IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 1, JANUARY 1999 163 Multiresolution Phase Unwrapping for SAR Interferometry Gordon W. Davidson, Member, IEEE, and Richard Bamler, Member, IEEE Abstract— An approach to two-dimensional (2-D) phase un- wrapping for synthetic aperture radar (SAR) interferometry is presented, based on separate steps of coarse phase and fine phase estimation. A technique called adaptive multiresolution is introduced for local fringe frequency estimation, in which difference frequencies between resolution levels are estimated and summed such that a sufficiently conservative phase gradient field is maintained. A coarse unwrapped phase of the full terrain height is then constructed using weighted least-squares based on coherence weighting. This coarse phase is used in a novel ap- proach to slope-adaptive spectral shift filtering and to reduce the phase variation of the interferogram. The resulting interferogram can be more accurately multilooked and unwrapped with any algorithm. In this paper, fine phase construction is done with weighted least-squares and with weights determined by simple morphological operations on residues. The approach is verified on a simulated complex interferogram and real SAR data. Index Terms— Multiresolution spectral estimation, phase un- wrapping, synthetic aperture radar (SAR) interferometry. I. INTRODUCTION T WO-DIMENSIONAL (2-D) phase unwrapping is a crit- ical step in the generation of digital elevation models (DEM’s) using synthetic aperture radar (SAR) interferometry. Given a complex-valued interferogram created from two reg- istered complex SAR images of the same scene, the terrain height is related to the absolute phase of the interferogram, whereas the measured phase is known only modulo 2 or is “wrapped.” Phase unwrapping consists of two steps: the first step is the estimation of phase gradients from the interferogram—either by multilooking and taking wrapped phase differences [1] or by local fringe frequency estimation [2]. The second step is the integration of the gradient estimates to obtain an unwrapped phase surface. The presence of noise and undersampling causes gradient estimates to be aliased so that the measured gradient field is unconservative, leading to a path-dependent integration and errors in the constructed phase surface. Aliasing errors occur between points of nonzero curl of the phase gradient field, or residues, and phase unwrapping algorithms attempt to exclude the aliasing errors from the integration process. This is done either by branch-cuts in a path-following integration [1] or by zero-weights in weighted least-squares or related algorithms [3]–[5]. In either case, Manuscript received January 6, 1997; revised February 17, 1998. G. W. Davidson is with MacDonald Dettwiler, Richmond, B.C., Canada V6V 2J3 (e-mail: gordon@mda.ca). R. Bamler is with the German Aerospace Center (DLR), German Re- mote Sensing Data Center, Oberpfaffenhofen, D-82234 Wessling, Germany (richard.bamler@dlr.de). Publisher Item Identifier S 0196-2892(99)00039-X. in noisy areas, the integration of aliasing errors tends to underestimate the terrain height [6]. Also when the residue density is high, much of the unwrapped phase is not obtained because much of the phase gradient information is excluded. Phase unwrapping can be improved if a coarse resolu- tion phase surface is available, such as from a DEM or an unwrapped phase surface obtained at another baseline or wavelength [7]. In this paper, we present a method to estimate a coarse phase surface from the data itself. This can be thought of as an extension of the common practice of removing the “flat earth” phase to reduce the phase variation to make phase unwrapping easier. Local estimation of fringe frequencies of the interferogram can provide the inputs to a least-squares construction of a coarse phase [2]. However, in the presence of terrain slope, the aliasing errors in the local frequency estimates have a nonzero mean, resulting in a slope bias that prevents construction of the full height of the phase surface [6], [8]. To solve this problem, we introduce a technique, called adaptive multiresolution frequency estimation, in which difference frequencies between resolution levels are estimated and summed such that a conservative phase gradient field is maintained. A least-squares construction gives a smooth unwrapped phase that is used for coherence estimation. Then using weighted least-squares based on coherence weighting, a coarse unwrapped phase surface that attains most of the terrain height is constructed. The coarse phase is used to preprocess the data to improve the quality of phase unwrapping. It provides information about local slope for use in slope-adaptive spectral shift filtering to improve coherence [9]. It is also used to reduce the interferogram phase variation, or “flatten” the interferogram. Then, noise filtering, or “multilooking” of the interferogram, can be performed more accurately, and the effect of phase slope on the aliasing error in phase gradient estimation is reduced. Also, since most of the terrain height is attained by the coarse phase, the overall phase unwrapping is more robust to errors in the fine phase construction. In a sense, the coarse phase construction, through the use of frequency estimation and coherence weighting, allows more information to be extracted from the data. Given the flattened, multilooked interferogram, the fine phase can be constructed more reliably with any phase un- wrapping algorithm. To complete the overall system here, we use a weighted least-squares algorithm for fine phase construction. The zero weights are determined by simple morphological dilation and erosion operations on an initial set of weights corresponding to residues and adjacent pixels. 0196–2892/99$10.00  1999 IEEE