Abstract

We propose a method for object localization in fluorescent tomography (FT) in the presence of a highly heterogeneous background. Existing approaches typically assume a homogeneous background distribution; thus, they are incapable of accurately accounting for the more general case of an unconstrained, possibly heterogeneous, background. The proposed method iteratively solves the inverse problem over a solution space partitioned into a background subspace and an object subspace to simultaneously estimate the background and localize the target fluorescent objects. Simulation results of this algorithm applied to continuous-wave FT demonstrate effective localization of target objects in the presence of highly heterogeneous background distributions.

© 2008 Optical Society of America

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2007 (2)

2006 (4)

A. Soubret and V. Ntziachristos, “Fluorescence molecular tomography in the presence of background fluorescence,” Phys. Med. Biol. 51, 3983-4001 (2006).
[CrossRef] [PubMed]

E. Chang, J. Sun, J. S. Miller, W. W. Yu, V. L. Colvin, J. L. West, and R. Drezek, “Protease-activated quantum dot probes,” Proc. SPIE 6191, 61911E.1-10 (2006).

R. Weissleder, “Molecular imaging in cancer,” Science 312, 1168-1171 (2006).
[CrossRef] [PubMed]

R. Roy, A. Godavarty, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical tomography of a large tissue phantom using point illumination geometries,” J. Biomed. Opt. 11, 044007 (2006).
[CrossRef] [PubMed]

2005 (6)

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: Study of the normalized Bornratio,” IEEE Trans. Med. Imaging 24, 1377-1386 (2005).
[CrossRef] [PubMed]

G. Zacharakis, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Fluorescent protein tomography scanner for small animal imaging,” IEEE Trans. Med. Imaging 24, 878-885 (2005).
[CrossRef] [PubMed]

A. K. Sahu, R. Roy, A. Joshi, and E. M. Sevick-Muraca, “Evaluation of anatomical structure and non-uniform distribution of imaging agent in near-infrared fluorescence-enhanced optical tomography,” Opt. Express 13, 10182-10199 (2005).
[CrossRef] [PubMed]

A. Godavarty, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Detection of single and multiple targets in tissue phantoms with fluorescence-enhanced optical imaging: Feasibility study,” Radiology 235, 148-154 (2005).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1-43 (2005).
[CrossRef]

A. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express 13, 9847-9857 (2005).
[CrossRef] [PubMed]

2004 (1)

R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

2003 (3)

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081-3094 (2003).
[CrossRef] [PubMed]

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

A. Godavarty, M. J. Eppstein, Z. Chaoyang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701-1720 (2003).
[CrossRef] [PubMed]

2002 (2)

K. Licha, “Contrast agents for optical imaging,” Top. Curr. Chem. 222, 1-29 (2002).
[CrossRef]

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

2001 (3)

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

R. Weissleder, “A clearer vision for in vivo imaging,” Nat. Biotechnol. 19, 316-317 (2001).
[CrossRef] [PubMed]

C. Bremer, C. H. Tung, and R. Weissleder, “In vivo molecular target assessment of matrix metalloproteinase inhibition,” Nat. Med. (N.Y.) 7, 743-748 (2001).
[CrossRef]

1998 (4)

1995 (1)

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

1993 (1)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Adibi, A.

Arridge, S. R.

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, 6696-6716 (2007).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1-43 (2005).
[CrossRef]

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

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Barbour, R. L.

Boas, D. A.

Bouman, C. A.

Boyd, S. P.

S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge U. Press, 2004).

Bremer, C.

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

C. Bremer, C. H. Tung, and R. Weissleder, “In vivo molecular target assessment of matrix metalloproteinase inhibition,” Nat. Med. (N.Y.) 7, 743-748 (2001).
[CrossRef]

Chang, E.

E. Chang, J. Sun, J. S. Miller, W. W. Yu, V. L. Colvin, J. L. West, and R. Drezek, “Protease-activated quantum dot probes,” Proc. SPIE 6191, 61911E.1-10 (2006).

Chang, J.

Chaoyang, Z.

A. Godavarty, M. J. Eppstein, Z. Chaoyang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701-1720 (2003).
[CrossRef] [PubMed]

Chen, S. S.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 20, 33-61 (1998).

Choe, R.

Colvin, V. L.

E. Chang, J. Sun, J. S. Miller, W. W. Yu, V. L. Colvin, J. L. West, and R. Drezek, “Protease-activated quantum dot probes,” Proc. SPIE 6191, 61911E.1-10 (2006).

Cong, A.

Corlu, A.

Delpy, D. T.

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

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Donoho, D. L.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 20, 33-61 (1998).

Drezek, R.

E. Chang, J. Sun, J. S. Miller, W. W. Yu, V. L. Colvin, J. L. West, and R. Drezek, “Protease-activated quantum dot probes,” Proc. SPIE 6191, 61911E.1-10 (2006).

Durduran, T.

Eftekhar, A. A.

Eppstein, M. J.

A. Godavarty, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Detection of single and multiple targets in tissue phantoms with fluorescence-enhanced optical imaging: Feasibility study,” Radiology 235, 148-154 (2005).
[CrossRef] [PubMed]

A. Godavarty, M. J. Eppstein, Z. Chaoyang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701-1720 (2003).
[CrossRef] [PubMed]

Foster, T. H.

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1-43 (2005).
[CrossRef]

Godavarty, A.

R. Roy, A. Godavarty, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical tomography of a large tissue phantom using point illumination geometries,” J. Biomed. Opt. 11, 044007 (2006).
[CrossRef] [PubMed]

A. Godavarty, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Detection of single and multiple targets in tissue phantoms with fluorescence-enhanced optical imaging: Feasibility study,” Radiology 235, 148-154 (2005).
[CrossRef] [PubMed]

A. Godavarty, M. J. Eppstein, Z. Chaoyang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701-1720 (2003).
[CrossRef] [PubMed]

Graber, H. L.

Graves, E. E.

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

Gurfinkel, M.

A. Godavarty, M. J. Eppstein, Z. Chaoyang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701-1720 (2003).
[CrossRef] [PubMed]

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1-43 (2005).
[CrossRef]

Hiraoka, M.

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

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Huang, J.

Hull, E. L.

Jiang, H.

Joshi, A.

Krishnaprasad, P. S.

Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition,” in Proceedings of the 27th Asilomar Conference on Signals Systems and Computers (IEEE, 1993), pp. 40-44.
[CrossRef]

Licha, K.

K. Licha, “Contrast agents for optical imaging,” Top. Curr. Chem. 222, 1-29 (2002).
[CrossRef]

Millane, R. P.

Miller, J. S.

E. Chang, J. Sun, J. S. Miller, W. W. Yu, V. L. Colvin, J. L. West, and R. Drezek, “Protease-activated quantum dot probes,” Proc. SPIE 6191, 61911E.1-10 (2006).

Milstein, A. B.

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081-3094 (2003).
[CrossRef] [PubMed]

A. B. Milstein, “Imaging of near-infrared fluorescence, absorption, and scattering in turbid media,” Ph.D. thesis, (Purdue University, 2004).

Mohajerani, P.

Nichols, M. G.

Ntziachristos, V.

A. Soubret and V. Ntziachristos, “Fluorescence molecular tomography in the presence of background fluorescence,” Phys. Med. Biol. 51, 3983-4001 (2006).
[CrossRef] [PubMed]

G. Zacharakis, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Fluorescent protein tomography scanner for small animal imaging,” IEEE Trans. Med. Imaging 24, 878-885 (2005).
[CrossRef] [PubMed]

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: Study of the normalized Bornratio,” IEEE Trans. Med. Imaging 24, 1377-1386 (2005).
[CrossRef] [PubMed]

R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

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

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

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

Oh, S.

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1989).

Pati, Y. C.

Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition,” in Proceedings of the 27th Asilomar Conference on Signals Systems and Computers (IEEE, 1993), pp. 40-44.
[CrossRef]

Rezaiifar, R.

Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition,” in Proceedings of the 27th Asilomar Conference on Signals Systems and Computers (IEEE, 1993), pp. 40-44.
[CrossRef]

Ripoll, J.

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: Study of the normalized Bornratio,” IEEE Trans. Med. Imaging 24, 1377-1386 (2005).
[CrossRef] [PubMed]

G. Zacharakis, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Fluorescent protein tomography scanner for small animal imaging,” IEEE Trans. Med. Imaging 24, 878-885 (2005).
[CrossRef] [PubMed]

R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

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

Rosen, M. A.

Roy, R.

R. Roy, A. Godavarty, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical tomography of a large tissue phantom using point illumination geometries,” J. Biomed. Opt. 11, 044007 (2006).
[CrossRef] [PubMed]

A. K. Sahu, R. Roy, A. Joshi, and E. M. Sevick-Muraca, “Evaluation of anatomical structure and non-uniform distribution of imaging agent in near-infrared fluorescence-enhanced optical tomography,” Opt. Express 13, 10182-10199 (2005).
[CrossRef] [PubMed]

Sahu, A. K.

Saunders, M. A.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 20, 33-61 (1998).

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1989).

Schnall, M. D.

Schulz, R. B.

R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

Schweiger, M.

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, 6696-6716 (2007).
[CrossRef] [PubMed]

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

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Sevick-Muraca, E. M.

R. Roy, A. Godavarty, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical tomography of a large tissue phantom using point illumination geometries,” J. Biomed. Opt. 11, 044007 (2006).
[CrossRef] [PubMed]

A. K. Sahu, R. Roy, A. Joshi, and E. M. Sevick-Muraca, “Evaluation of anatomical structure and non-uniform distribution of imaging agent in near-infrared fluorescence-enhanced optical tomography,” Opt. Express 13, 10182-10199 (2005).
[CrossRef] [PubMed]

A. Godavarty, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Detection of single and multiple targets in tissue phantoms with fluorescence-enhanced optical imaging: Feasibility study,” Radiology 235, 148-154 (2005).
[CrossRef] [PubMed]

A. Godavarty, M. J. Eppstein, Z. Chaoyang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701-1720 (2003).
[CrossRef] [PubMed]

Soubret, A.

A. Soubret and V. Ntziachristos, “Fluorescence molecular tomography in the presence of background fluorescence,” Phys. Med. Biol. 51, 3983-4001 (2006).
[CrossRef] [PubMed]

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: Study of the normalized Bornratio,” IEEE Trans. Med. Imaging 24, 1377-1386 (2005).
[CrossRef] [PubMed]

Sun, J.

E. Chang, J. Sun, J. S. Miller, W. W. Yu, V. L. Colvin, J. L. West, and R. Drezek, “Protease-activated quantum dot probes,” Proc. SPIE 6191, 61911E.1-10 (2006).

Theru, S.

A. Godavarty, M. J. Eppstein, Z. Chaoyang, S. Theru, A. B. Thompson, M. Gurfinkel, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical imaging in large tissue volumes using a gain-modulated ICCD camera,” Phys. Med. Biol. 48, 1701-1720 (2003).
[CrossRef] [PubMed]

Thompson, A. B.

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A. Godavarty, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraca, “Detection of single and multiple targets in tissue phantoms with fluorescence-enhanced optical imaging: Feasibility study,” Radiology 235, 148-154 (2005).
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Figures (10)

Fig. 1
Fig. 1

Spatial configuration of the simulation case study discussed in Section 5. The cylinder represents the phantom, and the filled (open) square marks indicate the source (detector) locations.

Fig. 2
Fig. 2

The z-slice images of (a) the object and (b) the background fluorophore distribution for the simulation case study discussed in Section 5. The fluorophore distribution inside the phantom consists of three spherical objects, each with a diameter of 5 mm and a background distribution consisting of two Gaussian functions, each with a standard deviation of 15 mm centered at z = 5 mm and + 5 mm .

Fig. 3
Fig. 3

The z-slice images of (a) the object and (b) the background distributions estimated using the proposed approach after 25 iterations; (c) the object distribution as estimated using an approach based on the assumption of homogeneity of the background distribution; (d) the object distribution estimated using Tikhonov regularization without any background compensation.

Fig. 4
Fig. 4

Background reduction gain in decibels versus the iteration number for the simulated case. The algorithm switches from the OMP mode to the BP mode at the 16th iteration. The high background reduction achieved in the OMP mode is maintained in the BP mode, where the object estimation is further improved.

Fig. 5
Fig. 5

Estimation error E i versus the iteration number the simulated case of Fig. 1. The estimation error is nonincreasing in both the OMP mode and the BP mode.

Fig. 6
Fig. 6

Object representation error versus object size using object and background subspaces for the measurement geometry of Fig. 1. For a given object location, the error is given as 100 × M o M o p M o , where M o and M o p are the object signal and its projection, respectively, into the corresponding subspace. The depicted error is averaged over many object locations.

Fig. 7
Fig. 7

Reconstruction of large and small objects coexisting in the phantom. (a) Actual object distribution consisting of a large object ( 2 cm in diameter) and a small object ( 0.5 cm in diameter) and (b) the actual background distribution. As seen in (c), the large object is reconstructed as a few point sources using the proposed method. In (d) the object is reconstructed using L 2 minimization after subtracting the background estimation obtained using the proposed method. The object distribution estimated using the uniform background model and L 2 minimization is shown in (e) for comparison.

Fig. 8
Fig. 8

Reconstructions for different object positions relative to the background. (a) Actual object distribution, (b) actual background surrounding the objects, (c) estimated background distribution using the proposed method, (d) estimated object distribution using the proposed method, and (e) estimated object distribution using the uniform background model. The reconstruction result for when objects are located on the opposite side of the phantom, shown in (f), is comparable to (d), which verifies the insensitivity of the proposed method to the location of objects. All slices are at z = 0 cm .

Fig. 9
Fig. 9

Effect of high object and background signal cross talk observed in the reconstructions. A large object (a) in the presence of the background (b) is reconstructed as a point source in the object domain (c), while significantly contributing to the background estimation (d). Arrows in (c) point to the contribution of the background fluorescence to the object reconstruction.

Fig. 10
Fig. 10

Reconstruction in the presence of optical absorption perturbation depicted in (a). The actual background and object distributions are shown in (b) and (c), respectively, and (d) shows the object distribution estimated using the proposed method. (e) Shows the object estimated using the uniform background model.

Tables (3)

Tables Icon

Table 1 Proposed Method for Estimation of Object and Background Distributions

Tables Icon

Table 2 Orthogonal Matching Pursuit

Tables Icon

Table 3 Iterative Method for Joint Estimation over Two Arbitrary Subspaces

Equations (25)

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M = Z X + η ,
X = X b + X o .
X b X ̃ b = i = 1 P Y b ( i ) B i ,
g ω : R R + ; g ω ( x ) = { ω sin ( x ω ) x x π ω 0 else ,
B i ω = [ g ω ( q 1 c i ) , , g ω ( q K c i ) ] ( i = 1 , , P ) ,
X ¯ b = { i = 1 P α i B i ω α i 0 } .
X ¯ o = { i I a i δ i a i R + , I N o , I { 1 , , K } } ,
R i = 20 log 10 ( M b M b M b ( i ) ) ,
S ¯ H ϵ = { x R + K H x ϵ x } .
{ φ i } = { i = 1 P a i ψ i a i > 0 } ,
i x i P i x i .
H f = H i a i ψ i = i a i H ψ i i a i H ψ i .
i a i H ψ i ϵ i a i ψ i .
H f ϵ i a i ψ i ϵ P i a i ψ i = ϵ P f ,
( ( a 1 , , a N o ) , I = ( i 1 , , i N o ) ) = arg min a j 0 , i j { 1 , , N } j = 1 N o a j V i j M ,
α 0 , x A ¯ α x A ¯ ,
α 0 , x B ¯ α x B ¯ .
( a o , b o ) = arg min a A ¯ , b B ¯ m a b .
x A ¯ ; m b i a i + 1 m b i x .
m b i a i + 1 m b i a i .
m a i + 1 b i + 1 m a i + 1 b i .
m a i + 1 b i + 1 m a i b i ,
E i + 1 E i .
ρ A ¯ B ¯ = sup a A ¯ , b B ¯ a , b a b .
a i + 1 = c i + λ i a i , b ̂ i .

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