# High Performance Computing in Satellite SAR Interferometry: A Critical Perspective
## 2 SAR Interferometric Processing
### Phase Unwrapping Operations  
The phase unwrapping (PhU) operation is the crucial stage in InSAR processing. It involves computing the (unknown) 2\(\pi\)-integer multiples that allow reconstructing the full phases (i.e., not restricted to the [\(-\pi\),\(\pi\)] range) from the measured wrapped phases. In high coherence regions and with moderate fringes, the PhU problem can be easily solved by integrating the phase differences between neighboring pixels of an interferogram, which can be correctly computed assuming that the modulus of the true gradient is only a fraction of \(\pi\). However, this assumption could generally break in some regions, leading to a wrongly computed phase gradient that would impact the subsequent integration step. In general, PhU techniques can be categorized in _local_ and _global_ approaches, based on how they perform gradient integration [70]. Local PhU algorithms exploit so-called path-following methods, which allow the computing of the unwrapped phase related to a given point in space by integrating the wrapped phase differences over connecting arcs (i.e., arcs between neighboring coherent points) and unwrapping each pixel locally, starting from a reference point. The phase of the reference point is either known a priori or assumed to be zero. [PERSON]'s algorithm [71] belongs to this class. The most reliable integration path is recovered by imposing that branch cuts obtained by connecting phase residues with the opposite sign are not crossed. Note that phase residues represent the curl of the wrapped phase differences over a spatial loop. In this way, the phases can be unwrapped consistently by integrating the wrapped phase difference along the selected path. The algorithm works well for images with high coherence values. However, areas trapped by branch cuts are inaccessible for the algorithm, resulting in spatially incomplete and/or disconnected PhU solutions. On the other hand, the algorithm fails if the branch cuts become very dense [72]. In addition, another class of path-following methods uses quality maps that synthetically describe the quality of individual pixels and connect spatial arcs. For example, the coherence is used to describe the quality of the phase, the signal-to-noise ratio, or the fringe pattern's spatial frequency. The integration path is then chosen in such a way that the high-quality pixels are unwrapped. [PERSON] and [PERSON] [73] present a review on reliability guided phase unwrapping algorithms. Global PhU algorithms [74; 75; 76; 77; 78] describe a global cost minimization problem. Historically, they have been developed to process independently single interferograms on regular and irregular grids by considering planar graphs connecting neighboring SAR pixels with medium-to-high coherence. The unwrapped phases relevant to every single arc of the connected graphs are computed from the (measured) wrapped phase differences by solving a weighted minimum \(L\)-\(p\) norm problem, where the weights are set considering the phase quality of the selected spatial arcs. Specifically, for \(p=2\) (LS optimization problem) relevant analytic solutions are known [79]. A drawback of the \(L\)-2 norm PhU solutions is that they tend to smooth discontinuities, thus not preserving the restriction that unwrapped and wrapped phases must differ by \(2\pi\)-integer multiples. For this reason, the \(L\)-1 norm is commonly preferred. The \(L\)-1 norm PhU problems was solved by [PERSON] [77] and [PERSON] [78; 80] by reformulating the PhU operation as the solution of a minimum cost flow (MCF) network optimization. The development of multi-temporal InSAR methods for the generation of ground deformation time-series from sequences of SAR images (see Section 2.6) has progressively moved the interest towards three-dimensional PhU algorithms that could generate a stack of multi-temporal unwrapped maps, taking profit from the knowledge of the spatial and temporal relationships among the computed interferograms.