## A class of solution-invariant transformations of cost functions for minimum cost flow phase unwrapping

JOSA A, Vol. 21, Issue 10, pp. 1975-1987 (2004)

http://dx.doi.org/10.1364/JOSAA.21.001975

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### Abstract

Phase unwrapping (PU) represents an important step in synthetic aperture radar interferometry (InSAR) and other interferometric applications. Among the different PU methods, the so called branch-cut approaches play an important role. In 1996 M. Costantini [Proceedings of the Fringe ’96 Workshop ERS SAR Interferometry (European Space Agency, Munich, 1996), pp. 261–272] proposed to transform the problem of correctly placing branch cuts into a minimum cost flow (MCF) problem. The crucial point of this new approach is to generate cost functions that represent the *a priori* knowledge necessary for PU. Since cost functions are derived from measured data, they are random variables. This leads to the question of MCF solution stability: How much can the cost functions be varied without changing the cheapest flow that represents the correct branch cuts? This question is partially answered: The existence of a whole linear subspace in the space of cost functions is shown; this subspace contains all cost differences by which a cost function can be changed without changing the cost difference between any two flows that are discharging any residue configuration. These cost differences are called strictly stable cost differences. For quadrangular nonclosed networks (the most important type of MCF networks for interferometric purposes) a complete classification of strictly stable cost differences is presented. Further, the role of the well-known class of node potentials in the framework of strictly stable cost differences is investigated, and information on the vector-space structure representing the MCF environment is provided.

© 2004 Optical Society of America

**OCIS Codes**

(120.0280) Instrumentation, measurement, and metrology : Remote sensing and sensors

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(280.6730) Remote sensing and sensors : Synthetic aperture radar

**History**

Original Manuscript: September 2, 2003

Revised Manuscript: March 30, 2004

Manuscript Accepted: March 30, 2004

Published: October 1, 2004

**Citation**

Michael Hubig, Steffen Suchandt, and Nico Adam, "A class of solution-invariant transformations of cost functions for minimum cost flow phase unwrapping," J. Opt. Soc. Am. A **21**, 1975-1987 (2004)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-10-1975

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