## Tumor localization using diffuse optical tomography and linearly constrained minimum variance beamforming

Optics Express, Vol. 15, Issue 3, pp. 896-909 (2007)

http://dx.doi.org/10.1364/OE.15.000896

Acrobat PDF (298 KB)

### Abstract

We present a tumor localization method for diffuse optical tomography using linearly constrained minimum variance (LCMV) beam-forming. Beamforming is a spatial filtering technique where signals from certain directions can be enhanced while noise and interference from other directions are suppressed. In our method, we tessellate the domain into small voxels and regard each voxel as a possible position of abnormality (e.g., tumor). We then design a spatial filter based on the linearly constrained minimum variance criterion and apply it to each voxel in the domain. The abnormality is localized by observing the peak in the filter output signals. We test our method using simulated 3D examples. We assume a cubic transmission geometry and consider different cases where the abnormality is an absorber, a scatterer, and both. We also give examples showing the resolution of our method and its performance under different perturbation levels and noise levels. Simulation results show that LCMV beamforming can localize the abnormality well with good computational efficiency. It can be used alone for tumor localization and also as an effective preprocessing tool for improving the image reconstruction performances of other inverse methods.

© 2007 Optical Society of America

## 1. Introduction

6. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**,R41–R93 (1999). [CrossRef]

8. R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. **45**,1051–1070 (2000). [CrossRef] [PubMed]

9. B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**,2950–2961 (1999). [CrossRef]

10. M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. **50**,2837–2858 (2005). [CrossRef] [PubMed]

11. J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffuse tomography,” IEEE Trans. Image Process. **10**,909–922 (2001). [CrossRef]

12. D. Grosenick, H. Wabnitz, H. Rinneberg, K. T. Moesta, and P. M. Schlag, “Development of a time-domain optical mammograph and first *in vivo* applications,” Appl. Opt. **38**,2927–2943 (1999). [CrossRef]

13. D. Grosenick, H. Wabnitz, K. T. Moesta, J. Mucke, M. Möller, C. Stroszczynski, J. Stöbel, B. Wassermann, P. M. Schlag, and H. Rinneberg, “Concentration and oxygen saturation of haemoglobin of 50 breast tumours determined by time-domain optical mammography,” Phys. Med. Biol. **49**,1165–1181, (2004). [CrossRef] [PubMed]

4. D. Grosenick, T. Moesta, H. Wabnitz, J. Mucke, C. Stroszcynski, R. Macdonald, P. Schlag, and H. Rinnerberg, “Time-domain optical mammography: Initial clinial results on detection and characterization of breast tumors,” Appl. Opt. **42**,3170–3186 (2003). [CrossRef] [PubMed]

14. B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE ASSP. Magazine **5**,4–24 (1988). [CrossRef]

16. K. Sekihara, S. S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita, “Reconstructing spatio-temporal activities of neural sources using an MEG vector beamforming technique,” IEEE Trans. Biomed. Eng. **48**,760–771 (2001). [CrossRef] [PubMed]

17. B. D. Van Veen, W. van Drongelen, M. Yuchtman, and A. Suzuki, “Localization of brain electrial activity via linearly constrained minimum variance spatial filtering,” IEEE Trans. Biomed. Eng. **44**,867–880 (1997). [CrossRef] [PubMed]

18. M. E. Spencer, R. M. Leahy, J. C. Mosher, and P. S. Lewis, “Adaptive filters for monitoring localized brain activity from surface potential time series,” in *Conference record of the twenty-sixth Asilomar conference on Signals, Systems and Computers* (Institute of Electrical and Electronics Engineers, Pacific Grove, CA, 1992), pp.156–161.

8. R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. **45**,1051–1070 (2000). [CrossRef] [PubMed]

## 2. Forward and measurement models

21. D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express **1**,404–413 (1997). [CrossRef] [PubMed]

### 2.1. Forward model

22. V. Ntziachristos, B. Chance, and A. G. Yodh, “Differential diffuse optical tomography,” Opt. Express **5**,230–242 (1999). [CrossRef] [PubMed]

*U*is the fluence (W/m

^{2}),

*μ*

_{a}the absorption coefficient (m

^{-1}),

*D*the diffusion coefficient (m

^{2}/s),

*c*the speed of light in the medium (m/s), and

*q*

_{0}the source (W/m

^{3}). The term

*D*is defined as

*D*=

*c*/3(

*μ*

_{a}+

*μ*

_{s}́), where

*μ*

_{s}́ = (1 -

*g*)

*μ*

_{s}is the reduced scattering coefficient (m

^{-1}) with

*μ*

_{s}representing the scattering coefficient (m

^{-1}), and

*g*the average of the cosine of the scattering angle. Since

*μ*

_{a}≪

*μ*

_{s}́ for most biological tissues [6

6. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**,R41–R93 (1999). [CrossRef]

*D*= 1/3

*μ*

_{s}́ in our calculations.

*μ*

_{a0}and

*D*

_{0}denote the homogenous background parts, and

*δμ*

_{a}(

*r*) and

*δD*(

*r*) the heterogenous parts. Based on this expansion, the DA can be solved using the Rytov approximation [7], where the photon fluence at position

*r*due to a source at

*r*is given as

_{s}*ϕ*

_{sc})

^{2}≪

*δμ*

_{a}/

*D*(i.e., the scattered field is slowly varying). In Eq. 4,

*U*

_{0}represents the homogenous photon density corresponding to

*μ*

_{a0}and

*D*

_{0}, and

*ϕ*

_{sc}the diffuse Rytov phase caused by

*δμ*

_{a}(

*r*) and

*δD*(

*r*). The task is then to find the relationship between

*δμ*

_{a}(

*r*)(

*δD*) and

*ϕ*

_{sc}(

*r*,

*r*

_{s}). After we discretize the whole domain into

*N*equivolume voxels and assume

*N*

_{s}source fibers and

*N*

_{d}detectors, the forward model can be expressed as [7]

^{2N}the change of optical parameters in each voxel,

*ϕ*

^{c}

_{sc}∈ C

^{Nsd}the Rytov phase,

*A*c ∈ C

^{Nsd×N}the weighting matrix, and

*N*

_{sd}=

*N*

_{s}×

*N*

_{d}the number of source-detector pairs. More specifically,

*i*

^{th}source on the

*j*

^{th}absorbing (scattering) voxel measured at the

*k*

^{th}detector, where

*G*(∙) denotes the Green function and

*h*

^{3}the volume of each voxel. Note that (i) Eq. (5) can be used for different simple geometries such as infinite, semi-infinite, or slab, by modifying the elements in

*A*

^{c}using the method of image sources [23

23. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” J. Appl. Opt. **28**,2331–2336 (1989). [CrossRef]

24. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A **12**,2532–2539 (1995). [CrossRef]

25. 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]

6. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**,R41–R93 (1999). [CrossRef]

*ϕ*

_{sc}= [

*R*

*ϕ*

_{cs}

^{cT}

*I*

*ϕ*

_{cs}

^{cT}]

^{T},

*A*= [

*RA*

^{cT}

*IA*

^{cT}]

^{T}, where

*R*(∙) takes the real part of the matrix element and

*I*(∙) the imaginary part; then (5) can be rewritten as

*ϕ*∈ R

^{Ntot},

*A*∈ R

^{Ntot×N}, and

*N*

_{tot}= 2×

*N*

_{sd}. We will use this expression as our forward model.

### 2.2. Measurement model

*y*∈ R

^{Ntot}denotes the measurement vector and

*e*∈ R

^{Ntot}the noise vector. In DOT the optical signals are most often corrupted by shot noise, which follows a Poisson distribution. However, when the number of photons is very large, a Poisson distribution can be well approximated by a Gaussian distribution [26]. Since the noise variance is expected to be proportional to the number of photons at the detector, we choose the noise covariance matrix as

## 3. Tumor localization using LCMV beamforming

*δμ*

_{a}or

*μ*

_{s}́ from the measurement

*y*. It is usually ill-posed, and different regularization techniques [9

9. B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**,2950–2961 (1999). [CrossRef]

10. M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. **50**,2837–2858 (2005). [CrossRef] [PubMed]

11. J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffuse tomography,” IEEE Trans. Image Process. **10**,909–922 (2001). [CrossRef]

*μ*

_{a}. Namely, we let

*A*

^{c}=

*A*

^{ca}and Δ = Δ

^{a}. From (5), we can see that

*ϕ*

_{sc}is actually a superposition of the Rytov phases resulting from many voxels. More specifically, the

*l*

^{th}column of matrix

*A*reflects the contribution of the

*l*

^{th}voxel in Δ on

*ϕ*

_{sc}. Since Δ contains only the change of

*μ*

_{a}compared with the background, then, assuming a localized abnormality, it is clear that ideally only a limited number of elements in Δ are nonzero, whose locations correspond to the abnormality’s position. Therefore, the inverse solution is essentially to find the elements in Δ that provide the greatest contribution to the measurements.

*ϕ*

_{sc}is composed of the fluence due to

*L*(non-zero valued) voxels; then (12) can be equivalently expressed as

**ind**{} denotes the set of indices for which Δ is nonzero and

*H*

_{ind{l}}the

**ind**{

*l*} column of

*A*. Based on this idea, we can think of solving the inverse problem as finding a spatial filter

*W*∈ R

^{Ntot}for each voxel location such that the filter output reflects the contribution of that voxel to the output. Mathematically, we are now looking for a vector

*W*so that by scanning the whole domain voxel by voxel, the filter output

*i*

^{th}element of Δ. For the ideal narrowband spatial filter, it should satisfy

*W*such that the variance of the filter output

*ϕ*̂ is minimized while satisfying the linear constraint (17). Physically, that is equivalent to minimizing the contribution to the filter output due to the signals in the stop band. Since the variance of a random signal is a measure of the strength, and for a vector case, it is defined as the sum of the variance of each component, the LCMV beamforming is mathematically expressed so as to find a filter

*W*,

_{i}*i*= 1,2,...,

*N*

_{tot}, for each voxel in the medium such that

*C*(∙) the covariance matrix. Using the method of Lagrange multipliers, we obtain the solution to

*W*as

17. B. D. Van Veen, W. van Drongelen, M. Yuchtman, and A. Suzuki, “Localization of brain electrial activity via linearly constrained minimum variance spatial filtering,” IEEE Trans. Biomed. Eng. **44**,867–880 (1997). [CrossRef] [PubMed]

*C*(

*y*). In practice,

*C*(

*y*) is generally unknown and needs to be estimated from the measurements. Assuming we have

*M*temporal observations,

*y*

_{1},

*y*

_{2},...,

*y*, each of them following Eq. (12), one common estimator is the unbiased sample covariance matrix

_{M}*y*represents the

_{m}*m*

^{th}temporal sample, and

*y*¯ = ∑

^{M}

_{m=1}

*y*/

_{m}*M*the sample mean. The validity of this estimation is based on two conditions: (i) we assume that the underlying

*μ*

_{a}distribution is constant for all measurements

*y*, and (ii) the number of observations

_{m}*M*must be larger than the dimension of the measurement vector

*y*, i.e.,

_{m}*N*

_{tot}. The second condition is for

*C*̂ to be nonsingular and typically,

*M*is chosen to be a few times larger than

*N*

_{tot}. When

*M*is smaller than or equal to

*N*

_{tot}, the sample covariance matrix (20) would not be a good approximation of the true covariance matrix. In this case, a novel shrinkage covariance estimator can be applied, which is well conditioned and always positive definite [27]. Since how to estimate the covariance matrix is not the main idea of beamforming (although it is an important factor), we just use (20) as our estimator for simplicity.

*N*will not impose a restriction for solving the inverse problem as in most methods. More specifically, looking at Eq. (19), we can see that the major computational load comes from calculating

*C*

^{-1}(

*y*), which has a computation complexity

*O*(

*N*

^{3}

_{tot}). Compared with typical Newton methods, which have either an

*O*(

*N*

^{3}) complexity if using direct matrix inversion or an

*O*(

*N*

^{2}) complexity for each iteration if some iterative approaches (e.g., steepest descent) are employed, our method offers faster implementation.

## 4. Numerical examples

*z*= 0 cm), and the sides are of length 8 cm, 8 cm, and 6 cm along the

*x*,

*y*, and

*z*directions, respectively. We place 25 sources on the bottom surface with 1.75 cm interdistance and 25 detectors on the top surface (

*z*= 6 cm) with 1.5 cm interdistance. We choose a source modulation frequency of 200MHz and wavelength λ = 750nm

### 4.1. Tumor localization results

*μ*

_{a0}= 0.05 cm

^{-1},

*μ*

_{s0}́ = 9.5 cm

^{-1}, and

*c*= 22 cm/ns [28

28. M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, “Three-dimensional diffuse optical mammography with ultrasound localization in a human subject,” J. Biomed. Opt. **5**,237–247 (2000). [CrossRef] [PubMed]

- (A) There is only one spherical absorbing abnormality centered at [-1.5,1.25,2.9] cm with radius
*R*= 1 cm and*δμ*_{a}(*r*) = 0.2 cm^{-1}(see 2(a)); - (B) There is only one spherical scattering abnormality centered at [2, -1.25,1.75] cm with
*R*= 1 cm and*δμ*_{s}́(*r*) = -4 cm^{-1}(see Fig. 3(a)); - (C) There is only one spherical abnormality that is both absorptive and scattering. It is centered at [-1.5,1.25,2.9] cm with
*δμ*_{a}(*r*) = 0.2 cm^{-1}and*δμ*_{s}́(*r*) = 2 cm^{-1}(see Fig. 4 (a)); - (D) Both abnormalities specified in (A) and (B) are shown in the domain (see Fig. 5(a)).

^{3}, leading to a total number of 4,800 voxels. We add a Gaussian noise with σ = 0.01 to obtain the measurements.

*δμ*

_{a}and

*δμ*

_{s}́, but it can be useful as an effective preprocessing tool. For example, the abnormality can first be localized preliminarily using LCMV beamforming, and then the

*a priori*information obtained can be used to initiate starting points for more accurate (and also more time-consuming) methods. We will illustrate this application in the next subsection.

### 4.2. Application of the LCMV beamforming as a preprocessing tool

*μ*

_{a}distribution is given in Fig. 2(a).

*δμ*

_{a}value is not correct as well, as it is very biased toward the initial searching vector. We then perform the reconstruction by initializing the Δ vector based on the information obtained from the LCVM beamforming results (Fig. 2(b)). More specifically, we set the

*δμ*

_{a}value to 0.2 at the positions where the LCMV filter output is more than half the peak value. The inverse results are shown in Fig. 7(b). We can see that the reconstruction result is greatly improved with a more localized abnormality.

### 4.3. Performance analysis

#### 4.3.1. Effect of the absorption perturbation level

*δμ*

_{a}= 0.2 cm

^{-1}was shown in Fig. 2(b). According to [7], the perturbation in the tumor’s absorption value ranges from 0.02 cm

^{-1}to 0.3 cm

^{-1}for most biological tissues. Therefore, we reduce

*δμ*

_{a}to 0.1 cm

^{-1}and 0.05 cm

^{-1}and applied our LCMV beamforming method. The results are shown in Fig. 8. Comparing Fig. 2(b) and Fig. 8, we observe that as the perturbation level decreases, the filter output signal becomes smaller as well. However, the abnormality can be reconstructed correctly with a pronounced contrast in all three cases, indicating the robustness of our method to small absorption perturbations.

#### 4.3.2. Effect of noise level

*δμ*

_{a}= 0.2 cm

^{-1}. We increase the noise level to σ = 0.1 and σ = 1, and the localization results are shown in Fig. 9. Comparing with Fig. 2(b) where σ = 0.01, we observe that as the noise level increases, the size of the recovered abnormality increases as well (i.e., the resolution worsens). Nevertheless, the center of the abnormality can still be recovered correctly and the tumor is detected with clear contrast.

#### 4.3.3. Resolution of the LCMV beamforming in DOT

*R*= 0.75 cm and

*δμ*

_{a}= 0.2 cm

^{-1}. In the first case, the centers of these two spheres are at [-1.5, 0.8, 2] cm and [1.5, -0.8, 4] cm, with a distance of 4 cm between two centers and 2.5 cm between the two closest points on the spheres. In the second case, the centers are moved to [-1, 0.5, 2.5] cm and [1, -0.5, 3.5] cm, and the distance becomes 2.45 cm between centers and 1 cm between the closet points. The localization results are shown in Fig. 10. We see that the abnormalities can be recovered with limited resolution when the two abnormalities are more than 2 cm apart (Fig. 10(a)). The contrast of the reconstructed image is not as good as for the single tumor case, but the two abnormalities can still be separated and the centers be recovered at the right places. When the two tumors are less than 1.5 cm apart, our LCMV beamforming can not differentiate them. Only one big sphere is shown in the reconstructed image (Fig. 10(b)), with no indication of whether this is from one big tumor or two small separated ones.

## 5. Conclusion

16. K. Sekihara, S. S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita, “Reconstructing spatio-temporal activities of neural sources using an MEG vector beamforming technique,” IEEE Trans. Biomed. Eng. **48**,760–771 (2001). [CrossRef] [PubMed]

29. M. Papazoglou and J. L. Krolik, “High resolution adaptive beamforming for three-dimensional acoustic imaging of zooplankton,” J. Acoust. Soc. Am. **100**,3621–3630 (1996). [CrossRef]

## Appendix

*W*and

_{i}*H*can be matrices instead of a row (column) vector as required by (18). For this reason, we omit the subscript

_{i}*i*in the following derivation.

*D*be a matrix of Lagrange multipliers; then finding the matrix

*W*satisfying (18) is equivalent to finding a

*W*that minimizes the function

*M*= tr

*M*for any square matrix

^{T}*M*, we have

*W*and the covariance matrix

*C*is positive definite, the minimum of

*L*(

*W*,

*D*) is obtained by setting the first term to zero; i.e.,

*D*is obtained by substituting (24) into the constraint

*W*=

^{T}H*I*, and we have

## Acknowledgments

## References and links

1. | A. H. Barnett, J. P. Culver, A. G. Sorensen, A. M. Dale, and D. A. Boas, “Bayesian estimation of optical properties of the human head via 3D structural MRI,” in |

2. | G. Strangman, D. Boas, and J. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry |

3. | A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. |

4. | D. Grosenick, T. Moesta, H. Wabnitz, J. Mucke, C. Stroszcynski, R. Macdonald, P. Schlag, and H. Rinnerberg, “Time-domain optical mammography: Initial clinial results on detection and characterization of breast tumors,” Appl. Opt. |

5. | X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. |

6. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. |

7. | M. A. O’Leary, “Imaging with diffuse photon density waves,” Ph.D. thesis, University of Pennsylvania (1996). |

8. | R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. |

9. | B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. |

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

11. | J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffuse tomography,” IEEE Trans. Image Process. |

12. | D. Grosenick, H. Wabnitz, H. Rinneberg, K. T. Moesta, and P. M. Schlag, “Development of a time-domain optical mammograph and first |

13. | D. Grosenick, H. Wabnitz, K. T. Moesta, J. Mucke, M. Möller, C. Stroszczynski, J. Stöbel, B. Wassermann, P. M. Schlag, and H. Rinneberg, “Concentration and oxygen saturation of haemoglobin of 50 breast tumours determined by time-domain optical mammography,” Phys. Med. Biol. |

14. | B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE ASSP. Magazine |

15. | K. Sekihara and S. S. Nagarajan, “Neuromagnetic source reconstruction and inverse modeling,” in |

16. | K. Sekihara, S. S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita, “Reconstructing spatio-temporal activities of neural sources using an MEG vector beamforming technique,” IEEE Trans. Biomed. Eng. |

17. | B. D. Van Veen, W. van Drongelen, M. Yuchtman, and A. Suzuki, “Localization of brain electrial activity via linearly constrained minimum variance spatial filtering,” IEEE Trans. Biomed. Eng. |

18. | M. E. Spencer, R. M. Leahy, J. C. Mosher, and P. S. Lewis, “Adaptive filters for monitoring localized brain activity from surface potential time series,” in |

19. | J.-F. Synnevag, A. Austeng, and S. Holm, “Minimum variance adaptive beamforming applied to medical ultrasound imaging,” in |

20. | A. C. Kak and M. Slaney, |

21. | D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express |

22. | V. Ntziachristos, B. Chance, and A. G. Yodh, “Differential diffuse optical tomography,” Opt. Express |

23. | M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” J. Appl. Opt. |

24. | R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A |

25. | 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 |

26. | S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. |

27. | J. Schäfer and K. Strimmer, “A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics,” Statist. Appl. Genet. Mol. Biol. 4, Article 32 (2005). |

28. | M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, “Three-dimensional diffuse optical mammography with ultrasound localization in a human subject,” J. Biomed. Opt. |

29. | M. Papazoglou and J. L. Krolik, “High resolution adaptive beamforming for three-dimensional acoustic imaging of zooplankton,” J. Acoust. Soc. Am. |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

**ToC Category:**

Image Processing

**History**

Original Manuscript: November 22, 2006

Revised Manuscript: January 15, 2007

Manuscript Accepted: January 16, 2007

Published: February 5, 2007

**Virtual Issues**

Vol. 2, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Nannan Cao and Arye Nehorai, "Tumor localization using diffuse optical
tomography and linearly constrained
minimum variance beamforming," Opt. Express **15**, 896-909 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-3-896

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### References

- A. H. Barnett, J. P. Culver, A. G. Sorensen, A. M. Dale, and D. A. Boas, "Bayesian estimation of optical properties of the human head via 3D structural MRI," in Photon Migration and Diffuse-Light Imaging, D. A. Boas, ed. Proc. SPIE 5138, 139-147 (2003). [CrossRef]
- G. Strangman, D. Boas, and J. Sutton, "Non-invasive neuroimaging using near-infrared light," Biol. Psychiatry 52, 679-693 (2002). [CrossRef] [PubMed]
- A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends Neurosci. 20, 435-442 (1997). [CrossRef] [PubMed]
- D. Grosenick, T. Moesta, H. Wabnitz, J. Mucke, C. Stroszcynski, R. Macdonald, P. Schlag, and H. Rinnerberg, "Time-domain optical mammography: Initial clinial results on detection and characterization of breast tumors," Appl. Opt. 42, 3170-3186 (2003). [CrossRef] [PubMed]
- X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. Yodh, and B. Chance, "In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green," Med. Phys. 30, 1039-1047 (2003). [CrossRef] [PubMed]
- S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999). [CrossRef]
- M. A. O’Leary, "Imaging with diffuse photon density waves," Ph.D. thesis, University of Pennsylvania (1996).
- R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, "A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient," Phys. Med. Biol. 45, 1051-1070 (2000). [CrossRef] [PubMed]
- B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, "Spatially variant regularization improves diffuse optical tomography," Appl. Opt. 38, 2950-2961 (1999). [CrossRef]
- M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005). [CrossRef] [PubMed]
- J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, "Nonlinear multigrid algorithms for Bayesian optical diffuse tomography," IEEE Trans. Image Process. 10, 909-922 (2001). [CrossRef]
- D. Grosenick, H. Wabnitz, H. Rinneberg, K. T. Moesta, and P. M. Schlag, "Development of a time-domain optical mammograph and first in vivo applications," Appl. Opt. 38, 2927-2943 (1999). [CrossRef]
- D. Grosenick, H. Wabnitz, K. T. Moesta, J. Mucke, M. M¨oller, C. Stroszczynski, J. St¨obel, B. Wassermann, P. M. Schlag, and H. Rinneberg, "Concentration and oxygen saturation of haemoglobin of 50 breast tumours determined by time-domain optical mammography," Phys. Med. Biol. 49, 1165-1181, (2004). [CrossRef] [PubMed]
- B. D. Van Veen and K. M. Buckley, "Beamforming: A versatile approach to spatial filtering," IEEE ASSP. Magazine 5, 4-24 (1988). [CrossRef]
- K. Sekihara and S. S. Nagarajan, "Neuromagnetic source reconstruction and inverse modeling," in Proceedings of IEEE EMBS Asian-Pacific Conference on Biomedical Engineering (Institute of Electrical and Electronics Engineers, Keihanna, Japan, 2003), pp. 20-22.
- K. Sekihara, S. S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita, "Reconstructing spatio-temporal activities of neural sources using an MEG vector beamforming technique," IEEE Trans. Biomed. Eng. 48, 760-771 (2001). [CrossRef] [PubMed]
- B. D. Van Veen, W. van Drongelen, M. Yuchtman, and A. Suzuki, "Localization of brain electrial activity via linearly constrained minimum variance spatial filtering," IEEE Trans. Biomed. Eng. 44, 867-880 (1997). [CrossRef] [PubMed]
- M. E. Spencer, R. M. Leahy, J. C. Mosher, and P. S. Lewis, "Adaptive filters for monitoring localized brain activity from surface potential time series," in Conference record of the twenty-sixth Asilomar conference on Signals, Systems and Computers (Institute of Electrical and Electronics Engineers, Pacific Grove, CA, 1992), pp. 156-161.
- J.-F. Synnevag, A. Austeng, and S. Holm, "Minimum variance adaptive beamforming applied to medical ultrasound imaging," in Proceedings of IEEE Ultrasonics Symposium, (Institute of Electrical and Electronics Engineers, Rotterdam, Netherlands, 2005), pp. 1199-1202.
- A. C. Kak and M. Slaney, Principles of computerized tomographic imaging (IEEE Press, 1988).
- D. A. Boas, "A fundamental limitation of linearized algorithms for diffuse optical tomography," Opt. Express 1, 404-413 (1997). [CrossRef] [PubMed]
- V. Ntziachristos, B. Chance, and A. G. Yodh, "Differential diffuse optical tomography," Opt. Express 5, 230-242 (1999). [CrossRef] [PubMed]
- M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," J. Appl. Opt. 28, 2331-2336 (1989). [CrossRef]
- R. Aronson, "Boundary conditions for diffusion of light," J. Opt. Soc. Am. A 12, 2532-2539 (1995). [CrossRef]
- 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]
- S. O. Rice, "Mathematical analysis of random noise," Bell Syst. Tech. J. 23, 282-332 (1944).
- J. Sch¨afer and K. Strimmer, "A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics," Statist. Appl. Genet. Mol. Biol. 4, Article 32 (2005).
- M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000). [CrossRef] [PubMed]
- M. Papazoglou and J. L. Krolik, "High resolution adaptive beamforming for three-dimensional acoustic imaging of zooplankton," J. Acoust. Soc. Am. 100, 3621-3630 (1996). [CrossRef]

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