Residue computation

Let \(\lfloor \cdot \rfloor\) represent the nearest integer function. Then a residue corresponding a loop of 3 pixels \(x\), \(y\) and \(z\) can be written in terms of wrapped phases as well as double differences depending on the unwrapping approach. We drop the superscript \(ij\) for simplicity here.

Using wrapped phases

\[ R_{xyz} = \lfloor \frac{\psi_{y} - \psi_{x}}{2\pi} \rfloor + \lfloor \frac{\psi_{z} - \psi_{y}}{2\pi} \rfloor + \lfloor \frac{\psi_{x} - \psi_{z}}{2\pi} \rfloor \]

Using double differences.

When wrapped phase differences are available

\[ R_{xyz} = \lfloor \frac{ \Delta \psi_{xy} + \Delta \psi_{yz} + \Delta \psi_{zx } }{2 \pi} \rfloor \]

When unwrapped phase differences are available

\[ R_{xyz} = \lfloor \frac{ \Delta \phi_{xy} + \Delta \phi_{yz} + \Delta \phi_{zx} }{2 \pi} \rfloor \]