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

Abstract

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.