A Combinatorical Approach to Vector Tomography for Doppler Spectral Data

Velocity spectra of a flow can be made by ultrasound Doppler measurements. Using only part of the information in these spectra, it is possible to reconstruct the solenoid part and the support of the flow. Here we demonstrate that using a local minimisation combinatorical algorithm and a certain neighbourhood structure it is possible to reconstruct all of the flow, hence also the divergence. In simulations the reconstructions are very close to the original full flow. 1 Spectrum theory Consider a stationary flow inside a bounded body described by a vector valued function . When a collimated ultrasonic wave of frequency ω0 and velocity c, a t eiω0t meets a particle of speed ν in the opposite direction of the wave, having the right acoustic properties, the reflected signal obtains a Doppler shift δ. If ν c, then δ kν with k 2ω0 c The sign is reversed if the particle moves in the same direction as the wave. If some simplifying physical assumptions are made, the reflected signal is b t λe ω0 kν t where λ is a proportionality constant. For simplicity we set λ 1. The problems with λ 1 will be discussed later. Let be a directed line with direction Θ Sn 1, where Sn 1 is the unit sphere in n . When having a stream of particles, superposition yields b t e ω0 kν σ ν dν where σ ν dν is a Radon measure, depending on the number of particles on with speed values between ν and ν dν along , σ ν dν m ν Θ ν dν where m is the Lebesgue measure on the line, or more precisely: σ ν dν Θ ! dl where denotes the push forward operation. The fundamental observation is that σ may be expressed as the Fourier transform, " , of b, σ ω # ω0 k $ c " b ω This means that the velocity spectrum σ ν along is obtained from a translated Fourier transform of the received signal. In Figure 1 a typical velocity spectrum i shown.