Huber Iterative Hard Threshold Algorithm in an Application Study of Fluorescence Molecular Tomography
Objective Fluorescence molecular tomography(FMT),which can observe the three-dimensional distribution of fluorescent probes in small animals via reconstruction algorithms,has become a promising imaging technology for preclinical studies.The strong scattering property of biological tissues and limited boundary measurements with noise have resulted in the FMT reconstruction problem being severely ill-posed.To solve the problem of FMT reconstruction,some studies have been conducted from different aspects,e.g.,the improvement of forward modeling and many regularization-based algorithms.Owing to the ill-posed nature and sensitivity to noise of the inverse problem,it is a challenge to develop a robust algorithm that can accurately reconstruct the location and morphology of the fluorescence source.Traditional reconstruction algorithms use the l2 error norm,which amplifies the influence of noise and leads to poor reconstruction results.Methods In this study,we applied the Huber iterative hard threshold(HIHT)algorithm to fluorescence molecular tomography.The HIHT algorithm modifies the l2 norm cost function into a robust metric function,and the inverse problem is modeled as a constrained optimization problem that is combinatorial in nature.The robust metric function combines the l1 and l2 loss functions to vary the robustness and efficiency of the algorithm by setting a user-defined tuning constant.In the presence of noise,the HIHT algorithm can effectively reduce the influence of noise and enhance the robustness of the algorithm.Results and Discussions Numerous numerical simulations and in vivo mouse experiments are conducted to evaluate the performance of the HIHT algorithm.The reconstruction performance of the HIHT algorithm is illustrated by the contrast-to-noise ratio(CNR),Dice coefficient,location error(LE),normalized mean square error(NMSE),and time.Quantitative and qualitative analyses show that the HIHT algorithm achieves the best reconstruction results in terms of the localization accuracy,spatial resolution of the fluorescent source,and morphological recovery,compared with the FISTA,Homotopy,and IVTCG algorithms(Figs.1,4).To further verify the robustness of the HIHT algorithm,we perform four sets of experiments with different Poisson and Gaussian noise intensities(Fig.2 and Fig.3).As the noise intensity increases,the NMSE of the HIHT algorithm is always the smallest,indicating that it has the highest reconstruction accuracy.At the same noise intensity,the HIHT algorithm has the smallest LE,indicating that it reconstructs the target closest to the position of the real source.When the noise intensity increases,the Dice coefficient of the HIHT algorithm is higher than those of the other three algorithms,which indicates that the HIHT algorithm has a better morphological reconstruction ability.The CNR fluctuation of the HIHT algorithm is smaller than the CNR variations of the other three algorithms in the 10％-25％noise range.The results show that when the noise level is lower than 25％,the HIHT algorithm still obtains satisfactory reconstruction results,compared with those of the other three algorithms.To further evaluate the reconstruction performance of the HIHT algorithm in practical applications,we also perform in vivo mouse experiments.The experimental results show that the HIHT algorithm has the smallest position error as well as the highest Dice coefficient,and the fluorescent bead reconstructed by the HIHT algorithm is the closest to the real fluorescent bead in terms of morphology,which further demonstrates the feasibility and robustness of the HIHT algorithm(Fig.5).The experimental results show that the HIHT algorithm not only achieves accurate fluorescence target reconstruction,but also improves the robustness to noise.Conclusions This study investigates the problem of insufficient algorithm robustness in FMT,and the HIHT algorithm reduces the impact of noise on the reconstruction performance by using the Huber loss function as the residual term.With the same noise intensity,compared with the other three algorithms,the HIHT algorithm obtains the smallest LE and NMSE values as well as the largest CNR and Dice coefficient values,indicating that the HIHT algorithm has the best reconstruction performance.As the noise intensity increases,the reconstruction performance of the HIHT algorithm outperforms the other three algorithms,and the performance is more superior in the Poisson noise test,which indicates that the HIHT algorithm has the best reconstruction accuracy and robustness.The experimental results are consistent with the theoretical description in Section 2.These results indicate that the HIHT algorithm is insensitive to noise and has good robustness.In summary,when the measurement data sets are disturbed by noise,unlike the algorithms based on the l2 norm residual term,the HIHT algorithm uses a robust loss function to reduce the influence of the noise.Therefore,the accuracy and robustness of the HIHT algorithm are significantly improved such that the position and shape of the fluorescence source can be reconstructed more accurately.Overall,the HIHT algorithm has the best robustness in the case of accurate reconstruction.Therefore,this study can promote the preclinical application of FMT.
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