Quantum Optimization for Phase Unwrapping in SAR Interferometry

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.