## 3D shape based reconstruction of experimental data in Diffuse Optical Tomography

Optics Express, Vol. 17, Issue 21, pp. 18940-18956 (2009)

http://dx.doi.org/10.1364/OE.17.018940

Acrobat PDF (829 KB)

### Abstract

Diffuse optical tomography (DOT) aims at recovering three-dimensional images of absorption and scattering parameters inside diffusive body based on small number of transmission measurements at the boundary of the body. This image reconstruction problem is known to be an ill-posed inverse problem, which requires use of prior information for successful reconstruction. We present a shape based method for DOT, where we assume *a priori* that the unknown body consist of disjoint subdomains with different optical properties. We utilize spherical harmonics expansion to parameterize the reconstruction problem with respect to the subdomain boundaries, and introduce a finite element (FEM) based algorithm that uses a novel 3D mesh subdivision technique to describe the mapping from spherical harmonics coefficients to the 3D absorption and scattering distributions inside a unstructured volumetric FEM mesh. We evaluate the shape based method by reconstructing experimental DOT data, from a cylindrical phantom with one inclusion with high absorption and one with high scattering. The reconstruction was monitored, and we found a 87% reduction in the Hausdorff measure between targets and reconstructed inclusions, 96% success in recovering the location of the centers of the inclusions and 87% success in average in the recovery for the volumes.

© 2009 Optical Society of America

## 1. Introduction

1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems **15**, 41—93 (1999).
[CrossRef]

3. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**, R1–R43(2005).
[CrossRef] [PubMed]

*a priori*) information to stabilize the reconstruction. Most commonly the prior information is included into the problem either in the form of regularisation techniques, which imply the specification of a prior probability distribution of voxel values based on geometric, local smoothness, or statistical asumptions[4

4. J. P. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inverse Problems **15**, 713–729 (1999).
[CrossRef]

5. A. Douiri, M. Schweiger, J. Riley, and S. R. Arridge, “Anisotropic diffusion regularisation methods for diffuse optical tomography using edge prior information,” Meas. Sci. Tech. **18**, 87—95 (2007).
[CrossRef]

7. C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. A **26**, no. 5, 1277–1290 (2009).
[CrossRef]

8. P. Hiltunen, D. Calvetti, and E. Somersalo, “An adaptive smoothness regularization algorithm for optical tomography,” Opt. Express **16**, 19957–19977 (2008).
[CrossRef] [PubMed]

15. B.A. Brooksby, H. Dehghani, B.W. Pogue, K.D. Paulsen, and KD. “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE Sel. Top. Quantum Electron. **9**, 199–209, (2003).
[CrossRef]

6. M. Schweiger and S.R. Arridge, “Optical Tomographic Reconstruction in a Complex Head Model Using a priori Region Boundary Information,” Phys. Med. Biol. **44**, 2703–2721 (1999).
[CrossRef] [PubMed]

11. B. W. Pogue and K. D. Paulsen, “High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information,” Opt. Lett. **23**, Is.21, 1716–1718, (1998).
[CrossRef]

13. M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. **50**, 2837–58, (2005).
[CrossRef] [PubMed]

14. X. Intes, C. Maloux, M. Guven, B. Yazici, and B. Chance, “Diffuse optical tomography with physiological and spatial a priori constraints,” Phys. Med. Biol. **49**, N155–63, (2004).
[CrossRef] [PubMed]

17. S. R. Arridge and J.C. Schotland, “Optical Tomography: Forward and Inverse Problems”, Inverse Problems **25**, 12, (2009).
[CrossRef]

18. O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Problems **22**, R67–R131, (2006).
[CrossRef]

19. O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Problems **14**, 1107–1130 (1998).
[CrossRef]

20. M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique,” Opt. Lett. **21**, no. 4, 471–473 (2006).
[CrossRef]

21. N. Naik, R. Beatson, J. Eriksson, and E. van Houten, “An implicit radial basis function based reconstruction approach to electromagnetic shape tomography,” Inverse Problems25, no. 2 (2009). [CrossRef]

22. D. Alvarez, P. Medina, and M. Moscoso, “Fluorescence lifetime imaging from time resolved measurements using a shape-based approach,” Opt. Express **17**, 8843–8855 (2009).
[CrossRef] [PubMed]

23. M. Soleimani, W. R. B. Lionheart, and O. Dorn, “Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data,” Inverse Problems in Sc. and Eng. **14**, 193–210 (2006).
[CrossRef]

24. M. Soleimani, O. Dorn, and W. R. B. Lionheart, “A narrow-band level set method applied to EIT in brain for cryosurgery monitoring,” IEEE Trans. Biomed. Eng. **53**, 2257–2264 (2006).
[CrossRef] [PubMed]

25. A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method,” Inverse Problems **22**, 1509–1532 (2006).
[CrossRef]

26. V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inverse Problems **15**, 1375–1391 (1999).
[CrossRef]

27. G. Boverman, E.L. Miller, D.H. Brooks, D. Isaacson, F. Qianqian, and D.A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging **27**, 752–765 (2008).
[CrossRef] [PubMed]

28. M. E. Kilmer, E. L. Miller, A. Barbaro, and D. Boas, Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt. **42**, 3129–3144 (2003).
[CrossRef] [PubMed]

*a priori*information about the topology, the approximate locations and shapes of the unknown subdomains and their optical properties can be incorporated in the inversion directly and the division of the body to different tissue types is obtained without post process segmentation.

26. V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inverse Problems **15**, 1375–1391 (1999).
[CrossRef]

25. A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method,” Inverse Problems **22**, 1509–1532 (2006).
[CrossRef]

## 2. Forward model

1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems **15**, 41—93 (1999).
[CrossRef]

*∂*Ω we consider the diffusion equation in the frequency case :

*D*and

*µ*

_{a}are the diffusion and absorption coefficients,

*ω*is the source modulation frequency, Φ is the photon density and c the speed of light in the medium. The boundary source distribution

*q*(

**r**;

*ω*) represents the number of photons at a boundary position

**r**; The diffusion coefficient is given by

*µ*′

_{s}denotes the reduced scattering coefficient. The appropriate boundary condition for our problem is of the Robin type

*ζ*models the refractive index mismatch at the boundary

*∂*Ω and

*ν*is the outward surface normal at

**r**.

**p**={

*µ*

_{a},

*D*} as

*𝓟*. The space of fields Φ will be denoted as

*𝓕*and the space of data y will be denoted as

*𝓩*. The data y will be calculated by the use of a measurement operator

*𝓕*:

*𝓕*→

*𝓩*in the form:

*∂*Ω where the detectors are located (one integral per detector). For our analysis it is sufficient to assume that this operator is linear and the adjoint measurement operator

*M** :

*𝓩*→

*𝓕*is also well defined. The adjoint operator will be used for defining artificial ‘adjoint sources’ for the calculation of the sensitivity functions in section 4.

*A*mapping from the parameter space to the measurement space

## 3. Discretization with FEM

26. V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inverse Problems **15**, 1375–1391 (1999).
[CrossRef]

25. A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method,” Inverse Problems **22**, 1509–1532 (2006).
[CrossRef]

^{N}

_{n=1}

*H*joined at

_{n}*i*=1…

*h*vertex nodes

*T*. In this paper we present the method in terms of tetrahedral elements, although it could easily be generalized for other types of elements. Given the set of

_{i}*T*nodes and a set of associated basis functions υ

_{i}_{i}(

**r**), the solution for the field Φ defined in Ω, can be approximated by a piecewise continuous function Φ(

**r**)=∑

^{h}

_{i=1}

*ϕ*υ

_{i}_{i}(

**r**), where Φ={

*ϕ*} is the h-dimensional vector of basis coefficients and υi are linear basis functions with limited support over the elements that have the node

_{i}*T*as one of their vertices. From previous work [29

_{i}29. S.R. Arridge, M. Schweiger, M. Hiraoka, and D.T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. **20**, 299–309 (1993).
[CrossRef] [PubMed]

34. M. Schweiger, S. Arridge, and I. Nissila, “GaussNewton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. **50**, 2365–2386 (2005).
[CrossRef] [PubMed]

*α*comes from the boundary conditions (3), and the system matrices K,C,E,B are given by:

*L*simply connected subdomains {Ω

_{ℓ}},

*ℓ*=1,…,

*L*, which are bounded by closed surfaces {Γ

_{ℓ}} and have constant optical parameters

*µ*

_{a}and

*D*(the remaining “background” domain is denoted by Ω

_{0}=Ω\∪

^{L}

_{ℓ=1}Ω

_{ℓ}). Since this is a three dimensional application we choose a real valued spherical harmonics parametric representation for the boundaries of the regions {Ω

_{ℓ}} as introduced in [25

**22**, 1509–1532 (2006).
[CrossRef]

_{ℓ}, of a surface Γ

_{ℓ}, are given by the representation

*C*}. Here, the basis functions

^{m}_{l}*Y*̃

*(ϑ,φ) are defined as*

^{m}_{l}*Y*are the (complex-valued) spherical harmonics functions, and

^{m}_{l}*W*is the maximum degree of spherical harmonics used for the particular representation. For simplicity we introduce the notation

*k*ranges over

*k*=1,⋯3

*W*

^{2}. {

*γ*} describes the finite set of spherical harmonics coefficients for the surface Γ

^{ℓ}_{k}_{ℓ}up to degree

*W*.

_{ℓ}with piecewise constant parameters,

*µ*

_{a}and

*D*can be expressed as

*χ*is the characteristic function for a region Ω

_{ℓ}_{ℓ}. Substituting in equation (7), and since only the matrices K and C are influenced by the parameters inside the domain, we get:

*υ*) represents the part of the domain where both the basis functions are non-zero, that is, the elements that contain both the

_{i}υ_{j}*T*and

_{i}*T*nodes.

_{j}*𝓟*: {

*γ,µ*

_{a},

*D*} to the space of data

*𝓩*. In the next subsections, we will present the methodology used to convert the FEM algorithm to implement this mapping in five stages.

### 3.1. Classification of FEM mesh nodes

*T*=(

_{i}*x*), of the finite element discretisation are inside a given closed surface Γ

_{i},y_{i},z_{i}_{ℓ}(

*θ,ϕ*). To reduce the computational cost we calculate first the bounding box for the surface Γ

_{ℓ}(

*θ,ϕ*) and work only with the nodes that are inside the bounding box. Using the Jordan surface theorem [33

33. R. Kopperman, P. Meyer, and R.G. Wilson, “A Jordan surface theorem for three-dimensional digital spaces,” Discrete Comput. Geom. **6**, 155–161 (1991).
[CrossRef]

*T*inside the bounding box with the surface. We discetise the surface Γ

_{i}_{ℓ}(

*θ,ϕ*) using the parametric mapping function, that was initially introduced in [25

**22**, 1509–1532 (2006).
[CrossRef]

_{ℓ}, with triangles

*G*and nodes

^{ℓ}_{k}*S*. We draw the segment

^{ℓ}_{m}*R*={(

_{i}*x*), (

_{i},y_{i},z_{i}*x*)}, where

_{i},y_{i}, z_{min}*z*is the lower z-coordinate of the volume and check for intersections with each of the triangles

_{min}*G*in the surface approximation Γ

^{ℓ}_{k}_{ℓ}. We calculate the bounding quadrilateral along the

*xy*-plane for each of the triangles in the surface, and check if the projection of the node

*T*along the xy-plane lies inside. For the triangles satisfying this check we further check for intersections with the line segment

_{i}*R*, using the [32] ray-triangle intersection algorithm.

_{i}### 3.2. Classification of mesh elements

*T*are classified as inside or outside the region bounded by a parametric surface Γ

_{i}_{ℓ}(

*θ,ϕ*), we proceed with the classification of the elements

*H*of the mesh using their respective nodes. We mark an element as

_{n}*outside*if all the nodes are outside the surface, and similarly we mark an element as

*inside*if all nodes are positioned internal to the region. In the case of an element with, one, two or three nodes inside a given region Γ

_{ℓ}we mark it as intercepted. Let

*I*(Γ

_{ℓ}) denote the intersecting elements to the surface Γ

_{ℓ}.

*outside*and the

*inside*elements are constant over the element, so that the elements contributions for the system matrix can be calculated using the equations (7), directly. In the case of an

*intercepted*element a more complicated procedure must be performed as we will see in the next subsections.

### 3.3. Determination of intersections of a surface with the elements of the FEM mesh

*H*∈

_{v}*I*(Γ

_{ℓ}(

*θ,ϕ*)), defined by the nodes {

*T*

^{v}_{1},

*T*

^{v}_{2},

*T*

^{v}_{3},

*T*

^{v}_{4}}, we need to find the intersection points

*k*along the edges of the tetrahedron. Using an element

^{v}*G*∈Γ

_{s}_{ℓ}, with its center the closest to the center of the element

*H*, and using the information from the mesh mapping function we get the spherical coordinates (

_{v}*ϑ*

_{0},

*φ*

_{0}) for the node

*S*

_{0}of the element

*G*. The intersection point

_{s}*k*of the parametrically defined surface with the edge of the tetrahedron between the nodes

_{v}*T*

^{v}_{1}and node

*T*

^{v}_{2}by definition is on that edge, so that:

*k*has to be on the parametrically defined surface. There exists a pair of unknown spherical coordinates

_{v}*k*given by (16), is done using a generic Gauss-Newton non-linear solver with initial values (

_{v}*ϑ*

_{0},

*φ*

_{0}). Since those initial values were constructed to be close to final solution the solution is rapid.

### 3.4. Subdivision of the tetrahedron in subtetrahedra

*intercepted*tetrahedron we can perform a subdivision of the element into subtetrahedra so that the boundary defined by the spherical harmonics surface Γ

_{ℓ}(

*θ,ϕ*) is accurately represented in the numerical scheme. For an intercepting element

*H*∈

_{v}*I*(Γ

_{ℓ}(

*θ,ϕ*)), defined by the nodes {

*T*

^{v}_{1},

*T*

^{v}_{2},

*T*

^{v}_{3},

*T*

^{v}_{4}} we classify two main cases for the intersection with a surface.

**• Case 1:**One of the nodes,

*T*

^{v}_{(+)}, is on a different side of the parametric surface from the other three {

*T*

^{v}_{(-, i)}}

*i*=1,2,3. In Fig. 1(a) we have denoted this partition of the nodes with the signed distance for each node from the surface : we denote the single node with (+) and the three nodes on the other side of the partition with (-). We should mention that the intersection is treated symmetrically and irrespective of which side of the surface is inside or outside. The node

*T*

^{v}_{(+)}together with the three intersection points {

*k*

^{v}_{1},

*k*

^{v}_{2},

*k*

^{v}_{3}}, calculated according to the previous subsection, form one subtetrahedron. The rest of the element that lies on the negative side of the surface has now the shape of a pentahedron and is split into seven subtetrahedra as shown in Fig. 1(c), where we have exploded the subelements for better comprehension.

**• Case 2**Two of the nodes,

*T*

^{v}_{(+, 1)}

*T*

^{v}_{(+, 2)}are on the positive side of the parametric surface and the other two {

*T*

^{v}_{(-,i)}}

*i*=1,…,2 on the negative. In Fig. 1(b) we denote two nodes with (+) and two with (-). Having found in the previous subsection the four intersection points {

*k*

^{v}_{1},

*k*

^{v}_{2},

*k*

^{v}_{3},

*k*

^{v}_{4}}, of the parametric surface with the edges of the element connecting each of the (+) nodes to the(-) nodes, we divide the tetrahedron into two prismatic solids of five faces each. In sequence the divided substructures are subdivided into three tetrahedra each producing the exploded subtetrahedra of Fig. 1(d).

### 3.5. Computation of the system matrix

*I*(Γ

_{ℓ}(

*θ,ϕ*)), that are intersected by the surface of a region we have to calculate the separate integrals ∫∇

*υ*·∇

_{i}*υ*

_{j}*d*

**r**and ∫υi·υj dr for each of their subelements. The integrals are mapped from the subtetrahedron to the local element and then evaluated with a Gaussian quadrature in the local element, according to a 3D version of the procedure described in [26

**15**, 1375–1391 (1999).
[CrossRef]

## 4. Inverse Problem

*γ*) by iterations of local linearization and Taylor expansion around the current estimate as

### 4.1. Jacobian calculation

*∂*Ω is considered known and the shape parameters do not influence the external boundary. The inclusions are also assumed mutually disjoint and disjoint of the exterior boundary. Therefore the derivative of S in respect to the shape parameters

*γ*are not dependent on the boundary term E or the frequency term B. The elements of the derivatives of S with respect to the shape parameters

_{k}*γ*are derived in equations (22)–(32)of [26

_{k}**15**, 1375–1391 (1999).
[CrossRef]

*δµ*

_{a}) and (

*δD*) denote the difference of

*µ*

_{a}and

*D*between inside and outside the boundary Γ. Using (25) on (24) we get the element

*J*, corresponding to the source

_{sd,k}*s*and detector

*d*for the shape parameter

*k*, of the Jacobian:

*γ*that describe that surface.

_{k}_{ℓ}of the regions Ω

*are necessary. For the calculation of the fields Φ*

_{ℓ}_{ℓ}and the adjoint fields Φ*

_{ℓ}on the nodes

*S*of the surface Γ

_{ℓ}_{ℓ}we find first the element

*E*of the volume discretisation where the node

_{p}*S*of the surface belongs to, and recall the shape functions of this element.

_{s}*l*ranges through the number of nodes in one element in the FEM discretisation (4 for a tetrahedra).

### 4.2. Scalings

34. M. Schweiger, S. Arridge, and I. Nissila, “GaussNewton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. **50**, 2365–2386 (2005).
[CrossRef] [PubMed]

## 5. Results

31. I. Nissilä, K. Kotilahti, K. Fallström, and T. Katila., Instrumentation for the accurate measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. **73**, 3306–3312 (2002).
[CrossRef]

_{2}particles and an infrared dye to provide scattering and absorption properties similar to that of biological tissue. The homogeneous optical coefficients of the background material were approximately

*µ*′

_{s}=1 ± 0.1 mm

^{-1}and

*µ*=0.01±0.001 mm

_{a}^{-1}at a wavelength of 800 nm. The speed of light for the phantom material was

*c*=0.19 mm ps

^{-1}. Two cylindrical inhomogeneities of diameter 9.5 mm and height 9.5 mm were located in the central plane

*z*=0 of the cylinder. The optical properties of the two targets relative to the background were set to (

*µ*,2

_{a}*µ*′

_{s}) and (2

*µ*,

_{a}*µ*′

_{s}), respectively. The geometry of the phantom is shown in Fig. 2. Optical fibres were used to transmit light from the source and to the detectors. In this experiment, 16 source and 16 detector sites arranged in two rings at a spacing of 12 mm were used. Filtering out the measurements from the detectors that are close to a source when the source is used to avoid light saturation, results in a final set of 192 amplitude and phase measurements in our data sets. To suppress amplitude scalings between the experimental and simulated datasets, data was collected in two sets, one with the blobs inside the homogeneous domain g

^{meas}and one where the blobs were absent g

^{ref}. The augmented residual of (17) becomes

*A*(

**p**

^{ref}) represents the solution of the numerical model without any blobs inside and homogeneous optical properties of

*µ*

_{a}=0.0078

*mm*

^{-1}and

*µ*′

_{s}=1.065

*mm*

^{-1}

34. M. Schweiger, S. Arridge, and I. Nissila, “GaussNewton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. **50**, 2365–2386 (2005).
[CrossRef] [PubMed]

^{target}

_{1}and Γ

^{target}

_{2}for the

*µ*

_{a}and the

*µ*′

_{s}targets, based on the assumed sizes and locations. To calculate the volume each object was divided into tetrahedra based on the mapped mesh and the volumes of all the tetrahedra were later added to calculate the total volume of the object. The desired value for this ratio is one and we notice that both the absorption and the scattering objects are performing quite well with the absorption reconstructed object being smaller than the expected target. The main reason for the smaller absorption target volume could be an inaccuracy in the experimental calculation of the “correct” values for both the absorption and scattering for the background and the targets. A study of the relation between the optical parameters and the volume was presented in [25

**22**, 1509–1532 (2006).
[CrossRef]

36. D.J. Cedio-Fengya, S. Moskow, and M.S. Vogelius, “Identification of conductivity imperfections of small diameter by bounday measurements. Continuous dependence and computational reconstruction,” Inverse Problems **14**, 553–595 (1998).
[CrossRef]

37. G. Bal, “Optical tomography for small volume absorbing inclusions,” Inverse Problems , **19**, 371–386 (2003).
[CrossRef]

*v*of Γ

*that is the farthest away from any node of the evolution surfaces Γ*

^{target}_{p}*at iteration n and is defined as*

^{n}_{p}*p*∈[1

1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems **15**, 41—93 (1999).
[CrossRef]

*z*=0 plane is accurate the sharp edges on the top and bottom of the cylindrical targets were not reconstructed equally well, as can be seen in Fig. 3(e) and Fig. 3(f). This is an expected result due to the diffusive nature of DOT, together with the positioning of the sources and detectors along the

*z*=0 plane that reduces the sensitivity of the the data in the z-direction.

## 6. Conclusion

**22**, 1509–1532 (2006).
[CrossRef]

38. V. Kolehmainen, S.R. Arridge, M. Vauhkonen, and J.P. Kaipio, “Recovery of constant coefficients in optical diffusion tomography,” Opt. Express **7**, 468–480 (2000).
[CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems |

2. | A. G. Yodh and D. A. Boas, “Functional imaging with diffusing light,” Biomedical photonics handbook(CRC Press, 2003). |

3. | A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. |

4. | J. P. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inverse Problems |

5. | A. Douiri, M. Schweiger, J. Riley, and S. R. Arridge, “Anisotropic diffusion regularisation methods for diffuse optical tomography using edge prior information,” Meas. Sci. Tech. |

6. | M. Schweiger and S.R. Arridge, “Optical Tomographic Reconstruction in a Complex Head Model Using a priori Region Boundary Information,” Phys. Med. Biol. |

7. | C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. A |

8. | P. Hiltunen, D. Calvetti, and E. Somersalo, “An adaptive smoothness regularization algorithm for optical tomography,” Opt. Express |

9. | S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” Journal of Physics: Conference Series 125,012001 (2008). |

10. | J. Chang, H.L. Graber, P. C. Koo, R. Aronson, S.-L. S. Barbour, and R. L. Barbour, “Optical Imaging of Anatomical Maps Derived from Magnetic Resonance Images Using Time-Independent Optical Sources,” IEEE Trans. |

11. | B. W. Pogue and K. D. Paulsen, “High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information,” Opt. Lett. |

12. | V. Ntziachristos, A. G. Yodh, M. D. Schnall, and B. Chance, “MRI-guided diffuse optical spectroscopy of malignant and benign breast lesions,” |

13. | M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. |

14. | X. Intes, C. Maloux, M. Guven, B. Yazici, and B. Chance, “Diffuse optical tomography with physiological and spatial a priori constraints,” Phys. Med. Biol. |

15. | B.A. Brooksby, H. Dehghani, B.W. Pogue, K.D. Paulsen, and KD. “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE Sel. Top. Quantum Electron. |

16. | S. R. Arridge, C. Panagiotou, M. Schweiger, and V. Kolehmainen, “Translational multimodality optical imaging, Chapter 5: Multimodal Diffuse Optical Tomography: Theory,” Artech Press, 101–124 (2008). |

17. | S. R. Arridge and J.C. Schotland, “Optical Tomography: Forward and Inverse Problems”, Inverse Problems |

18. | O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Problems |

19. | O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Problems |

20. | M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique,” Opt. Lett. |

21. | N. Naik, R. Beatson, J. Eriksson, and E. van Houten, “An implicit radial basis function based reconstruction approach to electromagnetic shape tomography,” Inverse Problems25, no. 2 (2009). [CrossRef] |

22. | D. Alvarez, P. Medina, and M. Moscoso, “Fluorescence lifetime imaging from time resolved measurements using a shape-based approach,” Opt. Express |

23. | M. Soleimani, W. R. B. Lionheart, and O. Dorn, “Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data,” Inverse Problems in Sc. and Eng. |

24. | M. Soleimani, O. Dorn, and W. R. B. Lionheart, “A narrow-band level set method applied to EIT in brain for cryosurgery monitoring,” IEEE Trans. Biomed. Eng. |

25. | A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method,” Inverse Problems |

26. | V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inverse Problems |

27. | G. Boverman, E.L. Miller, D.H. Brooks, D. Isaacson, F. Qianqian, and D.A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging |

28. | M. E. Kilmer, E. L. Miller, A. Barbaro, and D. Boas, Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt. |

29. | S.R. Arridge, M. Schweiger, M. Hiraoka, and D.T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. |

30. | A. Björck, “Numerical methods for least square problems,” SIAM, (1996). |

31. | I. Nissilä, K. Kotilahti, K. Fallström, and T. Katila., Instrumentation for the accurate measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. |

32. | T. Moller and B. Trumbore, |

33. | R. Kopperman, P. Meyer, and R.G. Wilson, “A Jordan surface theorem for three-dimensional digital spaces,” Discrete Comput. Geom. |

34. | M. Schweiger, S. Arridge, and I. Nissila, “GaussNewton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. |

35. | C.R. Vogel, “Computational methods for inverse problems,” |

36. | D.J. Cedio-Fengya, S. Moskow, and M.S. Vogelius, “Identification of conductivity imperfections of small diameter by bounday measurements. Continuous dependence and computational reconstruction,” Inverse Problems |

37. | G. Bal, “Optical tomography for small volume absorbing inclusions,” Inverse Problems , |

38. | V. Kolehmainen, S.R. Arridge, M. Vauhkonen, and J.P. Kaipio, “Recovery of constant coefficients in optical diffusion tomography,” Opt. Express |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(170.6960) Medical optics and biotechnology : Tomography

**ToC Category:**

Image Processing

**History**

Original Manuscript: July 21, 2009

Revised Manuscript: September 16, 2009

Manuscript Accepted: September 28, 2009

Published: October 8, 2009

**Virtual Issues**

Vol. 4, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Athanasios D. Zacharopoulos, Martin Schweiger, Ville Kolehmainen, and Simon Arridge, "3D shape based reconstruction of
experimental data in Diffuse Optical
Tomography," Opt. Express **17**, 18940-18956 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-21-18940

Sort: Year | Journal | Reset

### References

- S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, 41-93 (1999). [CrossRef]
- A. G. Yodh and D. A. Boas, "Functional imaging with diffusing light," Biomedical photonics handbook (CRC Press, 2003).
- A. P. Gibson, J. C. Hebden and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005). [CrossRef] [PubMed]
- J. P. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, "Inverse problems with structural prior information," Inverse Problems 15, 713-729 (1999). [CrossRef]
- A. Douiri, M. Schweiger, J. Riley and S. R. Arridge, "Anisotropic diffusion regularisation methods for diffuse optical tomography using edge prior information," Meas. Sci. Tech. 18, 87-95 (2007). [CrossRef]
- M. Schweiger and S.R. Arridge, "Optical Tomographic Reconstruction in a Complex Head Model Using a priori Region Boundary Information," Phys. Med. Biol. 44, 2703-2721 (1999). [CrossRef] [PubMed]
- C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy and S. R. Arridge, "Information theoretic regularization in diffuse optical tomography," J. Opt. Soc. Am. A 26, no. 5, 1277-1290 (2009). [CrossRef]
- P. Hiltunen, D. Calvetti and E. Somersalo, "An adaptive smoothness regularization algorithm for optical tomography," Opt. Express 16, 19957-19977 (2008). [CrossRef] [PubMed]
- S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger and A. Zacharopoulos, "Parameter and structure reconstruction in optical tomography," Journal of Physics: Conference Series 125,012001 (2008).
- J. Chang, H.L. Graber, P. C. Koo,R. Aronson,S.-L. S. Barbour and R. L. Barbour, "Optical Imaging of Anatomical Maps Derived from Magnetic Resonance Images Using Time-Independent Optical Sources," IEEE Trans. 16, 68-77,(1997).
- B. W. Pogue and K. D. Paulsen, "High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information," Opt. Lett. 23, 1716-1718, (1998). [CrossRef]
- V. Ntziachristos, A. G. Yodh, M. D. Schnall, and B. Chance, "MRI-guided diffuse optical spectroscopy of malignant and benign breast lesions," 4, 347-354, (2002).
- M. Guven, B. Yazici, X. Intes and B. Chance, "Diffuse optical tomography with a priori anatomical information, " Phys. Med. Biol. 50, 2837-58, (2005). [CrossRef] [PubMed]
- X. Intes, C. Maloux, M. Guven, B. Yazici and B. Chance, "Diffuse optical tomography with physiological and spatial a priori constraints," Phys. Med. Biol. 49, N155-63, (2004). [CrossRef] [PubMed]
- B.A. Brooksby, H. Dehghani, B.W. Pogue, K.D. Paulsen KD. "Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities," IEEE Sel. Top. Quantum Electron. 9, 199-209, (2003). [CrossRef]
- S. R. Arridge, C. Panagiotou, M. Schweiger and V. Kolehmainen, "Translational multimodality optical imaging, Chapter 5: Multimodal Diffuse Optical Tomography: Theory," Artech Press, 101-124 (2008).
- S. R. Arridge and J.C. Schotland, "Optical Tomography: Forward and Inverse Problems", Inverse Problems 25, 12, (2009). [CrossRef]
- O. Dorn and D. Lesselier, "Level set methods for inverse scattering," Inverse Problems 22, R67-R131, (2006). [CrossRef]
- O. Dorn, "A transport-backtransport method for optical tomography," Inverse Problems 14, 1107-1130 (1998). [CrossRef]
- M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos and V. Kolehmainen, "Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique," Opt. Lett. 21, no. 4, 471-473 (2006). [CrossRef]
- N. Naik, R. Beatson, J. Eriksson and E. van Houten, "An implicit radial basis function based reconstruction approach to electromagnetic shape tomography," Inverse Problems 25, no. 2 (2009). [CrossRef]
- D. Alvarez, P. Medina, and M. Moscoso, "Fluorescence lifetime imaging from time resolved measurements using a shape-based approach," Opt. Express 17, 8843-8855 (2009). [CrossRef] [PubMed]
- M. Soleimani, W. R. B. Lionheart and O. Dorn, "Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data," Inverse Problems in Sc. and Eng. 14, 193-210 (2006). [CrossRef]
- M. Soleimani, O. Dorn and W. R. B. Lionheart, "A narrow-band level set method applied to EIT in brain for cryosurgery monitoring," IEEE Trans. Biomed. Eng. 53, 2257-2264 (2006). [CrossRef] [PubMed]
- A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, "Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method," Inverse Problems 22, 1509-1532 (2006). [CrossRef]
- V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, "Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data," Inverse Problems 15, 1375-1391 (1999). [CrossRef]
- G. Boverman, E.L. Miller, D.H. Brooks, D. Isaacson, F. Qianqian and D.A. Boas, "Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography," IEEE Trans. Med. Imaging 27, 752-765 (2008). [CrossRef] [PubMed]
- M. E. Kilmer, E. L. Miller, A. Barbaro and D. Boas, Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography," Appl. Opt. 42, 3129-3144 (2003). [CrossRef] [PubMed]
- S.R. Arridge, M. Schweiger, M. Hiraoka, and D.T. Delpy, "A finite element approach for modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993). [CrossRef] [PubMed]
- A. Bj¨orck, "Numerical methods for least square problems," SIAM, (1996).
- I. Nissil¨a, K. Kotilahti, K. Fallstr¨om, and T. Katila., Instrumentation for the accurate measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 73, 3306-3312 (2002). [CrossRef]
- T. Moller and B. Trumbore, Fast, minimum storage ray-triangle intersection," J. Graphics Tools 2, 21-28 (1997).
- R. Kopperman, P. Meyer, and R.G. Wilson, "A Jordan surface theorem for three-dimensional digital spaces," Discrete Comput. Geom. 6, 155-161 (1991). [CrossRef]
- M. Schweiger, S. Arridge, and I. Nissila, "GaussNewton method for image reconstruction in diffuse optical tomography," Phys. Med. Biol. 50, 2365-2386 (2005). [CrossRef] [PubMed]
- C.R. Vogel, "Computational methods for inverse problems," Frontiers in Applied Mathematics Series, SIAM, 23 (2002).
- D.J. Cedio-Fengya, S. Moskow and M.S. Vogelius, "Identification of conductivity imperfections of small diameter by bounday measurements. Continuous dependence and computational reconstruction,"Inverse Problems 14, 553-595 (1998). [CrossRef]
- G. Bal, "Optical tomography for small volume absorbing inclusions," Inverse Problems, 19, 371-386 (2003). [CrossRef]
- V. Kolehmainen, S.R. Arridge, M. Vauhkonen, and J.P. Kaipio, "Recovery of constant coefficients in optical diffusion tomography," Opt. Express 7, 468-480 (2000). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.