Local vector tomography by use of wavelets

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.