Contemporary Mathematics Volume 653, 2015 Inversion of a Class of Circular and Elliptical Radon Transforms Gaik Ambartsoumian and Venkateswaran P. Krishnan Abstract. The paper considers a class of elliptical and circular Radon trans- forms appearing in problems of ultrasound imaging. These transforms put into correspondence to an unknown image function f in 2D its integrals Rf along a family of ellipses (or circles) . From the imaging point of view, of particu- lar interest is the circular geometry of data acquisition. Here the generalized Radon transform R integrates f along ellipses (circles) with their foci (centers) located on a fixed circle C. We prove that such transforms can be uniquely in- verted from radially incomplete data to recover the image function in annular regions. Our results hold for cases when f is supported inside and/or outside of the data acquisition circle C. 1. Introduction In various modalities of ultrasound imaging, the object under investigation is probed by sending through it acoustic waves and measuring the resulting wave reflections. In the bi-static setup of ultrasound reflection tomography (URT), one uses an emitter and a receiver separated from each other to send and receive acoustic signals at various locations around the body. In the mono-static case, the same device (a transducer) works both as an emitter and as a receiver. The mathematical model considered in this paper uses two simplifying assump- tions, which hold reasonably well in many applications. We assume that the speed of sound propagation in the medium is constant and the medium is weakly reflect- ing. The latter means that we ignore the signals that arrive at the receiver after reflecting more than once inside the object. Under these assumptions, the time delay t between sending and receiving the signal defines the distance d1 + d2 trav- eled by the wave from the emitter to the reflection location and from there to the receiver. Hence the measured signal is the superposition of all echoes generated by inclusions located at points x that have a constant sum d1 + d2 of distances to the emitter and the receiver. In the bi-static setup, this corresponds to the integral of 2010 Mathematics Subject Classification. Primary 44A12, 45D05, 92C55. Key words and phrases. xx. The work of both authors was supported in part by NSF grant DMS-1109417. The second author benefited from the support of the Airbus Group Corporate Foundation Chair in Mathe- matics of Complex Systems established at TIFR Centre for Applicable Mathematics and TIFR International Centre for Theoretical Sciences, Bangalore, India. The first author thanks the organizers of the 2013 Conference on Complex Analysis and Dynamical Systems VI in Nahariya, Israel, for hospitality and great organization. c 2015 G. Ambartsoumian, V. P. Krishnan 1
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