Phase Retrieval via Matrix Completion (1109.0573v2)

Abstract: This paper develops a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that any complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we introduce some theory showing that one can design very simple structured illumination patterns such that three diffracted figures uniquely determine the phase of the object we wish to recover.

• The paper introduces PhaseLift, a novel framework that reformulates phase retrieval as a matrix completion problem using structured illuminations.
• It employs trace-norm minimization with iterative reweighting to promote low-rank solutions and achieve high-quality reconstructions under noise.
• Theoretical analysis confirms unique recovery conditions for 1D and 2D signals, underscoring its practical viability in imaging applications.

Phase Retrieval via Matrix Completion

The paper "Phase Retrieval via Matrix Completion" by Emmanuel J. Candès, Yonina C. Eldar, Thomas Strohmer, and Vladislav Voroninski introduces a novel framework for addressing the phase retrieval problem. This problem, which is crucial in applications like X-ray crystallography, astronomical imaging, and diffraction imaging, entails recovering a signal from the magnitude of its Fourier transform. The inability to measure the phase directly leads to significant challenges in signal reconstruction. The authors present a method termed PhaseLift, combining structured illuminations with convex optimization, specifically trace-norm minimization inspired by matrix completion techniques, to effectively solve this problem.

Problem Formulation and Novel Approach

The phase retrieval problem is inherently ill-posed as multiple signals can yield the same magnitude spectrum. Traditional approaches often rely on specific constraints, such as positivity and support constraints, to resolve this ambiguity. In contrast, the PhaseLift methodology introduces a more generalizable approach utilizing structured illuminations and convex optimization.

Structured Illuminations:

• The technique involves capturing multiple diffraction images by modulating the illuminating wavefront striking the sample. Various strategies, including the use of masks, gratings, ptychography, and oblique illuminations, are discussed for generating distinct views of the object.

Convex Optimization and Matrix Completion:

• By lifting the phase retrieval problem into a higher-dimensional space, where the recovery of a signal is equivalent to finding a rank-one positive semidefinite matrix, the problem translates into a matrix completion task. The authors propose solving a trace-minimization SDP (Semi-Definite Program):

  $$\text{minimize} \quad \text{trace}(X) \quad \text{subject to} \quad A(X) = b, \quad X \succeq 0$$

• They further suggest a reweighting scheme to enhance the promotion of low-rank solutions, iteratively solving weighted trace-minimization subproblems that converge towards the solution.

Robustness to Noise

A significant contribution of this work is the formulation of noise-aware algorithms. The authors propose a penalized maximum likelihood approach to handle different noise models systematically. For example:

• Poisson Noise: Incorporating the Poisson likelihood function in the optimization problem.
• Gaussian Noise: Devising an objective using the squared error with an appropriate noise variance matrix.

These approaches make the PhaseLift method robust to realistic measurement noise prevalent in practical applications.

Theoretical Implications and Algorithmic Performance

The theoretical analyses provided in the paper establish conditions under which the proposed phase retrieval framework guarantees unique solutions. Specifically:

• For 1D signals, three structured illuminations are proven sufficient for unique reconstruction, assuming minimal frequency overlap and non-vanishing Fourier coefficients.
• In 2D scenarios, the conditions for unique recovery extend to structured modulations across multiple dimensions, ensuring coverage in both horizontal and vertical spatial domains.

The succcess of PhaseLift in empirical settings demonstrates its practical viability. Numerical experiments on both 1D and 2D signals, including complex-valued images and real-world X-ray diffraction data, showcase:

• High-quality reconstructions from sparse phase information with low mean-square error (MSE), even under significant noise.
• Superior performance over traditional holographic and iterative projection methods.

Potential and Limitations

Although the PhaseLift approach shows considerable promise, several avenues for further research and development are apparent:

1. Scalability: The computational complexity stemming from the high-dimensional lifting step necessitates efficient algorithmic strategies, possibly leveraging low-rank matrix approximations and randomized algorithms.
2. Experimental Validation: Real-world implementation and testing of the proposed structured illuminations in a laboratory setting would provide critical insights into the practical applicability of PhaseLift for various imaging modalities.
3. Theoretical Guarantees: Extending the theoretical guarantees of the algorithm to a broader range of signal classes and noise models will enhance its robustness and reliability.

Conclusion

The paper demonstrates that integrating structured illuminations with convex optimization via PhaseLift offers a compelling solution to the phase retrieval challenge. The empirical results underline its efficacy in both noise-free and noisy settings, while the theoretical groundwork lays a robust foundation for its application across various imaging fields. The research opens new directions in phase retrieval, suggesting substantial potential for future advancements and implementations in artificial intelligence and beyond.