INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS

Inverse Problems 17 (2001) 739–766 www.iop.org/Journals/ip PII: S0266-5611(01)17273-4

An efficient mollifier method for three-dimensionalvector tomography: convergence analysis andimplementation

Thomas Schuster

Fachbereich Mathematik, Geb. 36, Universitat des Saarlandes, 66041 Saarbrucken, Germany

E-mail: thomas.schuster@num.uni-sb.de

Received 20 September 2000, in final form 25 April 2001

AbstractWe consider the problem of three-dimensional vector tomography, that meansthe reconstruction of vector fields and their curl from line integrals over certaincomponents of the field. It is well known that only the solenoidal part of thefield can be recovered from these data. In this paper the method of approximateinverse is modified for vector fields and applied to this problem, leading to anefficient solver of filtered backprojection type. We prove convergence of thereconstructed solution, if the number of data tends to infinity, which means themethod is exact. Finally, numerical results are presented for a straight flowthrough a cylinder.

(Some figures in this article are in colour only in the electronic version; see www.iop.org)

1. Introduction

While classical computerized tomography deals with the reconstruction of scalar functions suchas density of human tissue from x-ray measurements, vector tomography means recoveringvector fields from integral information about certain components of the field. There exists awide area of interest in reconstructing vector fields in medical imaging as well as in industrialapplications. Yet in 1977 Wells et al [26] described a technique to distinguish blood flowof benign breast tumours from that of malign ones by ultrasound measurements. From thisstarting point Jansson et al [5] developed a tomographic method for detecting blood flow anda measurement system to obtain data from a flow phantom. However, vector tomography alsooccurs in the industrial field of interest. Sielschott [20] reconstructs the horizontal gas velocityin a furnace based on time-of-flight measurements. The problem of recovering velocity fieldsof conducting fluids from magnetic field and electric potential measurements is formulated byStefani and Gerbeth [23]. A solver for the arising inverse problem is also presented. Furtherapplications can be found in oceanography, photoelasticity, nuclear magnetic resonance orplasma physics (see Sparr and Stråhlen [21]).

We have three different ways to obtain the data in all these applications: acoustictransmission measurements (time-of-flight-data), Doppler backscattering and optical

0266-5611/01/040739+28$30.00 © 2001 IOP Publishing Ltd Printed in the UK 739

740 T Schuster

transmission measurements (interferometric measurements). For details we refer to Sparrand Stråhlen [21]. In this paper we consider only the case of Doppler backscattered data,which represents the most important case for detecting moving fluids. This technique requiresthe existence of scattering particles within the fluid, such as red blood cells, if we considerblood flow in the human body. To obtain the data we send an ultrasound signal onto the object� ⊂ R

3 under consideration, which we assume to be a bounded domain. If the test beam runsinto an obstacle, a part of the signal is transmitted; the rest is reflected. The difference betweenthe frequency of the reflected part η and the initial frequency η0 is the so-called Doppler shift�η caused by a particle moving with velocity ν in the direction opposite to the transmittedsignal. It can be calculated as

�η = η − η0 = 2 c η0 ν

c2 − ν2≈ 2 η0

cν = k ν,

where c denotes the velocity of sound within the medium. The approximation is justified sinceν � c. Thus, the Doppler shift �η is approximately proportional to the particle velocity ν.From such measurements, under the assumption �η = k ν, it is possible to calculate the sumalong a test beam L of the velocity components of a field in the direction of L,

Df (L) =∫L∩�

〈θ(L), f (x)〉 dσ (x) (1.1)

(see Sparr et al [22]). Here, θ(L) ∈ S2 means the direction of the line L; Sn−1 = {x ∈ Rn :

‖x‖2 = 1} is the unit sphere in Rn.

By the Helmholtz–Hodge decomposition theorem (see e.g. Chorin and Marsden [3,section 1.3]), we can write a field f as a sum of a solenoidal field f s, that means ∇ · f s = 0,and a potential part ∇p,

f = f s + ∇p. (1.2)

The decomposition becomes unique if we suppose �n · f s = 0 on the boundary ∂� with theouter normal vector �n in ∂�. It is a well known fact that the potential part of f lies in the nullspace of D. So the reconstruction of the solenoidal part f s of f is the best we can hope for.The potential field ∇p can be regained from (1.2) by solving a boundary value problem. If weknow the divergence ∇ · f of f in � and the flow �n · f on the boundary ∂�, then it followsfrom (1.2) that p is the solution of the potential equation

�p = ∇ · f in �,

∂p

∂ �n = �n · f in ∂�.

Therefore, if f has a nontrivial potential part ∇p, we have to solve both an integral equationand a boundary value problem to recover the field f completely.

Obviously the so-called Doppler transform (1.1) is similar to the two-dimensional Radontransform

Rf (L) =∫L∩�

f (x) dσ (x). (1.3)

In fact, there exists a relationship between the two transforms (see section 2). Hence a numberof algorithms for the two-dimensional case are based on this relation, leading to a method offiltered backprojection type for reconstructing the curl of f (see Stråhlen [24], Sparr et al [22],Winters and Rouseff [28]). Desbat [4] treats the two-dimensional vector field problem withdirect algebraic methods and gives some efficient parallel sampling schemes. The methodsmentioned before deal only with the two-dimensional case. Wernsdorfer [27] presented aniterative algorithm, which arises by applying the algebraic reconstruction technique (ART)

An efficient mollifier method for three-dimensional vector tomography 741

to the Doppler transform (1.1) assuming that the divergence of f is known. However theseiterative methods are rather slow.

We present in our article a novel method for the three-dimensional vector tomographyproblem, which is on the one hand quite efficient and on the other hand suitable forreconstructing both the curl of the field ∇ × f as well as the solenoidal part of the fieldf s itself.

The paper is organized as follows. In section 2 we parametrize the Doppler transform (1.1)in a suitable manner, taking into account the special scanning geometry, where we consideronly lines L which are parallel to one of the coordinate axes. This leads to a linear operatorbetween Hilbert spaces. We formulate some continuity results of D, which are proved inSchuster [19], as well as the relation to the Radon transform (1.3) mentioned above.

Section 3 consists of the presentation of our reconstruction method for bothf s (section 3.1)and ∇ × f (section 3.2). We use a modification of the approximate inverse for vector fields.Approximate inverse means a stable regularization scheme, which is based on the evaluationof scalar products of the given data with so-called reconstruction kernels. More precisely,assume A : L2(�) → Y to be a linear, bounded operator between L2(�), � ⊂ R

n, and aHilbert space Y . Furthermore let eγ (x, y) ∈ L2(Rn,Rn) be a mollifier, that means eγ (·, y) hasnormalized mean value and eγ (x, y) ≈ δx−y is an approximation to the Dirac delta distributionin some sense, which we will specify in section 3. Instead of a solution f from Af = g itself,we compute moments

fγ (x) = 〈f, eγ (x, ·)〉L2(�)

of f with eγ . Since we do not know the exact solution f , we accept

fγ (x) ≈ 〈Af,ψγ (x)〉Y = 〈g,ψγ (x)〉Y (1.4)

as an approximation to fγ , where the reconstruction kernel ψγ (x) solves

A∗ψγ (x) = PA∗eγ (x, ·)with PA∗ : L2(�) → L2(�) the orthogonal projection onto R(A∗), the range ofA∗. Obviouslywe have an equality in (1.4) if eγ (x, ·) ∈ R(A∗). Some properties of the approximate inversecan be found in Louis [10]. In Louis [11] it is verified that it acts like a regularization method.For a method to calculate reconstruction kernels we refer to Louis and Schuster [12]. Weshow how invariances of the underlying operator can be used to decrease the computationaleffort and present representations of the reconstruction kernels corresponding to the Gaussianfunction as mollifier.

In practical situations we have only finitely many data at hand. Thus, in section 4,we investigate this situation and formulate two convergence theorems, which describe thebehaviour of our reconstructed solution as the number of data tends to infinity. Finally, weshow that the implementation of the algorithm of filtered backprojection type can be achieved(section 5). This explains the high efficiency of the method. Numerical results with simulateddata of a straight flow are also given in this section. This paper includes a summary and someconclusions.

2. Mathematical setting

In this section we parametrize the Doppler transform (1.1) and summarize the most importantmathematical properties of this mapping, as done in Schuster [19].

We start with a permutation vj of the standard unit vectors vj ∈ R3. Let v1 = v3 =

(0, 0, 1)�, v2 = v1 = (1, 0, 0)� and v3 = v2 = (0, 1, 0)� be that permutation. For

742 T Schuster

Figure 1. Parameters of the Doppler transform D1.

j ∈ {1, 2, 3} we define the embedding Pj of R2 onto the subset v⊥

j , where v⊥j = {x ∈

R3 : 〈x, vj 〉 = 0}. More precisely, Pj : R

2 → v⊥j ⊂ R

3 is given by P1(x1, x2) = (x1, x2, 0)�,P2(x1, x2) = (0, x1, x2)

� and P3(x1, x2) = (x1, 0, x2)�. By 〈·, ·〉 and ‖ · ‖ we mean

always the Euclidean scalar product 〈·, ·〉Rn and the Euclidean norm ‖ · ‖Rn , respectively.The adjoint operators P∗

j : R3 → R

2 with respect to the inner product 〈·, ·〉 are givenby P∗

1 (x1, x2, x3) = (x1, x2), P∗2 (x1, x2, x3) = (x2, x3) and P∗

3 (x1, x2, x3) = (x1, x3).Furthermore we need mappings τj : R

3 → R, which are defined by

τj (x) = 〈x, vj 〉and have the adjoints τ ∗

j : R → R3 with τ ∗

j z = z vj . We suppose j ∈ {1, 2, 3} throughoutthis paper, if not mentioned otherwise.

A line Lj which is parallel to the coordinate plane {x ∈ R3 : 〈vj , x〉 = 0} is determined

by a direction ω = ω(ϕ) = (cosϕ, sin ϕ)� ∈ S1, the distance s from the vj -axis and thedistance a from the plane to the origin in the following way:

Lj(ω, s, a) = {x ∈ R3 : 〈x,Pjω〉 = s, τj (x) = a}.

For example, L1(ω, s, a) = {x ∈ R3 :

⟨(x1

x2

), ω

⟩ = s, x3 = a}. The different variables areillustrated in figure 1. We emphasize that ω is not a direction vector of the line Lj , but anormal vector. With these notations we obtain the following parametrization of the Dopplertransform D (1.1).

Definition 2.1. Let � ⊂ R3 be a bounded domain and f ∈ L2(�,R3) a square integrable

vector field. The mapping D = D1 ⊕ D2 ⊕ D3 with

Dj f (ω, s, a) =∫Lj (ω,s,a)∩�

〈vj × Pj (ω), f (x)〉 dσ (x) (2.1)

=∫ ∞

−∞〈θj (ω), f

(Pj (sω + tω⊥) + τ ∗j (a)

)〉 dt, (2.2)

An efficient mollifier method for three-dimensional vector tomography 743

where ω⊥(ϕ) = (− sin ϕ, cosϕ)� and θj (ω) = vj × Pj (ω) ∈ S2 is the vector of directionfrom Lj , is called the three-dimensional Doppler transform of f . In equation (2.2) we setf = 0 outside �.

Let K3 = {x ∈ R3 : ‖x‖∞ < 1} be the three-dimensional unit cube and �n = {x ∈ R

n :‖x‖ < 1} the open n-dimensional unit ball. By L2

�3(K3,R3) we denote the closed subspace

of L2(K3,R3), which consists of all square integrable vector fields f in K3, for which thesupport supp f is contained in the closed unit ball �3. More precisely,

supp f = {x ∈ K3 : f (x) �= 0} ⊂ �3.

From Schuster [19] we know that the mappings Dj : L2�3(K

3,R3) → L2(Q) and D :L2�3(K

3,R3) → ⊕3j=1L

2(Q), with Q := S1 × (−1, 1)2, are linear, bounded operators, wherethe adjoint D∗

j : L2(Q) → L2�3(K

3,R3) has the representation

D∗j g(x) =

Pj

(εj

∫ 2π

0g(ω(ϕ), 〈P∗

j (x), ω(ϕ)〉, τj (x)) sin ϕ dϕ,

−εj∫ 2π

0g(ω(ϕ), 〈P∗

j (x), ω(ϕ)〉, τj (x)) cosϕ dϕ

), x ∈ �3,

0, x ∈ K3\�3.

(2.3)

Here, ω(ϕ) = (cosϕ, sin ϕ)� ∈ S1, ε1 = ε2 = −1 and ε3 = 1.In several articles, for example Sparr et al [22], Juhlin [6], Schuster [19], a relationship

between the Doppler transform Dj and the two-dimensional Radon transform R : L2(�2) →L2(S1 × (−1, 1)),

Rf (ω, s) =∫

〈x,ω〉=sf (x) dσ (x),

is presented by

∂

∂sDj = (R ⊗ I ) dj . (2.4)

Here, dj : H 1(K3,R3) ∩ L2�3(K

3,R3) → L2(�2 × (−1, 1)) denotes the mapping

djf (x, a) = 〈vj ,∇ × f (Pj (x) + τ ∗j (a))〉, (2.5)

that means djf is the vj -component of the curl of f . The tensor product R ⊗ I is defined asoperator between the spaces L2(�2)⊗L2(−1, 1) ∼= L2(�2 × (−1, 1)) → L2(Q) as follows:

(R ⊗ I )f (ω, s, a) =∫

〈x,ω〉=sf (x, a) dσ (x).

That means R⊗I acts onL2(�2) like the two-dimensional Radon transform and onL2(−1, 1)like the identity. Thus, R⊗ I is a noncompact operator. (For further details of tensor productsof operators, see e.g. Aubin [2], Weidmann [25].)

Finally the Doppler transform Dj fulfils a smoothing property, which proves useful toshow the convergence results in section 4. For this we define for α, β > 0 the spaceH α,β := H

α,β

0 ((−1, 1)2 × (−1, 1),R3) as the closure of C∞c (�

3,R3) in L2(K3,R3) withrespect to the norm

‖f ‖Hα,β =( ∫

R2

∫R

(1 + ‖ξ‖2)α (1 + |ζ |2)β ‖f (ξ, ζ )‖2R3 dζ dξ

)1/2

.

Here, f (ξ, ζ ) means the three-dimensional Fourier transform of f (x, y) with respect to thevariables x and y. Note that we take the closure of Cc(�3,R3) and not of Cc(K3,R3), so

744 T Schuster

that H α,β is a subspace of L2�3(K

3,R3). By Hαp (Z) we denote the Sobolev space of periodic

functions defined on the rectangle Z = (0, 2π)× (−1, 1) of order α. Its norm is given by

‖f ‖Hαp

=( ∑

k∈Z

∑n∈Z

(1 + k2 + n2)α |fk,n|2)1/2

,

where

fk,n = 1

4π

∫Z

f (s, ϕ) e−i (πks+nϕ) dϕ ds

are the Fourier coefficients of f . The mentioned smoothing property is summarized in thefollowing theorem; the proof can be found in Schuster [19].

Theorem 2.2. The Doppler transform Dj is a continuous mapping between the spaces H α,β

and Hα+1/2p (Z)⊗Hβ

0 (−1, 1); moreover, we have

‖Dj f ‖Hα+1/2p (Z)⊗Hβ

0 (−1,1) � c ‖f ‖H α,β , (2.6)

where c > 0 is a constant which does not depend on f ∈ H α,β .

Remark 2.3. In theorem 2.2 we identified the direction ω in Dj f (ω, s, a) with the angle ϕin ω(ϕ) = (cosϕ, sin ϕ). The assertion says that Dj smooths with factor one-half w.r.t. twovariables and it acts like the identity operator on the third variable. As mentioned before wewill need this smoothing property to prove convergence of our algorithm in section 4.

3. Approximate inverse for vector fields

In Louis [10] a concept has been presented to develop stable regularization schemes for solvingequations Af = g, where A is a bounded operator between X = L2(�) with an open domain� ⊂ R

n and a separable Hilbert space Y . Using the notation of the introduction, we areinterested in solving equations of the type

A∗ψ(x) = PA∗eγ (x, ·), (3.1)

where A∗ is the adjoint of A with respect to the inner products of X and Y , PA∗ : X → R(A∗)is the orthogonal projection onto R(A∗) and eγ (x, y) is a mollifier. Let us first specify thisterm. For γ > 0 we call a function eγ (x, y) ∈ L2(R3 ×R

3) a mollifier if it fulfils the equations∫R3eγ (x, y) dx = 1 for every y ∈ R

3, (3.2)

limγ↘0

∫R3f (x) eγ (x, y) dx = f (y) a.e. (3.3)

So, we can see the mollifier eγ as an approximation to Dirac’s delta distribution δx−y and γ isa regularization parameter. We drop this subscript for better readability whenever possible. Inview of computational effort Louis [10] has shown how invariance properties of the underlyingoperator A can be used, so that we have to determine a reconstruction kernel ψ(x) for A onlyfor one single point x0 ∈ �. More precisely, if one kernel ψ(x0) is available and if there aremappings T x1 on X and T x2 , T x3 on Y by AT x1 = T x2 A and T x2 AA

∗ = AA∗T x3 , then ψ(x) isdetermined by ψ(x) = T x3 ψ(x0) provided that e(x, ·) = T x1 e(x0, ·) holds. Thus, the otherkernels ψ(x) are only transformations of ψ(x0). Thereby invariances of A as well as A∗

are used. We show that an invariance of A∗ is sufficient, when T x1 satisfies some additionalconditions.

An efficient mollifier method for three-dimensional vector tomography 745

Theorem 3.1. Let T x1 ∈ L(X) and T x2 ∈ L(Y ) be given with T x1 bijective, ‖T x1 ‖ � 1 andT x1 A

∗ = A∗T x2 . We assume furthermore that there exists for each x ∈ � an x∗ ∈ � with(T x1 )

−1 = T x∗

1 and that we have an x0 ∈ � with

e(x, y) = T x1 e(x0, y) (3.4)

for a mollifier e(x, y). Then the solution ψ(x) of (3.1) fulfils

ψ(x) = T x2 ψ(x0).

Proof. It suffices to show that

PA∗T x1 e(x0, ·) = T x1 PA∗e(x0, ·), (3.5)

since (3.5) implies

PA∗e(x, ·) = PA∗T x1 e(x0, ·) = T x1 PA∗e(x0, ·) = T x1 A∗ψ(x0) = A∗ T x2 ψ(x0)

and the assertion follows.Define X � f x := PA∗T x1 e(x0, ·). Then there is a sequence {gxn}n in Y with f x =

limn A∗gxn and

limn

‖A∗gxn − T x1 e(x0, ·)‖X = ming∈Y

‖A∗g − T x1 e(x0, ·)‖X. (3.6)

We have to verify

gxn = T x2 gx0n , (3.7)

because (3.7) yields

A∗gxn = T x1 A∗gx0

n

and thus f x = T x1 fx0 , which proves (3.5). Note that T x0

1 = I because of (3.4).To verify (3.7) suppose that (3.7) is not valid, which means the existence of a further

sequence {hxn}n in Y satisfying

‖A∗hxn − T x1 e(x0, ·)‖X < ‖A∗ T x2 gx0n − T x1 e(x0, ·)‖X (3.8)

for almost every n. The following chain of inequalities leads to a contradiction:

‖A∗ T x∗

2 hxn − e(x0, ·)‖X = ‖T x∗1 (A∗hxn − T x1 e(x0, ·))‖X � ‖A∗hxn − T x1 e(x0, ·)‖X

< ‖A∗ T x2 gx0n − T x1 e(x0, ·)‖X = ‖T x1 (A∗gx0

n − e(x0, ·))‖X� ‖A∗gx0

n − e(x0·)‖X,where we used ‖T x1 ‖ � 1 and (3.8). This would result in

limn→∞ ‖A∗ T x

∗2 hxn − e(x0, ·)‖X < lim

n→∞ ‖A∗gx0n − e(x0, ·)‖X = min

g∈Y‖A∗g − e(x0, ·)‖X,

which contradicts (3.6). So, (3.7) holds and the proof is complete. �

Theorem 3.1 is useful, if A is not injective and hence R(A∗) is not dense in X, since thenthe invariance T x1 A

∗ = A∗ T x2 alone is not enough to obtain ψ(x) = T x2 ψ(x0). Theorem 3.1says that this is only sufficient when T x1 is a bijective contraction on X with an additionalproperty related to a group structure. These facts will be important in the case A = Dj .

In Louis [10] the approximate inverse was only described for reconstructing scalarfunctions. So, if we want to apply this method to Dj , we have to modify it for vector fields.We explain a technique for both the reconstruction of the field f itself and the curl of the field∇ × f . From now on we suppose, throughout the paper, that ∇ · f = 0, i.e. f = f s is asolenoidal field. That means we consider only incompressible fluids.

746 T Schuster

3.1. Reconstruction of f

We define mollifier fields Ejγ ∈ L2(R3 × R

3,R3) by

Ejγ (x, y) = eγ (x, y) · vj , (3.9)

where vj ∈ R3 are the standard unit vectors. Given the data gj = Dj f for f ∈ L2

�3(K3,R3),

we calculate moments fγ of f with

fγ (x) :=( 〈f1, eγ (x, ·)〉L2(K3)

〈f2, eγ (x, ·)〉L2(K3)

〈f3, eγ (x, ·)〉L2(K3)

)(3.10)

instead of f itself. If there exists a function8jγ (x) ∈ L2(Q), called the reconstruction kernel,

fulfilling

D∗j8

jγ (x) = Ej

γ (x, ·), (3.11)

we have

fγ (x) =

〈f,E1γ (x, ·)〉L2

�3 (K3,R3)

〈f,E2γ (x, ·)〉L2

�3 (K3,R3)

〈f,E3γ (x, ·)〉L2

�3 (K3,R3)

=

〈D1f,8

1γ (x)〉L2(Q)

〈D2f,82γ (x)〉L2(Q)

〈D3f,83γ (x)〉L2(Q)

= 〈g1,8

1γ (x)〉L2(Q)

〈g2,82γ (x)〉L2(Q)

〈g3,83γ (x)〉L2(Q)

=: Sγ g(x). (3.12)

Since only in special cases does equation (3.11) have a solution, we minimize the defect‖D∗

j8jγ (x)− E

jγ (x, ·)‖L2(K3,R3) solving the normal equation

DjD∗j8

jγ (x) = DjE

jγ (x, ·), (3.13)

which is solvable whenever Ejγ (x, ·) ∈ R((D∗

j )†) = R(D∗

j )⊕ N(Dj ). Hence, we expect Sγ gto be a good approximation to the vector field f searched for and call Sγ g the approximateinverse of D. From (3.3), (3.12) and (3.13) we easily see that

(Sγ Df )j (x) = 〈f,PN(Dj )⊥Ejγ (x, ·)〉L2

�3 (K3,R3) −→ (PN(Dj )⊥f )j (x) (3.14)

for γ → 0, where PN(Dj )⊥ : L2�3(K

3,R3) → L2�3(K

3,R3) means the orthogonal projectoronto N(Dj )

⊥. Hence, if γ tends to zero, we expect to recover the parts of f , which areorthogonal to the null spaces N(Dj ). However, there will be inaccuracies if x is near theboundary ∂�3, since in general

∫�3 eγ (x, y) dx � 1 if y ∈ ∂�3. We will illustrate this in

section 5. The null spaces N(Dj ) have a special structure:

N(D1) ={(

∂

∂x1p,

∂

∂x2p, q

): p ∈ H 1

0 (�3), q ∈ L2

�3(K3)

}, (3.15)

N(D2), N(D3) analogously. From (3.15) we easily see that there exists a p ∈ H 10 (�

3) with((PN(D1)f )1, (PN(D2)f )2, (PN(D3)f )3)

� = ∇p, which implies

limγ→0

(f − Sγ Df ) → ∇p. (3.16)

Thus, if f = f s, as we supposed, the defect f−Sγ Df tends to an element which is orthogonalto f ,

limγ→0

(f − Sγ Df ) ⊥ f.

In Louis [11] it can be read that Sγ has the properties of a regularization operator.

An efficient mollifier method for three-dimensional vector tomography 747

Remark 3.2. The choice of the mollifier fields (3.9) is not the only one which results in asuitable approximation of f . Let Ej

γ = Ej,1γ + Ej,2

γ + Ej,3γ = eγ · vj , where the fields Ej,i

γ

fulfil Ej,iγ ⊥ vi as it does the range of D∗

i (see (2.3)). Then we have

〈fj , eγ (x, ·)〉L2(K3) =3∑i=1

〈Dif,8j,iγ (x)〉L2(Q),

if D∗i 8

j,iγ = E

j,iγ (x, ·). This setting of the mollifier fields will be the subject of further

investigations, since in view of section 4 we hope to derive better convergence results bychoosing the Ej

γ in this way.

To obtain a representation of a reconstruction kernel 8jγ we have to solve the normal

equation (3.13). The following lemma arises from equation (2.4) and characterizes thederivative of 8j

γ with respect to s.

Lemma 3.3. If the mollifier fields Ejγ lie in D((D∗

j )†) ∩ D(dj ), we obtain the formula

(R∗ ⊗ I )

(∂

∂s8jγ (x)

)= dj Eγ (x, ·) (3.17)

for the solution 8jγ of (3.13), where dj is defined as in (2.5).

Proof. The proof, which is based on formula (2.4) is achieved by differentiating the normalequation (3.13) and is outlined in Schuster [19]. �

We define translations T x1 , T x2,j for x ∈ �3 and show the identity

8jγ (x) = T x2,j 8

jγ (0),

whenever eγ (x, y) = T x1 eγ (y) holds for a eγ ∈ L2(R3).Let T x1 ∈ L(L2(R3)) and T x2,j ∈ L(L2(Q)) be given as

T x1 f (y) = 1

8f

(y − x

2

), (3.18)

T x2,j g(ω, s, a) = 1

8g

(ω,

s − P∗j x

2,a − τjx

2

). (3.19)

Then a straightforward calculation shows that the invariance

D∗j T

x2,j = T x1 D∗

j (3.20)

is valid. Here, we denote the extension of T x1 to L2�3(K

3,R3) again by T x1 . Together withtheorem 3.1 we derive the desired result.

Lemma 3.4. If there exists an eγ ∈ L2(R3) with eγ (x, y) = T x1 eγ (y) for a mollifier eγ , andthe appropriate mollifier field Ej

γ (x, ·) lies in D((D∗j )

†), then for x ∈ �3 the solution 8jγ (x)

of (3.13) is given as

8jγ (x) = T x2,j 8

jγ (0),

where 8jγ (0) is the solution of (3.13) for x = 0.

Proof. Letting T x1 f (y) = f (y − x) and T x2,j g(ω, s, a) = g(ω, s − P∗j x, a − τjx) we

find D∗j T

x2,j = T x1 D∗

j , ‖T x1 ‖L2→L2 � 1 and (T x1 )−1 = T −x

1 . Since we have Eγ (x, y) =T x1 {8−1 Eγ (0, y/2)} = T x1 Eγ (0, y), theorem 3.1 with x∗ = −x yields 8j

γ (x;ω, s, a) =T x2,j {8−1 8

jγ (0;ω, s/2, a/2)} = T x2,j8

jγ (0;ω, s, a). �

748 T Schuster

So, we have to solve (3.13) or (3.17) only for the single point x = 0, when our mollifier eγis a translated version of a function eγ . Unfortunately equation (3.17) is hard to solve explicitlyfor general mollifiers. In Schuster [19] a reconstruction kernel 8j

γ (0) has been computed fora special mollifier. Thereby eγ is given as the Gaussian function

eγ (x) = (2π)−3/2 γ−3 exp(−‖x‖2/(2γ 2)) (3.21)

and eγ (x, y) = T x1 eγ (y). From now on we always abbreviate 8jγ := 8

jγ (0) and Ej

γ (y) :=Ejγ (0, y). It is easily verified that eγ fulfils the conditions (3.2) and (3.3) and is in fact

a mollifier. The smoothness of eγ yields furthermore djEjγ (x, ·) ∈ R(R∗ ⊗ I ), where

Ejγ (x, y) = eγ (x, y) · vj . Note that R(R∗ ⊗ I ) is dense in L2(�2 × (−1, 1)). Finally,

we obtain an explicit expression for the reconstruction kernel 8jγ (x) of which the proof is

outlined in Schuster [19].

Theorem 3.5. Let eγ be given as in (3.21) and Ejγ (x, y) = T x1 Eγ (0, y). Then we have

8jγ (x) = T x2,j 8

jγ , where the unique solution 8j

γ of (3.13) in R(Dj ) is given as follows:

8jγ (ω(ϕ), s, a) = rj (ϕ) exp(−a2/(2γ 2))

(γ−4

4π3

∞∑k=0

cγ

k

(T2k+2(s)− 1

)+ π−1/2 κγ

). (3.22)

Here, r1(ϕ) = r2(ϕ) = sin ϕ, r3(ϕ) = cosϕ and Tk(s) = cos(k arccos s) denote theChebyshev polynomials of the first kind. The constants are defined by

κγ = γ−4

8π3

∞∑k=0

cγ

k (2π − zk)− γ−2

4π, c

γ

k = 2∫ 1

0τ exp(−τ 2/(2γ 2)) U2k+1(τ ) dτ,

z0 = −5π/3, zk = (−1)k+1 8πk + 1

4k2 + 8k + 3, k = 1, 2, . . . ,

where Uk(s) = sin((k + 1) arccos s)/ sin(arccos s) are the Chebyshev polynomials of thesecond kind.

For practical applications we have to truncate the series (3.22) after finitely many steps,leading to a kernel denoted by 8j,M

γ . A cutting criterion is given by choosing M so large that

|cγM | < ε (3.23)

for a small, positive number ε. Note that D∗j8

j,Mγ �→ E

jγ as M goes to infinity, since Dj has

a nontrivial kernel. In figure 2 a plot of −81,Mγ is displayed with M = 56, γ = 0.05, a = 0

and ε = 10−8 in (3.23).Thus, to apply the approximate inverse to D we have first to calculate the reconstruction

kernels 8j,Mγ . Then an approximate solution of Df = g is given by

fj (x) ≈ 〈gj , T x2,j8j,Mγ 〉L2(Q). (3.24)

Remark 3.6. As for example Juhlin [6] mentioned, it suffices to measure the data only in twofamilies of parallel planes. In fact we need only D1f , which involves the components f1

and f2 of the field f and D2f , which involves f2 and f3. For that we have to define 83γ as

a solution of D∗28

3γ (x) = PN(D2)⊥E

3γ (x, ·), that means we replace in (3.13) D3 and D∗

3 byD2 and D∗

2 , respectively. However, using this approach we get into trouble if D2f = 0 andf2 �= 0.

Let us make a few remarks about the reconstructed field using the special mollifier (3.21).Obviously we have

∂

∂ykEjγ (x, y) = − ∂

∂xkEjγ (x, y).

An efficient mollifier method for three-dimensional vector tomography 749

Figure 2. −81,Mγ for γ = 0.05,M = 56 and a = 0. The truncation indexM was chosen according

to (3.23) with ε = 10−8. The kernel is akin to the Shepp–Logan filter with an additional sine. Weplot the inverse of 81,M

γ to clarify the structure.

Figure 3. − ∂∂s8Mγ with γ = 0.07 and M = 32; the kernel is plotted for a = 0.

So, from (3.14) we obtain by a simple calculation

limγ→0

∇ · Sγ Df (x) =3∑

j=1

∂

∂xj(PN(Dj )⊥f )j (x) a.e. (3.25)

750 T Schuster

Together with (3.16) this yields

limγ→0

(∇ · f − ∇ · Sγ Df ) = �p

for a function p ∈ H 20 (�

3). Hence, the divergence of the reconstructed field tends to zero onlyif p is a harmonic function. Further inaccuracies in the reconstruction come from the truncationof the series (3.22). This part of the error is specified in the estimation of theorem 4.4.

Finally we give a suggestion of how to improve the reconstructed field Sγ Df adding acorrection term, which solves a boundary value problem. From (3.16) we know that

f − Sγ Df ≈ ∇pγ . (3.26)

Suppose pγ to be sufficiently smooth. We recall that f = f s is a solenoidal field. So, takinginto account that ∇ · f = 0 and �n · f = 0 on the boundary ∂�3, equation (3.26) results in aLaplace equation with Neumann boundary conditions:

�pγ = −∇ · Sγ Df in �3,

∂

∂ �n pγ = −�n · Sγ Df on ∂�3.

The field fγ = Sγ Df +∇pγ has the properties ∇·fγ = 0 and �n·fγ = 0 on the boundary. If wewant to obtain homogeneous Dirichlet boundary conditions, we have to replace the Neumannconditions by ∇pγ = −Sγ Df . A drawback of this method is the numerical difficulties fromdifferentiating Sγ Df , which enforces the noise in the reconstruction. We do not implementthis approach within this paper.

3.2. Reconstruction of ∇ × f

In this section, we describe how to obtain an approximation to ∇ ×f from the same set of datagj = Dj f just by using another reconstruction kernel. The main idea consists of applying (2.4)and well known techniques from inverting the two-dimensional Radon transform (see e.g.Rieder and Schuster [16]). The derivative with respect to the variable s will be transferred tothe reconstruction kernel by an integration by parts.

Again we assume eγ (x, y) = T x1 eγ (y), eγ ∈ L2(R3), to be a mollifier which is sufficientlysmooth. We will specify this degree of smoothness later. If the equation

(R∗ ⊗ I )8γ (x) = eγ (x, ·) (3.27)

has a solution 8γ (x) in L2(Q), the moments

(∇ × f )γ (x) :=( 〈(∇ × f )1, eγ (x, ·)〉L2(K3)

〈(∇ × f )2, eγ (x, ·)〉L2(K3)

〈(∇ × f )3, eγ (x, ·)〉L2(K3)

)(3.28)

can be calculated as follows:

(∇ × f )γ (x) =( 〈d2f, eγ (x, ·)〉L2(�2×(−1,1))

〈d3f, eγ (x, ·)〉L2(�2×(−1,1))

〈d1f, eγ (x, ·)〉L2(�2×(−1,1))

)=

〈(R ⊗ I )d2f, 8γ (x)〉L2(Q)

〈(R ⊗ I )d3f, 8γ (x)〉L2(Q)

〈(R ⊗ I )d1f, 8γ (x)〉L2(Q)

= 〈D2f,− ∂

∂s82γ (x)〉L2(Q)

〈D3f,− ∂∂s83γ (x)〉L2(Q)

〈D1f,− ∂∂s81γ (x)〉L2(Q)

=

〈g2,− ∂

∂s82γ (x)〉L2(Q)

〈g3,− ∂∂s83γ (x)〉L2(Q)

〈g1,− ∂∂s81γ (x)〉L2(Q)

=: Sγ g(x).Here we used (2.4) and integration by parts. Note that Dj f vanishes for s ∈ {−1, 1}.

An efficient mollifier method for three-dimensional vector tomography 751

The translation T x1 acts on L2(�2 × (−1, 1)) in a natural way. We define

T x1 f (y1, y2) = 1

8f

(y1 − P∗

1x

2,y2 − τ1x

2

).

A short computation proves the invariance

T x1 (R∗ ⊗ I ) = (R∗ ⊗ I ) T x2,1 (3.29)

with T x2,1 defined in (3.19). Since T x1 is again a contraction, theorem 3.1 ensures that it is

sufficient to determine 8γ (x) for one single point.

Lemma 3.7. If eγ ∈ R(R∗ ⊗ I ), then we have for x ∈ �3

8γ (x) = T x2,18γ , (3.30)

where 8γ := 8γ (0) is a solution of (3.27) for x = 0.

In the case of eγ /∈ R(R∗ ⊗ I ), the density of R(R∗ ⊗ I ) in L2(�2 × (−1, 1)), due to theinjectivity of R ⊗ I , guarantees the existence of 8ε

γ with

‖(R∗ ⊗ I )8εγ − eγ ‖L2(�2×(−1,1)) < ε

for an arbitrary small, positive number ε. From now on, we suppose eγ ∈ R(R∗ ⊗ I ) and eγ tobe smooth enough that the corresponding solution 8γ of (3.27) is differentiable with respectto s. This fact is important in view of the definition of Sγ . Mollifiers eγ which fulfil theseconditions are known, and the reconstruction kernels 8γ can be computed exactly from theinversion formula of R (see Rieder [15]). We describe a technique based on the singular-valuedecomposition (SVD) of R.

Let R@ be the adjoint of R as a mapping betweenL2(�2) andL2(S2 × (−1, 1), w−1)withthe weight w(s) = (1 − s2)1/2. The identity RR@uk,l = σ 2

k uk,l holds true, where the singularvalues σk and the singular functions uk,l ∈ L2(S2 × (−1, 1), w−1) are given by

σk = 2

√π

k + 1and uk,l(ω(ϕ), s) = 1

πw(s)Uk(s) ei l ϕ (3.31)

with k ∈ N0, |l| � k, l + k even. The singular functions uk,l form a complete orthonormalsystem of R(R). For more details we refer to Louis [9].

Theorem 3.8. Let eγ be a radial mollifier in R(R∗ ⊗ I ), that means eγ (x) = eγ (‖x‖) foreγ ∈ L2(R). Then,

8γ (ω(ϕ), s, a) = (2π2)−1∞∑k=0

(2k + 1) I γk (a)U2k(s) (3.32)

with

Iγ

k (a) =∫ 1

−1

∫R

eγ

(√τ 2 + t2 + a2

)dt U2k(τ ) dτ

is the unique solution of (3.27) in R(R ⊗ I ).

Proof. Since eγ ∈ R(R∗ ⊗ I ), the series

8γ (ω, s, a) =∞∑k=0

∑|l|�k

l+k even

σ−2k 〈(R ⊗ I )eγ (·, a), uk,l〉L2(S1×(−1,1),w−1) uk,l(ω, s) (3.33)

752 T Schuster

converges in L2(S1 × (−1, 1), w−1) with (R@ ⊗ I )8γ = eγ . We easily find that

(R ⊗ I )eγ (ω, s, a) =∫

R

eγ

(√s2 + t2 + a2

)dt,

which implies

〈(R ⊗ I )eγ (·, a), uk,l〉L2(S1×(−1,1),w−1)

= 1

π

∫ 1

−1

∫ 2π

0

∫R

eγ

(√τ 2 + t2 + a2

)dt Uk(τ ) ei l ϕ dϕ dτ

= 2∫ 1

−1

∫R

eγ

(√τ 2 + t2 + a2

)Uk(τ) dt dτ δl0,

where we used (3.31). Thus, we obtain

8γ (ω, s, a) = 2

π

∞∑k=0

σ−22k I

γ

k (a)w(s)U2k(s).

Taking into account the relation R∗(w−1 uk,l) = R@uk,l , we obtain

8γ (ω, s, a) = 2

π

∞∑k=0

σ−22k I

γ

k (a) U2k(s) (3.34)

as a solution of (3.27) in R(R ⊗ I ) and the proof is complete. �

Remark 3.9. Equation (3.32) shows that a radial mollifier eγ leads to a reconstruction kernel8γ which is, in contrast to the kernels 8j

γ (3.22), independent of ω. This fact is well knownin two-dimensional computerized tomography.

One possibility to choose eγ is again the Gaussian function (3.21). Then, we have

eγ (t) = (2π)−3/2 γ−3 exp(−t2/(2γ 2)) (3.35)

resulting in a reconstruction kernel which is arbitrarily often differentiable with respect to s.More exactly, putting (3.35) in (3.32), we obtain the kernel

8γ (s, a) = γ−2

4π3exp(−a2/(2γ 2))

∞∑k=0

cγ

k (2k + 1) U2k(s) (3.36)

with

cγ

k =∫ 1

−1e−τ 2/(2γ 2) U2k(τ ) dτ.

Again we will cut off the sum in (3.36) after finitely many steps according to a criterionas (3.23). More precisely, let

8Mγ (s, a) = γ−2

4π3exp(−a2/(2γ 2))

M∑k=0

cγ

k (2k + 1) U2k(s), (3.37)

where M > 0 is chosen so large that

|cγM | < ε (3.38)

for a given number ε > 0.The density of R(R∗ ⊗ I ) in L2(�2 × (−1, 1)) implies

(R∗ ⊗ I ) T x2,18Mγ = T x1 (R

∗ ⊗ I )8Mγ −→ T x1 eγ = eγ (x, ·) (3.39)

An efficient mollifier method for three-dimensional vector tomography 753

as M tends to infinity. Hence, we expect 8Mγ to be an accurate approximation to 8γ . Note

that the |cγk | decrease exponentially as k goes to infinity.Finally, the approximate inverse for recovering ∇ × f from data Dj f = gj appears as

follows:

(∇ × f )j (x) ≈⟨gp(j), T

x2,1

(− ∂

∂s8Mγ

)⟩L2(Q)

. (3.40)

Thereby, p = (3, 1, 2) ∈ S3 is an index permutation, M is chosen according to (3.38) and

− ∂

∂s8Mγ = −γ

−2

4π3exp(−a2/(2γ 2))

M∑k=0

cγ

k (2k + 1) U ′2k(s)

with

U ′2k(s) = s U2k(s)− (2k + 1) T2k+1(s)

1 − s2.

A plot of − ∂∂s8Mγ is shown in figure 3 for a = 0, γ = 0.07 and M = 32, where M is chosen

according to (3.38).

Remark 3.10. Juhlin [6] and Norton [14] showed how the vector field f can be reconstructedonly by knowing the curl ∇ × f , the divergence ∇ · f in the interior of a domain � and�n · f on the boundary ∂�. The field is then the solution of a system of differential equationswith Neumann boundary conditions. However, our aim was to present an algorithm of filteredbackprojection type for both the solenoidal field f = f s and the curl ∇ ×f . These algorithmsare significantly faster than the approach suggested by Juhlin or Norton. We have only tochange the reconstruction kernels and use the same sets of data. However, our reconstructedfield Sγ Df differs from f according to (3.16), but there are solenoidal fields resulting in anexact reconstruction for γ → 0 (see section 5).

4. Convergence results for finitely many data

In this section we will study the error in the approximations (3.24) and (3.40), if the algorithmsare applied in practical situations, that means we have only finitely many data. Our goal isto present estimations of accuracy in (3.24) and (3.40), if the number of data goes to infinity.For this reason we introduce sampling points ϕν = ν · hϕ , ν = 0, . . . , p − 1, hϕ = 2π/p,sD = D · hs , D = −q, . . . , q − 1, hs = 1/q, ak = k · ha , k = −r, . . . , r − 1, ha = 1/r and anobservation operator Ep,q,r : Yα := H

α+1/2p (Z)⊗Hα

0 (−1, 1) → Rn with α > 1/2, n = 4pqr

and

(Ep,q,ry)ν,D,k = ςν,D,k y(ϕν, sD, ak),

ν = 0, . . . , p − 1, D = −q, . . . , q − 1, k = −r, . . . , r − 1. (4.1)

The mappingEp,q,r represents the evaluation of the measured data Dj f at the sampling points(ϕν, sD, ak) and is continuous on Yα , since α > 1/2. The constants ςν,D,k will be specified laterin this section.

Remark 4.1. It is also possible to define the observation operatorEp,q,r as local averages. Thiswould be useful if we wanted to take into account a measurement error in the mathematicalmodel, but in this article we assume we have noise-free data.

From theorem 2.2 we know that Dj is a bounded operator between Xα := H α,α andYα . Thus the semi-discrete Doppler transform Ep,q,r Dj : Xα → R

n is also continuous for

754 T Schuster

α > 1/2 and the problem of reconstructing a vector field from its Doppler measurements maybe formulated as follows. Find f ∈ Xα for α > 1/2 with

Ep,q,r Dj f = gn (4.2)

for given data gn ∈ Rn. The reconstruction problem for ∇ × f consists in recovering ∇ × f

from the same set of data Ep,q,r Dj f .Obviously Ep,q,r Dj is an unbounded operator on L2

�3(K3,R3). Hence, the domain

of the adjoint (Ep,q,r Dj )∗ is a proper subspace of R

n (see e.g. Rudin [17]). In the worstcase D((Ep,q,r Dj )

∗) = {0} and the adjoint does not exist. Therefore, in view of (3.1), it isimpossible to compute reconstruction kernels for Ep,q,r Dj . To circumvent this dilemma wetake the point evaluations Ep,q,r T

x2,j8

j,Mγ and Ep,q,r (− ∂

∂sT x2,1)8

Mγ of the kernels from the

non-discrete problems and study the error when n → ∞. With this aim we follow the outlineof Rieder, Schuster [16], where these studies were done in an abstract frame.

We associate with each Ep,q,r a subspace Vp,q,r = Sϕ ⊗ Ss ⊗ Sa of L2(Q), where Sϕ , Ss ,Sa are piecewise constant spline space with respect to the knot sequences {ϕν}, {sD} and {ak},respectively. A basis of Vp,q,r is given by{Bp,ν ⊗ Bq,D ⊗ Br,k/ςν,D,k : ν = 0, . . . , p − 1, D = −q, . . . , q − 1, k = −r, . . . , r − 1

}.

(4.3)

The B-splines Bp,ν ∈ Sϕ , Bq,D ∈ Ss and Br,k ∈ Sa are defined by

Bp,ν = χ[ϕν,ϕν+1[, Bq,D = χ[sD,sD+1[, Br,k = χ[ak,ak+1[

with the characteristic function χ . The normalization factors ςν,D,k are the L2-norms of theB-splines

ςν,D,k := ‖Bp,ν ⊗ Bq,D ⊗ Br,k‖L2(Q),

ν = 0, . . . , p − 1, D = −q, . . . , q − 1, k = −r, . . . , r − 1. (4.4)

By this normalization we have that the basis (4.3) forms an L2(Q)-Riesz system. Furthermorewe define an interpolation operator Jp,q,r : Yα → Vp,q,r , α > 1/2, which presents a linkbetween the point evaluations Ep,q,r and L2(Q):

Jp,q,ry :=p−1∑ν=0

q−1∑D=−q

r−1∑k=−r

y(ϕν, sD, ak) Bp,ν ⊗ Bq,D ⊗ Br,k. (4.5)

A short calculation shows the mentioned link. For y1, y2 ∈ Yα we have

〈Ep,q,ry1, Ep,q,ry2〉Rn = 〈Jp,q,ry1,Jp,q,ry2〉L2(Q). (4.6)

The interpolation operators Jp,q,r have some further properties which are necessary to proveour convergence results.

Lemma 4.2. For y, y1, y2 ∈ Yα , α > 1/2 and h := max{hϕ, hs, ha} we obtain the followingestimations:

(a) ‖Jp,q,ry‖L2(Q) � c1 ‖y‖Yα ,(b) ‖Jp,q,ry − y‖L2(Q) � c2 h ‖y‖Yα as h → 0,(c) |〈Jp,q,ry1,Jp,q,ry2〉L2(Q) − 〈y1, y2〉L2(Q)| � c3 h ‖y1‖Yα ‖y2‖Yα as h → 0,

where ci > 0, i = 1, 2, 3, are constants which do not depend on y, y1 and y2, respectively.

An efficient mollifier method for three-dimensional vector tomography 755

Proof. Part (a) follows from (b), since

‖Jp,q,ry‖L2(Q) � ‖y‖Yα + ‖Jp,q,ry − y‖L2(Q) � (1 + c2 h) ‖y‖Yα .The proof of (b) is achieved as in Schumaker [18, Chapter 12, theorem 12.7].

Part (c) is stated using parts (a) and (b):

|〈Jp,q,ry1,Jp,q,ry2〉L2(Q) − 〈y1, y2〉L2(Q)|� ‖Jp,q,ry1 − y1‖L2(Q) ‖Jp,q,ry2‖L2(Q) + ‖Jp,q,ry2 − y2‖L2(Q) ‖y1‖L2(Q)

� c h ‖y1‖Yα ‖y2‖Yα .�

Lemma 4.3. Suppose y ∈ Yα , α � 0 and x ∈ �3. Then,

‖T x2,j y‖Yα � c ‖y‖Yα , (4.7)

with a constant c > 0 independent from x.Proof. For z ∈ �2, u ∈ (−1, 1), g ∈ L2(S1 × (−1, 1)) and h ∈ L2(−1, 1) we define

t z1g(ω, s) = 1

4g

(ω,

s − z�ω2

)and

tu2 h(a) = 1

2h

(a − u

2

).

Let y = y1 ⊗ y2 with y1 ∈ Hα+1/2p (Z) and y2 ∈ Hα

0 (−1, 1). Because of T x2,j = tP∗j x

1 ⊗ tτj x

2 weobtain

‖T x2,j y‖Yα = ‖tP∗j x

1 y1‖Hα+1/2p

‖t τj x2 y2‖Hα(−1,1)

� c ‖y1‖Hα+1/2p

‖y2‖Hα(−1,1),

where we used Rieder and Schuster [16, lemma 5.3]. Note that we identify ϕ ∈ [0, 2π) withω(ϕ) ∈ S1. Since span{y1 ⊗ y2 : y1 ∈ H

α+1/2p (Z), y2 ∈ Hα

0 (−1, 1)} is dense in Yα , theassertion follows. �

Given m points xi ∈ �3, i = 1, . . . , m, our aim is to approximate the moments fγ (3.10)and (∇ × f )γ (3.28) in the reconstruction points xi from data Ep,q,rDj f . That is why wedefine mappings Ej : L2

�3(K3,R3) → R

m

(Ej f )i = 〈f, T xi1 Ejγ 〉L2

�3 (K3,R3) = 〈fj , T xi1 eγ 〉L2(K3), i = 1, . . . , m, j ∈ {1, 2, 3},

(4.8)

where Ejγ = eγ · vj for a radial mollifier eγ .

By Ljp,q,r : R

n → Rm we denote the mapping

(Ljp,q,rv)i := 〈v,Ep,q,r T

xi2,j8

j,Mγ 〉Rn . (4.9)

Hence Ljp,q,r Ep,q,r Dj f is a discrete version of (3.24) for finitely many data:

(Ljp,q,r Ep,q,r Dj f )i = π

4pqr

r−1∑k=−r

p−1∑ν=0

q−1∑D=−q

Dj f (ω(ϕν), sD, ak)

×8j,Mγ

(ω(ϕν),

sD − (P∗j xi)

�ω(ϕν)

2,ak − τjxi

2

). (4.10)

We see that (4.10) can be implemented analogously to the well known filtered backprojectionalgorithm (FBA) from two-dimensional computerized tomography. We will state this fact moreprecisely in the next section. First we show that (4.10) is an approximate solution of (4.2) inthe sense that it approaches the moments Ej f as the number of data grows.

756 T Schuster

Theorem 4.4 (Convergence for the reconstruction of f ). Assume f ∈ Xα for 1/2 < α �3/2; letLj

p,q,r and Ej be defined as in (4.9) and (4.8), respectively, h := max{hϕ, hs, ha} andPN(Dj ) : L2

�3(K3,R3) → L2

�3(K3,R3) the orthogonal projection onto the null space N(Dj )

of Dj . Furthermore adopt the notations from theorem 3.5. Then there exists a constant c > 0,which does not depend on h, and a sequence {RM} with RM → 0 for M → ∞, that∥∥Lj

p,q,r Ep,q,r Dj f − Ej f∥∥

∞

� c ‖f ‖Xα

(h

∞∑k=0

((2k + 2)1+2α |cγk |

)+

∥∥PN(Dj )Ejγ

∥∥L2�3 (K

3,R3)+ RM

)(4.11)

as h → 0.

Proof. Let i ∈ {1, . . . , m}.|(Lj

p,q,r Ep,q,r Dj f )i − (Ej f )i | = |(Ljp,q,r Ep,q,r Dj f )i − 〈f, T xi1 E

jγ 〉L2

�3 (K3,R3)|

� |〈Ep,q,r Dj f,Ep,q,r Txi

2,j (8j,Mγ −8j

γ )〉Rn |+|〈Ep,q,r Dj f,Ep,q,r T

xi2,j8

jγ 〉Rn − 〈Dj f, T

xi2,j8

jγ 〉L2(Q)|

+|〈f, T xi1 (D∗j8

jγ − Ej

γ )〉L2�3 (K

3,R3)|. (4.12)

We estimate the three parts separately. First, we have

|〈Ep,q,r Dj f,Ep,q,r Txi

2,j (8j,Mγ −8j

γ )〉Rn | � c1 ‖Dj f ‖Yα ‖8jγ −8j,M

γ ‖Yα� c2 ‖f ‖Xα

‖8jγ −8j,M

γ ‖Yα ,where we used (4.6), lemmata 4.2, part (a), and 4.3, as well as the smoothing property (2.6).So, we have to bound ‖8j

γ −8j,Mγ ‖Yα . With representation (3.22) we easily see

‖8jγ −8j,M

γ ‖Yα � c3

∞∑k=M+1

|cγk | ‖T2k+2 − 1‖Hα+1/2(−1,1).

A short computation shows‖T2k+2−1‖L2(−1,1) = √3π/2. Formula (22.14.5) from Abramowitz

and Stegun [1] yields

|T2k+2 − 1|2H 1(−1,1) =∫ 1

−1|T ′

2k+2(s)|2 ds � 2 (2k + 2)4.

With Markov’s inequality for a polynomial Pk of degree k,

|P ′k(s)| � k2 max

−1�t�1|Pk(t)|, |s| � 1, (4.13)

see e.g. Lorentz [8], we obtain

|T2k+2 − 1|2H 2(−1,1) =∫ 1

−1|T ′′

2k+2(s)|2 ds � 2 (2k + 2)8,

which finally results in

‖T2k+2 − 1‖2H 1(−1,1) � 2 (2k + 2)4 + 3

2 π � 4 (2k + 2)4,

‖T2k+2 − 1‖2H 2(−1,1) � 2 (2k + 2)8 + 2 (2k + 2)4 + 3

2 π � 6 (2k + 2)8.

Using the interpolation inequality for Sobolev norms (see e.g. Lions and Magenes [7]), weobtain the estimate

‖T2k+2 − 1‖Hα+1/2(−1,1) � ‖T2k+2 − 1‖3/2−αH 1(−1,1) ‖T2k+2 − 1‖α−1/2

H 2(−1,1)

� c4 (2k + 2)2 (3/2−α) (2k + 2)4 (α−1/2) = c4 (2k + 2)1+2α.

An efficient mollifier method for three-dimensional vector tomography 757

Defining c5 := c2 c3 c4, this gives

|〈Ep,q,r Dj f,Ep,q,r Txi

2,j (8j,Mγ −8j

γ )〉Rn | � c5 ‖f ‖Xα

∞∑k=M+1

|cγk | (2k + 2)1+2α.

The second term in (4.12) is estimated with the help of (4.6), lemma 4.2, part (c), (4.7) and (2.6).

|〈Ep,q,r Dj f,Ep,q,r Txi

2,j8jγ 〉Rn − 〈Dj f, T

xi2,j8

jγ 〉L2(Q)|

= |〈Jp,q,r Dj f,Jp,q,r Txi

2,j8jγ 〉L2(Q) − 〈Dj f, T

xi2,j8

jγ 〉L2(Q)|

� c6 h ‖Dj f ‖Yα ‖T xi2,j8jγ ‖Yα

� c7 h ‖f ‖Xα‖8j

γ ‖Yα� c8 h ‖f ‖Xα

∞∑k=0

|cγk | (2k + 2)1+2α.

The last inequality follows according to the foregoing considerations for ‖8j,Mγ −8

jγ ‖Yα .

Because of ‖T xi1 ‖L2→L2 � 1 and D∗j8

jγ − E

jγ = PN(Dj )E

jγ due to (3.13), we obtain for

the third term in (4.12)

|〈f, T xi1 (D∗j8

jγ − Ej

γ )〉L2�3 (K

3,R3) � ‖f ‖Xα‖D∗

j8jγ − Ej

γ ‖L2�3 (K

3,R3)

= ‖f ‖Xα‖PN(Dj )E

jγ ‖L2

�3 (K3,R3).

Letting

RM :=∞∑

k=M+1

|cγk | (2k + 2)1+2α

and c := max{c5, c8, 1}, statement (4.11) follows. �Remark 4.5. The approximation error (4.11) consists of three parts. The first one is due to theapplication ofEp,q,r to the non-discrete reconstruction kernel8j,M

γ . The thus-arising vector inRn is not a reconstruction kernel for (4.2), but the reconstruction error becomes smaller when

the number of data grows, i.e. h → 0. The second one originates from the non-injectivity ofthe Doppler transform Dj . The term ‖PN(Dj )E

jγ ‖ can be eliminated if the mollifier field Eγ

is orthogonal to the null space N(Dj ). The Ejγ corresponding to the Gaussian function (3.21)

does not fulfil this criterion. However we are hopeful of obtaining such mollifier fields usingremark 3.2. The third part RM finally is caused by cutting the series in (3.22) after M steps.This term tends to zero as M goes to infinity.

Note that the series∞∑k=0

|cγk | (2k + 2)1+2α (4.14)

converges due to the exponential decay of |cγk | related to our special mollifier eγ (3.21). Werefer to Schuster [19] to illustrate that general mollifiers eγ have to fulfil∣∣∣∣

∫ 1

−1

∂

∂τ

[ ∫R

eγ

(√τ 2 + t2 + a2

)dt

]U2k+1(τ ) dτ

∣∣∣∣ = O(k−1−2α−η)

with η > 1 for guaranteeing the convergence of (4.14).

As for the reconstruction of ∇ × f we define analogously Ljp,q,r : R

n → Rm by

(Ljp,q,rv)i :=

⟨v,Ep,q,r T

x2,1

(− ∂

∂s8Mγ

)⟩Rn

. (4.15)

758 T Schuster

We see that Ljp,q,r Ep,q,r Dp(j)f is a discrete version of (3.40):

(Ljp,q,r Ep,q,r Dp(j)f )i = π

4pqr

r−1∑k=−r

p−1∑ν=0

q−1∑D=−q

Dj f (ω(ϕν), sD, ak)

×(− ∂

∂s8Mγ

) (1

2

(sl −

⟨(x1

x2

), ω(ϕν)

⟩),ak − x3

2

). (4.16)

We prove convergence of Ljp,q,r Ep,q,r Dp(j)f to the moments Ej (∇ × f ).

Theorem 4.6 (Convergence for the reconstruction of ∇×f ). Let f ∈ Xα , 3/2 � α <

5/2, Lp,q,r and Ej as in (4.15) and (4.8), respectively, h := max{hϕ, hs, ha} and eγ ∈Hβ

0 (�2)⊗Hα

0 (−1, 1), β > 4α + 10, a radial mollifier in R(R∗ ⊗ I ). Then there exists aconstant c > 0, which does not depend on h, and a sequence {RM} with RM → 0 forM → ∞, that

‖Lp,q,r Ep,q,r Dp(j)f − Ej (∇ × f )‖∞ � c ‖f ‖Xα‖eγ ‖Hβ(�2)⊗Hα(−1,1)

(RM + h

)(4.17)

as h → 0.

Proof. For i ∈ {1, . . . , m} we have

|(Ljp,q,r Ep,q,r Dp(j)f )i − (Ej (∇ × f ))i |

=∣∣∣∣⟨Ep,q,r Dp(j)f,Ep,q,r T

xi2,1

(− ∂

∂s

)8Mγ

⟩Rn

−〈(∇ × f )j , Txi

1 eγ 〉L2�3 (K

3,R3)

∣∣∣∣�

∣∣∣∣⟨Ep,q,r Dp(j)f,Ep,q,r T

xi2,1

(− ∂

∂s

)(8M

γ − 8γ )

⟩Rn

∣∣∣∣+

∣∣∣∣⟨Ep,q,r Dp(j)f,Ep,q,r

(− ∂

∂s

)8γ

⟩Rn

−〈dp(j)f, (R∗ ⊗ I ) T

xi2,18γ 〉L2(�2×(−1,1))

∣∣∣∣ (4.18)

where we used (3.29) and eγ ∈ R(R∗ ⊗ I ). We estimate∣∣∣∣⟨Ep,q,r Dp(j)f,Ep,q,r T

xi2,1

(− ∂

∂s

)(8M

γ − 8γ )

⟩Rn

∣∣∣∣ � c1 ‖Dp(j)f ‖Yα∥∥∥∥ ∂∂s (8M

γ − 8γ )

∥∥∥∥Yα

� c2 ‖f ‖Xα

∥∥∥∥ ∂∂s (8Mγ − 8γ )

∥∥∥∥Yα

and bound ‖ ∂∂s(8M

γ − 8γ )‖Yα by∥∥∥∥ ∂∂s (8Mγ − 8γ )

∥∥∥∥Yα

� c3

∞∑k=M+1

(2k + 1) ‖I γk ‖Hα(−1,1) ‖U ′2k‖Hα+1/1(−1,1) (4.19)

using representation (3.34). Due to |Uk(s)| � k+1 for |s| � 1 (see Abramowitz and Stegun [1,formula 22.14.6]) and Markov’s inequality (4.13), we obtain

‖U ′2k‖H 2(−1,1) � c4 (2k + 1)7 � c5 σ

−142k , ‖U ′

2k‖H 3(−1,1) � c6 (2k + 1)9 � c7 σ−182k .

Again, we apply the interpolation inequality for Sobolev norms to obtain

‖U ′2k‖Hα+1/2(−1,1) � ‖U2k‖5/2−α

H 2(−1,1) ‖U2k‖α−3/2H 3(−1,1)

� c8 σ−14 (5/2−α)2k σ

−18 (α−3/2)2k = c8 σ

−8−4α2k .

An efficient mollifier method for three-dimensional vector tomography 759

Note that 2 � α + 1/2 < 3.Next, we assume eγ = e1

γ ⊗ e2γ with e1

γ ∈ Hβ

0 (�2), e2

γ ∈ Hα0 (−1, 1). Then,

‖I γk ‖Hα(−1,1) = |〈Re1γ , w U2k〉L2((−1,1),w−1)| ‖e2

γ ‖Hα(−1,1)

= 12 |〈Re1

γ , u2k,0〉L2(S1×(−1,1),w−1)| ‖e2γ ‖Hα(−1,1)

= 12 σ2k |〈e1

γ , v2k,0〉L2(�2)| ‖e2γ ‖Hα(−1,1),

with the singular functions vk,l ∈ L2(�2) fulfilling R∗uk,l = σk vk,l . Since (R∗R)−β :Hβ

0 (�2) → L2(�2) is a continuous mapping (see Rieder and Schuster [16, lemma A.3]), this

yields together with (4.19) an estimate for ‖ ∂∂s(8M

γ − 8γ )‖Yα :∥∥∥∥ ∂∂s (8Mγ − 8γ )

∥∥∥∥Yα

� c9

∞∑k=M+1

σ−12k |〈e1

γ , v2k,0〉L2(�2)| ‖e2γ ‖Hα(−1,1) σ

−8−4α2k

= c9

∞∑k=M+1

σ−β2k |〈e1

γ , v2k,0〉L2(�2)| σβ−9−4α2k ‖e2

γ ‖Hα(−1,1)

� c9 ‖(R∗R)−β e1γ ‖L2(�2)

( ∞∑k=M+1

σ2β−8α−182k

)1/2

‖e2γ ‖Hα(−1,1)

� c10 ‖e1γ ‖Hβ(�2) ‖e2

γ ‖Hα(−1,1)

( ∞∑k=M+1

σ2β−8α−182k

)1/2

= c10 RM ‖eγ ‖Hβ(�2)⊗Hα(−1,1),

where we used Cauchy’s inequality and

RM :=( ∞∑k=M+1

σ2β−8α−182k

)1/2

< ∞,

iff 9 + 4α − β < −1 ⇔ β > 10 + 4α. Since the mollifiers eγ ∈ Hβ

0 (�2) ⊗ Hα

0 (−1, 1) aredense in Hβ

0 (�2)⊗Hα

0 (−1, 1), we have∣∣∣∣⟨Ep,q,r Dp(j)f,Ep,q,r T

xi2,1

(− ∂

∂s

)(8M

γ − 8γ )

⟩Rn

∣∣∣∣� c11 ‖f ‖Xα

RM ‖eγ ‖Hβ(�2)⊗Hα(−1,1) (4.20)

for an arbitrary eγ ∈ Hβ

0 (�2)⊗Hα

0 (−1, 1).Noting that 〈dp(j)f, (R

∗ ⊗ I ) Txi

2,18γ 〉L2(�2×(−1,1)) = 〈Dp(j)f,− ∂∂s8γ 〉L2(Q), we estimate

the second term of (4.18) accordingly with the help of lemma 4.2, (c):∣∣∣∣⟨Ep,q,r Dp(j)f,Ep,q,r

(− ∂

∂s

)8γ

⟩Rn

− 〈dp(j)f, (R∗ ⊗ I ) T

xi2,18γ 〉L2(�2×(−1,1))

∣∣∣∣� c12 h ‖f ‖Xα

∥∥∥∥ ∂∂s 8γ

∥∥∥∥Yα

� c13 h ‖f ‖Xα‖eγ ‖Hβ

0 (�2)⊗Hα

0 (−1,1)

( ∞∑k=0

σ2β−8α−182k

)1/2

, (4.21)

where ( ∞∑k=0

σ2β−8α−182k

)1/2

< ∞

760 T Schuster

whenever 9+4α−β < −1 ⇔ β > 10+4α. Putting (4.20) and (4.21) in (4.18), statement (4.17)

follows with c := max

{c11, c13

√∑σ

2β−8α−182k

}. �

Remark 4.7. Since R(R∗ ⊗ I ) is dense in L2(�2 × (−1, 1)), we have that the left-hand sidein (4.17), in contrast to (4.11), tends to zero, as h → 0 and M → ∞. Note that we even finda representation of 8γ by techniques described in Rieder [15], without using the SVD of R,which implies RM = 0. Hence, we obtain ‖Lp,q,r Ep,q,r Dp(j)f − Ej (∇ × f )‖∞ → 0 ash → 0.

5. Implementation and numerical results

We give a short description of how to implement the reconstruction formulae (4.10) and (4.16),respectively. Thereby we will use the special structure of the kernels (3.22) and (3.36) toaccelerate the algorithm.

Suppose thatm = (2N + 1)3 equally spaced reconstruction points xi ∈ K3, i = 1, . . . , m,are given by

xi =(i1

N,i2

N,i3

N

)�, −N � ij � N, j ∈ {1, 2, 3}. (5.1)

We easily see that the reconstruction formulae (4.10) and (4.16) can be implemented likethe well known FBA (see e.g. Natterer [13, section 5.1.1]) with an additional summation overthe parameters ak . To increase the efficiency we use the special structure of our reconstructionkernels. We have

8j,Mγ (ω(ϕ), s, a) = rj (ϕ) exp(−a2/(2γ 2)) p

γ

M(s),

8Mγ (s, a) = exp(−a2/(2γ 2)) p

γ

M(s)

with polynomials pγM ∈ J2M+2, pγM ∈ J2M of degree 2M + 2 and 2M , respectively. Astraightforward calculation proves the following statement.

Lemma 5.1. If h ∈ L2(−1, 1) satisfies h(−1) = h(1) = 0, then

limγ↘0

1

2√

2π γ

∫ 1

−1h(s) e−(s−t0)2/(8γ 2) ds = h(t0) for t0 ∈ [−1, 1]. (5.2)

Lemma 5.1 justifies for small values γ > 0 the approximation∫ 1

−1h(s) e−(s−t0)2/(8γ 2) ds ≈ 2

√2π γ h(t0). (5.3)

Thus, using (5.3), (3.24) results in

fj (x) ≈ 〈Dj f, Tx

2,j8j,Mγ 〉L2(Q)

= 1

8

∫ 2π

0

∫ 1

−1

∫ 1

−1Dj f (ω(ϕ), s, a) rj (ϕ) p

γ

M

(s − P∗

j (x)�ω(ϕ)

2

)

×e−(a−τj x)2/(8γ 2) da ds dϕ

= 1

8

∫ 2π

0

∫ 1

−1rj (ϕ) p

γ

M

(s − P∗

j (x)�ω(ϕ)

2

) ∫ 1

−1Dj f (ω(ϕ), s, a)

×e−(a−τj x)2/(8γ 2) da ds dϕ

An efficient mollifier method for three-dimensional vector tomography 761

≈ 1

4

√2π γ

∫ 2π

0

∫ 1

−1rj (ϕ) p

γ

M

(s − P∗

j (x)�ω(ϕ)

2

)

×Dj f (ω(ϕ), s, τjx) ds dϕ.

Choosing N = r implies the existence of a ki ∈ {−r, . . . , r} with

Dj f (ω(ϕν), sD, aki ) = Dj f (ω(ϕν), sD, τj (xi)) (5.4)

for each xi , i = 1, . . . , m, with the xi from (5.1). Due to these considerations we finally obtainthe following implementation of our reconstruction algorithm (4.10).APPINV

input: p, q, r = N , M , γ , Ep,q,r Dj f , pγM

output: Ljp,q,r Ep,q,r Dj f

begin

1. for ν = 0, . . . , p − 1

vν;µ,k = 1q

∑q−1D=−q Dj f (ω(ϕν), sD, ak) p

γ

M

(sµ−sD

2

)for µ = −q, . . . , q − 1 and k = −r, . . . , r − 1 end for

end for

2. for i = 1, . . . , m = (2N + 1)3

(Ljp,q,r Dj f )i = γ

√2π π

16p

∑p−1ν=0 [(1 − u) vν;µ,ki + u vν;µ+1,ki ] rj (ϕν),

where aki = τj (xi) according to (5.4)

and µ � q 〈P∗j xi, ω(ϕν)〉 < µ + 1, u = q 〈P∗

j xi, ω(ϕν)〉 − µ

end for

end

Thus, we obtain an approximation to fj with an algorithm of the same computationaleffort like the 2D-FBA from computerized tomography. In APPINV, however, we reconstructon a three-dimensional grid instead of a two-dimensional one. More exactly, to compute (4.10)on the grid given by (5.1), we need O(pq2r)+O(pN3) operations. In comparison, the iterativemethod of Wernsdorfer [27] needs O(pqrN2) operations for each iteration step.

Since 8Mγ has a structure similar to that of 8j,M

γ , the reconstruction of ∇ × f (4.16) canbe calculated analogously to APPINV just by changing the reconstruction kernel.

As a numerical experiment, we reconstruct a straight flow with APPINV. Let f ∈L2�3(K

3,R3) be defined as

f (x) = (1 − x22 − x2

3 ) · v1, f = 0 in K3\�3.

We have ∇ × f (x) = −2 x3 · v2 + 2 x2 · v3, ∇ · f = 0 and �n · f = 0 on ∂�3, whichimplies f = f s ⊥ N(D). Moreover, since ∂

∂x1f1 = 0, we obtain f ⊥ N(Dj ) for j ∈ {1, 3}

(see (3.15)). For j = 2 we regain the exact solution, since D2f = 0. Thus,

limγ→0

Sγ Df (x) = f (x) a.e. (5.5)

follows from (3.14), that means the reconstructed field Sγ Df converges to the exact solutionf in this case.

762 T Schuster

Figure 4. Reconstruction of f with APPINV from the data (Ep,q,rD1f,Ep,q,rD2f,Ep,q,rD3f )�

with p = 21, q = 11 and r = 4 on a grid with m = 93 reconstruction points. The regularizationparameter was γ = 0.005. The cut-off parameter in (3.22) was M = 56.

A short calculation yields

D1(ω(ϕ), s, a) = 2√

1 − s2 sin(ϕ)(a2 + s2 sin2(ϕ) + 1

3 (1 − s2) cos2(ϕ)− 1),

D2f = 0,

D3f = −D1f.

Figure 4 displays the reconstruction of f with the algorithm APPINV on a grid with m = 93

reconstruction points from data Ep,q,r Dj f . The data were sampled for p = 21, q = 11 andr = N = 4. The reconstruction was performed using the kernels 8j,M

γ with regularizationparameter γ = 0.005 and M = 56 corresponding to the cutting criterion (3.23). The numberof data and grid points was chosen so small because of visibility. Figure 5 shows an illustrationof L1

p,q,r Ep,q,r D1f for p = 153, q = 25, r = N = 50, γ = 0.0157 and M = 161. Thecomputing time was about 10 min on a Silicon Graphics O2.

In figure 6 a reconstruction of ∇ × f is plotted, where the parameters were chosen as infigure 4.

The reconstruction error of SγDf was about

‖SγDf − f ‖L2(K3,R3)

‖f ‖L2(K3,R3)

≈ 0.272,

with parameters chosen as in figure 5. Figure 7 illustrates that the main part of the reconstructionerror appears on the boundary of �3. The reason is the following. The field f , as well as themollifier eγ from (3.21), does not have compact supports in �3. Therefore,

8−1∫�3eγ ((x − y)/2) dx � 1

if y is near the boundary, and hence∫�3 f (x) T

y

1 eγ (x) dx �→ 0 for γ → 0 if y ∈ ∂�3.As for the reconstruction of ∇×f , we set again r = 50, p = 153, q = 25 and γ = 0.0174

An efficient mollifier method for three-dimensional vector tomography 763

2040

60

80

100

20

40

60

80

100

0.60.70.80.91

2040

60

80

Figure 5. Plot of L1p,q,rEp,q,rD1f with p = 153, q = 25, r = N = 50 and γ = 0.0157.

Figure 6. Reconstruction of ∇ × f with APPINV from the data (Ep,q,rD1f,Ep,q,rD2f,

Ep,q,rD3f )�. The parameters p, q, r , γ and m are chosen as in figure 4.

and the corresponding M = 114. The error in SγDf is about

‖SγDf − ∇ × f ‖L2(�2×(−1,1))

‖∇ × f ‖L2(�2×(−1,1))≈ 0.115. (5.6)

The fact that the relative error in SγDf is about half of the error in SγDf arises from thedivision by ‖∇ × f ‖L2 in (5.6) instead of ‖f ‖L2 and ‖∇ × f ‖L2 ≈ 2.14 · ‖f ‖L2 .

To show the stability of APPINV with respect to incorrect data, we add a noise εj ∈ Rn to

our exact data Ep,q,rDj f =: gj :

gj = gj + εj , ‖εj‖2 = ηj ‖gj‖2.

764 T Schuster

Figure 7. Error in the reconstruction of f1. The figure displays theslice x1 = 0 of the error; the parameters are chosen as in figure 5.

-1 -0.5 0.5 1

0.2

0.4

0.6

0.8

1

Figure 8. Reconstruction L1p,q,r g1 of f1 from noisy data. The reconstruction error becomes clear

if we draw a single line (x1 = x2 = 0). The dashed line shows the exact solution f1(0, 0, x3).

Thus we have a relative error‖gj − gj‖2

‖gj‖2= ηj

in our data. Figure 8 plots L1p,q,r g1 with η1 = 0.15 corresponding to strong noise. The

reconstruction error was about‖Sγ g − f ‖L2(K3,R3)

‖f ‖L2(K3,R3)

≈ 0.274.

Hence the reconstruction error is only dispensable higher than in the case with absenceof noise. This stability is due to the special structure of our reconstruction algorithm, whichis based on the evaluation of scalar products. An application of (4.6), lemmata 4.2, 4.3 andCauchy’s inequality gives

‖Ljp,q,r (gj − gj )‖2 � ηj

√m ‖8j,M

γ ‖Yα ,which proves that the error in the reconstruction has the same order as the noise.

6. Conclusions

We have presented in this article a novel and very efficient algorithm of filtered backprojectiontype for reconstructing three-dimensional vector fields and their curl. The necessary

An efficient mollifier method for three-dimensional vector tomography 765

reconstruction kernels can be calculated before the measurement process and without theinfluence of data noise (see equation (3.13)). Furthermore the special structure of our method,as well as of the scanning geometry, allows a parallel computation of the three components fjfrom the data sets Dj f .

It is easy to adapt the method for the two-dimensional case, which is considered forexample by Sparr et al [22], Jansson et al [5] and Sielschott [20]. For f ∈ L2(�2,R2),ω ∈ S1 and s ∈ (−1, 1)

Df (ω, s) =∫

〈x,ω〉=sω⊥ · f (x) dσ (x)

means the two-dimensional Doppler transform of f . Choosing

eγ (x) = (2π)−1 γ−2 exp(−‖x‖2/(2γ 2)), Ejγ (x) = eγ (x) · vj , j = 1, 2,

as mollifier, the same considerations as in Schuster [19] for the solutions 8jγ of

DD∗8jγ = DEj

γ

lead to

81γ (ω, s) =

√2π γ 81

γ (ω, s, a)∣∣a=0

,

82γ (ω, s) = −

√2π γ 83

γ (ω, s, a)∣∣a=0

with81γ and83

γ from theorem 3.5. Now SγDf is computed in the same way as in section 3.1.There are several locations in this paper where hints at future work are given. In this respect

the development of new mollifiers adjusted to R(D∗j ) for eliminating the factor ‖PN(Dj )E

jγ ‖

in the convergence theorem 4.4 is important. To do this, remark 3.2 may be helpful. A furtherstep should be to adapt the method to a more practical scanning geometry to accelerate themeasurement procedure. In section 3.1 we described how the boundary values of f can bemodelled to decrease the reconstruction error. This is only one application where a prioriinformation is necessary, so the use of such a priori information has to be investigated.

Acknowledgments

I am indebted to Professor Dr A K Louis for a careful review of the manuscript and for manyhelpful discussions. This work was supported by the Deutsche Forschungsgemeinschaft undergrant Lo310/4-1.

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