Comput Visual Sci (2008) 11:89–100 DOI 10.1007/s00791-007-0058-9 REGULAR ARTICLE A multilevel graph algorithm for two dimensional phase unwrapping Iddit Shalem · Irad Yavneh Received: 16 January 2006 / Accepted: 20 June 2006 / Published online: 28 April 2007 © Springer-Verlag 2007 Abstract Two-dimensional phase unwrapping is the problem of deducing unambiguous “phase” from val- ues known only modulo 2π . Many authors agree that the objective of phase unwrapping should be to find a weighted minimum of the number of places where adjacent (discrete) phase values differ by more than π (called discontinuities). This NP-hard problem is of considerable practical interest, largely due to its impor- tance in interpreting data acquired with synthetic aper- ture radar (SAR) interferometry. Consequently, many heuristic algorithms have been proposed. We present an efficient multi-level graph algorithm for the approx- imate solution of an equivalent problem—minimal res- idue cut in the dual graph. 1 Introduction Phase unwrapping is a key step in many applications such as magnetic resonance imaging, optical interfer- ometers and synthetic aperture radar interferometry. In all these applications a two-dimensional array of phase values relating to some physical quantity is obtained. Phase values are calculated by some inverse trigono- metric function (e.g. arctangent) providing only princi- pal values limited to the interval [-π , π). The true phase needs to be “unwrapped” before further processing of the physical quantity measured, as described below. We will use the synthetic aperture radar (SAR) interferom- Communicated by P. Hemker. I. Shalem (B ) · I. Yavneh Technion, Israel Institute of Technology, Haifa 3200, Israel e-mail: shalemi@cs.technion.ac.il etry as an example to introduce the phase unwrapping problem. See [1] for an elementary description. 1.1 Synthetic aperture radar Synthetic aperture radar is a valuable technique, pro- viding terrain imaging, with applications in mapping, reconnaissance and remote sensing. SAR works by illu- minating the Earth with coherent microwave radiation. This means that the transmitted signals share the same wavelength, amplitude and phase. The SAR interfer- ometry exploits coherence using two SAR images of the same surface, usually obtained from two spatially sepa- rated radar antennas, supplying two different look angle images of the same terrain. The pixel-wise phase differ- ence values are proportional to the height elevations of the imaged terrain. These measured phase values can only take values between -π and π , that is, the phase is ‘wrapped’. The phase values displayed as an image contain fringes that trace the topography. An example is shown in Fig. 1. These images were taken from the web site http://earth.esa.int/workshops/fringe_1996/fornaro/. The discontinuities in the wrapped phase image are caused by the wrapping. In order to compute the abso- lute phase difference, which is needed for extracting the topography, the 2π ambiguity inherent in the phase mea- surements must be eliminated using techniques of phase unwrapping. 1.2 Phase unwrapping Phase wrapping is a process whereby a real-valued sur- face, φ(s), is wrapped into the interval [-π , π) by the