## Singular value decomposition metrics show limitations of detector design in diffuse fluorescence tomography |

Biomedical Optics Express, Vol. 1, Issue 5, pp. 1514-1531 (2010)

http://dx.doi.org/10.1364/BOE.1.001514

Acrobat PDF (1854 KB)

### Abstract

The spatial resolution and recovered contrast of images reconstructed from diffuse fluorescence tomography data are limited by the high scattering properties of light propagation in biological tissue. As a result, the image reconstruction process can be exceedingly vulnerable to inaccurate prior knowledge of tissue optical properties and stochastic noise. In light of these limitations, the optimal source-detector geometry for a fluorescence tomography system is non-trivial, requiring analytical methods to guide design. Analysis of the singular value decomposition of the matrix to be inverted for image reconstruction is one potential approach, providing key quantitative metrics, such as singular image mode spatial resolution and singular data mode frequency as a function of singular mode. In the present study, these metrics are used to analyze the effects of different sources of noise and model errors as related to image quality in the form of spatial resolution and contrast recovery. The image quality is demonstrated to be inherently noise-limited even when detection geometries were increased in complexity to allow maximal tissue sampling, suggesting that detection noise characteristics outweigh detection geometry for achieving optimal reconstructions.

© 2010 OSA

## 1. Introduction

1. V. Ntziachristos and R. Weissleder, “Charge-coupled-device based scanner for tomography of fluorescent near-infrared probes in turbid media,” Med. Phys. **29**(5), 803–809 (2002). [CrossRef] [PubMed]

2. D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. **14**(3), 030501 (2009). [CrossRef] [PubMed]

3. D. Kepshire, N. Mincu, M. Hutchins, J. Gruber, H. Dehghani, J. Hypnarowski, F. Leblond, M. Khayat, and B. W. Pogue, “A microcomputed tomography guided fluorescence tomography system for small animal molecular imaging,” Rev. Sci. Instrum. **80**(4), 043701 (2009). [CrossRef] [PubMed]

## 2. Background material

### 2.1 Imaging system and geometry

3. D. Kepshire, N. Mincu, M. Hutchins, J. Gruber, H. Dehghani, J. Hypnarowski, F. Leblond, M. Khayat, and B. W. Pogue, “A microcomputed tomography guided fluorescence tomography system for small animal molecular imaging,” Rev. Sci. Instrum. **80**(4), 043701 (2009). [CrossRef] [PubMed]

*N*of photodetectors, separated by an angle

_{d}*θ,*and opposing the point of illumination. The laser and photodetectors are fixed on a gantry that can rotate 360° around the specimen with an angular separation providing a number

*N*of laser projections. In this theoretical study, fluorescence as well as transmission (excitation) measurements were simulated, allowing reconstructions to be performed based on fluorescence-to-transmission ratio data (Born normalization) [4

_{p}4. V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. **26**(12), 893–895 (2001). [CrossRef] [PubMed]

*i.e.,*CCD detection) was emulated by minimizing

*θ*and maximizing

*N*.

_{d}### 2.2 Forward modeling and image reconstruction

5. F. Leblond, H. Dehghani, D. Kepshire, and B. W. Pogue, “Early-photon fluorescence tomography: spatial resolution improvements and noise stability considerations,” J. Opt. Soc. Am. A **26**(6), 1444–1457 (2009). [CrossRef] [PubMed]

*D*is a column vector with

_{ρ}*N*

_{m}=

*N*

_{d}×

*N*

_{p}entries corresponding to the fluorescence-to-transmission ratio measurements obtained with the representative geometry shown in Fig. 1. The column vector

*C*

_{a}(

*a*= 1,2,…,

*N*

_{v}) is composed of entries representing the concentration (

*e.g.*, in units of Molars or μg/ml) of fluorophores in each of the

*N*

_{v}voxels contained in the discretized volume (

*e.g.*, an FEM mesh) associated with the interrogated specimen. Finally, the

*N*

_{m}×

*N*

_{v}matrix

*A*is derived from the light transport model of choice and relates the CW measurements vector to the fluorophore concentration vector as shown in Eq. (1). If the diffusion approximation to the radiative-transport equation (RTE) is used, the forward model matrix takes the form,

*Q*

_{F}is the quantum efficiency of the fluorophore,

*ε*

_{F}the extinction coefficient and

*τ*the lifetime. The vectors

*r*_{s},

*r*_{d}and

*r*_{a}(

*a*= 1,2,…,

*N*

_{v}) represent the location of a laser source point (labeled

*s*) projected on the specimen, a detection point (labeled

*d*) also projected on the specimen and the position of a voxel inside the interrogated medium, respectively. The functions Φ

*and Φ*

^{x}*are fluence fields at the excitation and the fluorescence emission wavelengths, respectively. These fields are computed by solving the diffusion equation on a discretized mesh either numerically, or analytically in cases where simple imaging geometries are considered. Fluorescence tomography involves the retrieval of the vector*

^{e}*C*in Eq. (1) − a volumetric fluorescence image − through the resolution of an inverse problem. An analysis of the inverse problem for guiding source-detector geometry optimization is discussed in the following section.

_{a}### 2.3 Singular value decomposition analysis for imaging geometry optimization

*a priori*assessment of the performance of various imaging geometry setups in diffuse optical tomography (DOT) as well as in diffuse FT [6

6. J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. **26**(10), 701–703 (2001). [CrossRef] [PubMed]

7. E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A **21**(2), 231–241 (2004). [CrossRef] [PubMed]

*A*into two sets of singular vectors and a finite number of singular values (Eq. (2).1) in Ref [8].),

*N*>

_{v}*N*,

_{m}*U*= (

*c*, …,

_{1}*c*) ∈ R

_{Nm}

^{Nm}^{x}

*and*

^{Nv}*V*= (

*d*, …,

_{1}*d*) ∈ R

_{Nv}

^{Nm}^{x}

*are orthonormal matrices, and where the diagonal matrix Σ = diag(*

^{Nv}*s*, …,

_{1}*s*) has non-negative elements appearing in non-increasing order (

_{Nm}*s*≥

_{1}*s*≥…≥

_{2}*s*≥0). A least-squares solution to the problem then takes the form,

_{m}*s*are the singular values,

_{i}*d*the singular data (SD) vectors and

_{i}*c*the singular image (SI) vectors. Each singular value is associated with one SD vector and one SI vector, all of which are labeled with the index

_{i}*i*running from 1 to

*N*, where

*N*is the maximal number of modes included in a reconstruction. Equation (4) represents a truncated singular value decomposition (TSVD), for which

*N*≤

*N*.

_{rank}*N*is the numerical rank of the matrix

_{rank}*A*, which is demarcated by the mode at which a significant and abrupt decrease in singular values is observed [8]. The singular value solution in Eq. (4) is roughly similar to the projection of a signal onto different spectral modes in Fourier analysis. According to this analogy, the vectors

*c*are the Fourier modes, the coefficients

_{i}*F*are regularization filter factors allowing for the relative contribution of the spectral components to be weighted, and the coefficients,

_{i}*D*.

## 3. Methods

2. D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. **14**(3), 030501 (2009). [CrossRef] [PubMed]

3. D. Kepshire, N. Mincu, M. Hutchins, J. Gruber, H. Dehghani, J. Hypnarowski, F. Leblond, M. Khayat, and B. W. Pogue, “A microcomputed tomography guided fluorescence tomography system for small animal molecular imaging,” Rev. Sci. Instrum. **80**(4), 043701 (2009). [CrossRef] [PubMed]

4. V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. **26**(12), 893–895 (2001). [CrossRef] [PubMed]

### 3.1 The difference between model mismatch errors and stochastic noise

*in vivo*(Fig. 3c below). For fluorescence tomography simulations, two 4 mm-diameter inclusions were inserted with a 5 mm separation between their center-of-masses, as shown in Fig. 3c (right-most image). The fluorescence contrast between the inclusions and the homogeneously fluorescent background was set to 10:1. For all geometries, data vectors were simulated using two-dimensional diffusion theory using the simulation software NIRFAST [9

9. S. C. Davis, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Contrast-detail analysis characterizing diffuse optical fluorescence tomography image reconstruction,” J. Biomed. Opt. **10**(5), 050501 (2005). [CrossRef] [PubMed]

*N*= 1979 nodes. In this first study, five data vectors were simulated: three fluorescence data vectors, two of which were simulated assuming an optically homogeneous background with and without 10% stochastic (Gaussian) noise added and one assuming an optically heterogeneous background and 0% noise. The last two simulated data vectors were transmittance vectors, one assuming an optically heterogeneous medium and one assuming a homogeneous medium (noise-free). The left-most image in Fig. 3c illustrates the different anatomical regions imposed to provide heterogeneity (1: brain, 2: bone, 3: other soft tissue). Table 2 summarizes the optical properties (absorption and reduced scattering coefficients) that were assigned to each region in the scope of two different types of simulations: heterogeneous and homogeneous media.

_{v}### 3.2 Singular value decomposition for optimizing imaging system geometries

7. E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A **21**(2), 231–241 (2004). [CrossRef] [PubMed]

10. T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. **11**(4), 389–399 (2007). [CrossRef] [PubMed]

*i*, of the singular order (Fig. 2 , Fig. 5 ); whereas, the shape evolution of the SD vectors follows a more complex pattern with increasing mode order (Fig. 6 ). The resolution (inverse of spatial frequency) of each individual SI vector mode was defined as the average distance, in millimeters, between two consecutive extrema (maxima or minima) along one-dimensional cross-sections of SI vector images. An example for this is the dotted line shown in Fig. 2a. More precisely, a numerical value has been assigned to each two-dimensional SI mode (Fig. 5) corresponding to the number of extrema,

*N*, along a one-dimensional cross-section divided by the length of the cross-section,

_{ex}^{SI}*L*(in millimeters),

*N*= 3,

_{ex}^{SI}*N*= 5 and

_{ex}^{SI}*N*= 9 for SI modes of singular order

_{ex}^{SI}*i*= 1,

*i*= 5 and

*i*= 30, respectively. Although Fig. 5 has been plotted using the one-dimensional cross-section (dotted line) shown in Fig. 2, numerical analysis (not presented) was performed demonstrating that the final conclusions derived from the mode resolution analysis in Section 4 are unchanged for different cross-sections. For this metric, favorable detection geometries were determined as those for which the modes available for reconstruction had the smallest possible spatial resolution.

*e.g.*, see Fig. 6a) was also characterized by a numerical criterion, in this case labeled

*mode frequency*(Fig. 6b). The frequency for each SD mode was computed by evaluating the number of extrema

*N*for each SD mode and dividing this number by the total number of measurements

_{ex}^{SD}*N*,,

_{m}*N*, was automatically computed using an algorithm similar to that described above for the SI vectors. While the SI mode resolution is an effective measure for the level of spatial resolution each image mode can contribute to a reconstructed FT image, the SD mode frequency does not lend itself to such a direct interpretation. However, as explained in Section 4, the numerical value of the SD mode frequency can be used as an indirect assessment of the propagation of stochastic noise and model mismatch error into FT images.

_{ex}^{SD}*F*corresponding to a Tikhonov regularization was also considered but did not provide significant image quality improvements.

*a priori*assessment of detector geometries included the steepness of the singular value vs. mode order and the number of singular modes above the noise floor (

*i.e.,*the modes available for image reconstruction).

### 3.3 Image quality: spatial resolution and recovered contrast

*e.g.*, spatial resolution and recovered contrast – images were reconstructed from the simulated data sets of all detection geometries (Table 1) under various levels of noise and with respect to forward model mismatch. For each of these images, an assessment of quality was determined, favoring images with: (1) higher contrast recovery (with respect to the target simulated image), (2) increased ability to spatially resolve multiple targets, and (3) the absence of image artifacts. These

*a posteriori*evaluations are then discussed in the context of the

*a priori*evaluations introduced in Section 3.2.

## 4. Results and discussion

### 4.1 The difference between model mismatch errors and stochastic noise

*in vivo*values for different organs are typically not available and in cases where they would be available, significant inter-specimen variations are expected. Additionally, different systems use different excitation and emission wavelengths, preventing use of generic values in the scope of routine experiments, and accurate co-registration with anatomical imaging modalities is difficult but required.

*homogenous medium - 10% noise*), illustrating the characteristics of noise propagation into the data sets. As expected, the difference between the two vectors consists of stochastic noise normally distributed around zero, with a maximum amplitude of 10% of the fluorescence measurement amplitude. This curve demonstrates that sources of noise such as photon shot noise will always contribute high-frequency components to a data vector. This is in opposition to the predominantly low frequency contributions associated with model mismatch. Therefore, the effect of both stochastic noise and model mismatch on fluorescence tomography can be empirically written as,

_{homo}is the vector that would result for an homogenous medium while D

_{high-f}and D

_{low-f}are vectors corresponding high-frequency (stochastic noise) and low-frequency (predominantly model mismatch) contributions. This parametric classification of data vector contributions is clearly not exact; however, it will be conceptually useful in the upcoming discussion relating to singular value decomposition of the matrix

*A*. It should also be noted here that model mismatch contributions associated with discretization in Eq. (2) will mainly contribute high-frequency components to the data vector in situations where the size of the individual mesh components is comparable or smaller than typical light mean free paths. On the other hand, the fact that most of the low frequency contributions is associated with model mismatch errors is a direct result of the diffusive nature of the light transport sensitivity functions, which leads to averaging of signals emerging from large regions within the interrogated medium.

### 4.2 Singular value analysis of image geometries

7. E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A **21**(2), 231–241 (2004). [CrossRef] [PubMed]

*N*in Eq. (4) are superior because more modes can be used to reconstruct images. Based on those criteria, inspection of Fig. 4 indicates that the wide-field detection geometry (V) is superior compared to the other geometries. It can also be deduced that the ranking of the most favorable geometries goes like: V, IV, II, III, I.

*A*,

*i.e.*,

*N*=

*N*

_{rank}. However, in practice the number of modes that can actually be used to reconstruct an image is limited by, for example, the presence of high-frequency noise in the data vector

*D*. The projection of stochastic noise contribution onto the data vector,

*D*

_{high-f}⋅

*d*, typically causes the coefficients

_{i}^{T}*K*to contribute disproportionally to reconstructed images for some value

_{i}*i*<

*N*

_{rank}, systematically leading to noisy reconstructed images. This effect might be perceived as a divergence in the singular coefficients caused by two competing effects: namely, monotonic decrease in singular values with increasing order and projection of high-frequency components of the data vector,

*e.g.*stochastic noise or volume discretization noise, onto the SD vectors, which is disproportionately amplified for some values of

*i*. For a better conditioned problem, sequentially smaller singular values (for

*i*<

*N*) will not cause the

_{rank}*K*’s to diverge since the monotonic increase of 1/

_{i}*s*is opposed by a continuously decreasing overlap of the SD and raw data vectors as

_{i}*i*increases. However, in the presence of high-frequency noise, the large

*i*behavior of the term

*D*

_{high-f}⋅

*d*introduces divergences, which hide the contribution of the lower frequency components,

_{i}^{T}*i.e.*, the modes containing the physically relevant biological information. In a nutshell, regularization in optical tomography amounts to minimizing the contribution of those singular vectors contributing disproportionally to reconstructed images due to the propagation of high-frequency noise. When using singular value decomposition to solve the inverse problem, regularization can be achieved by appropriately choosing the filter function

*F*

_{i}, or, by introducing a smoothing norm by using a generalized singular value decomposition (GSVD) [8].

### 4.3 Singular vector analyses of image geometries

### 4.4 Image quality

*A*) that can be retained for image reconstruction is 73% for 1% noise and 50% for 5% noise. In comparison, the wide-field detection geometry V allows only about 15% of its modes to be included for 1% and 5% noise. As for the other three geometries, they allow retention of between around 26-50% of the modes for 1% noise. In summary, more measurements do translate into more modes included in the images but in relative terms, high measurement density configurations may not be the most economical in terms of the percentage of useful modes.

*a posteriori*evaluation of the images based on image quality. For example, based on the singular value curve, geometry II is preferable to geometry III. However, based on the spatial resolution of the modes, geometry III is expected to perform better than geometry II. Similarly, based on singular values, the high density of measurements geometry IV is expected to be superior to geometry III, which is in disagreement with the prediction derived from the spatial resolution of the SI modes. Again, the prediction from the later criteria is more in tune with the ranking based on image quality.

### 4.5 Summary and conclusions

**21**(2), 231–241 (2004). [CrossRef] [PubMed]

*Scenario A*the contrast-to-background ratio remains constant (10:1) but the size of the tumor varies, in

*Scenario B*the size of the inclusion is constant but the contrast varies, and, in

*Scenario C*both the size and the contrast of the tumor vary. The tomography results for each scenario are presented in Fig. 9 where images were reconstructed using an homogenous matrix

*A*but with fluorescence-to-transmission data generated with the heterogeneous optical properties found in Table 2. For all the reconstructions, 1% stochastic noise has been added to the data vector. The results presented in Fig. 9 show that under the same noise conditions, a wide-field detection geometry provides only minor imaging gains when it comes to spatial resolution and recovered contrast.

### 4.6 Limitations and future work

*F*

_{i}in Eq. (4), an analysis of the problem based on a generalized SVD (GSVD) must be performed in order to allow penalty functions that would be consistent with,

*e.g.*, soft as well hard spatial priors. While the results presented in this paper provide useful information on the FT problem, they should be interpreted within the confines of those limitations. In fact, it is more commonplace in whole-body fluorescence tomography to use inverse problem resolution methods with regularization functions allowing certain aspects of the images to be favored as determined by prior knowledge (e.g., spectral and/or spatial) or pre-determined features that are expected for the images (e.g., edge preservation, absence of high spatial frequencies).

## Acknowledgments

## References and links

1. | V. Ntziachristos and R. Weissleder, “Charge-coupled-device based scanner for tomography of fluorescent near-infrared probes in turbid media,” Med. Phys. |

2. | D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. |

3. | D. Kepshire, N. Mincu, M. Hutchins, J. Gruber, H. Dehghani, J. Hypnarowski, F. Leblond, M. Khayat, and B. W. Pogue, “A microcomputed tomography guided fluorescence tomography system for small animal molecular imaging,” Rev. Sci. Instrum. |

4. | V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. |

5. | F. Leblond, H. Dehghani, D. Kepshire, and B. W. Pogue, “Early-photon fluorescence tomography: spatial resolution improvements and noise stability considerations,” J. Opt. Soc. Am. A |

6. | J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. |

7. | E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A |

8. | P. C. Hansen, |

9. | S. C. Davis, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Contrast-detail analysis characterizing diffuse optical fluorescence tomography image reconstruction,” J. Biomed. Opt. |

10. | T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. |

11. | F. Leblond, S. C. Davis, P. A. Valdés, and B. W. Pogue, “Pre-clinical whole-body fluorescence imaging: Review of instruments, methods and applications,” J. Photochem. Photobiol. B |

12. | B. W. Pogue, F. Leblond, V. Krishnaswamy, and K. D. Paulsen, “Radiologic and near-infrared/optical spectroscopic imaging: where is the synergy?” AJR Am. J. Roentgenol. |

**OCIS Codes**

(170.0110) Medical optics and biotechnology : Imaging systems

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(170.3660) Medical optics and biotechnology : Light propagation in tissues

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: September 7, 2010

Revised Manuscript: November 9, 2010

Manuscript Accepted: November 20, 2010

Published: November 29, 2010

**Citation**

Frederic Leblond, Kenneth M. Tichauer, and Brian W. Pogue, "Singular value decomposition metrics show limitations of detector design in diffuse fluorescence tomography," Biomed. Opt. Express **1**, 1514-1531 (2010)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-5-1514

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### References

- V. Ntziachristos and R. Weissleder, “Charge-coupled-device based scanner for tomography of fluorescent near-infrared probes in turbid media,” Med. Phys. 29(5), 803–809 (2002). [CrossRef] [PubMed]
- D. S. Kepshire, S. L. Gibbs-Strauss, J. A. O’Hara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14(3), 030501 (2009). [CrossRef] [PubMed]
- D. Kepshire, N. Mincu, M. Hutchins, J. Gruber, H. Dehghani, J. Hypnarowski, F. Leblond, M. Khayat, and B. W. Pogue, “A microcomputed tomography guided fluorescence tomography system for small animal molecular imaging,” Rev. Sci. Instrum. 80(4), 043701 (2009). [CrossRef] [PubMed]
- V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26(12), 893–895 (2001). [CrossRef] [PubMed]
- F. Leblond, H. Dehghani, D. Kepshire, and B. W. Pogue, “Early-photon fluorescence tomography: spatial resolution improvements and noise stability considerations,” J. Opt. Soc. Am. A 26(6), 1444–1457 (2009). [CrossRef] [PubMed]
- J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. 26(10), 701–703 (2001). [CrossRef] [PubMed]
- E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21(2), 231–241 (2004). [CrossRef] [PubMed]
- P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems (SIAM, 1998).
- S. C. Davis, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Contrast-detail analysis characterizing diffuse optical fluorescence tomography image reconstruction,” J. Biomed. Opt. 10(5), 050501 (2005). [CrossRef] [PubMed]
- T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. 11(4), 389–399 (2007). [CrossRef] [PubMed]
- F. Leblond, S. C. Davis, P. A. Valdés, and B. W. Pogue, “Pre-clinical whole-body fluorescence imaging: Review of instruments, methods and applications,” J. Photochem. Photobiol. B 98(1), 77–94 (2010). [CrossRef] [PubMed]
- B. W. Pogue, F. Leblond, V. Krishnaswamy, and K. D. Paulsen, “Radiologic and near-infrared/optical spectroscopic imaging: where is the synergy?” AJR Am. J. Roentgenol. 195(2), 321–332 (2010). [CrossRef] [PubMed]

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