Least-Squares with Periodic Gradients (algo_ls_poisson_pg)

This variant uses the same least-squares Poisson solve as the standard method, but it forces the wrapped gradients to be periodic before the FFT step. See Ghiglia and Pritt (1998).

API: xpunwrap.algo_ls_poisson_pg

Derivation

  1. Input: wrapped phase \(\phi_w\) in \([-\pi,\pi)\).

  2. Wrapped forward gradients:

  3. Periodic enforcement: The last column of \(g_x\) is set to the first column, and the last row of \(g_y\) is set to the first row, enforcing periodicity.

  4. Divergence:

    \[f = \nabla \cdot g\]

    where the discrete periodic divergence used by the FFT solver is:

    \[\nabla \cdot g = \left(g_x(x,y) - g_x((x-1)\bmod W,y)\right) + \left(g_y(x,y) - g_y(x,(y-1)\bmod H)\right)\]

  5. Least-squares / Poisson relation:

    The least-squares objective solved here is:

    \[\phi^\star = \arg\min_{\phi}\sum_{x,y}\lvert \nabla \phi - g \rvert^2\]

    with normal equation:

    \[\nabla^2 \phi = \nabla\cdot g = f\]

    (This is a PDE, not an ODE.)

  6. Periodic Poisson solve in the Fourier domain:

    \[\hat{\phi}(k_x,k_y) = \frac{\hat{f}(k_x,k_y)} {(2-2\cos(2\pi k_x)) + (2-2\cos(2\pi k_y))}\]

    Here \(k_x, k_y\) are normalized frequencies from \(\mathrm{fftfreq}\) (cycles/sample).

    The DC term is set to zero to fix the global offset.

  7. Sign convention: The implementation multiplies the result by \(-1\) to match the internal gradient sign convention.

  8. Inverse FFT returns the real-space unwrapped phase. If the input was 2D, the leading singleton dimension is removed.

Note

Boundary conditions and padding. Like Least-Squares Poisson (algo_ls_poisson), this solver assumes a periodic domain and applies no zero-padding before the FFT. The periodic-gradient enforcement (step 3) reduces the gradient discontinuity at domain edges compared to the standard method, but does not eliminate artifacts when the underlying phase is not truly periodic. For strongly non-periodic data, consider cropping the region of interest or using algo_ls_weighted().

References

  • D. C. Ghiglia and M. D. Pritt, “Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software,” Wiley, 1998.