input 
Initialize 
Electron micrography (EM) is a detection method for determining the structure of macromolecular complexes and biological specimens. However, some restrictions in the EM system, including poor signaltonoise, limited range of tilt angles, only a subregion subject to electron exposure and unintentional movements of the specimen, make the reconstruction procedure severely illposed. Because of these limitations, there may be severe artifacts in reconstructed images. In this paper, we first design an algorithm that can quickly calculate the radiological paths. Then we combine an iterative reconstruction algorithm using the MumfordShah model with an artifact reduction strategy. The combined method can not only regularize the illposedness and provide the reconstruction and segmentation simultaneously but also smooth additional artifacts due to the limited data. Also we improved the algorithm used for the calculation of radiological paths to accelerate the reconstruction. The proposed algorithm was translated into OpenCL programs and kernel functions to asynchronously and in parallel update the reconstructed image along rays by GPUs. We tested the method on both simulated and real EM data. The results show that our artifact reduced MumfordShah algorithm can reduce the noise and artifacts while preserving and enhancing the edges in the reconstructed image.
Citation: 
Figure 1.
Illustration of the calculation of 2d radiological paths. The variable
Figure 2.
Illustration of the calculation of 3d radiological paths.
Figure 3.
Reconstructions of 3D SheppLogan phantom using the alternating minimization scheme described in Algorithm 3 and artifact reduction strategy in Sec. 2.3. The size of SheppLogan phantom is
Figure 4.
Reconstructions and segmentations of the SheppLogan phantom
Figure 5. Reconstructions of cryospecimen using the alternating minimization scheme described in Algorithm 3 and artifact reduction strategy in Sec. 2.3. Left column is the reconstructed images, the middle column is the edge sets with our algorithm, and the right column is the reconstructed images with WBP algorithm in TOM toolbox. The images in the red block are zoomed in and put in the top right corner. The edges in the middle column are enhanced in the left column compared to the WBP reconstructions in the right column. Compared to the results of WBP in right column, the results in the left show the effect of noise reduction while the edges are preserved
Table Algorithm 1. Fast calculation of 2D radiological paths
input 
Initialize 
Table Algorithm 2. Fast calculation of 3D radiological paths
input 

Table Algorithm 3.
Alternate Minimization of
Table Algorithm 4. Image Minimization with Steepest Descent Method

Table Algorithm 5. Edge Minimization with Steepest Descent Method
[1]  L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B, 6 (1992), 105123. 
[2]  S. Bonettini, M. Prato and S. Rebegoldi, A cyclic block coordinate descent method with generalized gradient projections, Appl. Math. Comput., 286 (2016), 288300. doi: 10.1016/j.amc.2016.04.031. 
[3]  S. Bonettini, Inexact block coordinate descent methods with application to nonnegative matrix factorization, IMA Journal of Numerical Analysis, 31 (2011), 14311452. doi: 10.1093/imanum/drq024. 
[4]  D. Fanelli and O. Öktem. Electron tomography: A short overview with an emphasis on the absorption potential model for the forward problem, Inverse Problems, 24 (2008), 013001, 51 pp. doi: 10.1088/02665611/24/1/013001. 
[5]  J. Frikel and E. T. Quinto, Characterization and reduction of artifacts in limited angle tomography, Inverse Problems, 29 (2013), 20912128. doi: 10.1088/02665611/29/12/125007. 
[6]  A. Fujimoto, T. Tanaka and K. Iwata, Arts: Accelerated raytracing system, IEEE Computer Graphics and Applications, 6 (1986), 1626. 
[7]  A. Zampighi Guido, S. Cataldo, M. Zampighi Lorenzo, W. Michael, M. Wright Ernest and C. Brecha Nicholas, Conical tomography of a ribbon synapse: Structural evidence for vesicle fusion, Plos One, 6 (2011), e16944. 
[8]  G, T. Herman and J. Frank, Computational Methods for ThreeDimensional Microscopy Reconstruction, Springer New York, 2014. 
[9]  X. Huang, L. Sun, S. Ji, T. Zhao, W. Zhang, J. Xu, J. Zhang, Y. Wang, X. Wang, C. FranziniArmstrong, M. Zheng and H. Cheng, Kissing and nanotunneling mediate intermitochondrial communication in the heart, Proceedings of the National Academy of Sciences of the United States of America, 110 (2013), 28462851. 
[10]  M. Jiang, P. Maass, and T. Page, Regularizing properties of the mumfordshah functional for imaging applications, Inverse Problems, 30 (2014), 035007, 17 pp. doi: 10.1088/02665611/30/3/035007. 
[11]  Y. M. Jung, S. H. Kang and J. Shen, Multiphase image segmentation via modicamortola phase transition, SIAM Journal on Applied Mathematics, 67 (2007), 12131232. doi: 10.1137/060662708. 
[12]  E. Klann, A mumfordshahlike method for limited data tomography with an application to electron tomography, SIAM Journal on Imaging Sciences, 4 (2011), 10291048. doi: 10.1137/100817371. 
[13]  J. Klukowska, D. Ran and G. T. Herman, Snark09 a software package for reconstruction of 2d images from 1d projections, Computer Methods and Programs in Biomedicine, 110 (2013), 424440. 
[14]  A. Kucukelbir, F. J. Sigworth and H. D. Tagare, Quantifying the local resolution of cryoem density maps, Nature Methods, 11 (2014), 6365. 
[15]  S. Lanzavecchia, F. Cantele, P. L. Bellon, L. Zampighi, M. Kreman, E. Wright and G. A. Zampighi, Conical tomography of freezefracture replicas: A method for the study of integral membrane proteins inserted in phospholipid bilayers, Journal of Structural Biology, 149 (2005), 8798. 
[16]  Y. Li and J. Kim, Multiphase image segmentation using a phasefield model, Computers and Mathematics with Applications, 62 (2011), 737745. doi: 10.1016/j.camwa.2011.05.054. 
[17]  Y. K. Liu, H. Y. Song and B. $\rm{\ddot{Z}}$alik, A general multistep algorithm for voxel traversing along a line, In Computer Graphics Forum, (2008), 73–80. 
[18]  G. Luo, M. Jiang and P. Maass, Fpga acceleration by asynchronous parallelization for simultaneous image reconstruction and segmentation based on the mumfordshah regularization, 2015. 
[19]  R. Marabini, E. Rietzel, R. Schroeder, G. T. Herman and J. M. Carazo, Threedimensional reconstruction from reduced sets of very noisy images acquired following a singleaxis tilt schema: Application of a new threedimensional reconstruction algorithm and objective comparison with weighted backprojection, Journal of Structural Biology, 120 (1998), 363371. 
[20]  L. Modica and S. Mortola, Un esempio di $γ$convergenza, Bollettino Della Unione Matematica Italiana B, 14 (1977), 285299. 
[21]  D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variationalproblems, Communications on Pure and Applied Mathematics, 42 (1989), 577685. 
[22]  A. Munshi, The opencl specification, In 2009 IEEE Hot Chips 21 Symposium (HCS), (2009), 1–314. 
[23]  S. Nickell, F. Förster, A. Linaroudis, W. Del Net, F. Beck, R. Hegerl, W. Baumeister and J. M. Plitzko, Tom software toolbox: acquisition and analysis for electron tomography, Journal of Structural Biology, 149 (2005), 227234. 
[24]  O. Öktem, Mathematics of electron tomography, Handbook of Mathematical Methods in Imaging, Springer New York, New York, NY, (2015), 937–1031. 
[25]  E. T. Quinto, U. Skoglund and O. Oktem, Electron lambdatomography, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009), 218427. 
[26]  M. Radermacher, Weighted backprojection methods, Electron Tomography: Methods for ThreeDimensional Visualization of Structures in the Cell, Springer New York, New York, NY, (2006), 245–273. 
[27]  R. Ramlau and W. Ring, Regularization of illposed mumford shah models with perimeter penalization, Inverse Problems, 26 (2010), 115001, 25pp. doi: 10.1088/02665611/26/11/115001. 
[28]  L. Reimer, Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, SpringerVerlag, 1989. 
[29]  H. Rullgård, L.G. Öfverstedt, S. Masich, B. Daneholt and O. Öktem, Simulation of transmission electron microscope images of biological specimens, Journal of Microscopy, 243 (2011), 234256. 
[30]  W. Shang, F. Lu, T. Sun, J. Xu, L. L. Li, Y. Wang, G. Wang, L. Chen, X. Wang and M. B. Cannell, Imaging ca2+ nanosparks in heart with a new targeted biosensor, Circulation Research, 114 (2014), 412420. 
[31]  R. L. Siddon, Fast calculation of the exact radiological path for a threedimensional ct array, Medical Physics, 12 (1985), 252255. 
[32]  J. E. Stone, D. Gohara and G. Shi, Opencl: A parallel programming standard for heterogeneous computing systems, Computing in Science and Engineering, 12 (2010), 6672. 
[33]  E. Sundermann, F. Jacobs, M. Christiaens, B. De Sutter and I. Lemahieu, A fast algorithm to calculate the exact radiological path through a pixel or voxel space, Journal of Computing and Information Technology, 6 (1998), 8994. 
[34]  A. R. Webb, Statistical Pattern Recognition. 2nd ed, 2002. doi: 10.1002/0470854774. 
[35]  S. J. Wright, Coordinate descent algorithms, Math. Program., 151 (2015), Ser. B, 334. doi: 10.1007/s1010701508923. 
[36]  B. $\rm{\ddot{Z}}$alik, G. Clapworthy and ${\rm{\ddot{C}}}$. Oblon${\rm{\ddot{s}}}$ek, An efficient codebased voxeltraversing algorithm, Computer Graphics Forum, 16 (1997), 119128. 