Fluorescence diffuse optical tomography using the split Bregman method
Juan F. P. J. Abascal,Judit Chamorro-Servent,Juan Aguirre,Simon R. Arridge,Teresa Correia,Jorge Ripoll,Juan José Vaquero,Manuel Desco +7 more
TL;DR: This work proposes the use of the Split Bregman method to solve the image reconstruction problem for fDOT with a nonnegativity constraint that imposes the reconstructed concentration of fluorophore to be positive.
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Abstract: Purpose: Standard image reconstruction methods for fluorescence Diffuse Optical Tomography (fDOT) generally make use of L2-regularization. A better choice is to replace the L2 by a total variation functional that effectively removes noise while preserving edges. Among the wide range of approaches available, the recently appeared Split Bregman method has been shown to be optimal and efficient. Furthermore, additional constraints can be easily included. We propose the use of the Split Bregman method to solve the image reconstruction problem for fDOT with a nonnegativity constraint that imposes the reconstructed concentration of fluorophore to be positive. Methods: The proposed method is tested with simulated and experimental data, and results are compared with those yielded by an equivalent unconstrained optimization approach based on Gauss–Newton (GN) method, in which the negative part of the solution is projected to zero after each iteration. In addition, the method dependence on the parameters that weigh data fidelity and nonnegativity constraints is analyzed. Results: Split Bregman yielded a reduction of the solution error norm and a better full width at tenth maximum for simulated data, and higher signal-to-noise ratio for experimental data. It is also shown that it led to an optimum solution independently of the data fidelity parameter, as long as the number of iterations is properly selected, and that there is a linear relation between the number of iterations and the inverse of the data fidelity parameter. Conclusions: Split Bregman allows the addition of a nonnegativity constraint leading to improve image quality.
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Figures
FIG. 4. Performance of SB for simulated phantom data. (a) Behavior of the optimum iteration number that led to optimum results (in terms of the solution error norm) versus the inverse of the data fidelity parameter k, for a fixed value of the nonnegativity parameter (a 10 1). (b) Solution error norm versus k and a (same results as in Figs. 2 and 3 for SB). 
FIG. 3. Comparison of methods in terms of the negative relative part of the solution, ku u < 0ð Þk2= k uk2 (33), versus the iteration number, for simu lated phantom data (same results as Fig. 2). 
FIG. 5. Reconstructions of computer simulated phantom data with the different methods (same results as in Figs. 2 and 3). Axial, coronal, and sagittal slices (columns from left to right) for (a c) target in the reconstruction mesh, reconstructions with (d f) GN, (g i) GN P0, and (j l) SB. The negative part of images has been set to zero. 
FIG. 6. Profiles along y and z axes of reconstructed images (Fig. 5) using GN, GN P0 and SB. ![FIG. 7. Reconstruction of experimental phantom data with GN, for two data fidelity parameters (k 0.32 and k 0.52); and with SB, for a data fidelity param eter k 10 3 and a nonnegativity weighting parameter k 10 2. a) Signal to noise ratio (SNR), b) data misfit k Ju gk2, and c) the relative nonnegativity norm of the image versus the number of iterations [Eq. (33)].](/figures/fig-7-reconstruction-of-experimental-phantom-data-with-gn-4kruuq1q.png)
FIG. 7. Reconstruction of experimental phantom data with GN, for two data fidelity parameters (k 0.32 and k 0.52); and with SB, for a data fidelity param eter k 10 3 and a nonnegativity weighting parameter k 10 2. a) Signal to noise ratio (SNR), b) data misfit k Ju gk2, and c) the relative nonnegativity norm of the image versus the number of iterations [Eq. (33)]. ![FIG. 8. Axial, coronal and sagittal slices (columns from left to right) for the reconstruction of experimental phantom data. Reconstruction with GN for data fi delity parameters (a c) k 0.32 and (d f) k 0.52 [Fig. 7(b)]. (g l) Reconstructions with SB (for k 10 3 and a nonnegativity weighting parameter k 10 2) at iterations 37 and 61 that corresponded to the same data misfits than with GN in Fig. 7(b). The negative part of images has been set to zero.](/figures/fig-8-axial-coronal-and-sagittal-slices-columns-from-left-to-3o5ddwb3.png)
FIG. 8. Axial, coronal and sagittal slices (columns from left to right) for the reconstruction of experimental phantom data. Reconstruction with GN for data fi delity parameters (a c) k 0.32 and (d f) k 0.52 [Fig. 7(b)]. (g l) Reconstructions with SB (for k 10 3 and a nonnegativity weighting parameter k 10 2) at iterations 37 and 61 that corresponded to the same data misfits than with GN in Fig. 7(b). The negative part of images has been set to zero.
![FIG. 7. Reconstruction of experimental phantom data with GN, for two data fidelity parameters (k 0.32 and k 0.52); and with SB, for a data fidelity param eter k 10 3 and a nonnegativity weighting parameter k 10 2. a) Signal to noise ratio (SNR), b) data misfit k Ju gk2, and c) the relative nonnegativity norm of the image versus the number of iterations [Eq. (33)].](/figures/fig-7-reconstruction-of-experimental-phantom-data-with-gn-4kruuq1q.webp)
![FIG. 8. Axial, coronal and sagittal slices (columns from left to right) for the reconstruction of experimental phantom data. Reconstruction with GN for data fi delity parameters (a c) k 0.32 and (d f) k 0.52 [Fig. 7(b)]. (g l) Reconstructions with SB (for k 10 3 and a nonnegativity weighting parameter k 10 2) at iterations 37 and 61 that corresponded to the same data misfits than with GN in Fig. 7(b). The negative part of images has been set to zero.](/figures/fig-8-axial-coronal-and-sagittal-slices-columns-from-left-to-3o5ddwb3.webp)
Citations
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