Abstract

In fluorescence molecular tomography, the accurate and stable reconstruction of fluorescence-labeled targets remains a challenge for wide application of this imaging modality. Here we propose a two-step three-dimensional shape-based reconstruction method using graphics processing unit (GPU) acceleration. In this method, the fluorophore distribution is assumed as the sum of ellipsoids with piecewise-constant fluorescence intensities. The inverse problem is formulated as a constrained nonlinear least-squares problem with respect to shape parameters, leading to much less ill-posedness as the number of unknowns is greatly reduced. Considering that various shape parameters contribute differently to the boundary measurements, we use a two-step optimization algorithm to handle them in a distinctive way and also stabilize the reconstruction. Additionally, the GPU acceleration is employed for finite-element-method-based calculation of the objective function value and the Jacobian matrix, which reduces the total optimization time from around 10 min to less than 1 min. The numerical simulations show that our method can accurately reconstruct multiple targets of various shapes while the conventional voxel-based reconstruction cannot separate the nearby targets. Moreover, the two-step optimization can tolerate different initial values in the existence of noises, even when the number of targets is not known a priori. A physical phantom experiment further demonstrates the method’s potential in practical applications.

© 2012 Optical Society of America

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2012 (2)

M. Li, X. Cao, F. Liu, B. Zhang, J. Luo, and J. Bai, “Reconstruction of fluorescence molecular tomography using a neighborhood regularization,” IEEE Trans. Biomed. Eng. 59, 1799–1803 (2012).
[CrossRef]

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

2011 (1)

2010 (10)

M. Schweiger, O. Dorn, A. Zacharopoulos, I. Nissila, and S. R. Arridge, “3D level set reconstruction of model and experimental data in diffuse optical tomography,” Opt. Express 18, 150–164 (2010).
[CrossRef]

D. Wang, X. Liu, Y. Chen, and J. Bai, “In-vivo fluorescence molecular tomography based on optimal small animal surface reconstruction,” Chin. Opt. Lett. 8, 82–85 (2010).
[CrossRef]

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, 8630–8646 (2010).
[CrossRef]

K. Liu, X. Yang, D. Liu, C. Qin, J. Liu, Z. Chang, M. Xu, and J. Tian, “Spectrally resolved three-dimensional bioluminescence tomography with a level-set strategy,” J. Opt. Soc. Am. A 27, 1413–1423 (2010).
[CrossRef]

E. Alerstam, W. C. Y. Lo, T. D. Han, J. Rose, S. Andersson-Engels, and L. Lilge, “Next-generation acceleration and codeoptimization for light transport in turbid media using GPUs,” Biomed. Opt. Express 1, 658–675 (2010).
[CrossRef]

B. Zhang, X. Yang, F. Yang, C. Qin, D. Han, X. Ma, K. Liu, and J. Tian, “The CUBLAS and CULA based GPU acceleration of adaptive finite element framework for bioluminescence tomography,” Opt. Express 18, 20201–20214 (2010).
[CrossRef]

H. Gao, H. Zhao, W. Cong, and G. Wang, “Bioluminescence tomography with Gaussian prior,” Biomed. Opt. Express 1, 1259–1277 (2010).
[CrossRef]

D. Hyde, E. L. Miller, D. H. Brooks, and V. Ntziachristos, “Data specific spatially varying regularization for multi-modal fluorescence molecular tomography,” IEEE Trans. Med. Imaging 29, 365–374 (2010).
[CrossRef]

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, 77–94 (2010).
[CrossRef]

D. Wang, X. Liu, F. Liu, and J. Bai, “Full-angle fluorescence diffuse optical tomography with spatially coded parallel excitation,” IEEE Trans. Inf. Technol. Biomed. 14, 1346–1354 (2010).
[CrossRef]

2009 (4)

2008 (2)

G. Boverman, E. L. Miller, D. H. Brooks, D. Isaacson, Q. Fang, and D. A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef]

S. Wang, and A. P. Dhawan, “Shape-based multi-spectral optical image reconstruction through genetic algorithm based optimization,” Comput. Med. Imaging Graph. 32, 429–441 (2008).
[CrossRef]

2007 (6)

2006 (2)

M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography using a level set technique,” Opt. Lett. 31, 471–473 (2006).
[CrossRef]

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method,” Inverse Problems 22, 1509–1532(2006).
[CrossRef]

2005 (1)

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized born ratio,” IEEE Trans. Med. Imag. 24, 1377–1386 (2005).
[CrossRef]

2003 (2)

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

M. E. Kilmer, E. L. Miller, A. Barbaro, and D. Boas, “Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt. 42, 3129–3144 (2003).
[CrossRef]

2000 (1)

1995 (1)

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite elementmethod for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Adibi, A.

Ahn, S.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Alerstam, E.

Álvarez, D.

Andersson-Engels, S.

Arridge, S.

Arridge, S. R.

M. Schweiger, O. Dorn, A. Zacharopoulos, I. Nissila, and S. R. Arridge, “3D level set reconstruction of model and experimental data in diffuse optical tomography,” Opt. Express 18, 150–164 (2010).
[CrossRef]

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method,” Inverse Problems 22, 1509–1532(2006).
[CrossRef]

M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography using a level set technique,” Opt. Lett. 31, 471–473 (2006).
[CrossRef]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Deply, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A 17, 1659–1670 (2000).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite elementmethod for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Babaeizadeh, S.

S. Babaeizadeh and D. H. Brooks, “Electrical impedance tomography for piecewise constant domains using boundary element shape-based inverse solutions,” IEEE Trans. Med. Imaging 26, 637–647 (2007).
[CrossRef]

Bai, J.

M. Li, X. Cao, F. Liu, B. Zhang, J. Luo, and J. Bai, “Reconstruction of fluorescence molecular tomography using a neighborhood regularization,” IEEE Trans. Biomed. Eng. 59, 1799–1803 (2012).
[CrossRef]

D. Wang, X. Liu, F. Liu, and J. Bai, “Full-angle fluorescence diffuse optical tomography with spatially coded parallel excitation,” IEEE Trans. Inf. Technol. Biomed. 14, 1346–1354 (2010).
[CrossRef]

D. Wang, X. Liu, Y. Chen, and J. Bai, “In-vivo fluorescence molecular tomography based on optimal small animal surface reconstruction,” Chin. Opt. Lett. 8, 82–85 (2010).
[CrossRef]

X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express 15, 18300–18317 (2007).
[CrossRef]

Barbaro, A.

Boas, D.

Boas, D. A.

Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17, 20178–20190 (2009).
[CrossRef]

G. Boverman, E. L. Miller, D. H. Brooks, D. Isaacson, Q. Fang, and D. A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef]

Boverman, G.

G. Boverman, E. L. Miller, D. H. Brooks, D. Isaacson, Q. Fang, and D. A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef]

Brooks, D. H.

D. Hyde, E. L. Miller, D. H. Brooks, and V. Ntziachristos, “Data specific spatially varying regularization for multi-modal fluorescence molecular tomography,” IEEE Trans. Med. Imaging 29, 365–374 (2010).
[CrossRef]

G. Boverman, E. L. Miller, D. H. Brooks, D. Isaacson, Q. Fang, and D. A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef]

S. Babaeizadeh and D. H. Brooks, “Electrical impedance tomography for piecewise constant domains using boundary element shape-based inverse solutions,” IEEE Trans. Med. Imaging 26, 637–647 (2007).
[CrossRef]

Cao, X.

M. Li, X. Cao, F. Liu, B. Zhang, J. Luo, and J. Bai, “Reconstruction of fluorescence molecular tomography using a neighborhood regularization,” IEEE Trans. Biomed. Eng. 59, 1799–1803 (2012).
[CrossRef]

Chang, Z.

Chen, N.

Chen, Y.

Cherry, S. R.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Cong, W.

Davis, S. C.

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, 77–94 (2010).
[CrossRef]

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, 4066–4082 (2007).
[CrossRef]

Dehghani, H.

Deliolanis, N.

Delpy, D. T.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite elementmethod for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Deply, D. T.

Dhawan, A. P.

S. Wang, and A. P. Dhawan, “Shape-based multi-spectral optical image reconstruction through genetic algorithm based optimization,” Comput. Med. Imaging Graph. 32, 429–441 (2008).
[CrossRef]

Dorn, O.

Dutta, J.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Eftekhar, A. A.

Fang, Q.

Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17, 20178–20190 (2009).
[CrossRef]

G. Boverman, E. L. Miller, D. H. Brooks, D. Isaacson, Q. Fang, and D. A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef]

Feng, J.

Gao, H.

Graves, E. E.

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

Gulsen, G.

Han, D.

Han, T. D.

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite elementmethod for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Huang, J.

Hyde, D.

D. Hyde, E. L. Miller, D. H. Brooks, and V. Ntziachristos, “Data specific spatially varying regularization for multi-modal fluorescence molecular tomography,” IEEE Trans. Med. Imaging 29, 365–374 (2010).
[CrossRef]

N. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. 32, 382–384 (2007).
[CrossRef]

Isaacson, D.

G. Boverman, E. L. Miller, D. H. Brooks, D. Isaacson, Q. Fang, and D. A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef]

Jiang, S.

Kak, A.

A. Kak and M. Slaney, Computerized Tomographic Imaging (IEEE, 1987).

Kilmer, M. E.

Kolehmainen, V.

A. Zacharopoulos, M. Schweiger, V. Kolehmainen, and S. Arridge, “3D shape based reconstruction of experimental data in diffuse optical tomography,” Opt. Express 17, 18940–18956 (2009).
[CrossRef]

M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography using a level set technique,” Opt. Lett. 31, 471–473 (2006).
[CrossRef]

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method,” Inverse Problems 22, 1509–1532(2006).
[CrossRef]

Lasser, T.

N. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. 32, 382–384 (2007).
[CrossRef]

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

Leahy, R. M.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Leblond, F.

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, 77–94 (2010).
[CrossRef]

Li, C.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Li, M.

M. Li, X. Cao, F. Liu, B. Zhang, J. Luo, and J. Bai, “Reconstruction of fluorescence molecular tomography using a neighborhood regularization,” IEEE Trans. Biomed. Eng. 59, 1799–1803 (2012).
[CrossRef]

Lilge, L.

Lin, Y.

Liu, D.

Liu, F.

M. Li, X. Cao, F. Liu, B. Zhang, J. Luo, and J. Bai, “Reconstruction of fluorescence molecular tomography using a neighborhood regularization,” IEEE Trans. Biomed. Eng. 59, 1799–1803 (2012).
[CrossRef]

D. Wang, X. Liu, F. Liu, and J. Bai, “Full-angle fluorescence diffuse optical tomography with spatially coded parallel excitation,” IEEE Trans. Inf. Technol. Biomed. 14, 1346–1354 (2010).
[CrossRef]

Liu, J.

Liu, K.

Liu, X.

D. Wang, X. Liu, Y. Chen, and J. Bai, “In-vivo fluorescence molecular tomography based on optimal small animal surface reconstruction,” Chin. Opt. Lett. 8, 82–85 (2010).
[CrossRef]

D. Wang, X. Liu, F. Liu, and J. Bai, “Full-angle fluorescence diffuse optical tomography with spatially coded parallel excitation,” IEEE Trans. Inf. Technol. Biomed. 14, 1346–1354 (2010).
[CrossRef]

Lo, W. C. Y.

Luo, J.

M. Li, X. Cao, F. Liu, B. Zhang, J. Luo, and J. Bai, “Reconstruction of fluorescence molecular tomography using a neighborhood regularization,” IEEE Trans. Biomed. Eng. 59, 1799–1803 (2012).
[CrossRef]

Ma, X.

Medina, P.

Miller, E. L.

D. Hyde, E. L. Miller, D. H. Brooks, and V. Ntziachristos, “Data specific spatially varying regularization for multi-modal fluorescence molecular tomography,” IEEE Trans. Med. Imaging 29, 365–374 (2010).
[CrossRef]

G. Boverman, E. L. Miller, D. H. Brooks, D. Isaacson, Q. Fang, and D. A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef]

M. E. Kilmer, E. L. Miller, A. Barbaro, and D. Boas, “Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt. 42, 3129–3144 (2003).
[CrossRef]

Mohajerani, P.

Moscoso, M.

Nalcioglu, O.

Nissila, I.

Ntziachristos, V.

D. Hyde, E. L. Miller, D. H. Brooks, and V. Ntziachristos, “Data specific spatially varying regularization for multi-modal fluorescence molecular tomography,” IEEE Trans. Med. Imaging 29, 365–374 (2010).
[CrossRef]

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

N. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. 32, 382–384 (2007).
[CrossRef]

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized born ratio,” IEEE Trans. Med. Imag. 24, 1377–1386 (2005).
[CrossRef]

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

Paulsen, K. D.

Pogue, B. W.

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, 77–94 (2010).
[CrossRef]

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, 4066–4082 (2007).
[CrossRef]

Qin, C.

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Appl. Opt. (3)

Biomed. Opt. Express (2)

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S. Wang, and A. P. Dhawan, “Shape-based multi-spectral optical image reconstruction through genetic algorithm based optimization,” Comput. Med. Imaging Graph. 32, 429–441 (2008).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

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A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized born ratio,” IEEE Trans. Med. Imag. 24, 1377–1386 (2005).
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D. Hyde, E. L. Miller, D. H. Brooks, and V. Ntziachristos, “Data specific spatially varying regularization for multi-modal fluorescence molecular tomography,” IEEE Trans. Med. Imaging 29, 365–374 (2010).
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Figures (8)

Fig. 1.
Fig. 1.

Two-step optimization of ellipsoid-shape parameters.

Fig. 2.
Fig. 2.

Flowchart of the GPU-accelerated computation of shape-voxel mapping. One GPU block deals with only one voxel-ellipsoid pair with 256 threads. Two levels of discretization are used for faster speed. For each subvoxel, 64 threads are formed in one group to process it.

Fig. 3.
Fig. 3.

Simulation experiment sketch. A full-angle CCD camera-based imaging system is used for simulated data acquisition. The cylinder phantom object can be rotated to a different projection angle. A CCD camera with horizontal×vertical field of view (CCDHFOV×CCDVFOV) is used to capture excitation and fluorescence projection images at each projection angle. In the simulation, full-angle acquisitions were performed at 18 evenly distributed projection angles. For each projection angle, four projections were simulated with excitation source at different positions, as depicted by the red dots (in the left and middle parts). For each projection, the detectors, corresponding to selected detection points on the image plane, are within 1.8 cm detector horizontal FOV and 2.2 cm detector vertical FOV with 0.15 cm detector spacing.

Fig. 4.
Fig. 4.

Comparison results between the proposed method and voxel-based reconstruction. In slice images, the red circles denote the outer boundary of the imaged object, and the white lines denote the boundaries of the real inclusions. The slice images are of 3.0 cm height.

Fig. 5.
Fig. 5.

Reconstruction results with different initial estimates of dual small spheres. In slice images, the red circle denotes the boundary of the imaged object, and the white lines denote the boundaries of the real inclusions. The slice images are of 3.0 cm height.

Fig. 6.
Fig. 6.

Reconstruction results with the assumed target number larger than two. In slice images, the red circle denotes the boundary of the imaged object, and the white lines denote the boundaries of the real inclusions. The slice images are of 3.0 cm height.

Fig. 7.
Fig. 7.

Evaluation of GPU acceleration. The investigated targets were spheres for simplicity. (a)–(e) CPU and GPU computation of Jacobian matrix for different cases. The Jacobian matrix calculation is composed of two basic operations including shape-voxel mapping and matrix multiplication. (a) CPU execution time of shape-voxel mapping operation. (b) GPU speedup of shape-voxel mapping operation. (c) CPU execution time of matrix multiplication operation. (d) GPU speedup of matrix multiplication operation. (e) GPU speedup of whole Jacobian matrix computation. (f) GPU speedup of the total optimization process.

Fig. 8.
Fig. 8.

Physical phantom experiment. Two fluorescence inclusions were closed placed inside a cylinder phantom with 0.16 cm edge-to-edge distance. (a) Full-angle fluorescence molecular system. (b) Fluorescence projection images. (c) Reconstructed images. In slice images, the red circle denotes the boundary of the imaged object, and the white circles denote the inner (solid line) and outer (dashed line) boundaries of the real inclusions. The two slice images are at the same height slice, as depicted by the red circle in the 3D image. The images are shown with intensity range from 0 to their maximum, respectively.

Tables (3)

Tables Icon

Table 1. Parameters for Inclusions with Various Shapes for Cube or Cuboid Shapes, the Anisotropic Radii are Half its Edge Lengths, Respectively

Tables Icon

Table 2. Reconstructed Parameters for Inclusions with Various Shapes

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Table 3. Initial Estimates of Dual Small Spheres

Equations (18)

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Φbornmeas(r⃗s,r⃗d)=Φmmeas(r⃗s,r⃗d)/Φxmeas(r⃗s,r⃗d)=ΩG(r⃗s,r⃗)G(r⃗,r⃗d)/G(r⃗s,r⃗d)f(r⃗),
{[D(r⃗)G(r⃗s,r⃗)]+μa(r⃗)G(r⃗s,r⃗)=δ(r⃗r⃗s)r⃗Ω2AD(r⃗)G(r⃗s,r⃗)/n⃗+G(r⃗s,r⃗)=0r⃗Ω,
KG=Q,
f(r⃗)=f(x,y,z)=i=1nρiU(1XiTRiTΣiRiXi),
Xi=[xxc,iyyc,izzc,i],i=[1/rx,i1/ry,i1/rz,i],Ri=RαiRβiRγi,
Rαi=[cosαisinαisinαicosαi1],Rβi=[1cosβisinβisinβicosβi],Rγi=[cosγisinγisinγicosγi1].
argminζΨ(ζ)=12ΦbornmeasF(ζ)22,s.t.1)ρ0,2)rminrrmax,3)cxyrxry1/cxyrx,cyzryrz1/cyzry,czxrzrx1/czxrz,4)xcminxcxcmax,ycminycycmax,zcminzczcmax,5)π/2<βπ/2,6)SiSj=,i,j,ij,
Voverlap,i,j=Ω[(f(r⃗)i/ρi+f(r⃗)j/ρj)>1]=0,i,j,ij.
roverlap,i,j=[3Voverlap,i,j/(4π)1/3=0,i,j,ij,
ζk+1=ζk+argminδζ12Jδζ(ΦbornmeasF(ζk))22+C(ζk+δζ).
δζ=argminδζL(δζ,t)=12Jδζ(ΦbornmeasF(ζk))22+tj{min[0,cj(ζk+δζ)]}2.
(δζ)n+1=(δζ)n+(JTJ+δζ2C+λΛ)1{JT[Jδζ(ΦbornmeasF(ζk))]δζC}tn+1=2tn,
δζ2C6)=(δζC6))T(δζC6))/t.
F(ζ)(r⃗s,r⃗d)=i=1nVG(r⃗s,r⃗i)G(r⃗d,r⃗i)/G(r⃗s,r⃗d)ΔVf(r⃗i),
F(ζ)=W[f(r⃗1)f(r⃗nV)],
J=F(ζ)/ζ=W[f(r⃗1)f(r⃗nV)]/ζ.
f(r⃗i)/ζj=Δf(r⃗i)Δζj=Δf(r⃗i)(ζ+Δζj)Δf(r⃗i)(ζ)Δζj.
ξ=argminξ12ΦbornmeasWξ22s.t.ξ0,

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