Phase Unwrapping Using Least Squares Method

Least squares phase unwrapping is a critical technique in interferometric measurements and optical imaging, primarily addressing the 2π phase discontinuity issue in wrapped phase data.

When sensor-captured phase is constrained within the principal value range (-π to π), the unwrapping algorithm aims to restore the true continuous phase distribution. The least squares method achieves this by constructing a globally optimal solution, with its core concept transforming the unwrapping problem into minimizing phase gradient differences.

In practical implementation, the algorithm calculates gradient discrepancies between wrapped and unwrapped phases, then minimizes overall differences through least squares optimization. This approach involves solving a Poisson equation on discrete grids, where key computational steps include constructing gradient matrices and performing linear system solutions. Though computationally intensive, it effectively suppresses noise propagation and suits large-area continuous phase fields.

In image processing applications, this technique is widely used in synthetic aperture radar (SAR), optical interferometry, and medical imaging to reconstruct true surface topography or refractive index distributions. For phase maps with noise or undersampled regions, regularization methods are typically incorporated to enhance solution stability through additional constraints like Tikhonov regularization.

The essential understanding of least squares unwrapping lies in mastering Poisson equation solutions on discrete grids, along with computational acceleration techniques. Implementation often employs Fast Fourier Transform (FFT) for large-scale problems, where the phase unwrapping process can be optimized using FFT-based Poisson solvers that reduce complexity from O(n²) to O(n log n). Key functions in MATLAB implementations typically include fft2/ifft2 for Fourier transformations and discrete Laplacian operators for gradient calculations.