## Hierarchical Bayesian regularization of reconstructions for diffuse optical tomography using multiple priors |

Biomedical Optics Express, Vol. 1, Issue 4, pp. 1084-1103 (2010)

http://dx.doi.org/10.1364/BOE.1.001084

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### Abstract

Diffuse optical tomography (DOT) is a non-invasive brain imaging technique that uses low-levels of near-infrared light to measure optical absorption changes due to regional blood flow and blood oxygen saturation in the brain. By arranging light sources and detectors in a grid over the surface of the scalp, DOT studies attempt to spatially localize changes in oxy- and deoxy-hemoglobin in the brain that result from evoked brain activity during functional experiments. However, the reconstruction of accurate spatial images of hemoglobin changes from DOT data is an ill-posed linearized inverse problem, which requires model regularization to yield appropriate solutions. In this work, we describe and demonstrate the application of a parametric restricted maximum likelihood method (ReML) to incorporate multiple statistical priors into the recovery of optical images. This work is based on similar methods that have been applied to the inverse problem for magnetoencephalography (MEG). Herein, we discuss the adaptation of this model to DOT and demonstrate that this approach provides a means to objectively incorporate reconstruction constraints and demonstrate this approach through a series of simulated numerical examples.

© 2010 OSA

## 1. Introduction

1. 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]

_{2}and Hb respectively) within the brain using multiple wavelengths of light, which can potentially lead to the ability to discriminate blood flow and oxygen metabolism changes [2

2. T. J. Huppert, M. S. Allen, S. G. Diamond, and D. A. Boas, “Estimating cerebral oxygen metabolism from fMRI with a dynamic multicompartment Windkessel model,” Hum. Brain Mapp. **30**(5), 1548–1567 (2009) (PMCID: 2670946.). [CrossRef] [PubMed]

3. T. Wilcox, H. Bortfeld, R. Woods, E. Wruck, and D. A. Boas, “Using near-infrared spectroscopy to assess neural activation during object processing in infants,” J. Biomed. Opt. **10**(1), 011010 (2005). [CrossRef] [PubMed]

4. S. Perrey, “Non-invasive NIR spectroscopy of human brain function during exercise,” Methods **45**(4), 289–299 (2008). [CrossRef] [PubMed]

5. I. Miyai, H. C. Tanabe, I. Sase, H. Eda, I. Oda, I. Konishi, Y. Tsunazawa, T. Suzuki, T. Yanagida, and K. Kubota, “Cortical mapping of gait in humans: a near-infrared spectroscopic topography study,” Neuroimage **14**(5), 1186–1192 (2001). [CrossRef] [PubMed]

6. U. Sunar, S. Makonnen, C. Zhou, T. Durduran, G. Yu, H. W. Wang, W. M. Lee, and A. G. Yodh, “Hemodynamic responses to antivascular therapy and ionizing radiation assessed by diffuse optical spectroscopies,” Opt. Express **15**(23), 15507–15516 (2007). [CrossRef] [PubMed]

7. U. Sunar, H. Quon, T. Durduran, J. Zhang, J. Du, C. Zhou, G. Yu, R. Choe, A. Kilger, R. Lustig, L. Loevner, S. Nioka, B. Chance, and A. G. Yodh, “Noninvasive diffuse optical measurement of blood flow and blood oxygenation for monitoring radiation therapy in patients with head and neck tumors: a pilot study,” J. Biomed. Opt. **11**(6), 064021 (2006). [CrossRef] [PubMed]

8. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**(2), 14–93 (1999). [CrossRef]

9. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**(4), R1–R43 (2005). [CrossRef] [PubMed]

*a priori*choice of this weight or weights in the case of multiple priors.

10. J. Mattout, C. Phillips, W. D. Penny, M. D. Rugg, and K. J. Friston, “MEG source localization under multiple constraints: an extended Bayesian framework,” Neuroimage **30**(3), 753–767 (2006). [CrossRef] [PubMed]

12. R. W. Cox, “AFNI: software for analysis and visualization of functional magnetic resonance neuroimages,” Comput. Biomed. Res. **29**(3), 162–173 (1996). [CrossRef] [PubMed]

## 2. Theory

### The optical forward model

1. 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]

*I*is the intensity of light exiting the tissue and

*I*

_{o}is the light entering the tissue. G is a geometry dependent factor. In Eq. (1),

*υ*is an additive noise in the measurement space (e.g. instrument noise), which will be emphasized further in the context of the ReML model.

*L*is the optical measurement model obtained from estimation of the ensemble path of photons through the tissue and describes the summation of absorption values

_{ij}^{λ}*μ*

_{A}along the diffuse path traveled by the light going from a particular light emitter to a detector pair (i,j). Both

*μ*

_{A}and

*μ*

_{s}are vectors of the absorption and scattering values at each position in the volume and can be reshaped as an image of these changes.

*OD*is approximated by linearization of Eq. (1) around the baseline values of

*μ*

_{A}and

*μ*

_{s}and subtraction of the baseline absorption.

*Δμ*

_{A}is a vector of the changes in absorption at each position (voxel) in the underlying tissue. A

_{ij}^{λ}is the Jacobian of the optical measurement model. Equation (3), describes the optical forward model describing the change in optical signal caused by changes in the absorption in the underlying tissue for one particular wavelength of light and set of baseline optical properties. Typically in brain imaging studies, two or more wavelengths of light are used to provide an ability to distinguish changes in both oxy-hemoglobin (HbO

_{2}) and deoxy-hemoglobin (Hb). The overall absorption at each wavelength is a linear combination of the contributions from each chromophore and is given by the Beer-Lambert expressionwhere ε

^{λ}

_{HbX}is the molar extinction coefficient for oxy- or deoxy-hemoglobin at the particular wavelength and describes the wavelength specific absorption properties of these chromophores per molar unit of concentration. ω

_{HbX}is a second type of additive noise (or uncertainty error) term acting in the image (brain) space and is distinct from the measurement space noise (

*υ*). We will clarify this distinction later in the context of the ReML model. Again,

*Δμ*

_{A},

*Δ[HbX]*, and

*ω*are vectors representing these changes at each position in the tissue. Equation (4) can be substituted into the optical forward model to produce the optical measurement model with spectral priors (e.g. Li et al [13

13. A. Li, Q. Zhang, J. P. Culver, E. L. Miller, and D. A. Boas, “Reconstructing chromosphere concentration images directly by continuous-wave diffuse optical tomography,” Opt. Lett. **29**(3), 256–258 (2004). [CrossRef] [PubMed]

*) on baseline absorption and scattering coefficients will be no longer explicitly written (e.g. A*

_{i,j}^{λ}

_{i,j}= A

^{λ}

_{i,j}(

*μ*

^{λ}

_{A,}

*μ*

^{λ}

_{S}’) ). Changes in oxy- and deoxy-hemoglobin can be inferred from optical measurements at multiple (

*N*) wavelengths by means of solving a set of linear equations given by

_{i}denotes the

*i*

^{th}-wavelength. Hereafter, the optical forward model will be written in the more compact formwhere the new variable

*β*has been introduced to describe the unknown values of the combination of oxy- and deoxy-hemoglobin changes in the tissue, given by

_{2}and Hb in the tissue, and the changes in optical density as recorded on the surface between optical sources and detectors. It is this equation that must be inverted in order to reconstruct an image (volume) of the hemodynamic changes in the brain.

### The optical inverse problem

*Y*) than unknown parameters (

*β)*in the image to-be-estimated. This means that, in general, there is not enough information in the measurements alone to yield accurate and unique estimates of images of brain activity. There are two general approaches to solving this problem— regularization and Bayesian theories. In general, regularization theory (including Tikhonov regularization) has been most widely used and is more familiar to practitioners of the optical inverse problem. On the other hand, ReML and our current work are based on the Bayesian interpretation. For this reason, we will briefly attempt to reconcile these two theories noting that for a subset of regularization models in the class of linear-quadratic regularization (which includes many of the current optical inverse models), there is an equivalent Bayesian interpretation of the model.

_{0}) and is given by the minimization expression

_{0}). A typical assumption in the optical inverse model is that β

_{0}is zero, which results in what is called the minimum norm solution. The regularization model can be extended to add additional penalty terms. For example, Li

*et al*[13

13. A. Li, Q. Zhang, J. P. Culver, E. L. Miller, and D. A. Boas, “Reconstructing chromosphere concentration images directly by continuous-wave diffuse optical tomography,” Opt. Lett. **29**(3), 256–258 (2004). [CrossRef] [PubMed]

*M*specifies a binary mask of a predefined region-of-interest such that

_{1}and λ

_{2}), which applied penalties to the parameters inside and outside of the region-of-interest respectively. Alternative regularization models have been proposed to add low-pass or high-pass operators to impose smoothness on the solution. In regularization models, the L-curve technique and generalized cross-validation can be used to optimally select the hyperparameters of the model. However to date, many optical reconstruction methods have used λ as a manual tuning parameter allowing images to be adjusted in a subjective optimization. In general terms, the regularization hyperparameter (λ) is a weight that is assigned to that penalty term in the cost function.

_{0}). In the regularization model, these distance penalties (e.g. N and λ·P in Eq. (10) can be somewhat arbitrary provided that they are symmetric matrices. In contrast, the Bayesian model offers an alternative interpretation by suggesting that the optimal distance weight should be the inverse of a covariance matrix. For example, in Eq. (10), the weighted norm penalty N should be the inverse of the measurement noise covariance and from the second term, the value of λ·P should be the inverse of the parameter covariance. In terms of the optical inverse model, these two terms are the covariance of υ and ω respectively from Eq. (7).

_{N}and C

_{P}following the convention of the SPM software).

### Restricted Maximum Likelihood (ReML)

14. D. Harville, “Maximum likelihood approaches to variance component estimation and related problems,” J. Am. Stat. Assoc. **72**(358), 320–338 (1977). [CrossRef]

*MIXED*function. In the context of neuroimaging ReML was introduced by Friston

*et al*for the stabilization of the temporal deconvolution model used for analysis of brain activity images in functional MRI [15

15. K. J. Friston, W. Penny, C. Phillips, S. Kiebel, G. Hinton, and J. Ashburner, “Classical and Bayesian inference in neuroimaging: theory,” Neuroimage **16**(2), 465–483 (2002). [CrossRef] [PubMed]

16. K. J. Friston, D. E. Glaser, R. N. Henson, S. Kiebel, C. Phillips, and J. Ashburner, “Classical and Bayesian inference in neuroimaging: applications,” Neuroimage **16**(2), 484–512 (2002). [CrossRef] [PubMed]

10. J. Mattout, C. Phillips, W. D. Penny, M. D. Rugg, and K. J. Friston, “MEG source localization under multiple constraints: an extended Bayesian framework,” Neuroimage **30**(3), 753–767 (2006). [CrossRef] [PubMed]

15. K. J. Friston, W. Penny, C. Phillips, S. Kiebel, G. Hinton, and J. Ashburner, “Classical and Bayesian inference in neuroimaging: theory,” Neuroimage **16**(2), 465–483 (2002). [CrossRef] [PubMed]

16. K. J. Friston, D. E. Glaser, R. N. Henson, S. Kiebel, C. Phillips, and J. Ashburner, “Classical and Bayesian inference in neuroimaging: applications,” Neuroimage **16**(2), 484–512 (2002). [CrossRef] [PubMed]

_{N}and C

_{P}) from Eq. (14), the covariance models can be parameterized as a linear combination of covariance components. For example:where Q

_{N}and Q

_{P}are symmetric matrices that can be used to build up the covariance model. In the example of the optical model, C

_{N}represents the covariance of the measurement noise and thus, two (or more) diagonal covariance components (Q

_{N}) might be used with each representing the variance on one of the two (or more) measured optical wavelengths. In the methods section, we will further detail the selection of these components for the optical model. The hyper-parameters (Λ; upper-case lambda) in Eq. (15) adjust the weighting of these covariance components. Again, in the context of the two wavelength optical model, there would be two hyper-parameters allowing adjustment of the noise at the two wavelengths. Note in reference to the work by Friston et al [15

15. K. J. Friston, W. Penny, C. Phillips, S. Kiebel, G. Hinton, and J. Ashburner, “Classical and Bayesian inference in neuroimaging: theory,” Neuroimage **16**(2), 465–483 (2002). [CrossRef] [PubMed]

## 3. Methods

### Calculation of optical forward model

18. D. K. Joseph, T. J. Huppert, M. A. Franceschini, and D. A. Boas, “Diffuse optical tomography system to image brain activation with improved spatial resolution and validation with functional magnetic resonance imaging,” Appl. Opt. **45**(31), 8142–8151 (2006). [CrossRef] [PubMed]

### Wavelet reparameterization of DOT inverse model

*H*and the vector of unknowns

*β*. The matrix

*H*projects the hemoglobin concentration changes from points in the volume of tissue to the expected optical density changes measured at the surface by a particular grid of optical source-detector pairs. In a recent paper [19

19. F. Abdelnour, B. Schmidt, and T. J. Huppert, “Topographic localization of brain activation in diffuse optical imaging using spherical wavelets,” Phys. Med. Biol. **54**(20), 6383–6413 (2009) (PMCID: 2806654.). [CrossRef] [PubMed]

19. F. Abdelnour, B. Schmidt, and T. J. Huppert, “Topographic localization of brain activation in diffuse optical imaging using spherical wavelets,” Phys. Med. Biol. **54**(20), 6383–6413 (2009) (PMCID: 2806654.). [CrossRef] [PubMed]

*β*by first reparametrizing the model using orthogonal wavelet transform and then estimating the coefficients

*β*

_{w}of the wavelet transform of

*β*. The orthogonal wavelet transformation is a reversible rotation which can be expressed in a matrix notation such thatwhere W is the wavelet analysis model (transformation from image to wavelet space). Here, . In this work we use the Daubechies wavelet [20] generated by FIR orthogonal filters of length 2 coefficients and separable in the x-y (in-plane) dimension. Equation (7) is now restated for each layer in terms of wavelets as

_{W}) is now also in the wavelet domain. Since we intend to use a structure to the covariance components, which allows a non-white spatial frequency distribution (particularly to model systemic physiological noise), we will define the covariance components of the ReML model directly in the wavelet domain. The structure of the wavelet matrix W is shown in Fig. 3 . The low-pass, band-pass, and high-pass components map to regions of the matrix. Figure 3 shows the structure of this model for the one-dimensional with only 2 stages for simplicity. The actual model used a two-dimensional structure (in the x/y plane) with three stages.

### Example Covariance Components

*Minimum norm prior.*In order to compare against the Tikhonov regularization methods, the minimum norm (covariance) prior should take the form C

_{N}= Λ

_{1}·I and C

_{P}= Λ

_{2}·I where both oxy- and oxy-hemoglobin are modeled by a single covariance component and hyperparameter. C

_{N}and C

_{P}define the total noise model via Eq. (15). This produces the effect of a single hyperparameter to tune the model and is equivalent to the Tikhonov regularization modelwhere the ratio of Λ

_{1}and Λ

_{2}from the Bayesian model are replaced by the single regularization term λ.

*Measurement noise prior.*In general, optical recordings will have different noise depending on the wavelength. While this is less of a concern for systems with only two measured wavelengths, combining more than two wavelengths into estimates of oxy- and deoxy-hemoglobin via the modified Beer-Lambert law requires an estimate of the noise at each wavelength leading to the weighted least-squares model. In order to model this, C

_{N}is modeled by a separate component for each of the measurement types with unity values allow the corresponding diagonal elements for each wavelength type. Thus, the two-wavelength optical model, which we will use in this work, will have two hyper-parameters to define the measurement covariance (C

_{N}). While this paper is concerned with optical-only reconstructions, we note that this approach is amendable to multimodal data as well, for example, the joint image reconstruction of brain activity from concurrent optical and functional MRI data as shown in Huppert et al [21

21. T. J. Huppert, S. G. Diamond, and D. A. Boas, “Direct estimation of evoked hemoglobin changes by multimodality fusion imaging,” J. Biomed. Opt. **13**(5), 054031 (2008). [CrossRef] [PubMed]

*Depth specific spatial frequency priors.*As previously discussed, the reparameterization of the optical forward model via the wavelet transform allows statistical priors to impose relationships between the levels of spatial frequency. Namely, the variance of the corresponding low-pass, band-pass, and high-pass wavelet coefficients can be reweighted according to a priori assumptions, such as the expectation that superficial (systemic) signals will be low frequency. By weighting the variance between each frequency band, a covariance component acting as a low-pass filter can be constructed.

*Incorporating prior knowledge of location of ROI*. Finally, the covariance components of the ReML model can be used to impose a priori knowledge of regions-of-interest for the location of activation. Such prior information can be obtained for example from experience or from alternate modality such as functional MRI or atlas based priors. The resulting Q’s can then be given by the diagonal matrices for HbO

_{2}and Hb:

## 4. Results

### Comparison of ReML and L-curve

_{N}= Λ

_{1}·I and C

_{P}= Λ

_{2}·I). This model allows direct comparison to the L-curve approach to defining λ in Eq. (19) (λ = Λ

_{1}/Λ

_{2}). A single layered image with a depth of 1cm was generated (16 x 16 x 1 voxels [6.7mm6.7xmmx10mm]) and a colocalized oxy-hemoglobin [1μM; Fig. 4 (A1)] and deoxy-hemoglobin [−0.25μM; Fig. 4(B1)] perturbation was added. In Fig. 4, data was generated contrast-to-noise ratio of 100:1 by adding random zero-mean measurement noise and reconstructed using the ReML procedure [Fig. 4(A2) and 4(B2)]. A L-curve was generated and used to select the optimal regularization [Fig. 4(A3) and 4(B3)]. In Fig. 5 , the same model is shown but for a lower signal-to-noise level of 5:1. In the higher noise simulations, background noise is more clearly pronounced in the reconstructed images. As expected for this trivial case of minimum norm (covariance) prior, the L-curve and ReML estimation routines produced quantitatively similar reconstructions of both oxy- and deoxy-hemoglobin at both 50:1 and 5:1 signal-to-noise levels. In Fig. 6 , we further compare the performance of the L-curve and ReML models through a range of contrast-to-noise levels from 100,000:1 (little noise) to 1:10 (more noise than signal). Over the majority of this range, the two methods agree closely with each other and the theoretical optimal parameter. At very low single-to-noise levels, the L-curve tended to overestimate the regularization, which was the result of numerical instabilities in finding the corner of the L-curve. Nevertheless, we concluded that the two approaches were comparable over a large range of noise. This result was actually expected since discussion in Mattout et al [10

10. J. Mattout, C. Phillips, W. D. Penny, M. D. Rugg, and K. J. Friston, “MEG source localization under multiple constraints: an extended Bayesian framework,” Neuroimage **30**(3), 753–767 (2006). [CrossRef] [PubMed]

### Incorporation of physiological priors

*a priori*. In addition, oxy- and deoxy-hemoglobin changes are also subject to different noise contributions from superficial and systemic physiology; e.g. cardiac pulsation which preferentially contributes to noise in oxy-hemoglobin. In order to account for this, the covariance of oxy- and deoxy-hemoglobin parameters can be independently estimated through the inclusion of separate covariance components for each. In the context of our current simulations, this introduces a total of four covariance components (one per each of the two wavelengths measured and one per oxy- and deoxy-hemoglobin across the image).

*λ*is selected via the L-curve method to provide reasonable reconstruction of the oxy-hemoglobin component. However, because this

*λ*is also applied to the deoxy-hemoglobin component, the reconstructed doxy-hemoglobin image [Fig. 7(B3)] shows significant noise and artifacts of similar magnitude to

*λ*. In contrast, in the ReML method, because the regularization of oxy- and deoxy-hemoglobin is individually determined, a lower variance in the deoxy-hemoglobin model is adapted and the resulting artifacts are considerably lower. In the case of the EM model, the cross-talk in the deoxy-hemoglobin is close to negligible (<0.1%).

### Example of depth-specific regularization

22. D. A. Boas and A. M. Dale, “Simulation study of magnetic resonance imaging-guided cortically constrained diffuse optical tomography of human brain function,” Appl. Opt. **44**(10), 1957–1968 (2005). [CrossRef] [PubMed]

### Contamination from superficial noise

*a priori*that the upper layer has lower spatial frequency activities than in the lower layer, we impose covariance components that act as low-pass filters in this layer as described in Eq. (21). The wavelet coefficients for a given layer are increasingly attenuated as the spatial frequency increases. The two layers are assigned different attenuation rates, with the upper layer having σ = 2.2 voxels (15mm), and the lower layer with σ = 1 voxels (no low-pass filtering). This leads to a total of four covariance components, taking into account HbO

_{2}and Hb with the general form as given in Eq. (21).

### Incorporation of a priori region-of-interest information

_{2}and Hb for each layer (as shown previously). Figure 11(A1) shows the case of the image to be estimated with two distinct regions of activities, while Fig. 11(B1) shows the case of two regions of interest located near each other. Figures 11(A2) and 11(B2) depict the reconstructed images for the case where only covariance matrices separating the layers and HbO

_{2}and Hb for each layer. In Fig. 11(A2) the recovered image shows accurate recovery of the functional activities. In Fig. 11(B2) the spatially close regions of activities lead to a degree of ambiguity in the reconstructed image, where the boundaries of the two regions of activities tend to overlap. By using in addition covariance components describing the regions of interest for both oxy and deoxy-hemoglobin, visibly improved reconstruction is possible, as shown in Figs. 11(A3) and 11-B3 where in the latter it is possible to discriminate the two reconstructed activities.

## 5. Discussion

13. A. Li, Q. Zhang, J. P. Culver, E. L. Miller, and D. A. Boas, “Reconstructing chromosphere concentration images directly by continuous-wave diffuse optical tomography,” Opt. Lett. **29**(3), 256–258 (2004). [CrossRef] [PubMed]

**30**(3), 753–767 (2006). [CrossRef] [PubMed]

23. B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt. **11**(3), 033001 (2006). [CrossRef] [PubMed]

**29**(3), 256–258 (2004). [CrossRef] [PubMed]

21. T. J. Huppert, S. G. Diamond, and D. A. Boas, “Direct estimation of evoked hemoglobin changes by multimodality fusion imaging,” J. Biomed. Opt. **13**(5), 054031 (2008). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | 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 |

2. | T. J. Huppert, M. S. Allen, S. G. Diamond, and D. A. Boas, “Estimating cerebral oxygen metabolism from fMRI with a dynamic multicompartment Windkessel model,” Hum. Brain Mapp. |

3. | T. Wilcox, H. Bortfeld, R. Woods, E. Wruck, and D. A. Boas, “Using near-infrared spectroscopy to assess neural activation during object processing in infants,” J. Biomed. Opt. |

4. | S. Perrey, “Non-invasive NIR spectroscopy of human brain function during exercise,” Methods |

5. | I. Miyai, H. C. Tanabe, I. Sase, H. Eda, I. Oda, I. Konishi, Y. Tsunazawa, T. Suzuki, T. Yanagida, and K. Kubota, “Cortical mapping of gait in humans: a near-infrared spectroscopic topography study,” Neuroimage |

6. | U. Sunar, S. Makonnen, C. Zhou, T. Durduran, G. Yu, H. W. Wang, W. M. Lee, and A. G. Yodh, “Hemodynamic responses to antivascular therapy and ionizing radiation assessed by diffuse optical spectroscopies,” Opt. Express |

7. | U. Sunar, H. Quon, T. Durduran, J. Zhang, J. Du, C. Zhou, G. Yu, R. Choe, A. Kilger, R. Lustig, L. Loevner, S. Nioka, B. Chance, and A. G. Yodh, “Noninvasive diffuse optical measurement of blood flow and blood oxygenation for monitoring radiation therapy in patients with head and neck tumors: a pilot study,” J. Biomed. Opt. |

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

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

10. | J. Mattout, C. Phillips, W. D. Penny, M. D. Rugg, and K. J. Friston, “MEG source localization under multiple constraints: an extended Bayesian framework,” Neuroimage |

11. | K. J. Friston, Statistical parametric mapping: the analysis of functional brain images. 2007, London: Academic. vii, 647. |

12. | R. W. Cox, “AFNI: software for analysis and visualization of functional magnetic resonance neuroimages,” Comput. Biomed. Res. |

13. | A. Li, Q. Zhang, J. P. Culver, E. L. Miller, and D. A. Boas, “Reconstructing chromosphere concentration images directly by continuous-wave diffuse optical tomography,” Opt. Lett. |

14. | D. Harville, “Maximum likelihood approaches to variance component estimation and related problems,” J. Am. Stat. Assoc. |

15. | K. J. Friston, W. Penny, C. Phillips, S. Kiebel, G. Hinton, and J. Ashburner, “Classical and Bayesian inference in neuroimaging: theory,” Neuroimage |

16. | K. J. Friston, D. E. Glaser, R. N. Henson, S. Kiebel, C. Phillips, and J. Ashburner, “Classical and Bayesian inference in neuroimaging: applications,” Neuroimage |

17. | A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc., B |

18. | D. K. Joseph, T. J. Huppert, M. A. Franceschini, and D. A. Boas, “Diffuse optical tomography system to image brain activation with improved spatial resolution and validation with functional magnetic resonance imaging,” Appl. Opt. |

19. | F. Abdelnour, B. Schmidt, and T. J. Huppert, “Topographic localization of brain activation in diffuse optical imaging using spherical wavelets,” Phys. Med. Biol. |

20. | I. Daubechies, Ten Lectures On Wavelets. SIAM, 1992. |

21. | T. J. Huppert, S. G. Diamond, and D. A. Boas, “Direct estimation of evoked hemoglobin changes by multimodality fusion imaging,” J. Biomed. Opt. |

22. | D. A. Boas and A. M. Dale, “Simulation study of magnetic resonance imaging-guided cortically constrained diffuse optical tomography of human brain function,” Appl. Opt. |

23. | B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt. |

**OCIS Codes**

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

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

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: August 19, 2010

Revised Manuscript: October 2, 2010

Manuscript Accepted: October 2, 2010

Published: October 6, 2010

**Citation**

Farras Abdelnour, Christopher Genovese, and Theodore Huppert, "Hierarchical Bayesian regularization of reconstructions for diffuse optical tomography using multiple priors," Biomed. Opt. Express **1**, 1084-1103 (2010)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-4-1084

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### References

- 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]
- T. J. Huppert, M. S. Allen, S. G. Diamond, and D. A. Boas, “Estimating cerebral oxygen metabolism from fMRI with a dynamic multicompartment Windkessel model,” Hum. Brain Mapp. 30(5), 1548–1567 (2009) (PMCID: 2670946.). [CrossRef] [PubMed]
- T. Wilcox, H. Bortfeld, R. Woods, E. Wruck, and D. A. Boas, “Using near-infrared spectroscopy to assess neural activation during object processing in infants,” J. Biomed. Opt. 10(1), 011010 (2005). [CrossRef] [PubMed]
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