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Biomedical Optics Express

Biomedical Optics Express

  • Editor: Joseph A. Izatt
  • Vol. 5, Iss. 3 — Mar. 1, 2014
  • pp: 763–777
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Hadamard multiplexed fluorescence tomography

Ali Behrooz, Ali A. Eftekhar, and Ali Adibi  »View Author Affiliations


Biomedical Optics Express, Vol. 5, Issue 3, pp. 763-777 (2014)
http://dx.doi.org/10.1364/BOE.5.000763


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Abstract

Depth-resolved three-dimensional (3D) reconstruction of fluorophore-tagged inclusions in fluorescence tomography (FT) poses a highly ill-conditioned problem as depth information must be extracted from boundary data. Due to the ill-posed nature of the FT inverse problem, noise and errors in the data can severely impair the accuracy of the 3D reconstructions. The signal-to-noise ratio (SNR) of the FT data strongly affects the quality of the reconstructions. Additionally, in FT scenarios where the fluorescent signal is weak, data acquisition requires lengthy integration times that result in excessive FT scan periods. Enhancing the SNR of FT data contributes to the robustness of the 3D reconstructions as well as the speed of FT scans. A major deciding factor in the SNR of the FT data is the power of the radiation illuminating the subject to excite the administered fluorescent reagents. In existing single-point illumination FT systems, the source power level is limited by the skin maximum radiation exposure levels. In this paper, we introduce and study the performance of a multiplexed fluorescence tomography system with orders-of-magnitude enhanced data SNR over existing systems. The proposed system allows for multi-point illumination of the subject without jeopardizing the information content of the FT measurements and results in highly robust reconstructions of fluorescent inclusions from noisy FT data. Improvements offered by the proposed system are validated by numerical and experimental studies.

© 2014 Optical Society of America

1. Introduction

Fluorescence tomography (FT) aims at in vivo 3D localization and quantification of fluorescent contrast agents distributed in biological tissue [1

1. V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8(1), 1–33 (2006). [CrossRef] [PubMed]

, 2

2. V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1(2), 82–88 (2002). [CrossRef] [PubMed]

]. Fluorescent agents are used for in vivo tagging and tracking of inclusions or molecules of interest such as cancer lesions, test drugs, and protein expressions in small animals and human subjects [3

3. V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8(7), 757–761 (2002). [CrossRef] [PubMed]

, 4

4. A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15(11), 6696–6716 (2007). [CrossRef] [PubMed]

]. In FT, the subject is illuminated at a sequence of points on the skin by visible or near infrared (NIR) radiation from a laser or light emitting diode (LED). Photons from the illumination source diffuse through the tissue and excite the exogenously administered fluorescent agents that, in turn, emit visible or NIR fluorescent light at wavelengths higher than the excitation [1

1. V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8(1), 1–33 (2006). [CrossRef] [PubMed]

]. The fluorescent photons are collected and their intensity measured by optical detectors at various points on the surface of the subject. These surface intensity measurements are used in an inversion algorithm to reconstruct the 3D distribution of fluorescence in tissue. The 3D reconstruction of fluorescence distribution is a highly ill-posed problem as depth information must be extracted from diffuse boundary data. Consequently, noise and errors in the FT data and modeling can produce significant artifacts in the 3D reconstructions. FT inverse solvers utilize regularization techniques to provide robustness and stability against noise and errors [5

5. S. C. Davis, H. Dehghani, J. Wang, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization,” Opt. Express 15(7), 4066–4082 (2007). [CrossRef] [PubMed]

9

9. A. Behrooz, H. M. Zhou, A. A. Eftekhar, and A. Adibi, “Total variation regularization for 3D reconstruction in fluorescence tomography: experimental phantom studies,” Appl. Opt. 51(34), 8216–8227 (2012). [CrossRef] [PubMed]

]. However, as the level of noise and error contamination rises, the quality of regularized reconstructions deteriorates. Depending on the dynamics and nature of the inversion algorithms, different types of artifacts and errors arise in the 3D reconstructions when FT data are considerably noisy [9

9. A. Behrooz, H. M. Zhou, A. A. Eftekhar, and A. Adibi, “Total variation regularization for 3D reconstruction in fluorescence tomography: experimental phantom studies,” Appl. Opt. 51(34), 8216–8227 (2012). [CrossRef] [PubMed]

].

Improvements in the modeling and conditioning of the FT inverse problem can be of great help in enhancing the accuracy of the reconstructions. However, these improvements are not available in many FT scenarios, e.g., imaging of optically heterogeneous regions of small animals. In such cases, the level of modeling errors can be remarkably high and, depending on the bulk and shape of the animal, the associated inverse problem can often be extremely ill-conditioned. Alternatively, enhancing the signal-to-noise ratio (SNR) in the FT data can strongly contribute to the quality of the reconstructions and reduce noise-induced artifacts, irrespective of the inversion algorithm being used. The SNR of FT data is determined by several factors including the sensitivity of the data-acquisition system, e.g., charge-coupled device (CCD) camera, the absorption of the turbid medium, the quantum yield and absorption cross section of the administered fluorescent agent, and the radiative power of the illumination source. Existing FT systems are equipped with extremely costly, ultra-sensitive, cooled CCD cameras to guarantee a high data-acquisition SNR. Disadvantages of ultra-sensitive CCD systems are two-fold; they are only available at high costs, and they require lengthy integration times for extremely low-noise data acquisition, which result in extremely lengthy FT scan times. Absorption of the tissue sample being imaged and the properties of the administered fluorescent dye vary from experiment to experiment and cannot be controlled in FT systems.

The power of the light source used for illumination of the turbid medium and excitation of the administered fluorescent agents can be increased to raise the SNR of the FT data. However, the power of the illumination source must not exceed the threshold beyond which human skin and biological tissue get injured by the source radiation [10

10. D. Sliney and M. Wolbarsht, Safety with Lasers and Other Optical Sources, Plenum, New York (1980).

]. Therefore, for existing single-point illumination FT systems, the power entering the medium is bounded by the skin maximum permissible exposure (MPE) in the visible and NIR range (~2 mW/mm2) [11

11. ANSI Standard Z136.1, American National Standard for the Safe Use of Lasers, American National Standards Institute, Inc., New York (2000).

]. In this work, a multi-point illumination FT configuration is presented that allows for orders-of-magnitude increase in the source radiation entering the subject compared to existing single-point illumination FT systems. Since instead of single-point illumination, multiple points are illuminated simultaneously, more power can enter the subject without causing radiation injury. The only trade-off is that as the number of simultaneously illuminated points increases, the number of uncorrelated or minimally correlated measurements obtainable by changing the illumination points decreases. As an example, if all the sources were activated simultaneously, a tremendously high level of radiation would enter the subject and excite the administered fluorophores. However, the acquired data would not possess the same level of information obtainable through a series of single-point illumination measurements. As a result, a trade-off exists between the average number of simultaneously illuminated points per measurement, and the information content of the data acquired in the corresponding series of measurements. This trade-off can be optimized by applying the Hadamard transform to the illumination patterns in FT measurements.

In this paper, we introduce a multiplexed multi-point illumination architecture governed by the Hadamard transform to replace the existing single-point illumination architecture in FT for the purpose of increasing the FT data SNR and hence the robustness of the 3D reconstructions. We perform numerical studies to show the improvements offered by Hadamard-multiplexed FT over existing single-point illumination FT. Moreover, we present experimental results using a Hadamard-multiplexed FT system, which was developed in house in its entirety for this work, to demonstrate the advantages of multiplexed illumination over single-point illumination in FT.

2. Theory

2.1 Fluorescence tomography

The propagation of light in highly scattering turbid media such as tissue is modeled by the radiative transport equation (RTE) [12

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

]. It has been shown that a first-order approximation to RTE reduces the computational complexity and numerical burden of modeling while maintaining a relatively high level of accuracy sufficient for optical imaging purposes [13

13. S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42(5), 841–853 (1997). [CrossRef] [PubMed]

]. The first-order approximation, which is broadly used by the optical tomography community, results in a first-order partial differential equation (PDE) called the diffusion equation formulated as
D(r)Φ(r)+μa(r)Φ(r)=q(r),
(1)
where Φ(r) and q(r) represent the average light intensity and the illumination source flux at location r, respectively. Furthermore, μa and D represent the absorption coefficient and the diffusion coefficient, respectively. In FT, the propagation of excitation photons and fluorescent photons can be described by a pair of coupled diffusion equations as below
D(r)Φexc(r)+μa(r)Φexc(r)=qexc(r),
(2)
D(r)Φem(r)+μa(r)Φem(r)=ημflc(r)Φexc(r),
(3)
where Φexc(r) is the average intensity of excitation photons at location r; qexc(r) is the power density of the excitation source used for illumination of the tissue at location r (as a result, qexc(r) is zero inside the tissue and non-zero at the boundary source locations); Φem(r) is the average intensity of the fluorescent light at location r, η is the dimensionless quantum yield of the fluorescent dye; μfl is the per-molar fluorescent absorption coefficient of the fluorophores at the excitation wavelength, and c(r) is the molar concentration of the fluorescent dye at location r. In FT, the goal is to use the model in Eqs. (2) and (3) to estimate the fluorescence distribution c(r) by varying qexc(r) (through changing the location of the illumination source) and measuring Φexc(r) and Φem(r) on the boundary of the tissue while D(r), μa(r), η, and μfl are either known a priori or determined using diffuse optical tomography (DOT) measurements [14

14. D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001). [CrossRef]

].

Since analytical solutions are not available for the coupled PDEs formulated in Eqs. (2) and (3), numerical techniques such as the finite element method (FEM) must be used to discretize the coupled PDEs and numerically solve for the corresponding Green’s functions [15

15. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37(22), 5337–5343 (1998). [CrossRef] [PubMed]

]. The most common approach is the Galerkin formulation of the FEM where the volume of the turbid medium is discretized by a 3D tetrahedral mesh and spatial functions defined over the medium are transformed to discrete vectors defined over the voxels of the 3D mesh. As a result, Eqs. (2) and (3) are transformed to discrete matrix equations using which a linear relationship can be established between the fluorescence distribution vector (x) and the boundary measurements of fluorescent photons (y) through a system matrix (M) that depends on the geometry and optical properties of the tissue [15

15. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37(22), 5337–5343 (1998). [CrossRef] [PubMed]

]:

y=Mx.
(4)

2.2 Hadamard multiplexing for FT

The Hadamard transform is a linear transform used for SNR enhancement in multi-source or multi-input measurement systems [16

16. M. Harwit, and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, New York (1979).

18

18. R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Hadamard transform Raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19(2), 177–186 (1999). [CrossRef]

]. Instead of performing measurements with one source active at a time, measurements are acquired while multiple sources are active simultaneously. As a result, the radiation power entering the subject in a Hadamard multiplexed FT scan is increased in proportion to the number of sources that are on during the scan. The increase in the illumination power boosts the SNR of the data images acquired during the FT scan. As discussed in Section 1, a trade-off exists between the information content of the acquired data and the average number of sources active during each measurement. The theory of Hadamard transform provides the optimal multiplexing scheme for the enhancement of the SNR of multi-source measurements without lowering the information content of the measurements [16

16. M. Harwit, and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, New York (1979).

]. This optimal (0, 1)-weighing scheme is encoded in the Hadamard S-matrix that is a square matrix with entries that are either 0 or 1. The S-matrix codes offer the highest increase in SNR (or highest number of simultaneously active sources) while maintaining non-singularity and independence between successive measurements. From a theoretical perspective, the S-matrix coded measurements are A-optimal and D-optimal [19

19. V. V. Fedorov, Theory of Optimal Experiments, Academic Press, New York (1972).

]. The S-matrix is constructed such that each column (or row) has the maximum possible number of 1’s while the matrix maintains full rank and remains non-singular. Each column of the S-matrix is used as a multiplexing code for a corresponding measurement. The 1’s and 0’s in each column encode the sources that should be on and the sources that should be off, respectively, in the corresponding measurement. The number of columns of the S-matrix represents the number of FT measurements with distinct source distributions. As an example, a Hadamard S-matrix of size seven-by-seven is shown below:
S=[1110100110100110100110100111100111000111010111010].
(5)
The multiplexing scheme encoded in the first column of the S-matrix in Eq. (5) stipulates sources numbered 1, 2, 3, and 5 to be on and sources numbered 4, 6, and 7 to be off in the first measurement. The matrix has seven columns so a total of seven measurements can be obtained. Moreover, in the measurements obtained using this multiplexing scheme, an average of four sources are active during each measurement making the data SNR orders greater than that of single-source measurements. In general, in a measurement system with N sources, the Hadamard-multiplexing scheme increases the SNR by a factor of N approximately (more accurate for large N) [16

16. M. Harwit, and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, New York (1979).

]. Therefore, Hadamard multiplexing is extremely advantageous in measurement systems with a high number of sources.

In Fig. 1
Fig. 1 In a conventional FT system depicted in row (a), in each measurement one source illuminates the box-shaped turbid medium housing two fluorescent rods. In Hadamard-multiplexed FT, depicted in row (b), multiple sources (four out of seven) illuminate the medium in each measurement. A total of seven measurements are performed in each configuration. The S-matrix Hadamard encodings are based on the S-matrix formulated in Eq. (5).
, a graphical comparison of conventional single-point illumination FT architecture and Hadamard-multiplexed FT architecture is presented in which the multiplexing scheme of the S-matrix in Eq. (5) is applied to an FT system with seven sources that illuminate a slab-shaped turbid medium with two cylindrical fluorescent inclusions. As shown in Fig. 1, the radiative power entering the slab and exciting the fluorescent rods is considerably higher in the case of Hadamard-multiplexed FT. Meanwhile, Hadamard multiplexing does not require complex changes to the configuration of the conventional FT system. The only requirement is that the system must be modified so that simultaneous illumination of multiple points is possible.

The linear model of FT, as formulated in Eq. (1), is modified with Hadamard multiplexing as
y=WMx,
(6)
where W is the multiplexing matrix constructed from the Hadamard S-matrix entries as follows
W=[diagnd(S11)diagnd(S1ns)diagnd(Sns1)diagnd(Snsns)],
(7)
where diagnd(Sij) is a square diagonal matrix of size nd-by-nd with an scalar denoted by Sij populating all of its diagonal entries. Here, nd and ns represent the number of FT detectors and sources, respectively. Hence, the matrix W in Eq. (7) is a square matrix with a size of ndns-by-ndns. Also, the system matrix in the case of Hadamard-multiplexed FT becomes WM. Therefore, as predicted by the theory of Hadamard transform, the statistics of the noise in the FT data remain unchanged under multiplexing while the noiseless data vector, Mx, is amplified by W. This results in a boost in the FT data SNR.

3. Results

3.1 Numerical studies

To study the performance of Hadamard-multiplexed FT, we apply it to a 2D numerical study as shown in Fig. 2
Fig. 2 2D numerical studies with (a) 7, (b) 11, (c) 15, (d) 19, and (e) 23 sources illuminating a turbid medium with μs = 1 mm−1 and μa = 0.01 mm−1. The ground-truth fluorophore distribution is shown in rows labeled (i). The 2D reconstructions from (ii) conventional single-source illumination, and (iii) Hadamard-multiplexed data were performed using MLS-ART for data contaminated with noise levels equivalent to single-point illumination data SNRs of 60, 50, 40, 30, 20, 10, and 0 dB and are shown in corresponding rows ordered from left to right, respectively.
. A rectangular turbid medium of dimensions 60 mm by 80 mm with scattering and absorption coefficients of 1 mm−1 and 0.01 mm−1, respectively, houses two circular fluorophore inclusions and is illuminated by a varying number of equally spaced sources distributed on its boundary. The emitted fluorescent signal is collected on the boundary of the turbid medium by 23 equally spaced detectors. The propagation of the excitation and fluorescent photons in the 2D turbid medium is simulated by the FEM where a triangular mesh with 32305 nodes and 64000 elements is used to discretize the medium. To study the effect of Hadamard multiplexing on the quality of FT reconstructions, the simulated data is contaminated with various levels of additive white noise, which models the combined effects of read-out, dark-current, and shot noise, resulting in SNRs of 60, 50, 40, 30, 20, 10, and 0 dB in the single-point illumination configuration. Also, FT data are simulated for a varying number of sources increased in increments of 4 resulting in five FT configurations with 7, 11, 15, 19, and 23 sources illuminating the medium, as shown in rows labeled (i) of Figs. 2(a), 2(b), 2(c), 2(d), and 2(e), respectively. The results from these varying source configurations can reveal the effects of the number of illuminating sources on the advantages offered by the Hadamard-multiplexed FT architecture.

The reconstructions are performed by multi-level scheme algebraic reconstruction technique (MLS-ART) which is a fast commonly used inverse solver that does not require optimal parameter selection [20

20. R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29(3), 471–481 (1970). [CrossRef] [PubMed]

, 21

21. A. Behrooz, C. Kuo, H. Xu, and B. W. Rice, “Adaptive row-action inverse solver for fast noise-robust 3D reconstructions in bioluminescence tomography: theory and dual-modality optical/CT in vivo studies,” J. Biomed. Opt. 18(7), 076010 (2013). [CrossRef] [PubMed]

] unlike regularized least-squares techniques [5

5. S. C. Davis, H. Dehghani, J. Wang, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization,” Opt. Express 15(7), 4066–4082 (2007). [CrossRef] [PubMed]

, 6

6. P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. 46(10), 1679–1685 (2007). [CrossRef] [PubMed]

, 9

9. A. Behrooz, H. M. Zhou, A. A. Eftekhar, and A. Adibi, “Total variation regularization for 3D reconstruction in fluorescence tomography: experimental phantom studies,” Appl. Opt. 51(34), 8216–8227 (2012). [CrossRef] [PubMed]

]. Hence, the reconstruction algorithm (including its parameters) is the same for all data SNRs and source configurations (the relaxation parameter of MLS-ART is set to 1 in all cases). The rows labeled (ii) in Fig. 2 show the reconstructions by MLS-ART from conventional single-point illumination FT data, and the rows labeled (iii) in Fig. 2 show the reconstructions by MLS-ART from Hadamard-multiplexed FT data. For high SNRs (60-40 dB), the reconstructions from single-point illumination data and Hadamard-multiplexed data are similar and possess high accuracy. As the data SNR decreases below 40 dB, the reconstructions from single-point illumination data become inaccurate and contaminated with noise-induced impulses. However, the reconstructions from Hadamard-multiplexed data remain considerably accurate down to a data SNR of ~10 dB. Furthermore, as shown in the results presented in Fig. 2, the denoising power of Hadamard-multiplexed FT increases as the number of sources illuminating the medium increases. The reconstructions from Hadamard-multiplexed data presented in row (iii) of Fig. 2(e) (corresponding to the study with 23 sources) possess higher noise-robustness compared to those in row (iii) of Fig. 2(b) (corresponding to the study with 11 sources). To compare the results presented in Fig. 2 quantitatively, the relative mean-square error (MSE) corresponding to each reconstruction is plotted in Fig. 3
Fig. 3 The relative mean-square errors (ε) of the MLS-ART reconstructions presented in Fig. 2 versus the data SNR.
. This error is defined as
ε=x-x^2x2,
(8)
where x represents the actual ground-truth fluorescent distribution vector, and x^ represents the reconstructed fluorescent distribution.

As shown in Fig. 3, ε values higher than 5 are not included within the limits of the graph. Figure 3 clearly demonstrates that Hadamard multiplexing becomes considerably advantageous to single-point illumination as the data SNR decreases, and the number of sources increases. This advantage is expected based on the theoretical discussions presented in Sections 1 and 2.2. For high-SNR cases (>40 dB), the relative errors for both architectures are the same. However, for low-SNR studies, the error in the Hadamard-multiplexed cases remains below 1 for SNRs down to 10 dB, whereas in the single-point cases the relative errors grow larger than one for SNRs around or below 30 dB. It must be noted that we have used low number of sources to demonstrate the effect of Hadamard multiplexing on FT reconstructions. In practice, the number of FT sources is considerably higher resulting in remarkably higher practical advantage for Hadamard-multiplexed FT.

3.2 Experimental studies

In existing single-point illumination FT systems, the grid of source points is scanned sequentially by an optical fiber mounted on a translation stage [22

22. E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30(5), 901–911 (2003). [CrossRef] [PubMed]

]. In Hadamard-multiplexed FT, a different scheme must be used for multi-point illumination. In this work, we developed a simple non-contact illumination configuration that allows for simultaneously flooding light on multiple points in the source grid. As presented in Fig. 4
Fig. 4 Schematic of the non-contact Hadamard-multiplexed FT system. Visible or NIR radiation from a laser source is collimated and directed onto a lenslet array with an S-matrix mask mounted on it. The phantom is placed at the focal plane of the lenslet array. The non-masked lenslets form multi-point Hadamard S-matrix illumination patterns on the phantom. The radiation diffuses through the liquid phantom, excites the fluorescent inclusions (two rods) whose emission is imaged to a cooled CCD camera by an objective lens.
, after collimation, the visible or NIR radiation passes through a masked lenslet array. The mask grid blocks lenslets corresponding to the 0’s of the Hadamard S-matrix while allowing the radiation to pass through lenslets corresponding to the 1’s. The FT image acquisition configuration of this system is similar to existing non-contact FT systems where the excitation trans-illumination and fluorescent emission are imaged to a CCD by a lens and separated using a motorized filter wheel.

The experimental studies were carried out using an in-house developed Hadamard-multiplexed phantom-based FT system as shown in Fig. 5
Fig. 5 The phantom-based Hadamard-multiplexed FT system: a) Picture of the experimental system. b) A Hadamard S-matrix mask mounted on a lenslet array is illuminated with collimated beam of laser radiation. c) The S-matrix mask produces the desired excitation source pattern on the phantom surface.
. The collimated light beam of a 20-mW 635-nm He:Ne continuous-wave laser passes through an engineered diffuser and an opening that functions as an aperture to limit the beam waist arriving at the lenslet array. The 9-by-7 lenslet array focuses the light beam onto a grid of 63 points with a vertical pitch of 3 mm and horizontal pitch of 4 mm. The liquid phantom vessel is placed at the focal plane of the lenslet array so that its focal grid functions as a multi-point illumination pattern. As presented in Figs. 5(b) and 5(c), S-matrix masks are mounted on the lenslet array to create Hadamard-coded multi-point illumination patterns on the phantom. Given the number of source locations, 63 Hadamard codes are used sequentially for the multiplexed FT scan.

The liquid phantom used in the experimental studies is a water-based mixture of Intralipid-1% and India ink with scattering and absorption coefficients of 0.8 mm−1 and 0.05 mm−1 [23

23. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42(10), 1971–1979 (1997). [CrossRef] [PubMed]

, 24

24. S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12(5), 510–519 (1992). [CrossRef] [PubMed]

]. The mixture is poured into a rectangular vessel with transparent plexiglass sides and dimensions of 120 mm by 90 mm by 14 mm. The fluorescent dye used in the phantom experiments is a 100 µM dimethyl sulfoxide (DMSO)-based solution of Oxazine 750 Perchlorate whose emission peaks around 700 nm when excited at 635 nm. Two capillary glass tubes with an inner diameter of 1 mm are partially filled with the fluorescent dye to form a pair of fluorescent cylinders with 1 mm diameter and 10 mm height. The capillary tubes are made of relatively thin glass (thickness of around 100 microns). Hence, the error in the light diffusion model from the thin glass is negligible considering the dimensions of the slab phantom. The dye-filled tubes are suspended in the center of the liquid phantom by an optical post mounted on a translation stage for accurate positioning as depicted in Fig. 5(a). Using the translation stage, the dye-filled tubes are positioned at depths of 3 mm, 6 mm, and 9 mm from the front surface of the phantom vessel facing the camera. The trans-illumination and fluorescent emission are imaged from the front side of the phantom to a cooled CCD camera (SBIG® ST-10E) through a motorized filter wheel for separate acquisition of trans-illumination and emission images. The image acquisition is performed at a field of view (FOV) of 12 degrees with a binning factor of 4 and average exposure time of 15 sec∕image. The CCD camera is cooled down to around −10 °C to minimize the thermal noise. Dark-frame images (with laser off) are acquired in each measurement and subtracted from data images to correct for read-out noise, stray light effects, and other unwanted signals. Born normalization is performed on the acquired data images to facilitate quantification in the 3D reconstructions [25

25. 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]

].

To compare the performance of the Hadamard-multiplexed FT architecture with existing single-point illumination systems, the phantom experiments were repeated with single-point illumination architecture by replacing the Hadamard coded masks with single-element masks to keep the per-point radiative illumination power constant between experiments. 3D reconstructions were performed on both sets of experimental studies by MLS-ART (with 10 full iterations through the system of equations and a relaxation parameter of 1) on a tetrahedral mesh discretizing the phantom volume with 132,325 nodes and 634,149 voxels. The results are presented in Fig. 6
Fig. 6 Phantom-based experimental results: (i) the double-tube configuration of the fluorescent inclusions in the slab-shaped phantom. 3D reconstructions are performed by MLS-ART on (ii) conventional single-point illumination phantom FT data, and (iii) Hadamard-multiplexed FT data, where the depth of the pair of fluorescent tubes is (a) 3 mm, (b) 6 mm, and (c) 9 mm from the phantom surface facing the camera.
. The row labeled (i) in Fig. 6 shows the double-tube configuration of the phantom-based FT experiment. The reconstructions are presented in rows labeled (ii) and (iii) of Fig. 6. Columns labeled (a), (b), and (c) in Fig. 6 correspond to inclusion depths of 3, 6, and 9 mm, respectively. Reconstructions from single-point illumination FT are presented in the row labeled (ii), and those from Hadamard multiplexed FT in the row labeled (iii) in Fig. 6. As expected, the quality of the reconstructions deteriorates as the depth of the inclusions increases.

While reconstructions of shallow inclusions (3 mm) from both single-illumination and multiplexed data have a reasonable level of accuracy as shown in column (a) of Fig. 6, the advantage of Hadamard multiplexed FT in enhancing robustness becomes evident as the depth of inclusions increases as presented in columns (b) and (c) of Fig. 6. Similar to the numerical studies, it can be observed that Hadamard multiplexing adds considerable robustness to 3D reconstructions particularly for deeper inclusions as the data will be more noise-sensitive and the reconstructions more prone to noise-induced errors. To quantitatively verify the robustness offered by Hadamard multiplexing, the relative MSEs associated with the 3D reconstructions presented in Fig. 6 are plotted versus inclusion depth in Fig. 7
Fig. 7 The relative mean-square errors (ε) of the 3D MLS-ART reconstructions presented in Fig. 6 are plotted versus the depth of the fluorescent inclusions for single-point illumination and Hadamard multiplexed FT architectures.
. The errors in the reconstructions from single-point illumination and multiplexed data for the 3 mm inclusion depth are very close as shown in Fig. 7. The MSE increases with inclusion depth for both illumination architectures. However, the increase in the reconstruction error associated with multiplexed architecture is significantly lower than the single-point illumination architecture especially for the 9-mm deep inclusions. The quantitative results presented in Fig. 7 further validate the observed improvements offered by Hadamard multiplexing in the 3D reconstructions of Fig. 6.

4. Discussion and conclusions

In this work, we introduced a multiplexing scheme built upon Hadamard S-matrix codes to replace and improve the existing single-point illumination architecture in FT with multi-point illumination for the purpose of increasing the SNR and throughput in FT systems and reducing the required tomographic scan times. The high cost of wide-band tunable high-power light sources and per-area illumination power limitations in in vivo optical imaging pose considerable challenges for developing high-throughput high-SNR FT systems. Hadamard multiplexing allows us to overcome these challenges without over-complicating the architecture of the FT system or significant cost increase. Hadamard multiplexing provides an optimal trade-off between the throughput (SNR) and information content of a set of FT measurements. As discussed in Section 1, single-illumination FT measurements provide high information content because of their spatially disjoint sensitivity maps while suffering from low-throughput making them attractive to cases involving thin or low absorptive tissues. Hadamard multiplexed FT offers an optimal trade-off where without significantly jeopardizing the information content of the measurements, a boost in the measurement SNR and throughput is obtained.

As shown in Figs. 2 and 6, the 2D and 3D FT reconstructions indicate that for low-noise FT scenarios with shallow inclusions, the performance of single-point illumination architecture is not significantly different from Hadamard-multiplexed architecture. Due to changes in the system matrix, its condition number, and singular values, along with changes in the experimental setup, the reconstructions from single-point illumination data in both numerical and phantom studies differ from reconstructions from Hadamard-multiplexed data, even for low-noise FT scenarios, as presented in Figs. 2 and 6. The difference between the two, however, becomes more significant as the noise level and depth of the inclusions increase. The data from deeper inclusions is more diffuse and hence more sensitive to and affected by noise contamination. The accuracy of reconstructions of shallow sources from low-noise data is high and of the same order of magnitude for both architectures as shown in the reconstruction error plots of Fig. 3 and Fig. 7. This shows that though the condition of system matrix is affected by Hadamard multiplexing, the reconstruction accuracy is very little jeopardized if any. This is in part due to the low condition number of the Hadamard S-matrices. The condition numbers corresponding to S-matrices of sizes 7, 15, 23, 31, and 63 are 2.82, 4, 4.89, 5.65, and 8, respectively. These condition numbers are significantly low compared to the typical condition numbers of the system matrix in FT, which can range from around 1010 to above 1020 depending on the geometry and optical properties of the turbid medium. As a result, when multiplied by the multiplexing matrix, W, as formulated in Eqs. (6) and (7), the condition number and singular values of the FT system matrix M do not change significantly. Hence, the FT reconstruction accuracy is negligibly impaired by Hadamard multiplexing.

The advantage of Hadamard multiplexed FT becomes evident as the data noise level, inclusion depth, and number of FT sources increase. This can be observed in the comparative trend of 2D and 3D reconstructions and their relative errors in Figs. 2, 3, 6, and 7. In numerical studies, as the FT data SNR decreases to ~30 dB and below, the reconstructions from single-point illumination data completely lose their accuracy and become dominated by artifacts. Meanwhile, reconstructions from Hadamard-multiplexed data preserve their accuracy down to a noise level equivalent to ~10 dB single-point data SNR. In phantom studies, as the depth of the two fluorescent rods increases, the corresponding FT data becomes more diffuse, and hence the 3D reconstruction become more ill-posed and noise-sensitive. As a result, though the noise characteristics of the CCD remain approximately the same (dark-current, read-out, and image noise), artifact contamination in the reconstructions increases with depth. Hadamard multiplexing offers improved robustness in reconstructing the rods at 9 mm depth over single-point illumination architecture as shown in Figs. 6 and 7.

Consequently, Hadamard multiplexed FT can enhance the performance of FT systems especially when suffering from limited illumination power or in imaging scenarios dealing with highly absorbing organs, such as the liver or the lungs in small animals. In this work, full Hadamard S-matrix multiplexing was proposed and studied for FT systems. Nevertheless, partial Hadamard multiplexing of the FT illumination architecture can also offer benefits over existing systems. In partial multiplexing, the illumination grid points are divided into groups (e.g., each grid line forms a group of 5 points), and the S-matrix multiplexing is applied to these groups instead of individual illumination points. In FT systems with limited flexibility over modification of the illumination geometry and optics such as commercial FT systems that use translation stage-based illumination raster scan, partial Hadamard multiplexing can be used to boost the throughput by simply adding one source (and one stage) per each line of the illumination grid. Depending on the degree of partial multiplexing (the total number of multiplexed entities or groups), the data SNR and system throughput can be improved over single-point illumination systems. In other FT systems where optical fiber bundles are used for raster scanning the illumination points, full S-matrix multiplexing can be implemented by simply re-programming the illumination sequence of light sources coupled to the fibers.

Modifications to existing FT system architecture required for full or partial Hadamard multiplexing do not add considerable complexity or cost to these systems, unlike recently explored surface illumination FT architectures [26

26. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Non-contact fluorescence optical tomography with scanning patterned illumination,” Opt. Express 14(14), 6516–6534 (2006). [CrossRef] [PubMed]

29

29. V. Venugopal, J. Chen, and X. Intes, “Development of an optical imaging platform for functional imaging of small animals using wide-field excitation,” Biomed. Opt. Express 1(1), 143–156 (2010). [CrossRef] [PubMed]

]. While structured illumination FT systems can offer advantages over single-point illumination FT, they require more complex hardware that can pose challenges and complications for arbitrary non-flat subject geometries [30

30. D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30(11), 1354–1356 (2005). [CrossRef] [PubMed]

36

36. V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. Biomed. Opt. 18(3), 036006 (2013). [CrossRef] [PubMed]

]. The advantage of Hadamard multiplexed FT is that it offers high robustness and high-throughput wide-field illumination similar to structured-illumination FT without the complex hardware requirements. Hadamard multiplexed FT only requires simple modifications to single-point illumination FT. In FT systems that use fiber bundles for subject illumination, Hadamard multiplexing can be realized by modifying the illumination sequence of the source fibers. In scanning source FT systems, multiplexing can be realized either by a masked lenslet array configuration as presented in this work, or by adding extra source fibers to the system. Compared to structured-illumination FT systems where spatial light modulators (SLM) add significant complexity, cost, and power loss and are only optimal for flat or slab subject geometries, Hadamard multiplexing offers lower cost, complexity, and versatility. As a result, Hadamard multiplexed FT can offer improvements over existing FT systems.

In conclusion, it was shown that Hadamard multiplexed FT provides a versatile solution for improving the SNR, throughput, robustness, and speed of FT systems. The Hadamard-multiplexed FT architecture enhances the accuracy and robustness of FT reconstructions in low-SNR scenarios especially when the number of sources used for illumination is sufficiently high. Additionally, Hadamard multiplexing does not harm the quality of the reconstructions for high-SNR FT scenarios. It was shown that Hadamard multiplexed FT can be realized using hardware whose complexity and cost are not higher than existing single-point illumination FT architectures. These characteristics of Hadamard multiplexed FT, as demonstrated in this work, make it advantageous over existing FT systems.

References and links

1.

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8(1), 1–33 (2006). [CrossRef] [PubMed]

2.

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1(2), 82–88 (2002). [CrossRef] [PubMed]

3.

V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8(7), 757–761 (2002). [CrossRef] [PubMed]

4.

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15(11), 6696–6716 (2007). [CrossRef] [PubMed]

5.

S. C. Davis, H. Dehghani, J. Wang, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization,” Opt. Express 15(7), 4066–4082 (2007). [CrossRef] [PubMed]

6.

P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. 46(10), 1679–1685 (2007). [CrossRef] [PubMed]

7.

J. C. Baritaux, K. Hassler, and M. Unser, “An efficient numerical method for general Lp regularization in fluorescence molecular tomography,” IEEE Trans. Med. Imaging 29(4), 1075–1087 (2010). [CrossRef] [PubMed]

8.

D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express 18(8), 8630–8646 (2010). [CrossRef] [PubMed]

9.

A. Behrooz, H. M. Zhou, A. A. Eftekhar, and A. Adibi, “Total variation regularization for 3D reconstruction in fluorescence tomography: experimental phantom studies,” Appl. Opt. 51(34), 8216–8227 (2012). [CrossRef] [PubMed]

10.

D. Sliney and M. Wolbarsht, Safety with Lasers and Other Optical Sources, Plenum, New York (1980).

11.

ANSI Standard Z136.1, American National Standard for the Safe Use of Lasers, American National Standards Institute, Inc., New York (2000).

12.

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

13.

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42(5), 841–853 (1997). [CrossRef] [PubMed]

14.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001). [CrossRef]

15.

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37(22), 5337–5343 (1998). [CrossRef] [PubMed]

16.

M. Harwit, and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, New York (1979).

17.

L. Streeter, G. R. Burling-Claridge, M. J. Cree, and R. Künnemeyer, “Optical full Hadamard matrix multiplexing and noise effects,” Appl. Opt. 48(11), 2078–2085 (2009). [CrossRef] [PubMed]

18.

R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Hadamard transform Raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19(2), 177–186 (1999). [CrossRef]

19.

V. V. Fedorov, Theory of Optimal Experiments, Academic Press, New York (1972).

20.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29(3), 471–481 (1970). [CrossRef] [PubMed]

21.

A. Behrooz, C. Kuo, H. Xu, and B. W. Rice, “Adaptive row-action inverse solver for fast noise-robust 3D reconstructions in bioluminescence tomography: theory and dual-modality optical/CT in vivo studies,” J. Biomed. Opt. 18(7), 076010 (2013). [CrossRef] [PubMed]

22.

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30(5), 901–911 (2003). [CrossRef] [PubMed]

23.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42(10), 1971–1979 (1997). [CrossRef] [PubMed]

24.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12(5), 510–519 (1992). [CrossRef] [PubMed]

25.

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]

26.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Non-contact fluorescence optical tomography with scanning patterned illumination,” Opt. Express 14(14), 6516–6534 (2006). [CrossRef] [PubMed]

27.

A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys. 33(5), 1299–1310 (2006). [CrossRef] [PubMed]

28.

J. Dutta, S. Ahn, A. A. Joshi, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 55(10), 2961–2982 (2010). [CrossRef] [PubMed]

29.

V. Venugopal, J. Chen, and X. Intes, “Development of an optical imaging platform for functional imaging of small animals using wide-field excitation,” Biomed. Opt. Express 1(1), 143–156 (2010). [CrossRef] [PubMed]

30.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30(11), 1354–1356 (2005). [CrossRef] [PubMed]

31.

A. Mazhar, D. J. Cuccia, S. Gioux, A. J. Durkin, J. V. Frangioni, and B. J. Tromberg, “Structured illumination enhances resolution and contrast in thick tissue fluorescence imaging,” J. Biomed. Opt. 15(1), 010506 (2010). [CrossRef] [PubMed]

32.

N. Ducros, C. D’andrea, G. Valentini, T. Rudge, S. Arridge, and A. Bassi, “Full-wavelet approach for fluorescence diffuse optical tomography with structured illumination,” Opt. Lett. 35(21), 3676–3678 (2010). [CrossRef] [PubMed]

33.

C. D’Andrea, N. Ducros, A. Bassi, S. Arridge, and G. Valentini, “Fast 3D optical reconstruction in turbid media using spatially modulated light,” Biomed. Opt. Express 1(2), 471–481 (2010). [CrossRef] [PubMed]

34.

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15(1), 016006 (2010). [CrossRef] [PubMed]

35.

S. D. Konecky, A. Mazhar, D. Cuccia, A. J. Durkin, J. C. Schotland, and B. J. Tromberg, “Quantitative optical tomography of sub-surface heterogeneities using spatially modulated structured light,” Opt. Express 17(17), 14780–14790 (2009). [CrossRef] [PubMed]

36.

V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. Biomed. Opt. 18(3), 036006 (2013). [CrossRef] [PubMed]

OCIS Codes
(170.0110) Medical optics and biotechnology : Imaging systems
(170.3880) Medical optics and biotechnology : Medical and biological imaging
(170.6960) Medical optics and biotechnology : Tomography
(110.6955) Imaging systems : Tomographic imaging

ToC Category:
Image Reconstruction and Inverse Problems

History
Original Manuscript: October 21, 2013
Revised Manuscript: January 10, 2014
Manuscript Accepted: February 5, 2014
Published: February 18, 2014

Citation
Ali Behrooz, Ali A. Eftekhar, and Ali Adibi, "Hadamard multiplexed fluorescence tomography," Biomed. Opt. Express 5, 763-777 (2014)
http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-5-3-763


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References

  1. V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng.8(1), 1–33 (2006). [CrossRef] [PubMed]
  2. V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging1(2), 82–88 (2002). [CrossRef] [PubMed]
  3. V. Ntziachristos, C. H. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med.8(7), 757–761 (2002). [CrossRef] [PubMed]
  4. A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express15(11), 6696–6716 (2007). [CrossRef] [PubMed]
  5. S. C. Davis, H. Dehghani, J. Wang, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization,” Opt. Express15(7), 4066–4082 (2007). [CrossRef] [PubMed]
  6. P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt.46(10), 1679–1685 (2007). [CrossRef] [PubMed]
  7. J. C. Baritaux, K. Hassler, and M. Unser, “An efficient numerical method for general Lp regularization in fluorescence molecular tomography,” IEEE Trans. Med. Imaging29(4), 1075–1087 (2010). [CrossRef] [PubMed]
  8. D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express18(8), 8630–8646 (2010). [CrossRef] [PubMed]
  9. A. Behrooz, H. M. Zhou, A. A. Eftekhar, and A. Adibi, “Total variation regularization for 3D reconstruction in fluorescence tomography: experimental phantom studies,” Appl. Opt.51(34), 8216–8227 (2012). [CrossRef] [PubMed]
  10. D. Sliney and M. Wolbarsht, Safety with Lasers and Other Optical Sources, Plenum, New York (1980).
  11. ANSI Standard Z136.1, American National Standard for the Safe Use of Lasers, American National Standards Institute, Inc., New York (2000).
  12. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978).
  13. S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol.42(5), 841–853 (1997). [CrossRef] [PubMed]
  14. D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag.18(6), 57–75 (2001). [CrossRef]
  15. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt.37(22), 5337–5343 (1998). [CrossRef] [PubMed]
  16. M. Harwit, and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, New York (1979).
  17. L. Streeter, G. R. Burling-Claridge, M. J. Cree, and R. Künnemeyer, “Optical full Hadamard matrix multiplexing and noise effects,” Appl. Opt.48(11), 2078–2085 (2009). [CrossRef] [PubMed]
  18. R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Hadamard transform Raman imagery with a digital micro-mirror array,” Vib. Spectrosc.19(2), 177–186 (1999). [CrossRef]
  19. V. V. Fedorov, Theory of Optimal Experiments, Academic Press, New York (1972).
  20. R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol.29(3), 471–481 (1970). [CrossRef] [PubMed]
  21. A. Behrooz, C. Kuo, H. Xu, and B. W. Rice, “Adaptive row-action inverse solver for fast noise-robust 3D reconstructions in bioluminescence tomography: theory and dual-modality optical/CT in vivo studies,” J. Biomed. Opt.18(7), 076010 (2013). [CrossRef] [PubMed]
  22. E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys.30(5), 901–911 (2003). [CrossRef] [PubMed]
  23. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol.42(10), 1971–1979 (1997). [CrossRef] [PubMed]
  24. S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med.12(5), 510–519 (1992). [CrossRef] [PubMed]
  25. 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]
  26. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Non-contact fluorescence optical tomography with scanning patterned illumination,” Opt. Express14(14), 6516–6534 (2006). [CrossRef] [PubMed]
  27. A. Joshi, W. Bangerth, K. Hwang, J. C. Rasmussen, and E. M. Sevick-Muraca, “Fully adaptive FEM based fluorescence optical tomography from time-dependent measurements with area illumination and detection,” Med. Phys.33(5), 1299–1310 (2006). [CrossRef] [PubMed]
  28. J. Dutta, S. Ahn, A. A. Joshi, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol.55(10), 2961–2982 (2010). [CrossRef] [PubMed]
  29. V. Venugopal, J. Chen, and X. Intes, “Development of an optical imaging platform for functional imaging of small animals using wide-field excitation,” Biomed. Opt. Express1(1), 143–156 (2010). [CrossRef] [PubMed]
  30. D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett.30(11), 1354–1356 (2005). [CrossRef] [PubMed]
  31. A. Mazhar, D. J. Cuccia, S. Gioux, A. J. Durkin, J. V. Frangioni, and B. J. Tromberg, “Structured illumination enhances resolution and contrast in thick tissue fluorescence imaging,” J. Biomed. Opt.15(1), 010506 (2010). [CrossRef] [PubMed]
  32. N. Ducros, C. D’andrea, G. Valentini, T. Rudge, S. Arridge, and A. Bassi, “Full-wavelet approach for fluorescence diffuse optical tomography with structured illumination,” Opt. Lett.35(21), 3676–3678 (2010). [CrossRef] [PubMed]
  33. C. D’Andrea, N. Ducros, A. Bassi, S. Arridge, and G. Valentini, “Fast 3D optical reconstruction in turbid media using spatially modulated light,” Biomed. Opt. Express1(2), 471–481 (2010). [CrossRef] [PubMed]
  34. S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt.15(1), 016006 (2010). [CrossRef] [PubMed]
  35. S. D. Konecky, A. Mazhar, D. Cuccia, A. J. Durkin, J. C. Schotland, and B. J. Tromberg, “Quantitative optical tomography of sub-surface heterogeneities using spatially modulated structured light,” Opt. Express17(17), 14780–14790 (2009). [CrossRef] [PubMed]
  36. V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. Biomed. Opt.18(3), 036006 (2013). [CrossRef] [PubMed]

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