Error estimates for defect correction methods in Doppler tomography

T. Schuster 1
  • 1 Fachrichtung Mathematik, Geb. 36, Universität des Saarlandes, 66041 Saarbrücken, Germany.E-mail:

The aim of Doppler tomography is the reconstruction of velocity fields from ultrasonic measurements. The linearized model of this inverse problem is the Doppler transform which integrates projections of the fields along the lines the ultrasonic beams are sent off. Unfortunately the Doppler transform has a non-trivial null space. As a consequence only the solenoidal part of the field can be recovered from the data. This is the reason to develop and investigate methods of defect corrections, which are to improve the reconstruction accuracy. In [J. Inv. Ill-Posed Prob., 12(6), 597–626, 2004] and [Inv. Prob., 21,75–91, 2005] two approaches for the defect correction are considered which rely on the Helmholtz decomposition of vector fields. They consist of the solution of corresponding boundary value problems. This article is concerned with a comparison of these two approaches. We state error estimates based on orthogonal decompositions of vector fields, which indicate the improvement in accuracy we may hope for.

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