phase unwrapping

This paper investigates the stability of the PhaseLift algorithm for phase retrieval from coded diffraction patterns (CDPs) under additive noise. Addressing the limitation of existing error bounds—namely, their dependence on the ℓ²-norm of the noise vector (|mathbf{w}|_2) without reflecting average noise intensity—the work provides the first rigorous proof of Soltanolkotabi’s conjecture: the optimal error bound scales with the *average* noise magnitude, i.e., (|mathbf{w}|_2 / sqrt{m}). Leveraging tools from convex optimization, random matrix theory, and high-dimensional statistics, the authors derive an upper bound of (O(log n cdot |mathbf{w}|_2 / sqrt{m})) under adversarial noise and (O(sigma sqrt{n log^4 n / m})) under sub-Gaussian noise. Matching minimax lower bounds are established in both settings, differing only by logarithmic factors. These results close a fundamental theoretical gap in the stability analysis of PhaseLift for CDP-based phase retrieval.

This work addresses the channel covariance-driven continuous angular power spectrum (APS) recovery problem. We propose a weighted Fourier-domain affine projection reconstruction method. By establishing an exact energy identity, we derive, for the first time, an analytical expression for the APS reconstruction error and rigorously characterize identifiability: perfect recovery is achievable if and only if the true APS lies in a specific trigonometric polynomial subspace; otherwise, the method returns the minimum-energy consistent solution. The approach integrates weighted Fourier analysis, projection onto linear manifolds (PLM), closed-form solutions for positive-definite matrices, and trigonometric polynomial approximation, yielding a unique closed-form solution with an explicit error bound. Experiments demonstrate that the method achieves optimal APS estimation under covariance consistency constraints, significantly enhancing angular resolution and geometric interpretability.

This paper studies the 1-bit phase retrieval problem: reconstructing a real signal $mathbf{x} in mathbb{R}^n$—either dense or $k$-sparse—from $m$ 1-bit measurements $mathrm{sign}(|mathbf{a}_i^ op mathbf{x}| - au)$, where $mathbf{a}_i sim mathcal{N}(0,I_n)$. We establish the first information-theoretically optimal estimation error bounds, demonstrating that phase information is unnecessary in 1-bit compressed sensing. We propose a one-sided $ell_1$ gradient descent algorithm with spectral initialization, achieving linear convergence and near-optimal statistical accuracy. Key technical innovations include random hyperplane tessellation analysis, local restricted approximate invertibility characterization, and thresholded gradient updates. Our theory yields optimal error rates $O((n/m)log(m/n))$ for dense signals and $O((k/m)log(mn/k^2))$ for $k$-sparse signals. The algorithm attains sample complexities of $O(n)$ and $O(k^2 log n cdot log^2 (m/k))$, respectively—achieving both computational efficiency and statistical optimality.

This work addresses robust recovery of structured signals in phase-only compressive sensing (PO-CS). To handle the nonlinearity inherent in complex Gaussian phase measurements, we propose a linearization technique that embeds phase-only measurements into the standard compressive sensing framework, enabling uniform instance-optimal reconstruction for all sparse signals on the unit sphere. We establish, for the first time in a nonlinear sensing setting, a unified and robust instance-optimal theory: with near-optimal sample complexity, the reconstruction error satisfies ‖x♯ − x‖₂ ≤ Cσₛ(x)₁/√s. The method exhibits strong robustness against both front-end and back-end additive noise, as well as sparse corruptions—enabling stable recovery even when sparse perturbations coexist with dense noise—and further guarantees exact recovery in the presence of sparse corruption. The theoretical performance matches that of linear compressive sensing, providing a novel paradigm for nonlinear sparse inverse problems.

This paper addresses the affine phase retrieval problem—reconstructing an unknown signal from the magnitudes of affine measurements. To tackle this nonconvex, nonsmooth inverse problem, we propose a second-order optimization framework integrating Newton’s method and the Gauss–Newton method. Under signal priors (e.g., sparsity or structural constraints) and measurement models (e.g., Gaussian random or coded diffraction patterns), we establish, for the first time, a global quadratic convergence theory: we prove strong convexity of the objective function in a neighborhood of the solution and provide a unified convergence analysis for both second-order methods. The theoretical guarantees are initialization-free and hold in the noiseless setting. Numerical experiments demonstrate that our approach outperforms state-of-the-art first-order algorithms in reconstruction accuracy, convergence speed, and sampling efficiency—achieving exact recovery with measurements nearly attaining the information-theoretic lower bound, thus offering both computational efficiency and robustness.

This work addresses the sub-Nyquist reconstruction of multiband signals under single-channel modulo folding sampling. We propose a low-complexity, single-channel acquisition and reconstruction framework grounded in an unbounded sensing paradigm. Departing from conventional oversampling requirements, our approach tightens the bandpass sampling theorem by deeply integrating modulo folding sampling with sub-Nyquist reconstruction algorithms, while incorporating hardware-aware signal processing. Unlike traditional methods, the proposed scheme overcomes the dynamic range limitation inherent in modulo sampling, enabling high-fidelity recovery of up to six spectral bands using only a single hardware channel. Experimental results demonstrate a 13× improvement in effective dynamic range and a substantial reduction in sampling rate, thereby significantly alleviating hardware resource constraints.

This work resolves a long-standing open problem in structured phase retrieval by establishing the optimal sampling rate required for exact reconstruction of complex-valued signals from coded diffraction patterns. Under the standard random mask model, the authors prove for the first time that PhaseLift recovers any n-dimensional complex signal—up to a global phase—with only O(log n) random masks, achieving the information-theoretic lower bound Ω(log n). This yields a total sampling complexity of O(n log n). The analysis hinges on several key technical innovations: an approximate dual certificate constructed via an enhanced golfing scheme, adaptive mask allocation, and a dimension-independent truncation threshold. The proposed method attains this optimal sampling efficiency with a failure probability that decays polynomially in the signal dimension.

This work addresses the severe angular instability in object detection—particularly for square-like objects—caused by angle boundary discontinuity (ABD) and circular ambiguity (CA). To resolve these issues, the authors propose a lightweight, plug-and-play Fourier Series Encoder (FSC), which, for the first time, maps angles onto an orthogonal Fourier basis. By integrating geometric manifold constraints, phase unwrapping, and magnitude stabilization mechanisms, FSC establishes a continuous, invertible, and mathematically robust angle encoding–decoding paradigm that fundamentally eliminates boundary discontinuities and removes the need for heuristic truncation. Experiments demonstrate that FSC significantly enhances high-precision detection performance across three large-scale datasets, effectively suppresses angular bias, and exhibits strong noise robustness alongside seamless boundary continuity.

This paper addresses the stable recovery problem in phase retrieval under Poisson and heavy-tailed noise. We develop the first signal-energy-adaptive unified analysis framework for both nonconvex and convex least-squares estimators: at high signal-to-noise ratio (high energy), Poisson noise is modeled as sub-exponential; at low SNR (low energy), it is treated as heavy-tailed—thereby bridging the theoretical gap between these two noise regimes. Leveraging multiplier inequalities, empirical process theory, and random matrix analysis jointly, we derive tight risk bounds for the first time: the high-energy regime achieves the minimax-optimal rate $O(sqrt{n/m})$, while the low-energy regime attains $O(|x|^{2-1/4}(n/m)^{1/4})$; both rates remain optimal under heavy-tailed noise. The framework naturally extends to sparse phase retrieval and low-rank matrix reconstruction.

This work proposes the first polynomial-time algorithm for phase retrieval under heavy-tailed noise and simultaneous adversarial corruption of both sensing vectors and measurements. By integrating robust spectral initialization with state-of-the-art robust principal component analysis, the method leverages tools from high-dimensional statistical estimation and adversarially robust learning theory to accurately recover an $n$-dimensional signal up to a global phase, using only near-linear sample complexity ($O(n \log n)$). This approach overcomes the limitations of prior methods that relied on computationally intractable, exponential-time procedures, thereby achieving the first efficient and provably robust solution for phase retrieval in the challenging setting of heavy-tailed noise combined with dual adversarial contamination.