Principles of Modern Radar - Volume 2 1891121537

8.5 InSAR Processing Steps 369Along-Track (m)−6−4−2024696979899Range (m)(a)100101102−10−20−30−40−50−60−70Unwrapped IPD (radians)Along-Track (m)−6−4−2024610310496979899100101102103Range (m)the small linear patch of noisy IPD data at about 98.5 m of range. Such a patch could resultfrom low-reflectivity terrain, shadowing, or data corruption. Applying the residue test oneach 2 × 2 pixel loop in this image would reveal a number of nonzero residues of bothpositive and negative in the vicinity of this noise patch.Figure 8-20 shows the result of unwrapping the phase map of Figure 8-19b via pathfollowingtechniques. Figure 8-20a is the result obtained with a systematic path thatdisregards any possible residues. The IPD noise significantly degrades the unwrappedphase. Figure 8-20b used the GZW algorithm to determine branch cuts and then unwrappedalong a path that avoided branch cut crossings. In this case, the unwrapped phase isindistinguishable from the original IPD before wrapping except at two small holes atlocations that were inaccessible due to the pattern of branch cuts.(b)1040−10−20−30−40−50−60−70Unwrapped IPD (radians)FIGURE 8-20Phase unwrappingusing thepath-followingtechnique. (a) Resultignoring residues.(b) Results using theGZW algorithm.8.5.3.2 Least Squares MethodA second major class of two-dimensional phase unwrapping algorithms is least squaresmethods. The path-following techniques are local in the sense that they determine theunwrapped phase one pixel at a time based on adjacent values. In contrast, the leastsquares methods are global in the sense that they minimize an error measure over theentire phase map. A classic example of this approach is the Ghiglia-Romero algorithmdescribed in [48]. This technique finds an unwrapped phase function that, when rewrapped,minimizes the mean squared error between the gradient of the rewrapped phase functionand the gradient of the original measured wrapped phase. An efficient algorithm existsto solve this problem using the two-dimensional discrete cosine transform (DCT). Thesimplest version of the algorithm, called the unweighted least squares algorithm, beginsby defining the wrapped gradients of the M × N raw wrapped IPD data:⎧〈 ⎨ ˜φ difab [l + 1,m] − ˜φ difab˜ y [l,m] =〉2π [l,m] , 0 ≤ l ≤ L − 2, 0 ≤ m ≤ M − 1⎩0, otherwise⎧〈 ⎨ ˜φ difab [l,m + 1] − ˜φ difab˜ x [l,m] =〉2π [l,m] , 0 ≤ l ≤ L − 1, 0 ≤ m ≤ M − 2⎩0, otherwise(8.36)