LOCAL VECTOR TOMOGRAPHY BY USE OF WAVELETS

where mµ is the ordinary Lebesgue measure on weight µ. That is for A R mµ (A) =
Z
A
R with the
µ(x) dx ;
where µ is a function from R3 to R, assumed to be such that its restriction to A is well defined. The fundamental observation is that b may be expressed as the Fourier transform of σµ , σµ (f ; L;
where µ is a non-negative function describing the sensitivity to ultrasound. Let L be a directed line. When having a
Z
f (sΘ + y) d y =
Z
f (x) d x :
Θ?
x Θ=s
Vmபைடு நூலகம்=
m?1 j =?∞
M
Wj
and L2 (R) =
m2
M
Z
Wm :
We can see the subspaces Vm as containing ‘details up to level m’. The next level is reached by including the ‘detail difference’ space Wm . The function ϕ is called father wavelet or scaling function. A basis function ψ, such that
ABSTRACT Velocity spectra of a flow can be made by ultrasound Doppler measurements. Using only part of the information in two dimensions, the vectorial Radon transform appears, where integration is made over the components of the vector field along the ray of integration. It is well known that the solenoid (curl) part of the flow can be determined from the vectorial Radon transform. A problem is that the tomography reconstruction problem is not local in two dimensions, i.e. all measurement data is needed for the reconstruction in each point. However, it is possible to show by use of wavelets that, if it is sufficient to make reconstruction on a certain detail level and low frequency behaviour is of limited interest, the reconstruction can be made almost local, in the sense that only data for beams in the vicinity of the point of reconstruction are needed to make reliable reconstructions. Here we extend this theory to the vector tomography case. Vector tomography could be used to e.g. find tumours, in which the blood flow is more intense and irregular than in normal tissue. In this context the study of high frequency phenomena would suffice. 1. PRELIMINARIES Consider a stationary flow inside a bounded body described by a vector valued function f . 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 ob0 tains a Doppler shift. If jνj c, then δω = kν with k = 2ω c : The sign is reversed if the particle moves in the same direction as the wave. If some simplifying physical assumptions are made, see [1, 2], the reflected signal is b(t ; x) = µ(x)ei(ω0 +kν)t
A more detailed version of the discussion above can be found in [1]. Another more straightforward situation where such integrals (1) appear, is time-of-flight measurements [4], [5]. Yet another example is the probe direction transform, see [6]. We call (1) the vectorial Radon transform and it is denoted by V f (Θ; s) : The solenoid part of a Helmholtz decomposition of the flow can be determined from V f , whilst the potential part remains unknown, see e.g. [5, 1]. 2. LOCAL TOMOGRAPHY Sometimes it is only of interest to reconstruct part of the body being examined, and one would like to do so from a smaller set of data. In the vector tomography case one would like to ‘zoom in’ on intersecting regions, with highly oscillating curl. However, the tomographic reconstruction problem is not local for n = 2. Below we review a method showing that it is possible to make the reconstruction task into an almost local problem. This theory has been described in e.g. [7, 8, 9]. For a general reference about wavelets, see e.g. [10]. We show how the same approach with some modifications can be used to make almost local reconstructions of the curl of the flow from V f . 2.1. Wavelets The wavelet transform can be described as dividing e.g. a function or image into its frequency components and then study each frequency component with a spatial resolution adapted to its frequency. High frequency components are studied on a finer scale, while low frequency phenomena are studied on a coarser scale. The discrete wavelet transform can be regarded as a way of splitting up L2 = f f : j f j2 dx < ∞g into subspaces with certain properties.
LOCAL VECTOR TOMOGRAPHY BY USE OF WAVELETS Kent Stråhlén Centre for Mathematical Sciences, Mathematics Lund Institute of Technology P.O. Box 118, S-221 00 Lund, Sweden kent@maths.lth.se