The Simplest Spatial Phase Unwrapping Algorithm in Phase Measurement Profilometry

Phase Measurement Profilometry (PMP) is a technique that acquires 3D surface topography of objects through fringe projection and phase analysis. During phase measurement, since phase calculation typically returns wrapped phase (where phase values are constrained within [-π, π] or [0, 2π] range), phase unwrapping becomes necessary to restore the true continuous phase values.

The simplest spatial phase unwrapping algorithm operates under the assumption of phase continuity between adjacent pixels, meaning the actual phase difference between neighboring pixels is less than π. The algorithm's implementation follows this basic approach:

Starting Point Selection: Typically, a point in the image (such as the center point) is chosen as the initial reference point, assuming its true phase equals its wrapped phase value. This can be implemented in code as: reference_phase = wrapped_phase[center_y, center_x]

Row-wise or Column-wise Unwrapping: Beginning from the reference point, phase unwrapping proceeds pixel by pixel along rows or columns. For each current pixel, the phase difference with the previous pixel is calculated. If this difference exceeds π, the phase value is corrected by adding or subtracting integer multiples of 2π to maintain phase continuity. The core computational step can be expressed as: unwrapped_phase[i] = wrapped_phase[i] + 2π * round((unwrapped_phase[i-1] - wrapped_phase[i]) / (2π))

Global Expansion: Using a flood-fill approach starting from the initial point, the algorithm expands outward to ensure all pixels undergo phase unwrapping. This involves implementing a queue-based or stack-based traversal algorithm that processes neighboring pixels while maintaining phase continuity constraints.

The advantages of this method include simple implementation and fast computational speed, making it suitable for scenarios with low noise and gradual phase variations. However, its limitations are significant: - Sensitivity to noise, where noise artifacts can cause incorrect phase jumps - Inapplicability to objects with phase discontinuities or complex surface topologies - Path dependency, where errors in the unwrapping path can propagate to subsequent pixels

Despite its simplicity, this algorithm serves as fundamental groundwork for understanding more sophisticated phase unwrapping methods (such as least-squares approaches and quality-guided techniques), making it particularly suitable for educational purposes and experimental validation for beginners in the field.