Discrete Green's methods and their application to two-dimensional phase unwrapping.

Area of Science:

• Mathematics
• Computational Science
• Image Processing

Background:

• Classical calculus theorems (Stokes's, Gauss's, Green's) are foundational in continuous physics and mathematics.
• Discretizing these theorems often leads to approximations and loss of analytical rigor.
• The two-dimensional phase-unwrapping problem presents challenges in signal processing and image analysis.

Purpose of the Study:

• To present a fully self-contained discrete framework for calculus operators.
• To establish discrete equivalents of Stokes's, Gauss's, and Green's theorems.
• To demonstrate the framework's superiority over continuous methods for specific applications, particularly phase unwrapping.

Main Methods:

• Development of a discrete formulation analogous to continuous operators.
• Adherence to predefined rules for constructing discrete contours and domains.
• Application and validation of the discrete framework to the two-dimensional phase-unwrapping problem.

Main Results:

• The discrete framework precisely mirrors continuous theorems under specific construction rules.
• The method guarantees analytical rigor, unlike approximated continuous formulations on discrete grids.
• Demonstrated improved performance of the discrete framework for phase unwrapping on irregular domains, with undersampling and noise.

Conclusions:

• The proposed discrete framework provides a robust and analytically rigorous alternative to continuous methods.
• This discrete approach is particularly advantageous for complex problems like phase unwrapping in realistic scenarios.
• The framework offers a more reliable foundation for computational applications involving discrete data and complex geometries.