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Simultaneous reconstruction and segmentation with the Mumford-Shah functional for electron tomography
1. | LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China |
2. | Department of Mathematics, Tufts University, Medford, MA, USA |
3. | College of Life Sciences, Peking University, Beijing 100871, China |
4. | LMAM, School of Mathematical Sciences, Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China |
5. | China Cooperative Medianet Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, China |
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 signal-to-noise, limited range of tilt angles, only a sub-region subject to electron exposure and unintentional movements of the specimen, make the reconstruction procedure severely ill-posed. 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 Mumford-Shah model with an artifact reduction strategy. The combined method can not only regularize the ill-posedness 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 Mumford-Shah algorithm can reduce the noise and artifacts while preserving and enhancing the edges in the reconstructed image.
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On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B(7), 6 (1992), 105-123.
|
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S. Bonettini, M. Prato and S. Rebegoldi,
A cyclic block coordinate descent method with generalized gradient projections, Appl. Math. Comput., 286 (2016), 288-300.
doi: 10.1016/j.amc.2016.04.031. |
[3] |
S. Bonettini,
Inexact block coordinate descent methods with application to non-negative
matrix factorization, IMA Journal of Numerical Analysis, 31 (2011), 1431-1452.
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: 10.1088/0266-5611/24/1/013001. |
[5] |
J. Frikel and E. T. Quinto,
Characterization and reduction of artifacts in limited angle tomography, Inverse Problems, 29 (2013), 2091-2128.
doi: 10.1088/0266-5611/29/12/125007. |
[6] |
A. Fujimoto, T. Tanaka and K. Iwata,
Arts: Accelerated ray-tracing system, IEEE Computer Graphics and Applications, 6 (1986), 16-26.
|
[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 Three-Dimensional 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. Franzini-Armstrong, 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), 2846-2851.
|
[10] |
M. Jiang, P. Maass, and T. Page, Regularizing properties of the mumford-shah functional for imaging applications, Inverse Problems, 30 (2014), 035007, 17 pp.
doi: 10.1088/0266-5611/30/3/035007. |
[11] |
Y. M. Jung, S. H. Kang and J. Shen,
Multiphase image segmentation via modica-mortola phase transition, SIAM Journal on Applied Mathematics, 67 (2007), 1213-1232.
doi: 10.1137/060662708. |
[12] |
E. Klann,
A mumford-shah-like method for limited data tomography with an application to electron tomography, SIAM Journal on Imaging Sciences, 4 (2011), 1029-1048.
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), 424-440.
|
[14] |
A. Kucukelbir, F. J. Sigworth and H. D. Tagare,
Quantifying the local resolution of cryo-em density maps, Nature Methods, 11 (2014), 63-65.
|
[15] |
S. Lanzavecchia, F. Cantele, P. L. Bellon, L. Zampighi, M. Kreman, E. Wright and G. A. Zampighi,
Conical tomography of freeze-fracture replicas: A method for the study of integral membrane proteins inserted in phospholipid bilayers, Journal of Structural Biology, 149 (2005), 87-98.
|
[16] |
Y. Li and J. Kim,
Multiphase image segmentation using a phase-field model, Computers and Mathematics with Applications, 62 (2011), 737-745.
doi: 10.1016/j.camwa.2011.05.054. |
[17] |
Y. K. Liu, H. Y. Song and B. $\rm{\ddot{Z}}$alik, A general multi-step 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 mumford-shah regularization, 2015. |
[19] |
R. Marabini, E. Rietzel, R. Schroeder, G. T. Herman and J. M. Carazo,
Three-dimensional reconstruction from reduced sets of very noisy images acquired following a single-axis tilt schema: Application of a new three-dimensional reconstruction algorithm and objective comparison with weighted backprojection, Journal of Structural Biology, 120 (1998), 363-371.
|
[20] |
L. Modica and S. Mortola,
Un esempio di $γ$-convergenza, Bollettino Della Unione Matematica Italiana B, 14 (1977), 285-299.
|
[21] |
D. Mumford and J. Shah,
Optimal approximations by piecewise smooth functions and associated variational-problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685.
|
[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), 227-234.
|
[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 lambda-tomography, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009), 21842-7.
|
[26] |
M. Radermacher, Weighted back-projection methods, Electron Tomography: Methods for
Three-Dimensional Visualization of Structures in the Cell, Springer New York, New York,
NY, (2006), 245–273. |
[27] |
R. Ramlau and W. Ring, Regularization of ill-posed mumford shah models with perimeter penalization,
Inverse Problems, 26 (2010), 115001, 25pp.
doi: 10.1088/0266-5611/26/11/115001. |
[28] |
L. Reimer,
Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, Springer-Verlag, 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), 234-256.
|
[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), 412-420.
|
[31] |
R. L. Siddon,
Fast calculation of the exact radiological path for a three-dimensional ct array, Medical Physics, 12 (1985), 252-255.
|
[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), 66-72.
|
[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), 89-94.
|
[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, 3-34.
doi: 10.1007/s10107-015-0892-3. |
[36] |
B. $\rm{\ddot{Z}}$alik, G. Clapworthy and ${\rm{\ddot{C}}}$. Oblon${\rm{\ddot{s}}}$ek,
An efficient code-based voxel-traversing algorithm, Computer Graphics Forum, 16 (1997), 119-128.
|
show all references
References:
[1] |
L. Ambrosio and V. M. Tortorelli,
On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B(7), 6 (1992), 105-123.
|
[2] |
S. Bonettini, M. Prato and S. Rebegoldi,
A cyclic block coordinate descent method with generalized gradient projections, Appl. Math. Comput., 286 (2016), 288-300.
doi: 10.1016/j.amc.2016.04.031. |
[3] |
S. Bonettini,
Inexact block coordinate descent methods with application to non-negative
matrix factorization, IMA Journal of Numerical Analysis, 31 (2011), 1431-1452.
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: 10.1088/0266-5611/24/1/013001. |
[5] |
J. Frikel and E. T. Quinto,
Characterization and reduction of artifacts in limited angle tomography, Inverse Problems, 29 (2013), 2091-2128.
doi: 10.1088/0266-5611/29/12/125007. |
[6] |
A. Fujimoto, T. Tanaka and K. Iwata,
Arts: Accelerated ray-tracing system, IEEE Computer Graphics and Applications, 6 (1986), 16-26.
|
[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 Three-Dimensional 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. Franzini-Armstrong, 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), 2846-2851.
|
[10] |
M. Jiang, P. Maass, and T. Page, Regularizing properties of the mumford-shah functional for imaging applications, Inverse Problems, 30 (2014), 035007, 17 pp.
doi: 10.1088/0266-5611/30/3/035007. |
[11] |
Y. M. Jung, S. H. Kang and J. Shen,
Multiphase image segmentation via modica-mortola phase transition, SIAM Journal on Applied Mathematics, 67 (2007), 1213-1232.
doi: 10.1137/060662708. |
[12] |
E. Klann,
A mumford-shah-like method for limited data tomography with an application to electron tomography, SIAM Journal on Imaging Sciences, 4 (2011), 1029-1048.
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), 424-440.
|
[14] |
A. Kucukelbir, F. J. Sigworth and H. D. Tagare,
Quantifying the local resolution of cryo-em density maps, Nature Methods, 11 (2014), 63-65.
|
[15] |
S. Lanzavecchia, F. Cantele, P. L. Bellon, L. Zampighi, M. Kreman, E. Wright and G. A. Zampighi,
Conical tomography of freeze-fracture replicas: A method for the study of integral membrane proteins inserted in phospholipid bilayers, Journal of Structural Biology, 149 (2005), 87-98.
|
[16] |
Y. Li and J. Kim,
Multiphase image segmentation using a phase-field model, Computers and Mathematics with Applications, 62 (2011), 737-745.
doi: 10.1016/j.camwa.2011.05.054. |
[17] |
Y. K. Liu, H. Y. Song and B. $\rm{\ddot{Z}}$alik, A general multi-step 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 mumford-shah regularization, 2015. |
[19] |
R. Marabini, E. Rietzel, R. Schroeder, G. T. Herman and J. M. Carazo,
Three-dimensional reconstruction from reduced sets of very noisy images acquired following a single-axis tilt schema: Application of a new three-dimensional reconstruction algorithm and objective comparison with weighted backprojection, Journal of Structural Biology, 120 (1998), 363-371.
|
[20] |
L. Modica and S. Mortola,
Un esempio di $γ$-convergenza, Bollettino Della Unione Matematica Italiana B, 14 (1977), 285-299.
|
[21] |
D. Mumford and J. Shah,
Optimal approximations by piecewise smooth functions and associated variational-problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685.
|
[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), 227-234.
|
[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 lambda-tomography, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009), 21842-7.
|
[26] |
M. Radermacher, Weighted back-projection methods, Electron Tomography: Methods for
Three-Dimensional Visualization of Structures in the Cell, Springer New York, New York,
NY, (2006), 245–273. |
[27] |
R. Ramlau and W. Ring, Regularization of ill-posed mumford shah models with perimeter penalization,
Inverse Problems, 26 (2010), 115001, 25pp.
doi: 10.1088/0266-5611/26/11/115001. |
[28] |
L. Reimer,
Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, Springer-Verlag, 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), 234-256.
|
[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), 412-420.
|
[31] |
R. L. Siddon,
Fast calculation of the exact radiological path for a three-dimensional ct array, Medical Physics, 12 (1985), 252-255.
|
[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), 66-72.
|
[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), 89-94.
|
[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, 3-34.
doi: 10.1007/s10107-015-0892-3. |
[36] |
B. $\rm{\ddot{Z}}$alik, G. Clapworthy and ${\rm{\ddot{C}}}$. Oblon${\rm{\ddot{s}}}$ek,
An efficient code-based voxel-traversing algorithm, Computer Graphics Forum, 16 (1997), 119-128.
|






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