## Equivalence of cost generators for minimum cost flow phase unwrapping

JOSA A, Vol. 19, Issue 1, pp. 64-70 (2002)

http://dx.doi.org/10.1364/JOSAA.19.000064

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### Abstract

Phase unwrapping represents a crucial step in processing phase data obtained with techniques such as synthetic aperture radar interferometry, speckle interferometry, and magnetic resonance imaging. The so-called branch-cut approaches form an important class of phase unwrapping algorithms. In 1996, Costantini proposed to transform the problem of correctly placing branch cuts into a minimum cost flow problem [Proceedings of the Fringe ‘96 Workshop (European Space Agency, Munich, 1996), pp. 261–272]. The critical point of this new approach is to generate cost functions that have to represent all the *a priori* knowledge necessary for phase unwrapping. Any function transforming *a priori* knowledge into a cost function is called a cost generator. Several types of algorithms ranging from heuristic approaches to generators based on probability-theory interpretations were suggested. A problem arising from the growing diversity of algorithms is to find a criterion for the equivalence of different cost generators. Two cost generators are equivalent if they produce cost functions with the same minimal flow for every residue configuration on every image with all possible *a priori* knowledge. Comparing the results of different cost generators on test scenes can show only their nonequivalence. We solve this problem by proving the following mathematical classification theorem: Two cost generators are equivalent if and only if one can be transformed into the other by multiplication by a fixed constant.

© 2002 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: December 4, 2000

Revised Manuscript: May 3, 2001

Manuscript Accepted: May 3, 2001

Published: January 1, 2002

**Citation**

Michael Hubig, Steffen Suchandt, and Nico Adam, "Equivalence of cost generators for minimum cost flow phase unwrapping," J. Opt. Soc. Am. A **19**, 64-70 (2002)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-19-1-64

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### References

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