University of Kuopio Department of Applied Physics Report Series ISSN 0788-4672
Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data V. Kolehmainen, S. R Arridge, W. R. B. Lionheart, M. Vauhkonen and J. P. Kaipio June 22, 2000 Report No. 7/2000
A slightly modified version of this manuscript has appeared in Inverse Problems , vol. 15 , 1999
University of Kuopio • Department of Applied Physics P.O.Box 1627, FIN-70211 Kuopio, Finland
Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data ∗ V. Kolehmainen†, S. R Arridge ‡, W. R. B. Lionheart §, M. Vauhkonen and J.P. Kaipio June 22, 2000 ABSTRACT In this study we consider the recovery of smooth region boundaries of piecewise constant coefficients of an elliptic PDE −∇ · a∇Φ + bΦ = f from data on the exterior boundary ∂Ω. The assumption is that the values of the coefficients (a, b) are known a priori but the information about the geometry of the smooth region boundaries where a and b are discontinous is missing. For the full characterisation of (a, b) it is then sufficient to find the region boundaries separating different values of the coefficients. This results to a nonlinear ill-posed inverse problem. In this study we propose a numerical algorithm that is based on the finite element method and subdivision of the discretisation elements. We formulate the forward problem as a mapping from a set of coefficients representing boundary shapes to data on ∂Ω, and derive the Jacobian of this forward mapping. Then an iterative algorithm which seeks a boundary configuration minimising the residual norm between measured and predicted data is implemented. The method is first given for a general elliptic PDE and then applied to optical tomography where the goal is to find the diffusion and absorption coefficients of the object by transilluminating the object with visible or near infrared light. Numerical test results for this specific application are given with synthetic data.
1
Introduction
Let Ω ∈ R2 denote a bounded domain and let r = (x, y)T ∈ Ω. We consider an elliptic PDE −∇ · a∇Φ + bΦ ∂Φ cΦ + d ∂ν
=
f, r ∈ Ω
=
g,
(1)
r ∈ ∂Ω
in the special case of piecewise constant coefficients a and b. Assume that Ω is divided to L + 1 disjoint regions Ak Ω = ∪L (2) k=0 Ak , which are bounded by smooth closed boundary curves and have known constant values of coefficients {ak , bk }. Let χk (r) be the characteristic function of Ak and let C` ∈ Ω (` = 1, . . . , L) denote the smooth outer boundary of region A` . The outer boundary of the background region A0 is ∂Ω, see Fig. 1 for an example of the topology of the regions Ak . Assume that ∂Ω and the values {ak , bk } are known a priori but some of the geometrical information on the regions {Ak } is missing. This missing information may be for example the shapes, sizes, locations or in some cases even the number of the regions. In this kind of situation it is sufficient to find the region boundaries {C` } ∈ Ω (` = 1, . . . , L) for the full characterisation of the coefficients (a, b). ∗ Email:
[email protected] Kolehmainen, M. Vauhkonen and J.P. Kaipio are with the Department of Applied Physics, University of Kuopio, P.O.Box 1627, FIN-70211 Kuopio, Finland ‡ S. R. Arridge is with the Department of Computer Science, University College London, Gower Street, London, WC1E 6BT,UK § W. R. B. Lionheart is with the School of Computing and Mathematical Sciences, Oxford Brookes University, Gipsy Lane Campus, Oxford OX3 0BP, UK † V.
1
In this study we propose a numerical method for the recovery of the boundaries {C ` } of the regions {A` } from a set of measurements {zj } made on the exterior boundary ∂Ω arising from a set of internal sources {fj } and (Dirichlet, Neumann, Robin) boundary conditions {gj }. The forward operator, that is the projection operator Pj : a, b 7→ zj with given fj and gj is denoted here by zj = Pj (a, b), where operator Pj is of the form
(3)
Pj = M[Φj ]
and M is some measurement operator. Furthermore, let us denote by z := {zj } the whole boundary data set and let z = P(a, b) (4) denote the forward operator providing the whole data set z for given sets {f j } and {gj }. The goal is to approximate the shapes of the boundaries {C` } with a finite set γ of shape coefficients and then formulate the FEM discretization of the forward problem (4) as a mapping from the shape coefficients γ to boundary data z. The inverse problem is then to find the representation γ of the boundary configuration when z, {ak , bk }, {fj } and {gj } are given. This inverse problem is approached as an optimisation problem which seeks to minimise the residual norm between measured and predicted data. One attraction of this approach is that the dimension of the search space can be made smaller than with conventional pixelwise parametrization, which potentially leads to less ill-posed inverse problem. The recovery of unknown boundaries of piecewise constant coefficients of a PDE is of great interest in many physical measurement techniques, good examples being mine detection by electrostatic measurements, magnetic prospecting, crack detection from conducting body, determination of the shape of a scattering body and electrical impedance tomography (EIT). For example, in the most elementary form of EIT the coefficients of equation (1) are d = a = conductivity, b = f = c = 0 and g = source current density. In EIT the boundary data z is obtained with a measurement operator M which samples the potential field Φ on ∂Ω with respect to some reference potential [1, 2]. An interesting application in which boundary recovery methods have not been utilised to date is optical tomography. A widely used model for optical tomography is the diffusion approximation to the linear transport equation which is a PDE of the form (1) with the following terms: Φ is the photon density, a = diffusion, b = absorption, f is the distribution of internal sources, c = 1, d = 2ϑa ( ϑ is a coefficient due to the mismatch of speed of light in the domain Ω and the surrounding medium ) and g is the distribution of the boundary sources [3, 4]. In optical tomography the simplest possible measurement operator is of the form Γ = M[Φ] = −aν · ∇Φ [4, 5]. Different boundary recovery approaches have been applied successfully to elliptic PDE problems for example in [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. All these papers contain unique ideas and give methods for the boundary recovery problems. However, a common feature in these papers is that they approach boundary recovery by considering some specific elliptic PDE with certain boundary conditions and data on ∂Ω. In addition, most of these papers are concerned with the recovery of only one region or crack from the homogeneous background. In this study the proposed method is given in the general framework for any elliptic PDE problem. The approach we propose here is suitable for the recovery of several smooth outer boundaries {C` } of simply connected regions {A` } which are also allowed to be nested. The rest of this paper is organised as follows. In the next section we give the FEM discretization of the general case of elliptic PDE (1) and we also represent the coefficients (a, b) with respect to the set of shape coefficients γ approximating the shapes of the region boundaries {C ` }. The forward problem is then defined as a mapping from the shape coefficients to data, that is, P : γ 7→ z. In section 3 the inverse problem is stated as a problem of finding the region boundary configuration which optimises the objective function representing the residual norm between measured and predicted data. The derivation of the Jacobian of the mapping P : γ 7→ z is also given and then an iterative method for solving the inverse problem is implemented. In section 4 the proposed 2
method is applied to optical tomography and in section 5 we give some numerical test results for this application with synthetic data. In section 6 we give conclusions and address some suggestions for future work.
2
Discretization of the elliptic PDE with FEM
The alternatives for solving equation (1) include analytical methods, finite difference methods, boundary element methods (BEM) and finite element methods (FEM). Whereas analytical methods are restricted to very simple domains and finite difference methods are awkward for arbitrary boundary shapes, BE and FE methods offer great flexibility for arbitrary geometries and boundary shapes. The BEM approach has been considered for boundary recovery for example in [11, 13, 15, 12]. The BEM approach is in a way a natural choice since the boundaries that are discretized in this boundary integration approach are just the unknowns of the inverse problem. In addition, BEM approach is often justified over FEM with lower computational cost which is of practical importance in many applications. However, for us the FEM approach is advantageous in the sense that it allows more flexibility with respcet to the coefficients. For example, in some applications the situation may be that another of the coefficients (a, b) is fixed but not piecewise constant. This kind of situation can be handled easily with the FEM approach but not with the BEM. In addition, when there are many internal boundaries the dimension of the full BEM problem reaches rapidly the dimension of the FEM problem which results to the inversion of a sparse and symmetric system matrix. Although we consider FEM in this study only with triangular elements, the method we propose can be applied to any other elements as well. In the FEM approach the domain Ω is divided to P disjoint elements ∪P p=1 Ωp , joined at D vertex PD nodes Ni . The solution Φ is approximated with a piecewise polynomial function Φ h = i=1 φi ϕi ∈ fh , where fh is a finite dimensional subspace spanned by the basis functions ϕi (i = 1, . . . , D) which are chosen to have support over the elements Ωp which have node Ni as a vertex, that is, supp(ϕi ) = ∪p|Ni ∈Ωp Ωp . The problem of solving Φh becomes one of sparse matrix inversion for which standard methods such as Cholesky decomposition or conjugate gradients are readily available. By standard methods, equation (1) is expressed in the FEM framework as (K(a) + C(b) + R)Φ = G + F, where the entries of the system matrices are given by Z Kij = a∇ϕi · ∇ϕj dr Ω Z Cij = bϕi ϕj dr Z Ω ac ϕi ϕj dS Rij = ∂Ω d The terms on the right hand side of equation (5) are of the form Z Z a Fj = gϕj dS. f ϕj dr Gj = Ω ∂Ω d
(5)
(6)
(7)
For details, see for example [16, 3, 4, 2]. Let us now define piecewise constant coefficients a and b as a=
L X
ak χk (r),
b=
L X
k=0
k=0
3
bk χk (r).
(8)
By substituting equation (8) into equation (6), we obtain the elements of K and C as Kij =
L Z X
k=0
Cij =
supp(ϕi ϕj )∩Ak L Z X k=0
ak ∇ϕi · ∇ϕj dr
(9)
bk ϕi ϕj dr.
(10)
supp(ϕi ϕj )∩Ak
where supp(ϕi ϕj ) expresses the part of the domain Ω where both basis functions ϕi and ϕj are non-zero, that is, the union of the elements that contain both nodes Ni and Nj supp(ϕi ϕj ) := ∪t|{Ni ∈Ωt ,
Nj ∈Ωt } Ωt .
In this study we assume that the domains {A` } are simply connected. If the outer boundaries {C` } of the regions {A` } are sufficiently smooth, they can be approximated in the form {C` (s)} =
x` (s) y` (s)
=
Nθ X γnx` θnx (s) , γny` θny (s)
` = 1, . . . , L
(11)
n=1
where θn are periodic and differentiable basis functions with period 1. Let γ denote the vector of all boundary shape coefficients, that is, y L T y1 xL x1 . , . . . , . . . , γ1xL , . . . , γN , γ1yL , . . . , γN , γ1y1 , . . . , γN γ = γ1x1 , . . . , γN θ θ θ θ
The goal is now to express the discretization of the forward model as a mapping from the boundary coefficients γ to the data z, that is, given all the boundary shape coefficients and values a = (a0 , . . . , a` )T , b = (b0 , . . . , b` )T , compute the data on ∂Ω. The FEM implementation of this mapping is accomplished in 4 stages as follows: 1. Classification of mesh nodes as nodes inside or outside a given boundary C ` (s). This is accomplished by counting the number of boundary crossings λ of a horizontal line drawn through the node Ni = (xi , yi )T . Boundary crossings are obtained by solving equation 1 xi = C˜` , (12) +λ 0 yi where C˜` is a finely divided polygonal approximation of C` (s). The decision, whether Ni is inside or outside the boundary C` (s), is made on the basis of the number of λ > 0 or λ < 0. If either of these is odd, Ni is inside the boundary C` (s). It is sufficient to carry this test only for the nodes which are within the vertical and horizontal extent of C` . This test is known as Jordan curve lemma and in the field of computer graphics it is also known as the “odd-even” test. 2. Classification of mesh elements as inside, outside or intercepted by given region boundary C` (s). Let B(C` ) := {Ωm |Ωm ∩ C` 6= ∅} denote the set of elements intercepted by C` (s).
3. Determination of intersections of element edges with a given boundary C ` (s). The goal is to m find the exact intersection points {sm 1 , s2 } of C` (s) and the element edges for each element Ωm ∈ B(C` ). The intersection of C` (s) with the edge from Ni to Nj is obtained from xj − x i xi = C` (s) (13) +ε yj − y i yi with condition 0 < ε ≤ 1. We use a binary search algorithm for solving equation (13). An initial search space for s is obtained from C˜` by finding the polygon points that are just outside the vertical and horizontal extent of the element edge of interest. 4
4. Computation of the system matrices. Matrices K and C are constructed by using equations (9) and (10). Obviously, Rin the element R set {Ωm ∈ B(C` ) , Ωm ∈ supp(ϕi ϕj )} we have to compute the integrals ϕi ϕj dr and ∇ϕi · ∇ϕj dr over the intersections A` ∩ Ωm and Ar ∩ Ωm , see Fig. 2. When computing these integrals the region boundary in the intercepted element is approximated with a line drawn from C` (s1 ) to C` (s2 ), see the dashed line in Fig. 2b. The remaining part of the element is stillR divided to two R triangular parts by drawing a line from C` (s1 ) to Ni (Fig.2b). The integrals ϕi ϕj dr and ∇ϕi ·∇ϕj dr are then computed separately over each of the three subtriangles. The integrals are mapped from subtriangles to local element and then evaluated with a Gaussian quadrature in the local element [17]. Let T (γ) denote the sum T (γ) := (K(a) + C(b)),
(14)
Now the FEM equation (5) can be written in the form (T (γ) + R)Φ = G + F.
(15)
Equation (15) is formally solved by matrix inversion Φ = (T (γ) + R)−1 (G + F ). In order to relate the FEM discretisation to the forward problem (3) we define the discrete forward operator for the j:th source as Pj = Mj [Φj ], where Mj is the discretisation of the measurement operator and Φj is solution to equation (15) for the j:th source. Furthermore, let us denote by z = P(γ, a, b)
(16)
the discrete version of the forward operator (4) which gives the vector z = (z 1 , z2 , . . . , zS )T of samples of the whole data set for all S sources.
3
Inverse problem
If we denote by zmeas the vector of measurements for a given set of sources, the cost functional to be minimised can be written as Ξ(γ) = kzmeas − P(γ, a, b)k22 .
(17)
The inverse problem consists of finding the boundary configuration which minimises the cost functional (17), that is Find γ ∗ such that Ξ(γ ∗ ) = min kzmeas − P(γ ∗ , a, b)k22
(18)
A natural way to solve this nonlinear problem is to use Newton-type methods where we seek a zero of Ξ0 (γ) by an iterative method using Taylor expansion around the current estimate γ k .This leads to an algorithm γk+1 = γk + (Ξ00 (γk ))−1 Ξ0 (γk ) (19) where Ξ0 (γk ) = JkT (zmeas − P(γk , a, b)) and the inverse of Ξ00 (γk ) is usually approximated with (JkT Jk )−1 . However, in our case the Jacobian matrices Jk are usually ill-conditioned and therefore we stabilise the algorithm via the Levenberg-Marquardt method γk+1 = γk + (JkT Jk + λI)−1 JkT (zmeas − P(γk , a, b)), where λ is a control parameter. The Jacobian matrix J=
∂P(γ, a, b) ∂γ 5
(20)
is obtained as follows. By differentiating equation (15) with respect to the boundary coefficient γnα` (α is either x or y) , we obtain T
∂T ∂Φ Φ, α` = − ∂γn ∂γnα`
(21)
where we have assumed that the derivatives of R, G and F with respect to the boundary coefficients are zero. By definition T (γ + θnα (s)) − T (γ) ∂T (22) α` = lim →0 ∂γn where is a arbitrarily small perturbation in the basis function θnα (s) of either x` (s) or y` (s). Perturbation in one parameter γnα` changes the region configuration from {Ak } to {A˜k }. Let χ ˜k (r) denote the characteristic functions of the perturbed regions A˜k . Furthermore, let us define the − signed differences δA+ k and δAk as δA+ k
=
δA− k
=
(A˜k \ Ak ) ∩ A˜k (A˜k \ Ak ) ∩ Ak .
(23)
By using equation (23) we can write the relation A˜k = Ak + 4Ak , where
(24)
− 4Ak = δA+ k − δAk .
See Fig. 3 for a simple example of the perturbed region. By using equations (9),(10), (14) we obtain the matrix elements in equation (22) as ( L Z X ∂T 1 = lim ak ∇ϕi (r) · ∇ϕj (r)dr α` →0 ∂γn ij ˜ k=0 supp(ϕi ϕj )∩Ak ! L Z X + bk ϕi (r)ϕj (r)dr ˜ k=0 supp(ϕi ϕj )∩Ak L Z X ak ∇ϕi (r) · ∇ϕj (r)dr − k=0 supp(ϕi ϕj )∩Ak !) L Z X bk ϕi (r)ϕj (r)dr + k=0 supp(ϕi ϕj )∩Ak Z X 1 (b` − br ) = lim ϕi (r)ϕj (r)dr →0 m|Ωm ∈supp(ϕi ϕj ) 4A` ∩Ωm Z X ∇ϕi (r) · ∇ϕj (r)dr , + (a` − ar ) m|Ωm ∈supp(ϕi ϕj ) 4A` ∩Ωm
(25)
where (a` , b` ) are values of coefficients in the region A` , which has C` as the outer boundary, and (ar , br ) are the values in the (outer) neighboring region Ar , which has C` as the inner boundary. Let us next consider the evaluation of equation (25) in simplified notation. The problem is the evaluation of expressions of the form Z Z 1X f (x, y)dxdy, (26) lim →0 4A` ∩Ωm m 6
where f (x, y) is either ϕi ϕj or ∇ϕi · ∇ϕj . The area of integration depends on the perturbation as follows. We define a new positively oriented coordinate system (p, s) where s is the positively oriented parametrisation of the closed curve C` , and p multiplies a perturbation vector pointing outward from the region A` x(s) ξ(s) x` (s) p (27) = +p 7→ y(s) η(s) y` (s) s The boundary C˜` of the perturbed region A˜` can be expressed in the new coordinate system as x` (s) ξ(s) s 7→ + = C˜` (s). y` (s) η(s)
By utilising the coordinate transformation (27) into the equation (26) we obtain the form Z Z s2 1 ∂(x, y) lim dsdp f (x, y) →0 p=0 s=s ∂(p, s) 1 Z Z s2 d ∂(x, y) dsdp. = f (x, y) d =0 p=0 s=s1 ∂(p, s)
(28)
In the new coordinate system the region of integration is the rectangular strip [s 1 s2 ] × [0 ] in ∂(x,y) is obtained as each of the elements Ωm . The Jacobian determinant ∂(p,s) ∂x = ξ ∂p s ∂y = η ∂p s
and by evaluating this, we have Z Z d d =0
p=0
s2 s=s1
∂x = x˙ + pξ˙ ∂s p ∂y = y˙ + pη˙ ∂s p
˙ dsdp, f (x, y) ξ(y˙ + pη) ˙ − η(x˙ + pξ)
By evaluating equation (29) and using the result (28) we get the result Z Z s2 ∂(x, y) 1 dsdp f (x, y) lim →0 p=0 s=s ∂(p, s) 1 Z s2 = f (x, y)(ξ y˙ − η x)ds. ˙
(29)
(30)
s=s1
∂T ∂T are obtained by applying the result The final result for the elements of the matrices ∂γ x` and y ∂γn` n (30) into equation (25). When differentiating with respect to γnx` , that is perturbing θnx (s), we ∂T have ξ = θnx (s), η = 0 and the elements of ∂γ x` are obtained as n
∂T ∂γnx`
X
= ij
+
m| Ωm ∈B(C` ) ∩ Z s2
(a` − ar )
s1
supp(ϕi ϕj ) ∇ϕi ·
(b` − br )
Z
s2 s1
∇ϕj y˙ ` (s)θnx (s)ds
ϕi ϕj y˙ ` (s)θnx (s)ds
(31)
Correspondingly, when differentiating with respect to γny` we have η = θny (s), ξ = 0 and the ∂T elements of ∂γ y` are obtained as n
∂T ∂γny`
X
= ij
−
m| Ωm ∈B(C` ) ∩ Z s2
(a` − ar )
s1
supp(ϕi ϕj )
−(b` − br )
Z
s2 s1
∇ϕi · ∇ϕj x˙ ` (s)θny (s)ds . 7
ϕi ϕj x˙ ` (s)θny (s)ds (32)
The limits of integration s1 and s2 in equations (31) and (32) are the values of the curve parameter in the intersection points of the element edges and boundary curve C` in each element Ωm , see Fig. 3. By solving equation (21) for the j:th source ∂Φj −1 ∂T Φj α` = −T ∂γn ∂γnα` and by applying the discrete measurement operator Mj , we obtain the vector ∂Φj ρj,m = Mj ∂γnα`
(33)
which is the j:th block of the m:th column corresponding to the differentiation with respect to the coefficient γnα` in the Jacobian J. We have assumed here that the measurement operator M is independent of the configuration of the region boundaries {C` }. The Jacobian can be also constructed efficiently by utilising the adjoint method [5].
4
Application to optical tomography
By optical tomography we mean the use of visible or near infrared light in the wavelength range ∼ 700−1000nm, to probe highly scattering media in order to derive images of the optical properties of the object. Of the potential applications the most attention has been received by applications in medical imaging, such as detection of cancerous tumors from breast tissue and monitoring infant brain tissue oxygenation level [18]. In optical tomography the measurement system consists of S optic fibers placed on the source positions εj ∈ ∂Ω and M optic fibers placed in the measurement positions ζi ∈ ∂Ω. The methods of data acquisition in optical tomography can be divided to frequency domain methods and timeresolved methods. In the frequency domain methods light from a radio frequency modulated source is guided via the optic fibers usually to one source location εj at the time, and the phase shift and modulation amplitude of the transmitted light is measured on the measurement locations ζi , i = 1, . . . , M by using the optic fibers and light sensitive detectors [19]. This process is then repeated to all source locations εj , j = 1, . . . , S. Correspondingly, in the time resolved methods an ultra short input pulse (∼ 10ps) is triggered usually to one source location ζ i at the time, and the impulse response function Γi,j (t) of the object is measured on the detector locations εj , j = 1, . . . , S. Due to the multiple scattering events occuring in the highly scattering medium different photons arrive different times to the detector, and this causes the response function Γ(t) to be spread over several nanoseconds in width [3, 18, 20, 21, 22, 23]. The inverse problem in optical tomography is to reconstruct maps of the optical properties of the object Ω based on the set of boundary data {Γi,j }. The image reconstuction can be carried out by using the whole spatio-temporal data set, or alternatively, the reconstruction can be carried out by using some temporal integral transform of Γi,j (t) as data [24, 25]. In the results in this study we consider recovery of the region boundaries by using only the simplest datatype, namely the DC-intensity Z ∞
Ei,j = Γi,j (ω = 0) =
Γi,j (t) dt.
(34)
0
The use of alternative datatypes complicates the presentation of the method, so it is not discussed further here, although it is certainly of interest in optical tomography and may lead to better performance of our method [24, 25, 5]. Although light transport in scattering media is properly modelled with the radiative transfer equation (RTE), most of the current approaches to optical tomography use the diffusion approximation to the RTE as the forward model. In the case of the DC-intensity (34) the diffusion approximation is a PDE of the form (1) where Φ is the photon density, a = 1/(3(b + e0s ))
8
(35)
¯ s is the reduced scattering is the diffusion coefficient , b is the absorption coefficient, e0s = (1 − Θ)e ¯ is the mean cosine of the scattering phase function coefficient, es is the scattering coefficient, Θ and the rest of the coefficients in equation (1) are as given in section 1 [5, 4]. In other cases than DC-intensity the diffusion approximation contains an additional term with respect to equation (1). In the frequency domain case the additional term is iω c Φ, resulting in a complex field. In the time domain case the corresponding PDE is parabolic, but the datatypes of interest can be obtained by recursively solving a set of elliptic equations[5]. In either case, these additional terms do not affect the crucial parts of the boundary estimation algorithm given above. The diffusion approximation has been validated to give reasonable accuracy in scattering dominated media (e 0s b). For further details on the diffusion approximation to the RTE, see [26, 27, 28, 29, 30, 31, 5, 32, 33]. In the diffusion approximation model sources are usually modelled either as a collimated point source which is placed inside the domain Ω one mean free path from ∂Ω, or as a diffuse photon source on ∂Ω [4]. In the case of collimated point source we have non-zero distribution of internal sources fj 6= 0, and we set the boundary sources gj = 0 in equation (1). In the case of diffuse boundary sources, which is the source model we use in this study, we set f j = 0 and r ∈ εj Γs j gj = 0 r ∈ ∂Ω \ εj . The relation between the response function Γ and the photon density Φ is of the form Γi,j = −a(ζi )ν · ∇Φj (ζi )
(36)
in the diffusion approximation model. For details on the models of optical tomography and on the limitations of different models, see [31, 5, 34] and references therein.
5
Numerical results
In this section we consider some numerical examples of the proposed method applied to optical tomography. In order to be able to implement the numerical method given in sections 2 and 3 to the model of optical tomography which was given in section 4, we have to choose the basis functions θnx (s) and θny (s) for the curve parametrisation given in equation (11). In this study we use basis functions of the form θ1α (s) θnα (s) θnα (s)
=
1
n = sin 2π (s + φs ) , n = 2, 4, 6, . . . 2 n − 1 (s + φs ) , n = 3, 5, 7, . . . , = cos 2π 2
(37)
where φs is the phase of the curve parameter and α denotes either x or y. Before applying the boundary estimation method with this parametrisation of the curve C` we have to fix the phase of the curve parameter s in equation (11). Basically, the same curve C` can be represented with an infinite variety of different shape coefficients γ by adjusting the phase φ s with infinitesimal changes on the interval ]0 1[. If the phase is not fixed when utilising the boundary estimation method, we obtain a one dimensional nullspace N` per each curve C` when computing the Jacobian matrix. This null space is due to this phase ambiguity. In this study the phase is fixed with a very simple method, that is, by using unequal amount of sin and cos terms, i.e, we just set the number of basis functions Nθ to an even number. A clear advantage of this phase fixing method is that we do not necessarily have to know any interior point r0 ∈ A` when we utilise it for fixing the phase φs . The synthetic measurement system we use in this study consists of 16 sources and 16 detectors placed in equiangular positions with the order ε1 , ζ1 , ε2 , ζ2 , . . . , ε16 , ζ16 on the boundary ∂Ω of a circular domain Ω which has radius of 25mm. In the inverse computations the domain Ω was discretised to P = 6776 triangular elements Ωp and the number of nodes in this mesh was D = 3517. In the formulation of the numerical method given in sections 2 and 3 the conditions for the nodal basis functions are that ϕi is piecewise differentiable and has support supp(ϕi ) = ∪p|Ni ∈Ωp Ωp . In this study we use the first order basis functions ϕi . 9
The simulations were carried out by using the logarithm of the DC-intensity z i,j = log Ei,j as data. In the discrete mode the forward solution for the DC-intensity (34) is obtained by solving Φj from the FEM discretisation of an elliptic PDE in equation (15) [25, 5] ,and then applying Ej = M[Φj ] = DΦj where D is a linear discrete operator 1 2ϑ ϕn (ζi ), if node i ∈ Ωζi , i = 1, . . . , M n = 1, . . . , P. (38) Din = 0, otherwise Equation (38) is obtained by using equation (36) and the boundary condition in equation (1) [5] . The Jacobian for the data zj = log(Ej ) corresponding to the j:th source is obtained by utilising 0 (x) d equations (33), (38) and by utilising dx log(g(x)) = gg(x) . In the following simulations the values of the coefficients for the diffusion approximation were chosen from the range of interest in medical imaging [5]. For the absorption coefficient b we used values from the range 0.025mm−1 → 0.05mm−1 , for the scattering coefficient es we used values ¯ = 0.9 which from the range 20mm−1 → 40mm−1 and for the mean cosine we used a typical value Θ indicates strongly forward biased scattering [4]. Values of diffusion coefficient a were computed by using equation (35). Multiplicative random noise equivalent to 104 photons received at each detector was added to the synthetic data [35]. The mesh we used for the generation of the synthetic data consisted of P = 8600 triangular elements and D = 4429 nodes. In the data generation we used greater amount of trigonometric basis functions θnα (s) than in the inverse computations. In the following simulations the inverse problem was solved by using the Levenberg-Marquardt method (20). The stabilisation parameter λ and the number of iterations were chosen manually by performing a large number of different tests. The initial estimate γ0 was obtained by drawing an arbitrary polygon relatively near the true {C` } and then solving the coefficients γ0 by using least squares fit to the set of polygon points. In the inverse computations we used N θ = 8 basis functions of the form (37) for the curve representation (11). Results of the simulations are shown in Figs. 4a – 7a. The true boundary is represented with bold line, reconstructed boundary with thin line and initial estimate for the iteration with dotted line. Figs. 4b – 7b contain plots of the relative data error percentage kzmeas k−1 kzmeas −P(γk , a, b)k versus iteration index k. The result for the first test case is shown in Fig. 4a. The values of parameters are as follows: In the background A0 the values are b0 = 0.025mm−1 and es0 = 20mm−1 , and in the perturbation A1 values of b and es are twice the values of background, that is b1 = 0.05mm−1 and es1 = 40mm−1 . In this test case the stabilisation parameter was λ = 0.2. As it can be seen this simple perturbation A1 , which has good contrast with respect to background in both parameters, can be recovered with good accuracy from noisy data. The relative data error (%) versus iteration index k is shown in Fig. 4b. Results for the second test case are shown in Fig. 5. Parameter values are as follows: In the background A0 the values are b0 = 0.025mm−1 and es0 = 20mm−1 . The first target A1 which is located up and left from the center of Ω has contrast in both parameters (b1 = 0.05mm−1 ,es1 = 40mm−1 ) with respect to background. The second target A2 , which is located in the lower part of Ω, has contrast only in b with the values (b2 = 0.05mm−1 , es1 = 20mm−1 ), and the third target A3 has contrast only in es with the values (b3 = 0.025mm−1 , es3 = 40mm−1 ), respectively. In the reconstruction shown in Fig. 5 the stabilisation parameter was λ = 0.2. As it can be seen, all the three regions are found with good accuracy, and it seems that the small differences in the contrast do not affect much to the accuracy obtained in the recovery of the different regions. It is also worth of noting that in the region A3 the reconstuction is started inside of the true boundary C3 . Results for the third test case, which has some resemblance with a cross section of infant head, are shown in Fig. 6. The four nested regions Ak , k = 0, 1, 2, 3 of this simulated phantom are in respective order from the boundary ∂Ω towards the center of the domain Ω. The values of coefficients are as follows: b0 = 0.025mm−1 , es0 = 20mm−1 , b1 = 0.05mm−1 , es1 = 40mm−1 , b2 = 0.025mm−1 , es2 = 20mm−1 b3 = 0.03mm−1 and es3 = 30mm−1 . In Fig. 6 the stabilisation parameter was λ = 1.4. In this case, the boundaries C1 and C2 , that is the “skin-skull” and “skullbrain” interfaces, were found with good accuracy whereas the innermost region boundary C 3 is 10
found with poor accuracy. This can be expected since the amount of light propagating to the center of Ω is extraordinarily small due to the highly damping circular region A 1 . The proposed method relies on the assumption that we have some initial estimate for the boundaries. In the results shown in Figs. 4–6 we assumed that the number and approximative locations of the regions {A` } were known a priori when constructing the initial estimates. In these cases the boundaries {C` } were found with good accuracy and the convergence of the method was fast. In the fourth test case shown in Fig. 7 we assumed that the approximative location of the perturbation was known but we made error in the number of the perturbations. In Fig. 7 the true distribution has two perturbed regions A1 and A2 with the same contrast (b1,2 = 0.05mm−1 ,es1,2 = 40mm−1 ) with respect to the background values (b0 = 0.025mm−1 , es0 = 20mm−1 ). As an initial estimate we used a single perturbation surrounding both regions A1 and A2 . In the reconstruction shown in Fig. 7 the stabilisation parameter was λ = 5. Based on this result it seems that it could be possible in some cases to take the splitting and merging of the regions into account by utilising some supplementary decision algorithm which uses for example some minimum distance or crossing point criteria for the decision whether regions should be splitted or merged. We also conducted several test cases in which we tested the effects of poor initial estimate. In these tests we made large errors in the location of the target in the initial estimates. Based on these tests it seems that the proposed method does not perform well when the initial estimate is very far from the true boundaries. The poor performance of the method in these cases appeared as unpractically slow convergence and stability problems. However, in most situations we have reasonable good a priori information about the anatomy of the object, and thus we can construct reasonably good initial estimates. It is worth of noting that the implementation of the method is easier when one is considering equation of Rthe form −∇ · a∇Φ = f and using first order basis functions ϕj . In these cases the integration ∇ϕi · ∇ϕj dr over one element Ωm ∈ supp(ϕi ϕj ) in equation (9) does not have to be computed separately over the three splitted parts of the element, but is instead obtained as Z a` S` + a r Sr ∇ϕi · ∇ϕj dr, S` + S r Ωm where S` = A` ∩ Ωm and S` ∪ Sr = Ωm , see Fig. 2.
6
Conlusions
In this paper we proposed a new numerical method for the recovery of several internal region boundaries of piecewise constant coefficients of an elliptic PDE. The proposed method is based on the series expansion approximation of the smooth region boundaries and on the finite element method. The proposed method leads to an inverse problem which was solved iteratively using the Levenberg-Marquardt method. The proposed method was formulated in a general framework for elliptic PDEs, and then applied to optical tomography which is a field in which boundary recovery methods have not been utilised previously. Numerical results were given with synthetic data. Based on the test results it seems that the overall performance of the method is promising. A weak point of the method was that there are convergence and stability problems in cases in which the initial estimate was very far from the true boundaries. In these cases it would be possible to look for a good initial estimate by carrying out first a normal pixelwise reconstruction, or alternatively, by recovering the approximative locations and sizes of the regions by reconstructing only a few Fourier coefficients in the first stage. We made the following relatively weak assumptions on the topology in the proposed method: 1) Boundaries are closed and sufficiently smooth, 2) the regions enclosed by the boundaries are simply connected. In this study these boundaries were approximated with separate Fourier series for both coordinates of each of the boundary curves. A minor drawback of the chosen curve approximation is that it does not include the assumption 2) as a constraint. This caused stability problems in the test cases where the initial estimate was very far from the true situation. In these cases the Levenberg-Marquardt method exhibited tendency to produce self-intersecting boundaries. One possible solution to these problems with convergence and constraints on admissible solutions would 11
be to use some non-trivial object based prior for the boundary shapes. One possible method would be the subspace regularization method [2]. In addition regularization can be expected to improve the ill-posedness of the problem. The proposed method is flexible with respect to the curve approximation. The minimal conditions for the curve approximation are periodicity and differentiability, and for the regions the minimal requirement is that the regions have to be simply connected. Otherwise, the choice of the curve approximation depends on the topological conditions of the application and is on the user’s free choice. For example, if we can assume that the boundaries enclose starlike regions we can use a simpler curve approximation. Naturally, if we were using still the same amount of the basis functions θn , the number of parameters is reduced to half from the current version and this would lead to a less ill-posed problem. In this study we restricted ourselves to the case in which the values of the piecewise constant coefficients were known, and the only unknown information was on the geometry of the boundaries where the coefficients are discontinous. The method can be also expanded to the more complicated case in which the values of the coefficients are also considered as unknown quantities. However, this will lead to a more ill-posed inverse problem. Other aspects that could be investigated are the dependence of the performance on varying number of measurements and different data types such as frequency domain or time dependent measurements. Acknowledgements This work was supported by the Vilho, Yrj¨o and Kalle V¨ais¨al¨a Fund, the Saastamoinen Foundation, the Savo Foundation for Advanced Technology, the British Council and the Academy of Finland.
7
References
References [1] E. Somersalo, M. Cheney, and D. Isaacson. Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math., 52:1023–1040, 1992. [2] M. Vauhkonen, D. Vad´asz, P.A. Karjalainen, E. Somersalo, and J.P. Kaipio. Tikhonov regularization and prior information in electrical impedance tomography. IEEE Trans Med Imag, 17:285–293, 1998. [3] S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy. A finite element approach for modeling photon transport in tissue. Med. Phys., 20(2):299–309, 1993. [4] M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy. The finite element model for the propagation of light in scattering media: Boundary and source conditions. Med. Phys., 22(11):1779–1792, 1995. [5] S. R. Arridge. Optical tomography in medical imaging. Inverse Problems, 15(2):R41–R93, 1999. [6] F. Hettlich and W. Rundell. Iterative methods for the reconstruction of an inverse potential problem. Inverse Problems, 12(3):251–266, 1996. [7] F. Hettlich and W. Rundell. The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Problems, 14(1):67–82, 1998. [8] F. Hettlich and W. Rundell. Recovery of the support of a source term in an elliptic differential equation. Inverse Problems, 13(4):959–976, 1997. [9] G. Alessandrini and E. Rosset. The inverse conductivity problem with one measurement: Bounds on the size of the unknown object. SIAM J. Appl. Math., 58(4):1060–1071, 1998. [10] G. Alessandrini and L. Rondi. Stable determination of a crack in a planar inhomogeneous conductor. SIAM J. Math. Anal., 30(2):326–340, 1998. 12
[11] F. Santosa G. Alessandrini, E. Beretta and S. Vessella. Letter to the editor: Stability in crack determination from electrostatic measurements at the boundary – a numerical investigation. Inverse Problems, 11(5):L17–L24, 1995. [12] G. L. Chahine R. Duraiswami and K. Sarkar. Boundary element techniques for efficient 2-d and 3-d electrical impedance tomography. Chemical Engineering Science, 52(13):2185–2196, 1997. [13] M. Bonnet. Bie and material differentiation applied to the formulation of obstacle inverse problems. Engineering Analysis with Boundary Elements, 15:121–136, 1995. [14] D. Lesselier A. Litman and F. Santosa. Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set. Inverse Problems, 14(3):685–706, 1998. [15] F. Santosa P. G. Kaup and M. Vogelius. Method of imaging corrosion damage in thin plates from electrostatic data. Inverse Problems, 12(3):279–293, 1996. [16] S.C. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. Springer, 1994. [17] C. Cuvelier and A. Segal. Finite element methods and Navier-Stokes equations. D. Reidel Publishing Company, 1986. [18] S. R. Arridge and M. Schweiger. A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method. In C. Borgers and F. Natterer, editors, Computational Radiology and Imaging : therapy and Diagnosis, volume 110 of IMA Volumes in Mathematics and its Applications. IMA, Springer-Verlag, 1998. In Press. [19] H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson. Optical image reconstruction using frequency-domain data: Simulations and experiments. J. Opt. Soc. Am. A, 13:253–266, 1995. [20] D. T. Delpy, M. Cope, P. van der Zee, S. R. Arridge, S. Wray, and J. Wyatt. Estimation of optical pathlength through tissue from direct time of flight measurement. Phys. Med. Biol., 33:1433–1442, 1988. [21] R. Model, M. Orlt, M. Walzel, and R. H¨ unlich. Reconstruction algorithm for near-infrared imaging in turbid media by means of time-domain data. J. Opt. Soc. Am. A, 14(1):313–324, 1997. [22] S. S. Saquib, K. M. Hanson, and G. S. Cunningham. Model-based image reconstruction from time-resolved diffusion data. in Medical Imaging: Image Processing, K. M. Hanson, ed., Proc. SPIE, 3034:369–380, 1997. [23] S. B. Colak, D. G. Papaioannou, G. W. ’t Hooft, M. B. van der Mark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, and N. A. A. J. van Asten. Tomographic image reconstruction from optical projections in light-diffusing media. Appl. Opt., 36(1):180–213, 1997. [24] M. Schweiger and S. R. Arridge. Optimal data types in optical tomography. In Information Processing in Medical Imaging (IPMI’97 Proceedings), Lecture Notes in Computer Science, volume 1230, pages 71–84. Springer, 1997. [25] S. R. Arridge and M. Schweiger. Direct calculation of the moments of the distribution of photon time of flight in tissue with a finite-element method. Appl. Opt., 34(15):2683–2687, 1995. [26] S. Chandrasekhar. Radiative Transfer. Oxford University Press, London, 1950. [27] B. Davison. Neutron Transport Theory. Oxford University Press, 1957.
13
[28] A. M. Weinberg and E. P. Wigner. The physical theory of Neutron chain reactors. The university of Chicago press, 1958. [29] M. C. Case and P. F. Zweifel. Linear Transport Theory. Addison-Wesley, New York, 1967. [30] J. J. Duderstadt and W. R. Martin. Transport Theory. John Wiley & Sons, New York, 1979. [31] O. Dorn. A transport-backtransport method for optical tomography. Inverse Problems, 14(5):1107–1130, 1998. [32] R. T. Ackroyd. Finite element methods for particle transport: Applications to reactor and radiation physics. Research studies press Ltd., Taunton, 1997. [33] A. Ishimaru. Wave Propagation and Scattering in Random Media, volume 1. Academic, New York, 1978. [34] Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas. Imaging the diffusion coefficient in a parabolic inverse problem in optical tomography. Inverse Problems, 15(2):373–397, 1999. [35] S. R. Arridge, M. Hiraoka, and M. Schweiger. Statistical basis for the determination of optical pathlength in tissue. Phys. Med. Biol., 40:1539–1558, 1995.
14
A0
∂Ω
C1 A1
C2 A2
Figure 1: An example of a piecewise constant domain Ω = ∪k Ak with constant coefficients (ak , bk ), k = 0, 1, 2. The outer boundaries of regions A1 and A2 are denoted by C1 and C2 , respectively.
Nj
C` (s)
Nj
A` b` , a ` s2
s1
s1
s2
Ar br , a r a)
Ni
b)
Ni
Figure 2: a) A schematic representation of one FEM element Ωm intercepted by the region boundary C` (s). The pair (a` , b` ) correspond to the values of the coefficients of the equation (1) inside the region A` , and (ar , br ) are the values of the coefficients in the neighboring region Ar , respectively. C` (s1 ) and C` (s2 ) are the intersection points of the boundary C` (s) and the element edges. b) The division of the Ωm to three sub-triangles.
15
Nj
C` (s)
b` , a ` s1
C˜` (s)
s2 A` + 4A` Ar + 4Ar br , a r
Ni Figure 3: a) Perturbation in the boundary C` . Perturbed curve is of the form C˜` = C` + (θnx , 0)T when perturbation is in the x coordinate and C˜` = C` + (0, θny )T when it is in the − y-coordinate. In this simple case we have 4A` = δA+ ` − 0 and 4Ar = 0 − δAr .
8
6
4
2
0 2
4
6
8
10
12
Figure 4: a) Test case with a single perturbation A1 . Parameter values are (b0 , b1 , es0 , es1 ) = (0.025, 0.05, 20, 40)mm−1 . Bold line denotes the true boundary, thin line the reconstructed boundary and dotted line denotes the initial estimate. The dimension of the search space is 16. b) The relative data error percentage kzmeas k−1 kzmeas − P(γk , a, b)k · 100% versus iteration index k.
16
4
3
2
1
0 5
10
15
Figure 5: a) Test case with three perturbations. Target A1 is located up and left, A2 is located down from the center of Ω and A3 is located right from the center. Parameter values are (b0 , b1 , b2 , b3 , es0 , es1 , es2 , es3 ) = (0.025, 0.05, 0.05, 0.025, 20, 40, 20, 40)mm−1 . Bold lines denote the true boundaries, thin lines the reconstructed boundaries and dotted lines denote the initial estimate. The dimension of the search space is 48. b) The relative data error kzmeas k−1 kzmeas − P(γk , a, b)k · 100% versus k.
15
10
5
0 5
10
15
20
25
30
Figure 6: a) Test case with three nested boundaries. Subdomains A0 –A3 are in respective order from ∂Ω towards the center of Ω. Parameter values are (b0 , b1 , b2 , b3 , es0 , es1 , es2 , es3 ) = (0.025, 0.05, 0.025, 0.03, 20, 40, 20, 30)mm−1 . Bold lines denote the true boundaries, thin lines the reconstructed boundaries and dotted lines denote the initial estimate. The dimension of the search space is 48. b) The relative data error kzmeas k−1 kzmeas − P(γk , a, b)k · 100% versus k.
17
12 10 8 6 4 2 0 2
4
6
8
10
12
Figure 7: a) True phantom has two perturbation regions A1 and A2 . Parameter value are (b0 , b1 , b2 , es0 , es1 , es2 ) = (0.025, 0.05, 0.05, 20, 40, 40)mm−1 . Bold lines denote the true boundaries, thin line the reconstructed boundary and dotted line denotes the initial estimate. The dimension of the search space is 16. b) The relative data error kzmeas k−1 kzmeas − P(γk , a, b)k · 100% versus k.
18
