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TomoEED: fast edge-enhancing denoising of tomographic volumes

TomoEED: fast edge-enhancing denoising of tomographic volumes Summary: TomoEED is an optimized software tool for fast feature-preserving noise filtering of large 3D tomographic volumes on CPUs and GPUs. The tool is based on the anisotropic nonlinear diffu- sion method. It has been developed with special emphasis in the reduction of the computational demands by using different strategies, from the algorithmic to the high performance computing per- spectives. TomoEED manages to filter large volumes in a matter of minutes in standard computers. Availability and implementation: TomoEED has been developed in C. It is available for Linux plat- forms at http://www.cnb.csic.es/%7ejjfernandez/tomoeed. Contact: gmartin@ual.es or jj.fernandez@csic.es Supplementary information: Supplementary data are available at Bioinformatics online. with rI ¼ I ; I ; I being the gradient vector of the volume I and x y z 1 Introduction VQV denoting the eigen-decomposition of J. Electron tomography (ET) is an important imaging technique in AND follows the diffusion equation, I ¼ divðÞ D rI , where I t t molecular and cellular biology. ET allows three-dimensional (3D) denotes the derivative with respect to the time and div is the diver- analysis of the subcellular architecture at the nanometer scale (Lucic gence operator (Supplementary Material). The 3  3 matrix D is the et al., 2013). Nevertheless, interpretation of tomographic volumes is diffusion tensor and tunes the filtering according to the local struc- often hampered by the typically low signal-to-noise ratio (SNR), es- ture. D is built from the eigenvectors v of the structure tensor pecially under cryogenic conditions. Thus, noise reduction is usually (Equation 1) and its eigenvalues k (ranking in [0, 1]) define the applied as a post-processing step (Fernandez, 2012), or even during strength of the smoothing along the corresponding direction v . 3D reconstruction (Chen et al., 2016). Similar filtering needs arise in 2 3 k 00 other 3D electron microscopy techniques for visualization of subcel- 1 6 7 lular organization (Peddie and Collinson, 2014). 6 7 DJðÞ ¼ VLV ¼½ v v v  0 k 0 ½ v v v (2) 1 2 3 2 1 2 3 4 5 Anisotropic non-linear diffusion (AND) is currently the predominant 00 k technique in ET owing to its abilities to filter noise with feature preserva- tion (Fernandez and Li, 2003; Frangakis and Hegerl, 2001). It sets the For edge preservation, the smoothing along the maximum strength and direction of the filtering according to the local structure density variation direction (v ) is set as a monotonically decreasing around each voxel, as estimated by eigen-analysis of the structure tensor: function of the gradient. Typically, k ¼ 1:0  exp ð3:31488= 2 3 ðÞ jrIj=K Þ, where the parameter K acts as a gradient threshold that I I I I I x y x z defines edges. By contrast, k ¼ k ¼ 1 to highly filter along the two 2 3 6 7 T 2 T 6 7 I I I I I JðÞ I ¼ rI rI ¼ x y y z ¼ VQV (1) y directions with minimum change. 4 5 AND is, however, computationally expensive in terms of proc- I I I I I x z y z essing time and memory consumption, which hampers application V The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com 3776 TomoEED 3777 Table 1. Processing time (s), speedup factors and memory con- to large volumes. Its parallelization is not straightforward due to the sumption (GB) dependent stencils involved in the iterative process. Here, we intro- duce TomoEED, a tool for AND of 3D volumes that has been opti- Dataset Jacobi Analytic Speedup Memory mized for execution on standard computers, with reduced memory demands and response time. A GPU version is also included for com- size 1T 1T 16T GPU 1T 16T GPU consum. puters with NVIDIA graphics cards. 256 60.50 33.96 3.78 1.83 1.78 15.99 33.05 0.07 384 205.00 113.89 10.77 4.72 1.80 19.04 43.36 0.23 460 356.33 199.00 18.77 7.65 1.79 18.98 46.57 0.39 2 Implementation 512 498.46 275.61 24.18 9.79 1.81 20.61 50.90 0.53 640 961.34 530.20 45.23 19.11 1.81 21.65 50.31 1.02 2.1 Fast eigen-analysis of the structure tensors AND involves massive diagonalization of symmetric 3  3 matrices associated to the eigen-analysis of the structure tensor J operation by providing acceptable denoised solutions from which (Equations 1 and 2). This operation is required for all voxels in the manual refinement can follow (Fernandez et al., 2007). volume and as many times as iterations. Standard routines for ma- trix diagonalization are based on the accurate iterative Jacobi algo- rithm and are designed mainly for large matrices (Press et al., 2002). 3 Illustrative results Nevertheless, diagonalization of 3  3 matrices can be performed To illustrate the performance of TomoEED, we have applied it to data- much more efficiently by means of non-iterative analytical calcula- sets from different volume electron microscopy disciplines where noise tions, at the expense of limited numerical accuracy (Kopp, 2008). filtering is needed (Supplementary Material). Significant noise reduction TomoEED makes use of direct analytical calculation of the eigensys- and preservation of the main structural biological features are observed. tems to reduce processing time without practical influence in the We have also analyzed the processing time, scalability and mem- denoised results. Further details are in Supplementary Material. ory consumption with datasets of different cubic sizes (256, 384, 460, 512 and 640) on a computer with two octo-core processors Intel 2.2 High performance computing in TomoEED Xeon E5-2650 v2 and a NVIDIA GPU Tesla K80. Supplementary AND is a memory-bound application. Typical memory requirements Tables S1 and S2 presents a full report of the results. Table 1 summa- in standard implementations amount to eight copies of the volume. rizes the results. The processing times from 10 iterations of AND This is to hold the input/output volumes and the six components of obtained with TomoEED using analytical matrix diagonalization the symmetric tensor J (Equation 1), which are also shared (overrid- with 1 thread (1T), 16 threads (16T) and on the GPU are presented. den) by D (Equation 2)). For comparison, the results using the Jacobi algorithm optimized for TomoEED implements an efficient scheme where only one copy 3  3 matrices with 1 thread are included. It can be observed that the of the volume is held in memory and it is gradually updated by analytical diagonalization accelerates the computation in a factor Z-planes during the iterative process. An auxiliary sliding window is around 1.8 with respect to the Jacobi algorithm. The multithreaded used to maintain the data needed for the calculation of current execution on the 16-core machine further reduces the processing time Z plane: neighbouring Z-planes and their tensors (Supplementary and achieves a final speedup factor in the range 16–21, with higher Fig. S1). This optimized implementation allows making the most of values for larger volumes. This translates into computing times much the memory hierarchy and enables denoising of huge datasets in lower than a minute for all datasets. The GPU version achieves out- computers with modest amounts of memory. standing speedup factors (33–50), with times lower than 20s. For To exploit the power of modern multicore computers, comparison with standard programs, we applied AND within IMOD TomoEED runs in multithreaded mode. Here, the calculation of the (Kremer et al.,1996) and demonstrated that TomoEED is much current Z-plane is distributed among the threads running in parallel. faster, especially with analytical diagonalization, and requires 8 less Each thread processes one subset of Y-rows (Supplementary Figs S1 memory (Supplementary Table S3). and S2), which involves the calculation of J and D followed by the The limited accuracy of the analytical diagonalization does not iteration of the diffusion equation, with thread synchronizations in- produce noticeable visual differences in the denoised solutions. For between, for all Y-rows in their subset. quantitative assessment, the relative error between the solutions TomoEED is well suited to GPU processing as each voxel can be obtained with the accurate Jacobi algorithm and the analytical strat- processed independently (with synchronization points between egy was computed, and it turned out to be negligible (Supplementary iterations). A CUDA-based implementation is included that maps Table S4). Moreover, SNR and sharpness of the denoised solutions each voxel to a GPU thread for a massively parallel execution. confirmed that there are no practical differences between the two Additionally, it restructures the layout of J and D to increase mem- diagonalization strategies (Supplementary Table S5). ory performance on these architectures. 4 Conclusion 2.3 Automated parameter tuning The main parameter in AND, K, acts as a threshold on the gradient. TomoEED is a powerful and efficient software tool for fast feature- Voxels with higher gradient are considered edges to be preserved, preserving noise reduction in different volume electron microscopy thereby decreasing the filtering along the first eigen-direction. This disciplines. It is based upon anisotropic non-linear diffusion. Its parameter is dataset-dependent, its tuning is not trivial and it is mechanisms for automated parameter setup simplify user operation. usually set by trial-and-error. TomoEED adopts strategies for its Its optimized implementation enables its application to large data- automated, time-varying setup based on the average gradient of the sets on standard computers, with reduced turnaround times and whole 3D volume or a noise subregion. They facilitate user memory demands. 3778 J.J. Moreno et al. Fernandez,J.J. et al. (2007) Three-dimensional anisotropic noise reduction Funding with automated parameter tuning. Lect. Notes Comp. Sci., 4788, 60–69. Grants TIN2015-66680 and SAF2017-84565-R (AEI/FEDER, UE) and Frangakis,A.S. and Hegerl,R. (2001) Noise reduction in electron tomographic Fundacio ´ n Ramo ´ n Areces. reconstructions using nonlinear anisotropic diffusion. J. Struct. Biol., 135, 239–250. Conflict of Interest: none declared. Kopp,J. (2008) Efficient numerical diagonalization of hermitian 3  3 matri- ces. Int. J. Mod. Phys. C, 19, 523–548. Kremer,J. et al. (1996) Computer visualization of three-dimensional image References data using IMOD. J. Struct. Biol., 116, 71–76. Chen,Y. et al. (2016) FIRT: filtered iterative reconstruction technique with in- Lucic,V. et al. (2013) Cryo-electron tomography: the challenge of doing struc- formation restoration. J. Struct. Biol., 195, 49–61. tural biology in situ. J. Cell Biol., 202, 407–419. Fernandez,J.J. (2012) Computational methods for electron tomography. Peddie,C.J. and Collinson,L.M. (2014) Exploring the third dimension: volume Micron, 43, 1010–1030. electron microscopy comes of age. Micron, 61, 9–19. Fernandez,J.J. and Li,S. (2003) An improved algorithm for anisotropic diffu- Press,W.H. et al. (2002). Numerical recipes in C. In: The Art of Scientific sion for denoising tomograms. J. Struct. Biol., 144, 152–161. Computing, 2nd edn. Cambridge University Press, Cambridge. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bioinformatics Oxford University Press

TomoEED: fast edge-enhancing denoising of tomographic volumes

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Abstract

Summary: TomoEED is an optimized software tool for fast feature-preserving noise filtering of large 3D tomographic volumes on CPUs and GPUs. The tool is based on the anisotropic nonlinear diffu- sion method. It has been developed with special emphasis in the reduction of the computational demands by using different strategies, from the algorithmic to the high performance computing per- spectives. TomoEED manages to filter large volumes in a matter of minutes in standard computers. Availability and implementation: TomoEED has been developed in C. It is available for Linux plat- forms at http://www.cnb.csic.es/%7ejjfernandez/tomoeed. Contact: gmartin@ual.es or jj.fernandez@csic.es Supplementary information: Supplementary data are available at Bioinformatics online. with rI ¼ I ; I ; I being the gradient vector of the volume I and x y z 1 Introduction VQV denoting the eigen-decomposition of J. Electron tomography (ET) is an important imaging technique in AND follows the diffusion equation, I ¼ divðÞ D rI , where I t t molecular and cellular biology. ET allows three-dimensional (3D) denotes the derivative with respect to the time and div is the diver- analysis of the subcellular architecture at the nanometer scale (Lucic gence operator (Supplementary Material). The 3  3 matrix D is the et al., 2013). Nevertheless, interpretation of tomographic volumes is diffusion tensor and tunes the filtering according to the local struc- often hampered by the typically low signal-to-noise ratio (SNR), es- ture. D is built from the eigenvectors v of the structure tensor pecially under cryogenic conditions. Thus, noise reduction is usually (Equation 1) and its eigenvalues k (ranking in [0, 1]) define the applied as a post-processing step (Fernandez, 2012), or even during strength of the smoothing along the corresponding direction v . 3D reconstruction (Chen et al., 2016). Similar filtering needs arise in 2 3 k 00 other 3D electron microscopy techniques for visualization of subcel- 1 6 7 lular organization (Peddie and Collinson, 2014). 6 7 DJðÞ ¼ VLV ¼½ v v v  0 k 0 ½ v v v (2) 1 2 3 2 1 2 3 4 5 Anisotropic non-linear diffusion (AND) is currently the predominant 00 k technique in ET owing to its abilities to filter noise with feature preserva- tion (Fernandez and Li, 2003; Frangakis and Hegerl, 2001). It sets the For edge preservation, the smoothing along the maximum strength and direction of the filtering according to the local structure density variation direction (v ) is set as a monotonically decreasing around each voxel, as estimated by eigen-analysis of the structure tensor: function of the gradient. Typically, k ¼ 1:0  exp ð3:31488= 2 3 ðÞ jrIj=K Þ, where the parameter K acts as a gradient threshold that I I I I I x y x z defines edges. By contrast, k ¼ k ¼ 1 to highly filter along the two 2 3 6 7 T 2 T 6 7 I I I I I JðÞ I ¼ rI rI ¼ x y y z ¼ VQV (1) y directions with minimum change. 4 5 AND is, however, computationally expensive in terms of proc- I I I I I x z y z essing time and memory consumption, which hampers application V The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com 3776 TomoEED 3777 Table 1. Processing time (s), speedup factors and memory con- to large volumes. Its parallelization is not straightforward due to the sumption (GB) dependent stencils involved in the iterative process. Here, we intro- duce TomoEED, a tool for AND of 3D volumes that has been opti- Dataset Jacobi Analytic Speedup Memory mized for execution on standard computers, with reduced memory demands and response time. A GPU version is also included for com- size 1T 1T 16T GPU 1T 16T GPU consum. puters with NVIDIA graphics cards. 256 60.50 33.96 3.78 1.83 1.78 15.99 33.05 0.07 384 205.00 113.89 10.77 4.72 1.80 19.04 43.36 0.23 460 356.33 199.00 18.77 7.65 1.79 18.98 46.57 0.39 2 Implementation 512 498.46 275.61 24.18 9.79 1.81 20.61 50.90 0.53 640 961.34 530.20 45.23 19.11 1.81 21.65 50.31 1.02 2.1 Fast eigen-analysis of the structure tensors AND involves massive diagonalization of symmetric 3  3 matrices associated to the eigen-analysis of the structure tensor J operation by providing acceptable denoised solutions from which (Equations 1 and 2). This operation is required for all voxels in the manual refinement can follow (Fernandez et al., 2007). volume and as many times as iterations. Standard routines for ma- trix diagonalization are based on the accurate iterative Jacobi algo- rithm and are designed mainly for large matrices (Press et al., 2002). 3 Illustrative results Nevertheless, diagonalization of 3  3 matrices can be performed To illustrate the performance of TomoEED, we have applied it to data- much more efficiently by means of non-iterative analytical calcula- sets from different volume electron microscopy disciplines where noise tions, at the expense of limited numerical accuracy (Kopp, 2008). filtering is needed (Supplementary Material). Significant noise reduction TomoEED makes use of direct analytical calculation of the eigensys- and preservation of the main structural biological features are observed. tems to reduce processing time without practical influence in the We have also analyzed the processing time, scalability and mem- denoised results. Further details are in Supplementary Material. ory consumption with datasets of different cubic sizes (256, 384, 460, 512 and 640) on a computer with two octo-core processors Intel 2.2 High performance computing in TomoEED Xeon E5-2650 v2 and a NVIDIA GPU Tesla K80. Supplementary AND is a memory-bound application. Typical memory requirements Tables S1 and S2 presents a full report of the results. Table 1 summa- in standard implementations amount to eight copies of the volume. rizes the results. The processing times from 10 iterations of AND This is to hold the input/output volumes and the six components of obtained with TomoEED using analytical matrix diagonalization the symmetric tensor J (Equation 1), which are also shared (overrid- with 1 thread (1T), 16 threads (16T) and on the GPU are presented. den) by D (Equation 2)). For comparison, the results using the Jacobi algorithm optimized for TomoEED implements an efficient scheme where only one copy 3  3 matrices with 1 thread are included. It can be observed that the of the volume is held in memory and it is gradually updated by analytical diagonalization accelerates the computation in a factor Z-planes during the iterative process. An auxiliary sliding window is around 1.8 with respect to the Jacobi algorithm. The multithreaded used to maintain the data needed for the calculation of current execution on the 16-core machine further reduces the processing time Z plane: neighbouring Z-planes and their tensors (Supplementary and achieves a final speedup factor in the range 16–21, with higher Fig. S1). This optimized implementation allows making the most of values for larger volumes. This translates into computing times much the memory hierarchy and enables denoising of huge datasets in lower than a minute for all datasets. The GPU version achieves out- computers with modest amounts of memory. standing speedup factors (33–50), with times lower than 20s. For To exploit the power of modern multicore computers, comparison with standard programs, we applied AND within IMOD TomoEED runs in multithreaded mode. Here, the calculation of the (Kremer et al.,1996) and demonstrated that TomoEED is much current Z-plane is distributed among the threads running in parallel. faster, especially with analytical diagonalization, and requires 8 less Each thread processes one subset of Y-rows (Supplementary Figs S1 memory (Supplementary Table S3). and S2), which involves the calculation of J and D followed by the The limited accuracy of the analytical diagonalization does not iteration of the diffusion equation, with thread synchronizations in- produce noticeable visual differences in the denoised solutions. For between, for all Y-rows in their subset. quantitative assessment, the relative error between the solutions TomoEED is well suited to GPU processing as each voxel can be obtained with the accurate Jacobi algorithm and the analytical strat- processed independently (with synchronization points between egy was computed, and it turned out to be negligible (Supplementary iterations). A CUDA-based implementation is included that maps Table S4). Moreover, SNR and sharpness of the denoised solutions each voxel to a GPU thread for a massively parallel execution. confirmed that there are no practical differences between the two Additionally, it restructures the layout of J and D to increase mem- diagonalization strategies (Supplementary Table S5). ory performance on these architectures. 4 Conclusion 2.3 Automated parameter tuning The main parameter in AND, K, acts as a threshold on the gradient. TomoEED is a powerful and efficient software tool for fast feature- Voxels with higher gradient are considered edges to be preserved, preserving noise reduction in different volume electron microscopy thereby decreasing the filtering along the first eigen-direction. This disciplines. It is based upon anisotropic non-linear diffusion. Its parameter is dataset-dependent, its tuning is not trivial and it is mechanisms for automated parameter setup simplify user operation. usually set by trial-and-error. TomoEED adopts strategies for its Its optimized implementation enables its application to large data- automated, time-varying setup based on the average gradient of the sets on standard computers, with reduced turnaround times and whole 3D volume or a noise subregion. They facilitate user memory demands. 3778 J.J. Moreno et al. Fernandez,J.J. et al. (2007) Three-dimensional anisotropic noise reduction Funding with automated parameter tuning. Lect. Notes Comp. Sci., 4788, 60–69. Grants TIN2015-66680 and SAF2017-84565-R (AEI/FEDER, UE) and Frangakis,A.S. and Hegerl,R. (2001) Noise reduction in electron tomographic Fundacio ´ n Ramo ´ n Areces. reconstructions using nonlinear anisotropic diffusion. J. Struct. Biol., 135, 239–250. Conflict of Interest: none declared. Kopp,J. (2008) Efficient numerical diagonalization of hermitian 3  3 matri- ces. Int. J. Mod. Phys. C, 19, 523–548. Kremer,J. et al. (1996) Computer visualization of three-dimensional image References data using IMOD. J. Struct. Biol., 116, 71–76. Chen,Y. et al. (2016) FIRT: filtered iterative reconstruction technique with in- Lucic,V. et al. (2013) Cryo-electron tomography: the challenge of doing struc- formation restoration. J. Struct. Biol., 195, 49–61. tural biology in situ. J. Cell Biol., 202, 407–419. Fernandez,J.J. (2012) Computational methods for electron tomography. Peddie,C.J. and Collinson,L.M. (2014) Exploring the third dimension: volume Micron, 43, 1010–1030. electron microscopy comes of age. Micron, 61, 9–19. Fernandez,J.J. and Li,S. (2003) An improved algorithm for anisotropic diffu- Press,W.H. et al. (2002). Numerical recipes in C. In: The Art of Scientific sion for denoising tomograms. J. Struct. Biol., 144, 152–161. Computing, 2nd edn. Cambridge University Press, Cambridge.

Journal

Bioinformatics – Oxford University Press

Published: May 29, 2018

References

  • FIRT: filtered iterative reconstruction technique with information restoration
    Chen, Y.
  • Computational methods for electron tomography
    Fernandez, J.J.
  • An improved algorithm for anisotropic diffusion for denoising tomograms
    Fernandez, J.J.; Li, S.
  • Three-dimensional anisotropic noise reduction with automated parameter tuning
    Fernandez, J.J.
  • Noise reduction in electron tomographic reconstructions using nonlinear anisotropic diffusion
    Frangakis, A.S.; Hegerl, R.
  • Efficient numerical diagonalization of hermitian 3 × 3 matrices
    Kopp, J.
  • Computer visualization of three-dimensional image data using IMOD
    Kremer, J.
  • Cryo-electron tomography: the challenge of doing structural biology in situ
    Lucic, V.
  • Exploring the third dimension: volume electron microscopy comes of age
    Peddie, C.J.; Collinson, L.M.