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IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing

Quantum Optimization for Phase Unwrapping in SAR Interferometry

Kay Glatting, Jan Meyer, Sigurd Huber, Gerhard Krieger

Published Dec 17, 2024• 3 citations
Publisher

Abstract

Phase unwrapping is the reconstruction of a phase given its values mod 2$\pi$. Phase unwrapping is an important image processing technique used in synthetic aperture radar interferometry, in the context of height estimation and ground deformation. By formulating the phase unwrapping problem as a global minimization problem,$L^{p}$norm techniques form the backbone of many established phase unwrapping algorithms. The$L^{0}$norm is commonly agreed to produce the best unwrapping solutions, but the necessary brute-force calculations are infeasible for the phase unwrapping of SAR interferograms. Developments in variational quantum algorithms suggest computational advantages, when compared to classical approaches, for a multitude of applications. We therefore investigate the application of quantum algorithms to the phase unwrapping problem and introduce and derive an approximate$L^{0}$norm optimization solver utilizing the quantum approximate optimisation algorithm. This hybrid algorithm uses classical nonconvex parameter optimization to produce optimal parameters for a quantum circuit encoding the$L^{0}$phase unwrapping. Subsequent execution of this quantum circuit allows for an estimate of the solution of the phase unwrapping problem with advantages compared to brute force approaches. We validate our approach for small topographies, by demonstrating improved results when compared to established$L^{1}$optimization.

DOI
10.1109/jstars.2024.3519701

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Phase unwrapping is the reconstruction of a phase given its values mod 2π\piπ. Phase unwrapping is an important image processing technique used in synthetic aperture radar interferometry, in the context of height estimation and ground deformation. By formulating the phase unwrapping problem as a global minimization problem,LpL^{p}Lpnorm techniques form the backbone of many established phase unwrapping algorithms. TheL0L^{0}L0norm is commonly agreed to produce the best unwrapping solutions, but the necessary brute-force calculations are infeasible for the phase unwrapping of SAR interferograms. Developments in variational quantum algorithms suggest computational advantages, when compared to classical approaches, for a multitude of applications. We therefore investigate the application of quantum algorithms to the phase unwrapping problem and introduce and derive an approximateL0L^{0}L0norm optimization solver utilizing the quantum approximate optimisation algorithm. This hybrid algorithm uses classical nonconvex parameter optimization to produce optimal parameters for a quantum circuit encoding theL0L^{0}L0phase unwrapping. Subsequent execution of this quantum circuit allows for an estimate of the solution of the phase unwrapping problem with advantages compared to brute force approaches. We validate our approach for small topographies, by demonstrating improved results when compared to establishedL1L^{1}L1optimization.
Phase unwrapping is the reconstruction of an absolute phase given its values modulo 2π. It is a commonly applied post-processing technique in synthetic aperture radar interferometry applications, such as topographic height and deformation estimation. Established InSAR phase unwrapping approaches focus on large scene phase retrieval, aiming for good overall performance and relying on auxiliary data, such as additional acquisitions, to correct the resulting large local deviations. Given recent progress in quantum algorithm development for optimization problems and phase unwrapping, we are further exploring the feasibility of leveraging gate-based quantum computers and hybrid quantum algorithms to improve small-scale high-accuracy phase unwrapping. We consider a new quantum approach to L1-norm phase unwrapping and a multistage combined approach leveraging the potential advantages of L0quantum and L1phase unwrapping.
Phase unwrapping is the reconstruction of an absolute phase given its values modulo 2π2\pi2π. This post-processing step is commonly used in synthetic aperture radar interferometry (InSAR) for applications such as estimating topographic height and surface deformation. Conventional InSAR unwrapping techniques are designed for large-scale phase correction, focusing on overall consistency and often relying on auxiliary data, such as additional acquisitions or ground data, to compensate for significant local height changes. Motivated by recent advances in quantum algorithms for optimisation and phase retrieval, we explore the potential of gate-based quantum computing and hybrid quantum approaches for addressing the phase unwrapping problem. We provide an overview of the fundamental and practical limitations of quantum computing, with particular emphasis on the constraints imposed by the current noisy intermediate-scale quantum era. Furthermore, we highlight three promising directions for improved phase unwrapping: accelerating existing L1L^1L1 optimisation methods, enabling direct access to L0L^0L0 formulations, and enhancing the performance of machine learning-based approaches.
The focus of this work is to explore the use of quantum annealing solvers for the problem of phase unwrapping of synthetic aperture radar (SAR) images. Although solutions to this problem exist based on network programming, these techniques do not scale well to larger sized images. Our approach involves formulating the problem as a quadratic unconstrained binary optimization (QUBO) problem, which can be solved on a quantum annealer. Given that present embodiments of quantum annealers remain limited in the number of qubits they possess, we decompose the problem into a set of subproblems that can be solved individually. These individual solutions are close to optimal up to an integer constant, with one constant per subimage. In a second phase, these integer constants are determined as a solution to yet another QUBO problem. This basic idea is extended to several passes, where each pass results in an image which is subsequently decomposed to yet another set of subproblems until the resulting image can be accommodated by the annealer at hand. Additionally, we explore improvements to the method by decomposing the original image into overlapping subimages and ignoring the results on the overlapped (marginal) pixels. We test our approach with a variety of software-based QUBO solvers and on a variety of images, both synthetic and real. Additionally, we experiment using D-wave systems’ quantum annealer, the D-wave 2000Q_6 and developed an embedding method which, for our problem, yielded improved results. Our method resulted in high quality solutions, comparable to state-of-the-art phase-unwrapping solvers.
The focus of this work is to explore the use of quantum annealing solvers for the problem of phase unwrapping of synthetic aperture radar (SAR) images. Although solutions to this problem exist based on network programming, these techniques do not scale well to larger-sized images. Our approach involves formulating the problem as a quadratic unconstrained binary optimization (QUBO) problem, which can be solved using a quantum annealer. Given that present embodiments of quantum annealers remain limited in the number of qubits they possess, we decompose the problem into a set of subproblems that can be solved individually. These individual solutions are close to optimal up to an integer constant, with one constant per sub-image. In a second phase, these integer constants are determined as a solution to yet another QUBO problem. We test our approach with a variety of software-based QUBO solvers and on a variety of images, both synthetic and real. Additionally, we experiment using D-Wave Systems's quantum annealer, the D-Wave 2000Q. The software-based solvers obtain high-quality solutions comparable to state-of-the-art phase-unwrapping solvers. We are currently working on optimally mapping the problem onto the restricted topology of the quantum annealer to improve the quality of the solution.
The focus of this work is to explore the use of quantum annealing solvers for the problem of phase unwrapping of synthetic aperture radar (SAR) images. Our approach involves formulating the problem as a quadratic unconstrained binary optimization (QUBO) problem, which can be solved on a quantum annealer. Given that present embodiments of quantum annealers remain limited in the number of qubits they possess, we decompose the problem into a set of subproblems that can be solved individually. These individual solutions are close to optimal up to an integer constant. In a second phase, these integer constants are determined as a solution to yet another QUBO problem. In this work, we present a margin-based extension of the approach discussed above. We also present its performance under a variety of QUBO solvers including DWave’s 2000Q 6.