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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 2 — Jan. 27, 2003
  • pp: 141–150
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Spatial location weighted optimization scheme for DC optical tomography

Jun Zhou, Jing Bai, and Ping He  »View Author Affiliations


Optics Express, Vol. 11, Issue 2, pp. 141-150 (2003)
http://dx.doi.org/10.1364/OE.11.000141


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Abstract

In this paper, a spatial location weighted gradient-based optimization scheme for reducing the computation burden and increasing the reconstruction precision is stated. The method applies to DC diffusion-based optical tomography, where otherwise the reconstruction suffers slow convergence. The inverse approach employs a weighted steepest descent method combined with a conjugate gradient method. A reverse differentiation method is used to efficiently derive the gradient. The reconstruction results confirm that the spatial location weighted optimization method offers a more efficient approach to the DC optical imaging problem than unweighted method does.

© 2002 Optical Society of America

1. Introduction

In recent years there has been an increased interest in using diffuse optical tomography (OT) for imaging the human breast and brain [1

1. A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-Based Iterative Image-Reconstruction Scheme for Time-Resolved Optical Tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999) [CrossRef]

5

5. S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213 [CrossRef] [PubMed]

]. This novel medical imaging technique attempts to reconstruct the spatial distribution of optical properties (absorption and transport scattering coefficients, µa and µ’s) within the body from the measurements of transmitted and reflected near-infrared light. Three kinds of techniques are employed to obtain the information used to reconstruct the image: time resolved method [1

1. A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-Based Iterative Image-Reconstruction Scheme for Time-Resolved Optical Tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999) [CrossRef]

,3

3. F. E. W. Schmidt, Development of a Time-Resolved Optical Tomography System for Neonatal Brain Imaging, Ph. D thesis (1999), University College London

,5

5. S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213 [CrossRef] [PubMed]

], frequency domain method [2

2. R. Roy and E. M. Sevick-MUraca, “Active constrained truncated Newton method for simple-bound optical tomography,” J. Opt. Soc. Am. A 17, 1627–1641 (2000) [CrossRef]

,4

4. R. Roy and E. M. Sevick-MUraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express 4, 353–371 (1999), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-10-353 [CrossRef] [PubMed]

], and continuous intensity method (DC case) [6

6. A. Y. Bluestone, G. Abdoulaev, C. H. Schmitz, R. L. Barbour, and A. H. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express 9, 272–286 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-6-272 [CrossRef] [PubMed]

,7

7. Y. Pei, Optical Tomographic Imaging Using the Finite Element Method, Ph. D. Thesis (1999), Polytechnic University.

]. In time resolved case, an ultra-short pulsed laser is used as the light source and the temporal distribution of the light emerging from the tissue surface is detected using either a synchronous scan streak camera with picosecond time resolution or a time-correlated single photon counting (TCSPC) system. In frequency domain method, a radio frequency modulated light source is used to illuminate an object and the phase and modulation depth of the transmitted light are measured, usually using a heterodyne detection method. Both the time resolved and frequency domain schemes exhibit high temporal resolutions (in the picosecond range for the time resolved and nanosecond for the frequency domain) [3

3. F. E. W. Schmidt, Development of a Time-Resolved Optical Tomography System for Neonatal Brain Imaging, Ph. D thesis (1999), University College London

]. However, tomographic measurements made with the time resolved technology require data-acquisition time on a time frame of minutes to derive data having acceptable low noise levels [8

8. C. H. Schmitz, H. L. Graber, H. Luo, I. Arif, J. Hira, Y. Pei, A. Bluestone, S. Zhong, R. Andronica, I. Soller, N. Ramirez, S. S. Barbour, and R. L. Barbour, “Instrumentation and calibration protocol for imaging dynamic features in dense-scattering media by optical tomography,” Appl. Opt. 9, 6466–6486 (2000) [CrossRef]

]. In addition, they have the disadvantages of high cost and complexity in the equipment. In DC case, a point laser source with constant intensity is used to continuously illuminate the object and the amplitude of the outgoing light on the boundary is measured. The continuous intensity system has the advantage of low cost and high dynamic range, as well as a relative high signal to noise ratio (SNR) [8

8. C. H. Schmitz, H. L. Graber, H. Luo, I. Arif, J. Hira, Y. Pei, A. Bluestone, S. Zhong, R. Andronica, I. Soller, N. Ramirez, S. S. Barbour, and R. L. Barbour, “Instrumentation and calibration protocol for imaging dynamic features in dense-scattering media by optical tomography,” Appl. Opt. 9, 6466–6486 (2000) [CrossRef]

]. The drawbacks include the difficulty in simultaneous unique recovery of diffusion and absorption coefficients [9

9. S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23, 882–884 (1998) [CrossRef]

] (though a normalized-constraint algorithm for minimizing inter-parameter crosstalk in DC optical tomography has been developed by Y. Pei. [10

10. Y. Pei, H. L. Graber, and R. L. Barbour, “Normalized-constraint algorithm for minimizing inter-parameter crosstalk in DC optical tomography,” Opt. Express 9, 97- (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-2-97 [CrossRef] [PubMed]

]) and inability to probe objects embedded deep inside the medium [3

3. F. E. W. Schmidt, Development of a Time-Resolved Optical Tomography System for Neonatal Brain Imaging, Ph. D thesis (1999), University College London

,11

11. S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part2: Zinite-element-method calculations,” Appl. Opt. 34, 8026–8037 (1995) [CrossRef] [PubMed]

].

Various reconstruction methods for OT have been proposed, which include the perturbation based [7

7. Y. Pei, Optical Tomographic Imaging Using the Finite Element Method, Ph. D. Thesis (1999), Polytechnic University.

,12

12. Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour. “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Amer. A 14, 325–342 (1997) [CrossRef]

,13

13. K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995) [CrossRef] [PubMed]

] and the gradient based methods [1

1. A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-Based Iterative Image-Reconstruction Scheme for Time-Resolved Optical Tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999) [CrossRef]

,2

2. R. Roy and E. M. Sevick-MUraca, “Active constrained truncated Newton method for simple-bound optical tomography,” J. Opt. Soc. Am. A 17, 1627–1641 (2000) [CrossRef]

,4

4. R. Roy and E. M. Sevick-MUraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express 4, 353–371 (1999), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-10-353 [CrossRef] [PubMed]

6

6. A. Y. Bluestone, G. Abdoulaev, C. H. Schmitz, R. L. Barbour, and A. H. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express 9, 272–286 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-6-272 [CrossRef] [PubMed]

,14

14. A. J. Davies, D. B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Advances in Engineering Software 28, 217–221 (1997) [CrossRef]

]. The advantage of the gradient-based iterative image reconstruction scheme as compared with the perturbation methods has been discussed in detail by A. H. Hielscher [1

1. A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-Based Iterative Image-Reconstruction Scheme for Time-Resolved Optical Tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999) [CrossRef]

]. The gradient based method contains two important components: the efficient calculation of the gradient of the objective function and the optimization algorithm used for minimizing the objective function. The adjoint source [5

5. S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213 [CrossRef] [PubMed]

] and adjoint differentiation [14

14. A. J. Davies, D. B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Advances in Engineering Software 28, 217–221 (1997) [CrossRef]

] methods, both called adjoint models, are widely employed in the gradient calculation. The efficiency of different adjoint models have been analyzed by A. D. Klose [15

15. A. D. Klose and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer — Part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transfer 72, 715–732 (2002) [CrossRef]

] extensively. The Conjugate Gradient (CG) method [1

1. A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-Based Iterative Image-Reconstruction Scheme for Time-Resolved Optical Tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999) [CrossRef]

,5

5. S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213 [CrossRef] [PubMed]

] can be adopted for the optimization of the objective function. Arridge [5

5. S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213 [CrossRef] [PubMed]

] have made a comparison between the CG method, Steepest Descent (SD) method, and the Algebraic Reconstruction Technique (ART) for a time resolved case, which shows that the CG method is preferable. The gradient-based method has been successfully applied to both the time resolved scheme and frequency domain scheme [1

1. A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-Based Iterative Image-Reconstruction Scheme for Time-Resolved Optical Tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999) [CrossRef]

,5

5. S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213 [CrossRef] [PubMed]

]. For the DC case, A. J. Davies et [14

14. A. J. Davies, D. B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Advances in Engineering Software 28, 217–221 (1997) [CrossRef]

] demonstrated, using simple simulated data, that the CG method was time consuming and sometimes did not converge.

2. Theory review

2.1 Forward and inverse problems in OT

a) Forward model

Light propagation in a tissue medium can be described by the well-known diffusion equation [16

16. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 9

]. For a domain Ω having a boundary ∂Ω, this is represented by the following expression:

·[D(r)∇Φ(r)]μa(r)Φ(r)=δ(rrs),rΩ
(1)

Γ(ξ)=D(ξ)n̂·Φ(ξ)
(2)

where n^⃑ is the outer normal vector of ∂Ω at ξ. The collimated source incident at ζ∈∂Ω is commonly represented by a diffuse point source located at a transport mean free path ltr beneath the surfaces [17

17. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994) [CrossRef]

].

b) The inverse formulation

Let us consider an experimental setting that include S point light sources located at ξj∈∂Ω (j = 1, …, S), and Mj measurement positions ςj,i∈∂Ω (i = 1, …, Mj) for each source j, and define the following objective function:

E=12j=1Si=1Mj((Γj,i)me(Γj,i)c)2
(3)

where Γj,i represents the photon intensity measured at position ς j,i with the incident light source located at ξj. The subscript c denotes the values calculated by the forward simulation and the subscript me denotes the experimental (or in this case synthetically generated) values. The optical inverse problem can be solved by minimizing the objective function.

2.2 The gradient calculation

To solve the optimization problem using a conjugate gradient method, the gradient of the objective function needs to be calculated. Arridge et al [5

5. S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213 [CrossRef] [PubMed]

] have developed an adjoint source scheme to calculate the gradient. They represent the gradient vector z⃑ as the following form:

z=E=JTb
(4)

where the Jacobian J of the forward problem is a c×NTOT matrix, where MTOT=j=1sMj is the total measurement number and NTOT is the number of the coefficients to be reconstructed. b⃑ is the residual data error for the measurement. Since the use of matrix J explicitly to derive the gradient ∇E is prone to be too computationally demanding (requires 2MTOT forward computations), they developed a much more efficient way to compute ∇E directly using two forward computations for source vector and weighted measurement vector (requires only 2S forward computation) [18

18. I. W. Kwee, Towards a Bayesian Framework for Optical Tomography, Ph. D. Thesis (1999), University College London.

]. However, the computation of J is still of interest for sensitivity analysis of the imaging problem. The essential calculation of the J is based on the establishment of the PMDF (Photon Measurement Density Function, as defined in [11

11. S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part2: Zinite-element-method calculations,” Appl. Opt. 34, 8026–8037 (1995) [CrossRef] [PubMed]

]). In [11

11. S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part2: Zinite-element-method calculations,” Appl. Opt. 34, 8026–8037 (1995) [CrossRef] [PubMed]

], the Jacobian calculation utilizing the reciprocity principle in the FEM (Finite Element Method) framework has been described in detail. It has also been shown that the PMDF has maxima close to the source and detector positions and falls off toward the interior of the object for a homogeneous background, especially for DC case. Since the residual data error b⃑ in Eq. (4) is independent of the points being reconstructed, the gradient ∇E, which is used for parameters updating, will favor the points near to the surface. When a conjugate gradient method is applied, the convergence would be very slow for objects embedded in deeper layers.

A more clear interpretation to the problem just mentioned can be found when we take the reverse differentiation scheme (also called adjoint differentiation), which is used by A. H. Hielscher [1

1. A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-Based Iterative Image-Reconstruction Scheme for Time-Resolved Optical Tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999) [CrossRef]

] and A. J. Davies et al [14

14. A. J. Davies, D. B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Advances in Engineering Software 28, 217–221 (1997) [CrossRef]

], to derive the gradient. In [14

14. A. J. Davies, D. B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Advances in Engineering Software 28, 217–221 (1997) [CrossRef]

], the derivative of the objective function Eq. (3) with respect to the optical properties D (can either be absorption or diffusion coefficient) is given by:

Dm*=eli,j(Kel*)i,j(Kel)i,jDm
(5)

3. The spatial location weighted reconstruction scheme

3.1 Steepest descent directions: A-norm of z⃑

We apply a steepest descent method in our new algorithm for the first iteration. Instead of adopting a descent direction d⃑ in Euclidean measurement, as indicated by the following equation:

d=ZZ2
(6)

we seek a descent direction in A-measurement. This is accomplished by constructing a diagonal positive definite matrix A of NTOT×NTOT whose elements are in direct proportion to the distance from the points to the surface.

A=[a11a220aii0aNN]
aii=Liβ,{i=1,2,,NTOTβ0}
(7)

Whereas a variety of weighting methods could be adopted, we have chosen one that has the form shown in Eq. (7). For a circular object, Li can be the radial distance between the point i and the surface. β is a weighting factor.

Using this new method, the minimizing problem is defined as:

minimizeZT·d
subject todT·A·d1
(8)

The solution can be expressed as:

d=A1·z(ZT·A1·Z)0.5
(9)

3.2 The choice of the weighting factor β

Algorithm 1. Spatial location weighted optimization method
Set initial guess x⃑(0)
Calculate the gradient of the initial guess Z⃑(x (0))
Set initial β = 0
Repeat
Construct the matrix A
Calculate the steepest descent direction d⃑(β) in A-measurement.
Find γ(β) that minimizes E (β) = E(x (0) + γ(β)d⃑(β))
β = β + 1
Until E (β) < E (β - 1)
x (1) = x (0) + r (β-1)d⃑(β-1)
Use x (1) as the initial guess for the conjugate-gradient method

4. Simulation results

To investigate the efficiency and the accuracy of Algorithm 1, we used finite element method (FEM) to simulate a circular object having a radius of 25 mm and one or three absorbing objects embedded in a homogeneous background, each with a radius of 2.5 mm, as shown in Table 1. The finite-element grid used comprised 958 elements for one absorbing object (see Fig. 1), and 4880 for three absorbing objects. The optical properties of the background medium used for the initial guess were µa = 0.025 mm-1 and µ’ s = 2.0 mm-1 (D = 0.16461 mm). All reconstructions started from a homogeneous background.

Table 1. Optical parameters of the circular test object

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The forward data used in the reconstruction of these simulations were calculated for 36 source and 36 detector positions, equally spaced along the circumference every 10 degree, using a FEM light transport model in continuous intensity mode [20

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

]. The FEM forward engine was based on the FEMLAB2.1 (COMSOL Inc., Burlington, MA 01803, USA, a FEM solution in MATLAB (The MathWorks, Inc.)), which provided a powerful interactive environment for modeling and solving scientific and engineering problems based on partial differential equations (PDEs). A model of Helmholtz’s equation, which has the form:

·(cu)+au=f
(10)

was adopted to solve Eq. (1). The coefficient c and a in Eq. (10) can be determined according to Eq. (1). ƒ is a point source with a unit amplitude. All the codes were programmed in MATLAB6.0.

The objective of these simulations is to demonstrate that the spatial location weighted optimization technique is more efficient, accurate, and computationally less expensive than the unweighted optimization method. Furthermore, this method accelerates convergence.

Fig. 1. Reconstructed images after several iterations using conjugate gradient method (col. 1) and weighted steepest descent method with different weighting factors (col. 2), geometry and mesh of the simulation, target image, and the objective function E as a function of weighting factor (col. 3)

In the above simulation, the embedded object is placed at 10 mm beneath the surface. Two other simulations, with the embedded object located at 5 mm and 15 mm beneath the surface separately, have also been conducted. The results of the three different cases are summarized in Table 2.

Table 2. Summaries of the results for the three simulation cases

table-icon
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where O: Origin of the embedded object

P: coordinate of the reconstructed peak µa value

Dev: Deviation of P from O

WSD: Weighted steepest descent method

M: Mean square of relative errors

N: Number of iteration

From Table 2, we will find that the WSD method exhibits an extraordinary fast convergence rate. In all three cases, the Dev (The deviation of the location of the peak µa value from the center of the original location of the embedded object) and M of the WSD method are smaller than that of the CG method, indicating that the WSD method recovers more precise position information of the embedded object than the CG method does. None of the methods recovers the absolute values of the embedded objects’ coefficients very well, but the new method gives better results. Though the error functions of the WSD method are larger than that of the CG method, it does not mean that their final results are inferior. Both algorithms are more sensitive to the perturbation closer to the boundary, but the new method can recover more deep heterogeneity compared to the conventional CG method.

Figure 2 shows the results from a more complex case. Three embedded objects with different parameters (see Table 1) are placed in a homogeneous background. It is only the bottom left object that has perturbations in both coefficients, while the top and bottom right objects have perturbations in only µa and only µ’ s, respectively. The two images in the left column of Fig. 2 show the original target images of the embedded objects. The middle column shows the results using the conventional CG method after 100 iterations. The right column is the results using the WSD method with the optimal weighting factor (β = 7). It is worth noting that both the conventional CG method and the WSD method present cross-talk in the recover images. The increase in an object’s absorption coefficient (or scattering coefficient) is recovered as a decrease in its scattering coefficient (or absorption coefficient). So the intensity measurements alone are insufficient to distinguish effects of absorption from scattering, as Arridge [9

9. S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23, 882–884 (1998) [CrossRef]

] have claimed (Y. Pei [10

10. Y. Pei, H. L. Graber, and R. L. Barbour, “Normalized-constraint algorithm for minimizing inter-parameter crosstalk in DC optical tomography,” Opt. Express 9, 97- (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-2-97 [CrossRef] [PubMed]

] has reported a normalized-constraint algorithm for minimizing inter-parameter crosstalk in DC optical tomography. This method was not used in this article).

Fig. 2. Target image (col. 1), reconstructed images after 100 iterations using CG method (col. 2), and WSD method with optimal weighting factor (col. 3). In all cases, the top image is µa, and the bottom image is µ’s.

From the simulations conducted above, we can find that the WSD method can give more accurate location of the embedded target. However, the proper determination of the optimal weighting factor is dependent on the target locations. To further validate the practical applicability of the algorithm, we conducted simulation using multiple targets that are located both near-boundary and deep inside (see Fig. 3). Figure 3(a) shows the original target images of the embedded objects. Figure 3(b) shows the result using conventional CG method after 100 iterations. Figure 3(c) shows the result using one step WSD method with optimal weighting factor (β = 5). We find that this weighting factor can efficiently localize the near-boundary target, but fail to recover the other two objects. Since the near-boundary object exercises more influence on the surface measurements, the calculated gradient vector has large values for the points around the near-boundary objects. To reduce this effect, we first take 30 iterations of conventional CG method, which can greatly decrease the gradient values for the points around the near-boundary object. And then we perform another WSD step to find the second optimal weighting factor, as shown in Fig. 3(d) (β = 8). However, both the conventional CG method and the WSD method failed to recover the deepest object.

Fig. 3. Simulation results for multiple targets that are located with different depth. (a) target image. (b) reconstruction result using conventional CG method after 100 iterations. (c) reconstruction result using one step of WSD method and (d) final result using another WSD step.

5. Conclusions

In this paper we have presented a method for image reconstruction in DC diffusion-based optical tomography. By imposing a spatial location weighted steepest descent method, we can precisely locate the embedded objects. In addition, the computation burden can also be greatly reduced. We presented the reconstruction of both absorption and scattering coefficient in steady state. In fact, this method can also be applies to other cases such as reconstruction of absorption and diffusion coefficients in frequency domain or time dependent case.

We have made a comparison between the conventional CG method and our new method using the simulated data. The new method can effectively locate the perturbations in absorptions and diffusions. Future work in our laboratory will involve experimental studies for clinical imaging problems.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China and Tsinghua University.

References and links

1.

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-Based Iterative Image-Reconstruction Scheme for Time-Resolved Optical Tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999) [CrossRef]

2.

R. Roy and E. M. Sevick-MUraca, “Active constrained truncated Newton method for simple-bound optical tomography,” J. Opt. Soc. Am. A 17, 1627–1641 (2000) [CrossRef]

3.

F. E. W. Schmidt, Development of a Time-Resolved Optical Tomography System for Neonatal Brain Imaging, Ph. D thesis (1999), University College London

4.

R. Roy and E. M. Sevick-MUraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express 4, 353–371 (1999), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-10-353 [CrossRef] [PubMed]

5.

S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-2-6-213 [CrossRef] [PubMed]

6.

A. Y. Bluestone, G. Abdoulaev, C. H. Schmitz, R. L. Barbour, and A. H. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express 9, 272–286 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-6-272 [CrossRef] [PubMed]

7.

Y. Pei, Optical Tomographic Imaging Using the Finite Element Method, Ph. D. Thesis (1999), Polytechnic University.

8.

C. H. Schmitz, H. L. Graber, H. Luo, I. Arif, J. Hira, Y. Pei, A. Bluestone, S. Zhong, R. Andronica, I. Soller, N. Ramirez, S. S. Barbour, and R. L. Barbour, “Instrumentation and calibration protocol for imaging dynamic features in dense-scattering media by optical tomography,” Appl. Opt. 9, 6466–6486 (2000) [CrossRef]

9.

S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23, 882–884 (1998) [CrossRef]

10.

Y. Pei, H. L. Graber, and R. L. Barbour, “Normalized-constraint algorithm for minimizing inter-parameter crosstalk in DC optical tomography,” Opt. Express 9, 97- (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-2-97 [CrossRef] [PubMed]

11.

S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part2: Zinite-element-method calculations,” Appl. Opt. 34, 8026–8037 (1995) [CrossRef] [PubMed]

12.

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour. “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Amer. A 14, 325–342 (1997) [CrossRef]

13.

K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995) [CrossRef] [PubMed]

14.

A. J. Davies, D. B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Advances in Engineering Software 28, 217–221 (1997) [CrossRef]

15.

A. D. Klose and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer — Part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transfer 72, 715–732 (2002) [CrossRef]

16.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 9

17.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994) [CrossRef]

18.

I. W. Kwee, Towards a Bayesian Framework for Optical Tomography, Ph. D. Thesis (1999), University College London.

19.

S. G. Nash and A. Sofer, Linear and nonlinear programming (McGraw-Hill, New York, 1996)

20.

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

OCIS Codes
(170.3010) Medical optics and biotechnology : Image reconstruction techniques
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.3880) Medical optics and biotechnology : Medical and biological imaging
(170.6960) Medical optics and biotechnology : Tomography

ToC Category:
Research Papers

History
Original Manuscript: December 12, 2002
Revised Manuscript: January 16, 2003
Published: January 27, 2003

Citation
Jun Zhou, Jing Bai, and Ping He, "Spatial location weighted optimization scheme for DC optical tomography," Opt. Express 11, 141-150 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-2-141


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References

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  11. S. R. Arridge, M. Schweiger, �??Photon-measurement density functions. Part2: Finite-element-method calculations,�?? Appl. Opt. 34, 8026-8037 (1995) [CrossRef] [PubMed]
  12. Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, R. L. Barbour. �??Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,�?? J. Opt. Soc. Am. A 14, 325-342 (1997) [CrossRef]
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  15. A. D. Klose and A. H. Hielscher, �??Optical tomography using the time-independent equation of radiative transfer �?? Part 2: inverse model,�?? J. Quant. Spectrosc. Radiat. Transfer 72, 715-732 (2002) [CrossRef]
  16. A Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 9
  17. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, �??Boundary conditions for the diffusion equation in radiative transfer,�?? J. Opt. Soc. Am. A 11, 2727-2741 (1994) [CrossRef]
  18. I. W. Kwee, Towards a Bayesian Framework for Optical Tomography, Ph. D. Thesis (1999), University College London.
  19. S. G. Nash, A. Sofer, Linear and nonlinear programming(McGraw-Hill, New York, 1996)
  20. S. R. Arridge, M. Schweiger, M. Hiraoka, D.T. Delpy, �??A finite element approach for modeling photon transport in tissue,�?? Med. Phys. 20, 299-309 (1993). [CrossRef] [PubMed]

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