Reconstruction of Flows from Doppler Tomography Measurements using a Local Minimisation Algorithm

Velocity spectra of a ow 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 ow. Here we show that using a local minimisation combinator-ical algorithm and a certain neighbourhood structure it is possible to reconstruct all of the ow, hence also the divergence. In simulations the reconstructions are very close to the original full ow. 1 Spectrum theory Consider a stationary ow inside a bounded body described by a vector valued function f. When a colli-mated ultrasonic wave of frequency ! 0 and velocity c, a(t) = e i!0t ; meets a particle of speed in the opposite direction of the wave, having the right acoustic properties, the reeected signal obtains a Doppler shift. If jj c, then = k with k = 2!0 c : The sign is reversed if the particle moves in the same direction as the wave. If some simplifying physical assumptions are made, the reeected signal is b(t; x) = e i(!0+k)t ; where is a proportionality constant. For simplicity we set = 1. The problems with 6 = 1 will be discussed later. Let L be a directed line with direction 2 S n?1 , where S n?1 is the unit sphere in R n. When having a stream of particles, superposition yields b(t) = Z e i(!0+k)t (f; L;) dd ; where (f; L;) dd is a Radon measure, depending on the number of particles on L with speed values between and + dd along L, (f; L;) dd = m fx 2 Lj < f(x) + ddg ; where m is the Lebesgue measure on the line, or more precisely: (f; L;) dd = ((f(x)) dl ; where denotes the push forward operation. The fundamental observation is that b may be expressed as the Fourier transform of , k) = cFb(!) : This means that the velocity spectrum (f; L;) along L is obtained from a translated Fourier transform of the received signal.