The application of tomography to the reconstruction of vector fields is considered. Two different types of interaction between the radiation used for measurement and the investigated vector fields are discussed. A complete reconstruction of a vector field can only be obtained if measurements based on both types of interaction are available. Otherwise, only curl-free or source-free components of the field can be reconstructed. In the case of nonhomogeneous boundary conditions, additional harmonic field components occur. Some experimental results are presented showing the reconstruction of flow velocity fields based on ultrasonic time-of-flight measurements. >

Tomographic reconstruction of vector fields

Summary

We examine the problem of reconstructing a 2-D vector field v(x,y) throughout a bounded region D from the line integrals of v(x, y) through D. This problem arises in the 2-D mapping of fluid-flow in a region D from acoustic travel-time measurements through D. For an arbitrary vector field, the reconstruction problem is in general underdetermined since v(x, y) has two independent components, vx(x, y) and vy(x, y). However, under the constraint that v is divergenceless (▿ v = 0) in D, we show that the vector reconstruction problem can be solved uniquely. For incompressible fluid flow, a divergenceless velocity field follows under the assumption of no sources or sinks in D.



A vector central-slice theorem is derived, which is a generalization of the well-known ‘scalar’ central-slice theorem that plays a fundamental role in conventional tomography. the key to the solution to the vector tomography problem is the decomposition of the field v(x, y) into its irrotational and solenoidal components: v =▿φ+▿×ψ, where φ(x, y) and ψ(x, y) are scalar and vector potentials. We show that the solenoidal component ▿ x ψ can be uniquely reconstructed from the line integrals of v through D, whereas the irrotational component ▿φ cannot. However, when the field is divergenceless in D, the scalar potential φ solves Laplace's equation in D and can be determined by the values of v on the boundary of D. an explicit formula for φ from the boundary values of v is derived. Consequently, v(x, y) can be uniquely recovered throughout the region of reconstruction from the following information: line-integral measurements of v through this region and v measured on the boundary of this region.

/pdf/tomographic-reconstruction-of-2-d-vector-fields-application-53m112a4wx.pdf

Tomographic Reconstruction of 2-D Vector Fields: Application to Flow Imaging

The problem of reconstructing a vector field v(r) from its line integrals (through some domain D) is generally undetermined since v(r) is defined by two component functions. When v(r) is decomposed into its irrotational and solenoidal components, it is shown that the solenoidal part is uniquely determined by the line integrals of v(r). This is demonstrated in a particularly simple manner in the Fourier domain using a vector analog of the well-known projection slice theorem. In addition, under the constraint that v(r) is divergenceless in D, a formula for the scalar potential phi (r) is given in terms of the normal component of v(r) on the boundary D. An important application of vector tomography, i.e., a fluid velocity field from reciprocal acoustic travel time measurements or Doppler backscattering measurements, is considered. >

Unique tomographic reconstruction of vector fields using boundary data

Acoustic tomography is a type of inverse problem. The idea of estimating physical quantities that influence sound propagation by measuring the parameters of propagation has proven to be successful in many practical domains, including temperature and wind estimation in the atmosphere. However, in most of the previous work in this area, the algorithms used have not been proven mathematically to provide the correct solution to the inverse problem. This paper considers the problem of reconstructing 2D temperature and wind fields by using acoustic tomography setups. Primarily, it shows that the classical time-of-flight measurements are not sufficient to reconstruct wind fields. As a solution, an additional set of measurements related solely to the parameters of sound propagation—more precisely, to the angles of departure/arrival of sound waves—is suggested. To take the full benefit of this additional information, the bent-ray model of sound propagation is introduced. In this work, it is also shown that, when a temperature and a source-free 2D wind field are observed on bounded domains, the complete reconstruction is possible using only measurements of the time of flight. Conversely, the angles of departures/arrivals are sufficient to reconstruct a temperature and a curl-free 2D wind fields on bounded domains. Further, an iterative reconstruction algorithm is proposed and possible variations to the main scheme are discussed. Finally, the performed numerical simulations confirm the theoretical results, demonstrate fast convergence, and show the advantages of the adopted bent-ray model for sound propagation over the straight-ray model.

/pdf/acoustic-tomography-for-scalar-and-vector-fields-theory-and-51mux2merh.pdf

Acoustic tomography for scalar and vector fields: theory and application to temperature and wind estimation

Tensors of diffusion, deformation (stress and strain), and conductivity are physical quantities of biological tissue, which can be used to characterize particular disease states. Tensor tomography may be a useful imaging technique for eliciting these tensor quantities when applied in conjunction with an imaging modality such as magnetic resonance imaging (MRI). This paper presents a method for reconstructing a 2/spl times/2 second-order tensor field from scalar projection measurements of the tensor field. The reconstruction of the four components of a 2/spl times/2 tensor may require as many as four distinct scalar measurements for each projection ray through the tensor field. Fourier projection theorems have been developed for the reconstruction of a tensor field which is decomposed into solenoidal and irrotational components. Results of a computer simulation that demonstrate the validity of the mathematical formulations are presented. A method is also proposed to obtain a diffusion tensor field tomographically from MRI projection measurements of the diffusion tensor field. The method is evaluated experimentally and results of an MRI phantom study are presented.

Tensor tomography

A theory with supporting experimental evidence is presented for reconstructing the three-dimensional fluid velocity vector field in a moving medium from a set of measurements of the acoustic propagation time between a multiplicity of transmitter and receiver locations on a stationary boundary surface. The inversion of the integrals’relating the acoustic propagation path to the propagation time measurements is affected by linearization and discrete approximation of the integrals and application of an algebraic reconstruction technique (ART). The problem of the presence of certain invisible fluid flow functions is treated. Since this technique does not require the presence of scattering centers or the optical transparency of the medium, it may be applied in many cases (i.e., turbid, opaque, or chemically pure media) where Doppler or optical (e.g., laser holography) methods fail.

C. R. Hansen

Papers

Reconstructing three-dimensional fluid velocity vector fields from acoustic transmission measurements