection alone resolve. Mature theoretical analyses provide convergence, sample complexity, and error guarantees across a spectrum of models, and practical implementations consistently demonstrate superior empirical performance in compressed sensing, imaging, communications, spectroscopy, and deep inverse problems (Thao et al., 13 May 2025, Ma et al., 2014, Ma et al., 2013, Mukhopadhyay et al., 2020, Guo et al., 2017, Zheng et al., 2022, Perlmutter et al., 2019, Chen et al., 2013, Xia et al., 20 Dec 2025, Gadde et al., 2015).