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

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.     

Automatic Extraction of Traffic Flows Using TerraSAR-X Along-Track Interferometry

Spaceborne synthetic aperture radar (SAR) offers great potential for the measurement of ground traffic flows. A SAR with multiple receiving apertures aligned in flight direction repeatedly images the same ground area with a short time lag. This allows for an effective detection of moving ground objects, whose range variation translates into an interferometric phase signal between the receiving channels. The high-resolution German SAR satellite TerraSAR-X offers several ways to create multiple along-track apertures. We exploit this to demonstrate satellite-based traffic-flow measurements using along-track interferometry (ATI) and Displaced Phase Center Array techniques. In this paper, we address the usage of different TerraSAR-X ATI modes for data acquisition and describe an automatic near-real-time processing chain for the extraction of traffic information. The performance of this TerraSAR-X traffic processor is significantly driven by incorporating a priori knowledge of road networks. We present examples of automatic traffic detection as well as empirical evaluations thereof using different kind of reference data.      

Equivalence of cost generators for minimum cost flow phase unwrapping.

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 non-equivalence. 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.