Introduction

Reconstructing a scalar function from line integrals is a commonly found problem in medical imaging, e.g., MRI and computed tomography (CT) and other fields, e.g., seismic tomography. The roots of the mathematical groundwork for tomography lie in the breakthrough work of Radon, who indicated that, using line integrals, any two-dimensional object can be uniquely reconstructed [180]. The equation

I_L= Z

L

f(x)|dx|, (3.1.1)

with f(x) = f(x, y) denoting a continuous function and L denoting straight lines scanning the function’ s domain, is the basis for scalar tomography. However, other applications require the reconstruction and visualization of vector fields from integral information, in which case methods for VFT need to be applied. Such applications in- clude acoustic tomography for recovering the temperature (scalar) and wind (vector) fields [100], optics, for flow monitoring [153,154,229], optical polarization tomography, for estimating electric fields in Kerr materials [3, 89, 230], plasma science, for heavy particles velocity field reconstruction [62], Schlieren tomography, for estimating the intersecting with the beam component of the index of refraction field gradient [29], and polarimetric tomography, for magnetic field imaging in a tokamak [193]. Among the disciplines of biomedicine, VFT has been used in the spectrum of imaging tech- nologies, e.g., for blood flow imaging using Doppler backscattering [102, 203], and for recovering the three-dimensional diffusion tensor fields with MRI [54].

VFT constitutes methods for reconstructing a vector field using integral data over projections of the field in different directions. The integrals more commonly used in the VFT applications are the longitudinal and/or the transversal integrals, expressed as

I_L= Z

L

f·rdr and IL⊥ = Z

L

f·r⊥dr, (3.1.2)

respectively, where we have considered a vector field fand a tomographic line L that scans the domain whereflies. rstands for the unit vector parallel to the scanning line while r⊥ is the unit vector that is orthogonal to the line. The longitudinal integral

CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 31

coincides with the Radon transform in two dimensions and with the ray transform in three dimensions, as the Radon transform is defined using integrals over hyper-planes in higher dimensions.

Depending on the application, VFT methods try to reconstruct different kinds of fields. In fact, according to the fundamental theorem of vector calculus, or Helmholtz decomposition [88], any smooth, rapidly decaying vector field can be decomposed as the sum of an irrotational vector field and a solenoidal vector field. Solenoidal fields are called divergence-free or incompressible fields, they satisfy the condition ∇ ·f = 0 and they can be written as the curl of another vector field A as f = ∇ ×A. In a bounded domain, a solenoidal field is a field without sources or sinks. Irrotational fields are curl-free fields that satisfy the condition∇×f= 0 and, in a simply connected region, can be written as the negative gradient of a scalar function f=−∇Φ. Hence, according to the Helmholtz decomposition,

f=∇ ×A− ∇Φ, (3.1.3)

which holds for infinite or bounded spaces with zero values on the boundary.

A key study on VFT is Norton’ s work, who demonstrated that in a two-dimensional bounded domain only the solenoidal component could be recovered from line integrals, while the recovery of both components was tractable in the case of a divergenceless field [153], [152]. In particular, assuming a vanishing irrotational component, they showed that in a two-dimensional domain V, the Fourier transform of the longitu- dinal line integral equals the Fourier transform of the solenoidal component. The solenoidal component can be estimated using traditional approaches of scalar tomog- raphy such as filtered back-projection [148], or algebraic reconstruction techniques such as ART [103]. Then, if the field is divergence-free, the irrotational field can be estimated using Laplace’s equation [67], which can be then solved using finite differences methods [22], BEM of FEM.

These ideas were extended in [178] and [158] for the recovery of three-dimensional fields, where by the use of a generalized inner product formula, namely, the probe

CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 32

transform, a model for retrieving the solenoidal and irrotational components was de- veloped. Before that, Juhlin outlined a model for the recovery of the solenoidal part of a three dimensional divergenceless field of fluids when particular boundary field values are known [101], later advanced by Sparr et al. to further investigate cases where the flow of the field occurs in narrow channels [198]. Extensive work on the three dimensional case has been carried out by Schuster [191, 192], who employed the method of approximate inverse, introduced by Louis and Maas [132], a numer- ical inversion formula that uses reconstruction kernels. The latter were computed with mollifiers. A convergence theorem was also provided in the work of Rieder and Schuster [182].

Petrou, Giannakidis and Koulouri, in their research [78–81, 112, 172], dealt with the discrete reconstruction of two dimensional irrotational fields in a problem for- mulation that cannot be resolved using the continuous Radon transform approach.

In particular, the method suggests the discretization of a square bounded domain in tiles and the reconstruction of the vector field in specific sampling points, that are the centers of the tiles. Ideal point sensors are placed on the middle of the outward edges of the boundary tiles. The line integrals that connect all the couples of sensors apart from the ones lying on the same edge are also sampled, and the discrete points on the lines are assigned to the nearest predefined reconstruction points. Thus, the integral data may be viewed as weighted sums of the local vector field’s Cartesian components. The unknown field components are finally estimated from the solution of a linear equations system. On this ground, sampling theory was employed in order to determine sampling limitations for the Radon domain parameters, according to which a second uniform positioning of virtual sensors was proposed, whose values are calculated by interpolation, given the values of the first experimental setup. Hence, the dense and uniform sampling of the Radon parameters, prerequisites for accurate image reconstruction, is accomplished. An improved implementation was further presented, in which the former regular configuration of the sensors along the domain boundaries is used, whereas probabilistic weights are applied to recompense for the non-uniformity of the Radon parameters distribution [81]. Finally, it was proven that the discretization technique serves as a means of regularization to the ill-conditioned

CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 33

inverse problem, as the numerical solution error, caused by the approximation and the sampling procedures, is estimated to be bounded under conditions [112].