## Algorithmic depth compensation improves quantification and noise suppression in functional diffuse optical tomography |

Biomedical Optics Express, Vol. 1, Issue 2, pp. 441-452 (2010)

http://dx.doi.org/10.1364/BOE.1.000441

Acrobat PDF (1152 KB)

### Abstract

Accurate depth localization and quantitative recovery of a regional activation are the major challenges in functional diffuse optical tomography (DOT). The photon density drops severely with increased depth, for which conventional DOT reconstruction yields poor depth localization and quantitative recovery. Recently we have developed a depth compensation algorithm (DCA) to improve the depth localization in DOT. In this paper, we present an approach based on the depth-compensated reconstruction to improve the quantification in DOT by forming a spatial prior. Simulative experiments are conducted to demonstrate the usefulness of this approach. Moreover, noise suppression is a key to success in DOT which also affects the depth localization and quantification. We present quantitative analysis and comparison on noise suppression in DOT with and without depth compensation. The study reveals that appropriate combination of depth-compensated reconstruction with the spatial prior can provide accurate depth localization and improved quantification at variable noise levels.

© 2010 OSA

## 1. Introduction

1. A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. **20**(10), 435–442 (1997). [CrossRef] [PubMed]

2. D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage **23**(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

3. C. K. Lee, C. W. Sun, P. L. Lee, H. C. Lee, C. Yang, C. P. Jiang, Y. P. Tong, T. C. Yeh, and J. C. Hsieh, “Study of photon migration with various source-detector separations in near-infrared spectroscopic brain imaging based on three-dimensional Monte Carlo modeling,” Opt. Express **13**(21), 8339–8348 (2005). [CrossRef] [PubMed]

**A**, is ill-posed along depth. In conventional DOT reconstruction algorithm using Tikhonov regularization [2

2. D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage **23**(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

4. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**(2), R41–R93 (1999). [CrossRef]

**A**matrix results in a significant error in depth localization [2

2. D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage **23**(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

5. H. Niu, F. Tian, Z. J. Lin, and H. Liu, “Development of a compensation algorithm for accurate depth localization in diffuse optical tomography,” Opt. Lett. **35**(3), 429–431 (2010). [CrossRef] [PubMed]

**M**, to counterbalance the severe sensitivity decay of

**A**matrix along depth. The original

**A**matrix is replaced by a depth-compensated matrix,

**AM,**in reconstruction. The improved accuracy in depth localization by the depth compensation algorithm has been demonstrated in both laboratory phantom experiments and

*in vivo*human brain study [6]. Thus, it is expected that this algorithm will also improve the quantification in DOT with a reasonable accuracy. However, this improvement cannot be achieved directly. It is because the matrix

**AM**used in depth-compensated reconstruction is not in compliance with the actual measurement. Although the reconstructed image with depth compensation can reflect the actual location and size of local absorption perturbation, the quantity of the image does not reflect the actual quantity of the absorption perturbation. Thus, it is necessary to further develop an approach, which is based on the depth-compensated reconstruction, to improve the quantification in DOT. In this paper, we present such an approach which improves the quantification of local absorption perturbation by forming a spatial constraint or prior in reconstructed image. Simulative experiments are conducted to confirm the usefulness of this approach.

**23**(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

*α*is used in Tikhonov regularization to stabilize and optimize the solution of an ill-posed inverse problem [4

4. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**(2), R41–R93 (1999). [CrossRef]

## 2. Methods

### 2.1 Reconstruction and depth compensation

7. M. Cope, D. T. Delpy, E. O. Reynolds, S. Wray, J. Wyatt, and P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. **222**, 183–189 (1988). [PubMed]

**23**(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

*OD*(

*λ*,

*t*) is the change in optical density (OD) attained from a source-detector pair (measurement) at wavelength

*λ*and time

*t*, Δ

*μ*is the absorption perturbation in the

_{a,j}*j*th voxel of the medium, and

*L*is the effective path length of detected photons through the

_{j}*j*th voxel.

*λ*and

*t*):where

**y**denotes the vector of Δ

*OD*(

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

*N*) from all the measurements,

_{meas}**x**denotes the vector of Δ

*µ*(

_{a,j}*j*= 1 ...

*N*) from all the voxels in the 3D medium, and

_{vox}**A**is a

*N*×

_{meas}*N*sensitivity matrix of

_{vox}*L*and can be derived from the photon diffusion equation using the Rytov approximation [4

_{i,j}4. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**(2), R41–R93 (1999). [CrossRef]

**23**(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

**I**is the identity matrix,

*s*

_{max}is the maximum eigen-value of

**AA**

*, and*

^{T}*α*is the regularization parameter to stabilize the solution and to suppress the noise existing in a real measurement.

**A**the elements from superficial layers are significantly greater in magnitude than those from deeper layers due to severe attenuation in photon density with increased depth. In depth-compensated reconstruction, a weight matrix,

**M**, is created appropriately [5

5. H. Niu, F. Tian, Z. J. Lin, and H. Liu, “Development of a compensation algorithm for accurate depth localization in diffuse optical tomography,” Opt. Lett. **35**(3), 429–431 (2010). [CrossRef] [PubMed]

**A**in order to counterbalance the severe decay along depth. Specifically, the maximum singular value of layered sub-matrix

**A**

*(*

_{k}*k*= 1 ...

*N*), denoted by

_{layer}*s*for the

_{max,k}*k*th depth, is inversely arranged to form matrix

**M**:where power

*γ*is the parameter to adjust the weight of

**M**. An optimal value of

*γ*exists and has been determined empirically [5

5. H. Niu, F. Tian, Z. J. Lin, and H. Liu, “Development of a compensation algorithm for accurate depth localization in diffuse optical tomography,” Opt. Lett. **35**(3), 429–431 (2010). [CrossRef] [PubMed]

**A**=

^{#}**AM**is used to replace the actual sensitivity matrix

**A**in Eq. (3), leading towhere

*s'*

_{max}is the maximum eigen-value of (

**AM**)(

**AM**)

*.*

^{T}### 2.2 Quantification approach based on depth-compensated reconstruction

**M**; the second step is to define an appropriate region of interest (ROI) in the reconstructed image to form a spatial hard prior or constraint.

*(1) Determination of scaling parameter:*By introducing the weight matrix

**M**, the depth compensation algorithm actually derives the solution from an artificial forward model

**35**(3), 429–431 (2010). [CrossRef] [PubMed]

*K*. Thus, Eq. (2) can be rewritten as:where

**y**is attained from actual measurements,

**A**is the original sensitivity matrix and can be obtained through the calculation of forward problem,

*K*is unknown and can be solved in principle. In this paper the optimal estimation of

*K*is carried out by linear regression across all the measurements. After

*K*value is determined,

*(2) Formation of spatial prior*: A general problem in DOT is the poor spatial resolution, which is partially attributed to the diffuse nature of photons and partially attributed to the limited number of measurements as compared to the huge number of voxels to be quantified in the medium. As a result, the outline of the absorber is blurred, and the recovered quantities of absorption perturbation are reduced in the reconstructed DOT image. A widely used approach to improve such a problem is to combine DOT with a spatial prior or constraint of abnormal structures that can be obtained from other high-resolution imaging modalities, such as CT and MRI [8

8. H. Dehghani, B. W. Pogue, J. Shudong, B. Brooksby, and K. D. Paulsen, “Three-dimensional optical tomography: resolution in small-object imaging,” Appl. Opt. **42**(16), 3117–3128 (2003). [CrossRef] [PubMed]

9. G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. **50**(17), 3941–3956 (2005). [CrossRef] [PubMed]

*μ*. Based on these assumptions, Eq. (2) after the spatial prior is applied can be rewritten as:

_{a_ROI}*μ*. The optimal estimation of Δ

_{a_ROI}*μ*is then determined via linear regression across all measurements.

_{a_ROI}### 2.3 Simulative experiments

10. E. R. Hom, http://www.nmr.mgh.harvard.edu/PMI/toolbox/index.html

^{−1}, respectively. A 5 × 5 optode array (13 sources and 12 detectors) with an interval of 1.4 cm was arranged on the surface of medium, producing 132 measurements when the 1st to 4th nearest source-detector pairs were used (separation ranged from 1.4 to 5.0 cm). No noise was added to the data during the calculation of forward problem. In accordance with the coordinate system shown in Figs. 1(a) to 1(c), images of absorption perturbation were reconstructed in a 3D volume under the probe with x = −3 to 3 cm, y = −3 to 3 cm and z = −0.4 to −3 cm. The voxel size was 0.1 × 0.1 × 0.1 cm

^{3}. In each experiment, one (or two) cylindrical absorber(s) was (were) embedded into the medium with one circular side facing up to mimic the absorption perturbation due to brain activation(s). The absorber(s) had an identical diameter of 1.6 cm and thickness of 0.8 cm. The reduced scattering coefficient of the absorber(s) was the same as the background medium.

^{−1}(i.e., the actual perturbation due to the presence of the absorber, Δ

*μ*, is 0.2 cm

_{a}^{−1}).

^{−1}and the other one was 0.3 cm

^{−1}.

^{−1}.

*γ*ranges from 1.0 to 1.6. In this study a medium

*γ*value of 1.3 was used. Although no noise was added in the forward simulation, we used α = 10

^{−3}in regularization which was a representative value we had used in tissue-like phantom experiments. To evaluate the quantification accuracy of the reconstructed image, we used two parameters: the first parameter was the maximum absorption perturbation, Δ

*μ*, in the scale-corrected image (i.e.,

_{a_max}*μ*.

_{a_ROI}^{−3}) and evaluation process was repeated using the conventional DOT reconstruction algorithm [2

**23**(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

### 2.4 Noise suppression

*α*, seen in Eqs. (3) and (5) is used to suppress the noise existing in real measurements. The optimal

*α*value for DOT is usually determined by an L-curve algorithm [11,12

12. P. C. Hansen and D. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. (USA) **14**(6), 1487–1503 (1993). [CrossRef]

*α*value has to be used for data with a higher noise level to suppress the noise more severely. However, a bigger

*α*simultaneously suppresses more subtle but true signals that are likely to stem from deep tissues and consequently results in a bigger localization error in depth. When the depth compensation algorithm is applied, this suppression becomes more complex since another parameter,

*γ*, is introduced to adjust the power of depth compensation. These two parameters may have interaction or crosstalk. Based on the setup of simulative experiment I, which is shown in Fig. 1(a), we investigated the potential interaction between

*α*and

*γ*in the following three aspects:

*Noise interference:*Two data sets were generated when calculating the forward problem: the first data set was noise-free and the second one was added with 1% random noise. For both data sets, images of the absorber were reconstructed with and without depth compensation, respectively. In each reconstructed image the location of voxel with maximum absorption perturbation was identified to represent the central location of the absorber. Since the regularization parameter

*α*was responsible for noise suppression, we altered the

*α*value in reconstruction from 1 to 10

^{−6}at a logarithm step size of 10

^{−1}(1, 10

^{−1}, 10

^{−2}, ..., 10

^{−6}) and then compared the results from data with and without random noise. In this way, the sensitivity of respective reconstruction algorithm (with or without depth compensation) to noise can be seen clearly.

*Depth localization:*For depth-compensated reconstruction, we varied the

*γ*value from 1.1, 1.3 to 1.5 simultaneously while modifying

*α*values to investigate if

*γ*had any cross effect with

*α*on depth localization. It is noteworthy that the conventional DOT algorithm without depth compensation is equivalent to the case of

*γ*= 0 when depth compensation is applied. For each pair of

*α*and

*γ*, the location of the reconstructed absorber was again represented by the location of voxel with maximum value of absorption perturbation.

*Quantification:*Using the data set with 1% random noise, values of Δ

*μ*and Δ

_{a_max}*μ*of the absorber were also quantified from the reconstructed images at variable

_{a_ROI}*α*and

*γ*values, and then compared between reconstruction algorithms with and without depth compensation.

## 3. Results

### 3.1 Reconstruction and quantification

*Experiment I:*Figs. 2(a) to 2(c) show depth cross sections (y-z plane, x = 0) of the actual absorber, the reconstructed DOT images without and with depth compensation from the simulation setup shown in Fig. 1(a). The reconstructed image without depth compensation has a significant localization error in depth. Its maximum value, Δ

*μ*, is 0.059 cm

_{a_max}^{−1}and located at about z = −1.4 cm, as seen in Fig. 1(b). By applying the half-maximum threshold, the recovered absorption perturbation within ROI, Δ

*μ*, remains to be 0.059 cm

_{a_ROI}^{−1}, which is about 30% of the actual value. After using depth compensation but without spatial prior, the reconstructed image recovers Δ

*μ*to be 0.070 cm

_{a_max}^{−1}and locates at z = −1.9 cm, as shown in Fig. 1(c). If half-maximum thresholding is utilized to define ROI as a spatial prior, the recovered absorption perturbation with depth-compensated reconstruction is dramatically improved to be 0.122 cm

^{−1}, about 61% of the actual value. It is mainly because the identified ROI in reconstructed image with depth compensation can retrieve the location and size of the actual absorber very well, as illustrated by the dash circle in Fig. 3(c) .

*Experiment II:*Figs. 3(a) and 3(b) show a depth cross section along the diagonal plane at y = x [shown by the dashed line in Fig. 3(b)] and a lateral cross section in the x-y plane at z = −2.0 cm for the two simulated absorbers [see Fig. 1(b)], respectively. Both absorbers are at z = −2.0 cm. The actual Δ

*μ*values of the two absorbers are 0.1 and 0.2 cm

_{a}^{−1}, respectively. Correspondingly, Figs. 3(c) and 3(d) show the depth and lateral cross sections of the reconstructed DOT images without depth compensation. Both of the absorbers are untruthfully projected to a depth of z = −1.4 cm. Therefore, z = −1.4 cm was chosen to plot the lateral cross section, as shown in Fig. 3(d). Without either depth compensation or spatial prior applied, the Δ

*μ*values for the two absorbers are 0.024 and 0.048 cm

_{a_max}^{−1}, about 24% of the actual values. In this case since the two absorbers can be clearly separated in the reconstructed images, two individual ROIs are identified in two sub regions using their respective half-maxima. The determined Δ

*μ*values for the two absorbers are 0.022 and 0.048 cm

_{a_ROI}^{−1}, as low as the Δ

*μ*values.

_{a_max}*μ*values for the two absorbers are 0.027 and 0.058 cm

_{a_max}^{−1}, having a recovery rate of 27-29% of the actual values. Then, after two sub ROIs (described above) are appropriately identified, the reconstructed values of Δ

*μ*for the two absorbers are improved to be 0.047 and 0.104 cm

_{a_ROI}^{−1}, with a recovery rate of 47-52%.

*Experiment III:*Fig. 4(a) shows a depth cross section (along diagonal plane at y = x) of the two simulated absorbers. The actual absorbers are located at z = −2.2 and −1.8 cm with the same Δ

*μ*value of 0.2 cm

_{a}^{−1}. Correspondingly, Fig. 4(b) shows the depth cross section of the reconstructed absorber without using depth compensation. Both absorbers are untruthfully projected to z = −1.4 cm, regardless of their actual depths. Without either depth compensation or ROI-based spatial prior applied, the recovered Δ

*μ*values for the two absorbers are 0.024 and 0.053 cm

_{a_max}^{−1}, largely different from one another and from the single true value (0.3 cm

^{−1}). After two separate ROIs are identified in two sub regions, using the same method as Experiment II, the determined Δ

*μ*values are 0.020 and 0.055 cm

_{a_ROI}^{−1}for the respective absorbers, as low and wrong as the Δ

*μ*values.

_{a_max}*μ*values for the two absorbers are 0.020 and 0.052 cm

_{a_max}^{−1}. Then after two spatial priors are applied in two sub regions, the recovered Δ

*μ*values of the respective absorbers are improved to be 0.075 and 0.091 cm

_{a_ROI}^{−1}, with recovery rate of 38-46% of the actual values. It is also noteworthy that the Δ

*μ*values become less different between the two absorbers.

_{a_ROI}*μ*is the actual value of absorption perturbation, Δ

_{a}*μ*and Δ

_{a_max}*μ*are the recovered values without and with the use of ROI-based spatial prior, respectively.

_{a_ROI}### 3.2 Noise suppression

*Noise interference:*Figs. 5(a) and 5(b) show the reconstructed depths of the absorber at variable

*α*and

*γ*values, using the data without and with noise, respectively. It is clearly seen in Fig. 5(b) that if 1% random noise is added in the data, for the two reconstruction algorithms with and without depth compensation, the recovered depths of absorber are stable/consistent only when

*α*is 10

^{−3}and bigger (marked by the dashed line in that figure). Thus, two reconstruction algorithms have approximately same sensitivity to noise.

*Depth localization:*Figs. 5(a) and 5(b) also illuminate how the depth localization is affected when both

*α*and

*γ*vary. For all

*α*values which can stabilize the inverse solutions (between 1 and 10

^{−6}for noise-free data, and between 1 and 10

^{−3}for data with 1% random noise), with the use of conventional DOT without depth compensation (

*γ*= 0), a smaller

*α*value results in a better accuracy in depth, approaching to, but never reaching the actual depth at z = −2 cm. When a big

*α*-value is used (which in principle should be used for more noisy data), the reconstructed image without depth compensation has a very significant depth error as the absorber is untruthfully projected toward the surface. When depth-compensated reconstruction is applied, the reconstructed depth of absorber is relatively stable, within ± 3 mm of its actual depth, rather independent of either

*α*or

*γ*(between 1.1 and 1.5). The overall results in Fig. 5 demonstrate that when depth compensation is used in DOT reconstruction, any value of

*γ*in the range of 1.1 to 1.5 is able to provide reliable depth localization with a 3-mm possible deviation for data at variable noise levels, even at a high noise level which may mandate the regularization parameter

*α*to be a large value of 0.1 or 1.

*Quantification:*Using the data set with 1% random noise, values of Δ

*μ*and Δ

_{a_max}*μ*of the absorber were also quantified at variable

_{a_ROI}*α*and

*γ*values. Figures 6(a) and 6(b) show the quantified Δ

*µ*and Δ

_{a_max}*µ*values of the absorber within

_{a_ROI}*α*≥ 10

^{−3}since the reconstruction becomes unstable when

*α*< 10

^{−3}[see Fig. 5(b)]. Note that each of these two figures includes the data obtained both without (

*γ*= 0) and with depth compensation. In addition, Fig. 6(c) shows the volumes of the identified ROI,

*V*(in cm

_{ROI}^{3}), for respective cases and within

*α*≥ 10

^{−3}. Without the ROI-based spatial prior, Fig. 6(a) plots Δ

*µ*values at four different

_{a_max}*γ*values, none of which can recover the actual absorption perturbation in a reasonable range. The best recovery rate is ~28% when

*α*= 10

^{−3}.

*µ*are in a range of 25-64% recovery with respect to the actual value of the absorber (Δ

_{a_ROI}*µ*= 0.2 cm

_{a}^{−1}) while both

*α*and

*γ*values are varied. Specifically, a smaller

*α*value seems to lead to an improved Δ

*µ*, as expected. At

_{a_ROI}*α*= 10

^{−3}, for example, the Δ

*µ*values are 52% and 62% recovered with

_{a_ROI}*γ*= 1.1 and 1.5, respectively. In contrast, Δ

*µ*shows a mere 5-27% recovery rate when depth-compensated reconstruction is not used (

_{a_ROI}*γ*= 0), even though the ROI-based spatial prior are formed appropriately.

*V*, on

_{ROI}*α*for all cases (without and with depth compensation in reconstruction), illustrating a decreasing pattern as

*α*values become smaller. This trend can account for the increasing trend in Δ

*µ*with reduced

_{a_ROI}*α*value, as seen in Fig. 6(b). At

*α*= 10

^{−3}, the values of

*V*approach the actual volume of the absorber, which is marked by the dash line in Fig. 6(c), for variable

_{ROI}*γ*values.

## 4. Discussion and conclusion

**35**(3), 429–431 (2010). [CrossRef] [PubMed]

*µ*, of absorption perturbations in each experiment are comparable to those obtained without using any depth compensation. But, none of them can recover well the true values of absorption perturbation. Then after defining an appropriate ROI by half-maximum thresholding in the reconstructed image with depth compensation, the accuracy in quantification is improved greatly in each of the three experiments. More significant improvement has been observed for deeper absorbers, as demonstrated in experiment III (Fig. 4). The results given throughout all three experiments are reasonable and expected since the spatial priors with depth-compensated reconstruction are relatively close to the actual locations and sizes of the absorber(s) in each experiment. Since the spatial prior utilized in this approach also depends on the spatial resolution of reconstructed image, we expect that the accuracy of quantification using this approach may vary somewhat for different probe geometries. Such expectation needs to be studied in future.

_{a_max}*α*and

*γ*were varied. The reconstruction outputs from the data without and with 1% random noise illustrate that the two reconstruction algorithms, i.e., reconstruction with and without depth compensation, have approximately same sensitivity to noise. For all

*α*values which can stabilize the inverse solutions, however, the recovered DOT images without applying depth compensation cannot exhibit the correct depth (z = −2 cm) of the absorber, regardless of any

*α*value used. On the other hand, if depth compensation is employed, the reconstructed depths are much closer to the expected value (z = −2 cm) without any restriction in

*α*values for noise-free data and with

*α*larger than 10

^{−3}for the data having 1% random noise.

*α*value is employed for more noisy data, conventional DOT algorithm (

*γ*= 0) has a very significant depth error. On the other hand, when depth compensation algorithm is combined in the DOT reconstruction, the recovered depth of the absorber is relatively stable across a broad range of

*α*values when an optimal

*γ*range is selected (between 1.1 and 1.5). It is a strong support that the depth compensation algorithm can provide reliable and accurate depth localization for data at variable noise levels. This finding is practically important with the consideration of the physiological noises to be encountered during

*in vivo*human brain measurements [2

**23**(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

## Acknowledgement

## References and links

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

2. | D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage |

3. | C. K. Lee, C. W. Sun, P. L. Lee, H. C. Lee, C. Yang, C. P. Jiang, Y. P. Tong, T. C. Yeh, and J. C. Hsieh, “Study of photon migration with various source-detector separations in near-infrared spectroscopic brain imaging based on three-dimensional Monte Carlo modeling,” Opt. Express |

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

5. | H. Niu, F. Tian, Z. J. Lin, and H. Liu, “Development of a compensation algorithm for accurate depth localization in diffuse optical tomography,” Opt. Lett. |

6. | H. Niu, Z. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive Investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed. Opt. in press. |

7. | M. Cope, D. T. Delpy, E. O. Reynolds, S. Wray, J. Wyatt, and P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. |

8. | H. Dehghani, B. W. Pogue, J. Shudong, B. Brooksby, and K. D. Paulsen, “Three-dimensional optical tomography: resolution in small-object imaging,” Appl. Opt. |

9. | G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. |

10. | E. R. Hom, http://www.nmr.mgh.harvard.edu/PMI/toolbox/index.html |

11. | L. Wu, “A parameter choice method for Tikhonov regularization,” Electron. Trans. Numer. Anal. |

12. | P. C. Hansen and D. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. (USA) |

13. | B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U.S.A. |

**OCIS Codes**

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(170.6960) Medical optics and biotechnology : Tomography

(170.2655) Medical optics and biotechnology : Functional monitoring and imaging

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: June 8, 2010

Revised Manuscript: July 26, 2010

Manuscript Accepted: July 26, 2010

Published: August 2, 2010

**Virtual Issues**

Optical Imaging and Spectroscopy (2010) *Biomedical Optics Express*

**Citation**

Fenghua Tian, Haijing Niu, Sabin Khadka, Zi-Jing Lin, and Hanli Liu, "Algorithmic depth compensation improves quantification and noise suppression in functional diffuse optical tomography," Biomed. Opt. Express **1**, 441-452 (2010)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-2-441

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### References

- A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. 20(10), 435–442 (1997). [CrossRef] [PubMed]
- D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage 23(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]
- C. K. Lee, C. W. Sun, P. L. Lee, H. C. Lee, C. Yang, C. P. Jiang, Y. P. Tong, T. C. Yeh, and J. C. Hsieh, “Study of photon migration with various source-detector separations in near-infrared spectroscopic brain imaging based on three-dimensional Monte Carlo modeling,” Opt. Express 13(21), 8339–8348 (2005). [CrossRef] [PubMed]
- R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]
- H. Niu, F. Tian, Z. J. Lin, and H. Liu, “Development of a compensation algorithm for accurate depth localization in diffuse optical tomography,” Opt. Lett. 35(3), 429–431 (2010). [CrossRef] [PubMed]
- H. Niu, Z. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive Investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed. Opt. in press.
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