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

Abstract

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, v,(x, y) and v,(x, y). However, under the constraint that v is divergenceless ( V -v = O ) 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 = V@ + V x I , where @(x, y) and Y(x, y ) are scalar and vector potentials. We show that the solenoidal component V X Y can be uniquely reconstructed from the line integrals of v through D , whereas the irrotational component V@ cannot. However, when the field is divergenceless in D , the scalar potential @ solves Laplace's equation in D and can be determined by the values of v on the boundary of D. An explicit formula €or @ 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.

Full Tomographic Reconstruction of 2D Vector Fields using Discrete Integral Data

“…Several applications of vector field tomography have been considered in the literature. These include: blood flow imaging [4,5]; fluid mesoscale velocity imaging in ocean acoustic tomography [6][7][8]; fluid-flow imaging [9][10][11][12][13][14][15][16]; electric field imaging in Kerr materials [17][18][19]; imaging of the component of the gradient of the refractive index field, which is transversal to the beam, in Schlieren tomography [14]; velocity field imaging of heavy particles in plasma physics [20]; density imaging in supersonic expansions and flames in beam deflection optical tomography [21]; non-destructive stress distribution imaging of transparent specimens in photoelasticity [22,23]; determination of temperature distributions and velocity vector fields in furnaces [24]; and magnetic field imaging in Tokamak in polarimetric tomography [25].…”

3D reconstruction of tensors and vectors

“…In fact, physical problems are often defined on bounded domains and it is the boundary that creates or partially defines the field. Norton [16,21], Braun and Hauck [20], and Osman and Prince [35] were all concerned about the vector field tomography problem on bounded domains. Norton showed that the longitudinal measurements with boundary conditions reconstruct a divergence-free vector field composed of a solenoidal component that satisfies homogeneous boundary conditions and an irrotational component defined by the gradient on the boundary.…”

“…History of Vector Field Tomography Tensor tomography builds on much of the work already accomplished in vector field tomography [2,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. Applications have involved acoustic flow imaging using time-of-flight measurements in medicine [7], non-destructive evaluation [11], and ocean tomography [9,14].…”

“…In one of the earliest studies Johnson et al [7] used ultrasound to reconstruct velocity vector fields in blood vessels from acoustic time-offlight measurements. An iterative algebraic reconstruction technique (ART) was used to compute what was later understood by Norton [16,21] to be the divergence-free component of the vector field. Using the Helmholtz decomposition and the Fourier central-slice theorem, Norton [16,21] derived a reconstruction method for the velocity field and showed that the reconstruction of the acoustic time-of-flight (longitudinal projection [20]) measurements produced only the divergence-free component of the vector field.…”

“…An iterative algebraic reconstruction technique (ART) was used to compute what was later understood by Norton [16,21] to be the divergence-free component of the vector field. Using the Helmholtz decomposition and the Fourier central-slice theorem, Norton [16,21] derived a reconstruction method for the velocity field and showed that the reconstruction of the acoustic time-of-flight (longitudinal projection [20]) measurements produced only the divergence-free component of the vector field. Before Norton's important contribution, Kramer and Lauterbur [10] developed a hybrid filtered backprojection algorithm to reconstruct flow vector fields using NMR, and also showed that some flow components could not be recovered from only longitudinal projection measurements.…”