Comparative Study of Projection/Back-projection Schemes in Cryo-EM Tomography

Transcription

1 Comparative Stuy of Projection/Back-projection Schemes in Cryo-EM Tomography Yu Liu an Jong Chul Ye Department of BioSystems Korea Avance Institute of Science an Technology, Daejeon, Korea ABSTRACT In the cryo-em tomography, the projection an back-projection are essential steps in reconstruction the 3D structure of the virus an macromolecules. Distance riven metho (DD) is the latest projection /backprojection algorithm originally employe for x-ray compute tomography. This paper is mainly concerne about employing this algorithm to the cryo-em tomography for reconstruction performance improvement. Existing algorithms use in cryo-em are pixel-riven an ray riven projection/backprojection, etc. These methos are generally quite time consuming because of their high computational complexity. Furthermore, interpolation artifacts are usually noticeable when the sufficient view an etector samples are not available. The DD is originally propose to overcome these rawbacks. The interpolation process in DD is one by calculating the overlap area between the etector an pixel bounaries. This proceure largely removes the interpolation artifacts, an reuces the computational complexity significantly. Furthermore, it guarantees that the projection an backprojection are ajoint to each other a esire property to guarantee the convergence of the iterative reconstruction algorithm. However, unlike the x-ray compute tomography, the cryo-em tomography problem generally has limite number of the projections, an projection angles are ranomly istribute over 4pi steraian. Therefore, the conventional DD shoul be moifie. Rather than computing the bounary overlap in the previous 3-D DD metho, we propose a novel DD algorithm base on volume overlap. CCMV virus moel is use as testing example. Results are visualize using AMIRA software. Analysis is mae upon the avantages an rawbacks of both the existing approaches an istance riven metho. Keywors: Cryo-EM, Projection, Back-projection, Pixel-riven interpolation, Distance-riven interpolation 1. INTRODUCTION Single particle reconstruction using cryo-electron microscopy (cryo-em) is a very active area of research in structural biology ue to its avantages over the x-ray crystallography [1]. In the Cryo-EM tomographic reconstruction algorithms such as weighte backprojection (WBP)[7], or algebraic reconstruction technique (ART) [5], the projection an backprojection step are essential components. The process which transforms the 3D image to D sinograms is calle projection. Physically, this process is the line integration along the electron beams irection. An the backprojection transfers multiple views of D projection onto a 3D volume back through the view irection. In iscrete implementation, the continuous line integral shoul be approximate on a iscrete lattice. Depening on the approximation, ifferent type of artifact coul be expecte. Conventionally, the most popular algorithms in this fiel are pixel riven (PD) an ray riven (RD) methos []. They are wiely use in the compute tomography area uring the past 0 years. Pixel riven approach traces the line through the centers of voxels, an the ray riven metho traces the line through the etector centers. The istance riven (DD) approach by De Man an Basu [3] is ifferent from the PD an RD in that it uses so-calle kernel operation, an a common axis or the common plane for 3D case parallel to the pixel cells. The interpolation process in DD is one by calculating the overlap area between the etector an pixel bounaries. Due to the low mathematical complexity an high sequential memory access pattern, istance riven approach achieves faster computational spee over the conventional methos. Furthermore, the istance riven metho result in better performance in eliminating artifacts of reconstructe 3D image. Yu Liu: liuyu0817@kaist.ac.kr, Telephone: +8 (0) Jong Chul Ye: corresponing autor, jong.ye@kaist.ac.kr Telephone: +8 (0) Image Reconstruction from Incomplete Data IV, eite by Philip J. Bones, Michael A. Fiy, Rick P. Millane, Proc. of SPIE Vol. 6316, , (006) X/06/$15 oi: / Proc. of SPIE Vol Downloae From: on 09/8/016 Terms of Use:

2 Distance riven approach suits well for the meical compute tomography implementation. Meical compute tomography is generally a fan beam in x-y irection with limite beamwith along z-irection. This results in an accurate calculation of the bounary overlap. However, unlike the x-ray compute tomography, the cryo-em tomography problem generally have limite number of the projections ranomly istribute over 4pi steraian This intrinsic ifference makes it quite complicate to get a sufficiently accurate approximation of D overlap using the conventional istance riven metho. Therefore, the conventional DD shoul be moifie. For the Cryo-EM tomography, we propose to use the common plane parallel to the etector plane. Furthermore, each voxel is approximate as a sphere. Then, the projection of the voxel correspons to the circle in the common plane, an the circle to circle intersection area is calculate. This propose scheme offer better performance in the reconstruction quality over conventional DD approaches using a few set of fixe common planes. In orer to spee up the circle to circle overlap area calculation, a fast approximation metho is propose using a bouning rectangle. Significant spee up of the interpolation was obtaine with negligible performance egraation. row x (mn) (k+1,p) (k,p) (k-i,p) (a) (b) (c) Fig 1. (a) D emonstration of pixel riven projection an backprojection approach. It traces the electron beam from the focal spot through the center of the pixel of interest to the etector. (b) D ray riven projection an backprojection. it traces the electron beam from the focal spot through the image to the center of the etector cell of interest. (c) D istance riven projection an back projection, the bounaries of pixel cells an etector cells are mappe to common axis.. REVIEW OF CONVENTIONAL METHODS.1 Pixel Driven an Ray Driven Approach in -D As shown in Figure 1, the pixel riven projection an backprojection traces the electron beam from the focal spot through the center of the pixel of interest to the etector. The center of the pixel is mappe to the etector row. In the projection process, the etector value is calculate base on the interpolation of the pixel value using Eq. (1): r( k, p) r( k, p) + f ( m, n) (1) 1 + where f(m,n) enotes the pixel value at (m,n) of the image space, an r(k,p) is the upate sinogram values at (k,p) in the raon space, respectively. Here, 1 an value enotes the istance from ajacent etector cell centers. In the backprojection, the pixel value is upate an accumulate by aing interpolate values of the etecter cells as shown in Eq.(): 1 f ( m, n) f ( m, n) + r( k, p) + r( k + 1, p) () Proc. of SPIE Vol Downloae From: on 09/8/016 Terms of Use:

3 Ray riven projection an backprojection trace the electron beam from the focal spot through the image to the center of the etector cell of interest. For each image row, the intersection is calculate. In the projection process, the linear interpolate value of the ajacent pixel values is calculate, which is then accumulate at the etector cell as shown in Eq. (3) r( k, p) r( k, p) + f ( m, n) + f ( m 1, n) (3) In the backprojection step, relative istances from the center of the ajacent pixels are calculate an the etector values are istribute accoring to the weighting as shown in Eq. (5) an Eq. (6). 1 3 f ( m, n) f( m, n) + r( k, p) (5) 1+ 3 f ( m 1, n) f( m 1, n) + r( k, p) (6) 1+ The intrinsic ifferences between the pixel riven an ray riven approaches are from the fact whether the line is connecte through the center of pixels or the center of the etector cells. Due to these ifferences, istinct types of artifact are observe.. Distance Driven Approach in -D The istance riven approach propose by De Man an Basu [] uses a common axis for the D case uring interpolation step. Fig. 1 (c) shows the -D istance riven projection/backprojection steps. Here, x axis is the common axis, to which the bounaries of pixel cells as well as the etector bounaries are uniquely mappe. Then, the interpolation process is one base on so-calle kernel operation; i.e. the intersection lengths are calculate on this axis, an the estination value is compute base on the normalize overlap length. More specifically, the projection step is given by Eq. (7) whereas the back-projection part is given by Eq. (8). x x x x r( k, p) r( k, p) + f ( m+ 1, n) f ( m, n) 3 x4 x x4 x x x x x f m, n f( m, n) + p k 1, p 1 + p k, p x x x x ( ) ( ) ( ) (8).3 3D Distance Driven Approach for x-ray Compute Tomography The basic iea of 3D istance riven approach has no significant ifferences from the D application. Instea of using common axis, common plane is use for 3D application. As shown in Fig., the common plane is parallel to the XZ plane, the voxel bounary an the etector cell bounaries are mappe to the common plane. In practice, they only map the vertical an horizontal bounaries of the voxels an etector cells to the common plane an approximate them as rectangles. Then, the estination values(etector value for the projection process an pixel value for the backprojection process) are compute base on the normalize overlap area. This metho suits well for the meical compute tomography application, where the common plane is parallel to the XZ plane. More specifically, when mapping the vertical an horizontal bounaries of the voxels to the common plane, the resultant shape on the common plane is a quite accurate approximation of the 3D voxel bounary mapping to common plane. Besies, the shape of the etector cell bounary on the common plane is close to a rectangle, so we can approximate it as a rectangle without sacrificing the accuracy of the interpolation. 3. MODIFIED DISTANCE DRIVEN ALGORITHM FOR 3D CRYO-EM TOMOGRAPHY 3.1 Circle base metho However, the intrinsic ifferences of Cryo-EM tomography from x-ray compute tomography make it impractical to use the conventional istance riven metho. Unlike the x-ray compute tomography, in which the projecting angles are within small range, parallel beam projections use in cryo-em tomography are ranomly istribute over 4pi steraian. Suppose we assign the XY plane as the common plane similar to the conventional istance riven approach. As shown (7) Proc. of SPIE Vol Downloae From: on 09/8/016 Terms of Use:

4 in Fig.3, when we map the bounaries of etector cells to the common plane, the resulting shape will be parallelogram; hence, it is very inaccurate if we approximate this parallelogram as a rectangle suggeste by Deman an Basu []. (a) (b) Fig. 3D implementation of istance riven approach [3]: (a) vertical an horizontal bounaries of voxels an etector cells are mappe to the common plane. The overlap lengths along x axis an z axis are calculate an then multiplie together to get the overlap area. (b) approximate the overlap area as rectangle. In orer to overcome the rawbacks, our new algorithm uses the common plane parallel to the etector plane or, equivalently, we can consier the etector plane is just the common plane. This approach appears somewhat similar to the pixel-riven approach in some sense; however, it is funamental ifferent from the pixel-riven approach in that the moifie DD uses the area overlap as a weighting factor uring the interpolation rather than the istance from the center in the PD. Then, the technical issue in this case is how to get the shape of the voxels mappe to the etector plane an how to calculate the overlap between the pixel bounaries(from voxels) an etector bounaries in a computationally efficient way. (a) (b) Fig 3. For the case of the common plane parallel to the XY plane (basically that means parallel to the slices of the 3D image to be projecte). (a) the etector plane(real line )an common plane in 3D. The ashe gri is the pixel bounaries mappe to the common plane. (b) The etector cells are mappe to the common plane, the shape is parallelogram, this will make it ifficult to calculate the overlap area with pixel bounaries(ashe line), an it will be very inaccurate if we approximate this parallelogram as a rectangle(real line)with sies parallel to the x an y coorinates. Proc. of SPIE Vol Downloae From: on 09/8/016 Terms of Use:

5 Instea of consiering each voxel as a cube voxel, we propose a new metho to approximate it as a sphere. Then, for each etector cell, it is consiere as a circle regarless of the view angle. Fig 4. We efine the common plane parallel to the etector plane, that means the projecting irection is always perpenicular to the common plane. Here, the voxel is approximate as a sphere. When the sphere is mappe to the common plane, the shape is always circle regarless of the rotation angle. We also approximate that the etector elements as a circle. Then, the overlap computation accuracy benefits greatly from this approach, because no matter what view angle is applie, when voxel is mappe to the etector plane, the shape is circle. So the problem turns to be the circle an circle overlap area calculation. 'N Fig 5. Overlap situation between the one pixel cell (big circle) an the etector cells (small circles) on the common plane. More specifically, the projection operation can be escribe by ri (, j, θ) ri (, j, θ) + f(, lmn, ) O A (9) where ri (,, jθ ) enotes the (i,j)-th etector value at the view angle θ, f(l,m,n) is the voxel value at the (l,m,n)-th voxel, O enotes the overlap area between the circles, an A is the etector cell area. Eq. (9) shoul be compute for all voxel f(l,m,n) whose overlap area O is not zero. Similarly, the back-projection operation can be escribe by Proc. of SPIE Vol Downloae From: on 09/8/016 Terms of Use:

6 f(, lmn, ) f(, lmn, ) + ri (, j, θ ) O (10) A where O is the overlap area between two circles, an A is the projecte voxel area on the etector plane, Note that Eq. (9) an (10) are the kernel equation for our approach, an they are ajoint operation to each other. In the moifie istance riven approach, the overlap area between the etector circles an pixel circles can be calculate accoring to the formula Eq. (11). = r = Fig 6. Overlap area computation 1 + r R 1 + R r O= r cos + R cos r R 1 ( + r+ R) ( + r R) ( r+ R)( + r+ R) (11) 3. Bouning box for algorithm simplification However, the intrinsic complexity of Eq. (11) mainly ue to the calculation of arcos() makes this algorithm somewhat time consuming. Hence, we introuce the bouning box metho to simplify this approach. For the simulation results an ifferences of the circle base approach an simplifie one, the etaile iscussion will be presente later in this paper. N / Fig 7. Circle an its bouning box Fig 8. Square overlapping Proc. of SPIE Vol Downloae From: on 09/8/016 Terms of Use:

7 For each circle, incluing pixel circles an etector circles, we make a bouning box with the same center, an the sie length equal to the iameter of the circle. Then, the implementation is exactly the same as Eq. (9)(10) except that the overlap area is calculate by O = lx ly (1) where lx is the overlap length in x axis irection, ly is the overlap length in y axis irection. Similarly, the area A is calculate as the size of the bouning box. Compare to the circle to circle overlapping, this simplifie metho greatly reuce the computing complexity. 4. IMPLEMENTATION AND RESULTS 4.1 Quality an artifacts We use synthetic icosaherons moel of CCMV virus for testing, as shown in Fig 9. Icosaheron is very common structure of viruses. We will make etail comparison between the pixel riven results an istance riven results. The results of pixel riven approach are generate by Xmipp package[8], which is popular software package in electron microscopy area. We implement our own approach base on Linux OS with GCC 4.0 compiler, an moerate optimization is applie. The computation is one on a computing server with ual Xeon 3. GHz CPU an GB RAM. (a) (b) Fig 9. 3D icosaherons moel: (a) 3D visualization from AMIRA[1]. (b) visualization by slices Fig 10. Projection sinograms: upper row shows the results from pixel riven approach, bottom row shows the results from istance riven approach. The two figures at same column are generate with same euler angle. 100 by 100 for each image size. Proc. of SPIE Vol Downloae From: on 09/8/016 Terms of Use:

8 We applie 3 euler angles to escribe the rotation of the moel (or the projection view). We uses same set of rotation angles for ifferent approaches, the Euler angles are uniforme ranomly istribute. Fig 10 shows the results from pixel riven approach (from Xmipp package) an the moifie istance riven approach. Even though the ifferences are not very obvious, still we can see the istance riven approach provie sharper projection image with high contrast. For reconstruction, we use WBP (weighte back projection) for both approaches. Here, the same ramp filter is applie before the back projection. (a) (b) (c) Fig 11. Artifacts comparisons between same slice(the reconstruction results in (b) an (c) are from 64 projections): (a) from original 3D image (b) from Xmipp package (c) our approach base on circle overlapping As shown in Fig. 11(b), which is the result generate by Xmipp package, the image backgroun is noisy, but in (c), which is the result from our approach, the image quality is better, the noises are eliminate an the contrast looks higher. Our approach provier clearer structure. (a) (b) Fig 1 (a) circle base istance riven metho (b) simplifie metho base on bouning box Fig 1 (b) is generate by the simplifie approach using bouning box metho. Compare to Fig 1 (a) from circle overlapping base approach, this simplifie metho provies very similar result. This result prove that this simplifie approach is more practical for implementation. Fig 13 an Fig. 14 shows the slice-by-slice view of the reconstructe CCMV virus moel using XMIPP an the moifie istance riven metho, both are base on 1000 projections. For large number of view samples, similar reconstruction results are observe as expecte. Fig 19 shows the 3D visualization result of the reconstruction results using istance riven back projection approach. Accurate 3D structure is obtaine.. Proc. of SPIE Vol Downloae From: on 09/8/016 Terms of Use:

9 Fig 13. Reconstruction result from Xmipp WBP (1000 projections) Fig 14. Reconstruction result from the simplifie istance riven approach (boun box metho with 1000 projections) 4. Computational Performance In orer to compare the computational buren of the moifie algorithm compare to the XMIPP, 64 by 64 by 64 icosaherons moel is use for testing. Size of projection image is 100 by projection images are generate in projection process an use for back projection. The computations of projection an backprojection take very similar time. Compare to the Xmipp package, in the projection process, our approach takes 5secon, an for the same process, Xmipp package take 108 secon. This means the computation complexity of our approach has much better computational performance that the pixel riven metho in Xmipp package. 5. CONCLUSION The overlapping shape approximation is essential issue in istance riven approach. In Cryo-EM tomography, projection angles are ranomly istribute over 4pi steraian. But in x-ray compute tomography, large number of Proc. of SPIE Vol Downloae From: on 09/8/016 Terms of Use:

10 projecting angles are istribute within a small range. Hence, the irect use of conventional istance riven approach results in inaccurate approximation an the reconstruction artifacts. The moifie istance riven metho successfully overcomes the problem with better result than that of pixel riven approach. The low arithmetic complexity in the simplifie metho with bouning box make the new istance riven approach provie high performance in computation spee. We can conclue that in cryo-em tomography, the moifie istance riven algorithm is one of goo approaches for accurate reconstruction with high computation performance. Fig 15. 3D Visualization by AMIRA (result from 1000 projection using our approach) ACKNOWLEDGEMENT This work was supporte by a grant No. R from the Basic Research Program of the Korea Science & Engineering Founation. REFERENCES [1] Joachim Frank, Single-particle imaging of macromolecules by cryo-electron microscopy, Annual Review of Biophysics an Biomolecular Structure, Vol. 31: , 00 [] Herman G, Image Reconstruction from Projections, Acaemic Press, 1980, [3] Bruno De Man an Samit Basu, Distance-riven projection an backprojection in three imensions, Physics in Meicine an Biology, , 005 [4] De Man B an Basu S, Distance-riven projection an backprojection, IEEE Nuclear Science Symp. Meical Imaging Conf. (Norfolk), 004 [5] Goron R, Bener R, Herman GT. Algebraic reconstruction techniques (ART) for three-imensional electron microscopy an x-ray photography, J Theor Biol, Dec;9(3):471-81, 1970 [6] A.H. Anersen an A.C. Kak, Simultaneous Algebraic Reconstruction Technique (SART): a superior implementation of the ART algorithm, Ultrason. Img., vol. 6, pp , [7] Joachim Frank. Three-Dimensional Electron Microscopy of Macromolecular Assemblies. Acaemic Press, 006 [8]C.O.S. Sorzano, R. Marabini, J. Velazquez-Muriel, J.R. Bilbao-Castro, S.H.W. Scheres, J.M. Carazo, A. Pascual-Montano, XMIPP: a new generation of an open-source image processing package for Electron Microscopy, Journal of Structural Biology 148() pp , 004 [9]A. C. Kak an M. Slaney, Principles of computerize tomographic imaging, IEEE Press, New York, [10]Crowther, R. A., DeRosier, D. J., an Klug, A, The reconstruction of a three-imensional structure from projections an its application to electron microscopy, Proc. R. Soc. Lonon Sect A 317, , [11]Salvatore Lanzavecchia, Pier Luigi Bellon, an Michael Raermacher, Fast an accurate three-imensional reconstruction from projections with ranom orientations via Raon transforms, Journal of Structural Biology 18, , 1999 [1] Proc. of SPIE Vol Downloae From: on 09/8/016 Terms of Use: