PARALLEL ALGORITHMS FOR SEGMENTATION OF CELLULAR STRUCTURES IN 2D+TIME AND 3D MORPHOGENESIS DATA

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1 Proceedings of ALGORITMY PARALLEL ALGORITHMS FOR SEGMENTATION OF CELLULAR STRUCTURES IN 2D+TIME AND 3D MORPHOGENESIS DATA KAROL MIKULA, MICHAL SMÍŠEK AND RÓBERT ŠPIR Abstract. In this aer we resent a robust method to segment 3D and 2D+time biological data. Our method uses distance function and the Generalized Subjective Surfaces GSUBSURF) level-set segmentation algorithm. We focus on memory-efficient imlementation of distance function comutation, a novel way of finite volume method discretization of GSUBSURF PDE and the otimized arallelization of our code via the MPI library, enabling us to take the advantage of high-erformance comuting servers. Several exeriments, including otimized arallelization test, validation of arallel GSUBSURF imlementation and visual check of method erformance on real data, are resented as well. Key words. distance function, level-set segmentation, biological image analysis, arallelization, high-erfomance comuting AMS subject classifications. 35K65, 35L60, 65M06, 65M08, 65Y05 1. Introduction and roblem definition. In this aer we resent algorithms for solution of two roblems: First of them is to segment tubular structures of cell evolution from a 2D video of a fruit fly Drosohila) in ual state. Our second task is, from a 3D image of cells of a zebrafish Danio Rerio) in embryogenesis, to extract the volume and shae of the whole embryo. Although these tasks are different in nature, technically the solutions arising for both of them tend to be very similar. The similarity of these roblems is caused by treating the temoral dimension in the 2D+t roblem as a third, artificial, satial dimension. Then we can use the same algorithms in 3D domain to solve these roblems. The inut to both our roblems is the intensity function in defined the domain Ω and the set of oints denoting cell ositions. We call them the cell identifiers. The inut intensity function over the domain is a 2D video of cellular evolution in the first roblem and a 3D image of cell membranes in the second roblem. Same holds for the dimensionality of cell identifier sets. The visualization of intensity functions is to be seen in fig The suggested algorithms take into account the intrinsic noise introduced by the confocal laser microscoe imaging technology and their goal is to be robust against imaging imerfections. Therefore, we accet that some of the cell identifiers could be missing, and some cells could have more than one cell identifier. We obtain cell identifiers directly from our data using the level-set center detection LSCD) algorithm [10]. The aim of LSCD algorithm is to yield a set of oints identifying the aroximate centers of mass of each cell. From the segmentation oint of view, the cell identifiers are to be used as seeds to create the initial segmentation rofile. The outut to both roblems is the segmentation of the domain into logical subdomains, reresenting cell volumes. In the first case, it is the segmentation of 2D+t This work was suorted by Grant No.: APVV Deartment of Mathematics and Descritive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinskeho 11, Bratislava, Slovakia mikula@math.sk, smisek@math.sk, sir@math.sk). 416

PARALLEL ALGORITHMS FOR SEGMENTATION OF CELLULAR STRUCTURES 417 Fig. 1.1. Data examle.

Lower left - slice of 512x512x190 voxel 3D data of zebrafish embryo, view from to, lower right - slices of zebrafish data, view from sides.

This segmentation is to be rocessed further, to obtain 2D+t trajectories of cells by a backward tracking algorithm as in [2, 12], in order to obtain the cell lineage.

2 PARALLEL ALGORITHMS FOR SEGMENTATION OF CELLULAR STRUCTURES 417 Fig Data examle. Uer left frames of a 100x100 ixel video of drosohila evolution stacked ato each other, uer right - 50th frame of drosohila evolution video. Lower left - slice of 512x512x190 voxel 3D data of zebrafish embryo, view from to, lower right - slices of zebrafish data, view from sides. domain into satio-temoral air-of-ants-like structures reresenting the evolution of cells in time. This segmentation is to be rocessed further, to obtain 2D+t trajectories of cells by a backward tracking algorithm as in [2, 12], in order to obtain the cell lineage. In the second case, the outut is the segmentation of 3D domain reresenting the shae of the embryo. We use this information further to calculate volume of the embryo and some other characteristics, like the density of cells er unit volume or the density of cell divisions [3, 13]. In comarison with the revious aers [2, 3, 13], we resent here the arallel strategy to solve these tasks. Our roosed arallel algorithmic aroach consists of these stes: A. comute distance function DF) from inut set of oints by an efficient arallel algorithm, B. construct in arallel the initial segmentation function ISF) from DF, C. use a arallel level-set algorithm, in our case the Generalized Subjective Surfaces GSUBSURF) to obtain the segmented domain. The organization of this aer is following: in chaters 2 and 3, we discuss the design and imlementation of these algorithmic stes. In chater 4, we describe arallelization of the code, using MPI. And finally, in chater 5, we discuss numerical results in the study of the exerimental order of convergence, and the exeriments erformed on hantom and real data.

3 418 KAROL MIKULA, MICHAL SMÍŠEK AND RÓBERT ŠPIR 2. Distance function. To comute the distance function from cell identifiers, we solve the time relaxed Eikonal equation d t + d = 1, 2.1) using the method from [4, 13], based on the so-called Rouy-Tourin scheme [14]. Let Ω be the solution domain and let be a voxel in this domain: Ω. Let the set of 6 neighbouring voxels to the voxel, i.e. voxels sharing common face, be N = {e, n, t, w, s, b} according to east - north - to - west - south - bottom ointsof-comass labeling. We denote a articular neighbour of by q. Let the voxels of our domain be uniform cubes. Let the length of its side be h D. For each, let the aroximate value of the solution d at time ste n, in the center of, be d n. Let us define M q, q N, as [14] d n+1 = d n + τ D τ D h D max M e, M w M q = mind n q d n, 0)) 2. The scheme for solving the eq. 2.1) in the three-dimensional sace then reads as follows: ) n, M s ) t, M b )) 1/2 2.2) where τ D is the time ste size. Scheme is stable for τ D h D /2 [13]. To comute the distance function with otimal memory requirements, we calculate new time stes in a chessboard-like manner often called the Red-Black Scheme in numerical mathematics) where all red elements have only black neighbours and all black elements have only red neighbours. In each time ste we first udate all red elements and then all black elements. We can redefine M q for red and black elements as M q R, = mindn B,q d n R,, 0)) 2 M q B, = mindn+1 R,q dn B,, 0)) 2, and the scheme 2.2) has the form of a two-ste rocedure d n+1 R, = dn R, + τ D τ D h D max M e R,, M w R, d n+1 B, = dn B, + τ D τ D h D max M e B,, M w B, ) n R,, MR, s ) t R,, MR, b )) 1/2, ) n B,, MB, s ) t B,, MB, b )) 1/2, where indices B and R denote black or red elements. Using this aroach we can use only one array and overwrite it directly by new iterations as they come, thus reducing memory requirements by half. 3. Generalized Subjective Surfaces. The artial differential equation for GSUBSURF method is ) u u t w a g u w c g u = 0, 3.1) u

4 PARALLEL ALGORITHMS FOR SEGMENTATION OF CELLULAR STRUCTURES 419 and comes from the level set formulation of the geodesic active contour model [5, 11, 15, 6, 13]. In the model 3.1), g is an edge detector function, for which we use gs) = 1 1+Ks, 2 where K is the edge detection sensibility arameter and s = I, where I is the inut image intensity function. Parameters w a and w c are weights for the advection and curvature terms of the model, resectively. The choice of a boundary condition deends on the roblem solved: In our first roblem, where we reconstruct the 2D+t structures, we assume the cell image continues even beyond the image domain, so we use zero Neumann boundary condition. For our second roblem, where the 3D embryo is well searated from the boundary of the image domain, we choose zero Dirichlet boundary condition. We create the initial segmentation function ISF) from the DF d x) comuted by the method from section 2, where d x) reresents distance in R 3 to a set of oints reresenting cell identifiers. We can use an isosurface δ > 0 of DF and have a iecewise constant ISF: { 1 if d x) δ, u 0 x) = 0 else, or, we can comute ISF by considering u 0 x) = d x). The reason to comute ISF from the DF, and thus lacing the comutational burden on the DF itself, is to minimize amount of inter-rocess communication. This makes it more suitable for arallel comuting. In the former, non-arallel aroach[13], the communication had to take lace for each voxel and for each cell identifier. Now, when the DF is known, there is no inter-rocess communication and the formulation of the ISF is a local roblem. In order to derive the discretization of 3.1), we use the semi-imlicit finite volume method FVM) based on [9] for advective art, and on [8] for the curvature art. Image voxel serves as a natural choice for FVM control volume. Similarly to the revious section, let be the voxel of our interest and N = {e, n, t, w, s, b} be the set of neighbours in the oints-of-comass labeling. Let u n be the value of the solution u in voxel at the time ste n. The voxel is a box in R 3 and we assume its uniform edge size h. The voxel faces are denoted by σ and their measure is mσ) = h 2. Let denote the boundary of this domain and let n denote the outer normal to this boundary. Volume of the voxel is denoted by m) = h 3. Let σ q be the common boundary of and q and let the line connecting the center of voxel with the center of the face σ be denoted by d σ. Its measure is h/2. First, we aroximate the time derivative by the backward difference: u t u n+1 u n τ, where τ is the size of the time ste. Then, let us use the semi-imlicit aroach in time, and integrate 3.1) over the finite volume : u n+1 u n ) u dx w a g u n+1 dx w c g u n n+1 τ u n dx = ) The aroximation of the first term in eq. 3.2) is u n+1 u n dx m) un+1 u n. 3.3) τ τ

5 420 KAROL MIKULA, MICHAL SMÍŠEK AND RÓBERT ŠPIR To disretize the second term in eq. 3.2), we first define velocity v = w a g. Then, w a g u n+1 dx = v u n+1 dx = vu n+1 )dx u n+1 v dx, and, using Green s theorem in both terms and the constant reresentation of a solution in the finite volume in the second term, we obtain vu n+1 )dx u n+1 v dx vu n+1 ) n dx u n+1 v n dx. 3.4) We define an aroximate gradient of the edge detector function in finite volume using central differences: g = G e, G n, G t ) = G w, G s, G b ), where G w = G e g e g w 2h G s = G n g n g s 2h G b = G t g t g b 2h and g q is the value of the edge detector function in q N. Then, we define the integrated flux through a voxel side by v q = v n ds = w a g n ds w a G q mσ q ). σ q σ q Using the integral fluxes, we can define inflows and outflows through voxel sides as v in q = minv q, 0), v out q = maxv q, 0). We then aroximate 3.4) by using the uwind rincile as in [9] and obtain the result: vu n+1 ) n dx u n+1 v n dx v in q u n+1 q + v out q N q N v in q u n+1 q u n+1 ). q N q u n+1 v in q u n+1 q N,,, vq out u n+1 = q N To discretize the third term in eq. 3.2), we use the aroach built in [8] for aroximation of the mean curvature term and we get ) u n+1 w c g u n ε u n dx ε w c g u n ε q N un+1 q u n+1 h 2 u n ε + u n q ε mσ q ),

6 PARALLEL ALGORITHMS FOR SEGMENTATION OF CELLULAR STRUCTURES 421 be the ε-regularized a- where we use the Evans-Sruck regularization [7]. Let f n roximation of gradient modulus, f n u n ε, given by u n ε f n = ε mσ q) m) d σ q N ) u n σ u n 2, 3.5) which gives the exact value if u is a linear function and ε = 0 [8]. In 3.5), u n σ is the aroximate value of u in oint x σ, which is the intersection of σ q and a line connecting centers of voxels and q, at the time ste n. It is comuted as follows [8]: u n+1 σ fq n + u n+1 q f n + fq n = un+1 f n. 3.6) The full linear system reads as follows: m) un+1 w c g f n u n + τ q N un+1 q N v in q u n+1 h q u n+1 q u n+1 ) 2 mσ) f n + f n q = 0 3.7) From the initial condition, one can obtain u 0 and u 0 σ. We set n = 0 and f n is comuted by 3.5). Then, the system 3.7) is solved, and new u n+1 σ are comuted by 3.6). 4. Parallelization. To seed u the comutation and reduce the rocess memory consumtion we arallelize the algorithms using MPI Message-assing interface) [1]. We divide the whole volume along the volume edge into N subvolumes, the same number as total count of arallel rocesses. Of these N subvolumes, N-1 has always the same size and the last subvolume may be the same size or smaller, deending on how the volume can be divided. Each rocess then erforms the calculation only on aroriate subvolume. The only inter-rocess communication is needed at the following stes of arallel imlementation: at the beginning, the cell identifiers are read by first rocess and then broadcasted to all other rocesses. The neighbouring slices with udated data are exchanged between rocesses during the comutation of DF and also during the solution of the linear system after each half-iteration of the red-black SOR algorithm. The aroriate art of the inut image is read from inut file in arallel by each rocess and communication is not needed. We tested the arallel comutation seedu on a high-erformance comuting HPC) server with 4x octo-core AMD Oteron 6134 with 256GB of RAM using real data volume with dimensions 512x512x120 voxels. We achieved nearly linear seedu while running u to eight rocesses and slightly lower seedu after that tab. 4.1). During the initial testing we observed a curious henomenon where there was a massive slowdown when the rogram was run on certain number of rocessors. Interestingly, this roblem was deendent not on the number of rocessors, but on the number of the subvolume slices. We verified this fact by running the calculation on fourteen rocesses with 37, 38 or 39 slices on each subvolume. Then the calculation took 3223, 2815 and 2664 seconds resectively, see also eaks in the uer curve in fig Logically, the calculation should be faster on smaller subvolumes, but on

7 422 KAROL MIKULA, MICHAL SMÍŠEK AND RÓBERT ŠPIR Table 4.1 Parallel calculation seedu. No. of rocesses Calculation time Seedu the contrary, it was slower. Deending on the number of slices, arrays of variables in algorithms stored in memory are not otimally aligned for the rocessor caches, so with certain sizes there is heavy cache trashing and calculation is slowed down significantly. To overcome this roblem, we restructure the rogram memory organization so when the variables are used together, we are not using searate arrays for each of them, but we store them in a single array of structures. Then they are stored near each other in memory. For examle, when we are calculating u σ for north, south, west, east, bottom and to voxel faces, we store them all in single structure containing 6 double variables for each voxel and we are using single array of these structures for whole image. Likewise we are storing other variables in our rograms. With this memory organization the hardware data refetching on rocessors works in an otimal way. This change not only eliminates the aforementioned roblem, but also seeds u the serial code by 40% as seen on fig. 4.1 starting oints of the curves). The slight slowdowns in otimized code after multiles of eight rocesses are caused by the fact, that each four cores of the server are sharing memory access. If we run eight rocesses, each rocess has dedicated memory access, but when we run nine rocesses, two rocesses are sharing memory access, so there is slight slowdown. In the tab. 4.1 we can see that we achieved nearly linear seedu while running u to eight rocesses. After that, the main bottleneck in the calculation is the seed of memory access, so the seedu is lower. We verified this by running the calculation with sixteen rocesses on two servers, eight rocesses on each. With this configuration the memory access by rocesses is used otimally. The calculation time, desite the need of inter-rocess communication through local network, was faster than running sixteen rocesses on a single server, where each two rocesses share single memory access 1814 seconds versus 2078 seconds). To minimize the amount of data needed for communication, we first rotate the volume so it is always divided along the longest edge - then the amount of data sent during the communication is minimal. This does not bring a notable seedu when running the calculation on single server, because communication through shared memory inside the server is sufficiently fast even for larger amount of data. On the other hand, in the worst-case scenario, when each rocess runs on a different server connected together by 1Gbit local area network, the seedu was considerable. With our testing volume, when divided along the edge with the size of 120 voxels, we are sending slice with the size of 512x512 tye double variables 2MB of data). The calculation on eight rocessors took 5520 seconds. When we divided it along the 512 voxels edge, the amount of data for communication was reduced to 0.5MB and whole calculation took 3750 seconds, which is faster by 33%. 5. Numerical exeriments Phantom exeriments. To erform the basic correctness test of our imlementations, we created simle hantom data. For the first roblem, we created an artificial image of the evolution of three cells, one of which undergoes a cell division

PARALLEL ALGORITHMS FOR SEGMENTATION OF CELLULAR STRUCTURES 423 Fig. 4.1. Comarison of duration of the calculation before and after the otimization deending on the number of used rocessors.

The uer curve reresents calculation time before otimizations and the lower curve reresents time after the otimization. in the middle of the video.

For the second roblem, we created an artificial hemishere as an aroximation of the embryo shae.

8 PARALLEL ALGORITHMS FOR SEGMENTATION OF CELLULAR STRUCTURES 423 Fig Comarison of duration of the calculation before and after the otimization deending on the number of used rocessors. X-axis: number of rocessors, y-axis: time in seconds, note that y-axis has logarithmic scale. The uer curve reresents calculation time before otimizations and the lower curve reresents time after the otimization. in the middle of the video. Then, using randomly generated cell identifiers, we constructed ISF. Visually we can see that the segmentation algorithm correctly retrieves satiotemoral shaes of cellular evolution, cf. fig For the second roblem, we created an artificial hemishere as an aroximation of the embryo shae. From randomly laced cell identifiers we comuted DF, constructed the ISF and we were able to reconstruct the embryo hantom using the GSUBSURF algorithm, cf. fig Fig D+time hantom image segmentation. From left to right: first - hantom image of three cells, where the central one undergoes a cell division during its evolution, second - hyerbolic initial segmentation function, visualized as a slice, third - segmentation result after 1000 GSUBSURF time stes, visualized as a slice, fourth - the same segmentation result, visualized as an isosurface Study of the exerimental order of convergence. We tested our arallel imlementation of 3.5)-3.7) for the curvature art of GSUBSURF, so we let w a = 0 and w c = 1 and g = 1 and we considered the exact solution of the roblem in the form [6] u x, y, z, t) = x 2 + y 2 + z 2 1 ) /4 + t. 5.1) We solve the roblem in the satial domain Ω = [ 1.25, 1.25] [ 1.25, 1.25]

424 KAROL MIKULA, MICHAL SMÍŠEK AND RÓBERT ŠPIR Fig. 5.2. 3D hantom image segmentation.

result Table 5.1 This table shows EOC analysis of full domain solution. NT S-number of time stes, err - error in L 2 Ω)-norm at the time T = 0.16. n h NT S τ err EOC 10 0.25 4 0.04 0.00582 20 0.

9 424 KAROL MIKULA, MICHAL SMÍŠEK AND RÓBERT ŠPIR Fig D hantom image segmentation. Left - hantom image of an embryo of hemisherical shae, center - initial segmentation function, iecewise-constant thresholding of DF from randomly generated cell identifiers, right - segmentation result Table 5.1 This table shows EOC analysis of full domain solution. NT S-number of time stes, err - error in L 2 Ω)-norm at the time T = n h NT S τ err EOC [ 1.25, 1.25] over the time interval T [0, 0.16]. We divide the domain subsequently into n 3 finite volumes, where n = 10, 20, 40, 80, 160 with edge size h = 2.5/n. The length of time ste τ is roortional to h 2. We measured the exerimental order of convergence EOC) with resect to grid refinement. The results of arallel comutations are shown in tab The obtained results are indeendent on the number of rocessors used for calculations Real data exeriment. Finally, we erformed a segmentation of real data. In fig. 5.3 we show isosurfaces of the ISF left) and isosurfaces of the corresonding segmentation result, showing segmented satio-temoral tubular structures. These reresent cellular evolution of aroximately 30 cells from the mono-layered eithelium of a Drosohila ua in morhogenesis. In fig. 5.4 we resent segmentation of the overall shae of the zebrafish embryo during the first stages of the develoment. The segmentation is visualized as one horizontal and two vertical slices, where we lot original data together with a segmentation isosurface. REFERENCES [1] Y. Aoyama, J. Nakano, RS/6000 SP: Practical MPI Programming, IBM, Poughkeesie, New York, 1999 [2] Y. Bellaïche, F. Bosveld, F. Graner, K. Mikula, M. Remešíkova, M. Smíšek, New Robust Algorithm for Tracking Cells in Videos of Drosohila Morhogenesis Based on Finding an Ideal Path in Segmented Satio-Temoral Cellular Structures, Proceeding of the 33rd Annual International IEEE EMBS Conference, Boston Marriott Coley Place, Boston, MA, USA, August 30 - Setember 3, 2011, IEEE Press, 2011

PARALLEL ALGORITHMS FOR SEGMENTATION OF CELLULAR STRUCTURES 425 Fig. 5.3. Drosohila real data analysis. Left - initial segmentation function, right - final segmentation. Fig. 5.4. Zebrafish real data analysis.

Sarti, 4D embryogenesis image analysis using PDE methods of image rocessing, Kybernetika, Vol. 46, No. 2 2010). 226-259 [4] P. Bourgine, P. Frolkovic, K. Mikula, N. Peyrie ras, M.

Scale Sace and Variational Methods in Comuter Vision, Voss, Norway, June 1-5,2009), Sringer 2009). 38-49 [5] V. Caselles, R. Kimmel, G.

10 PARALLEL ALGORITHMS FOR SEGMENTATION OF CELLULAR STRUCTURES 425 Fig Drosohila real data analysis. Left - initial segmentation function, right - final segmentation. Fig Zebrafish real data analysis. Left - segmentation slice viewed from to, right - segmentation slices, viewed from sides. [3] P. Bourgine, R. C underlı k, O. Drblı kova, K. Mikula, N. Peyrie ras, M. Remes ı kova, B. Rizzi, A. Sarti, 4D embryogenesis image analysis using PDE methods of image rocessing, Kybernetika, Vol. 46, No ) [4] P. Bourgine, P. Frolkovic, K. Mikula, N. Peyrie ras, M. Remes ı kova, Extraction of the intercellular skeleton from 2D microscoe images of early embryogenesis, in Lecture Notes in Comuter Science 5567 Proceeding of the 2nd International Conference on Scale Sace and Variational Methods in Comuter Vision, Voss, Norway, June 1-5,2009), Sringer 2009) [5] V. Caselles, R. Kimmel, G. Sairo, Geodesic active contours, International Journal of Comuter Vision ), [6] S. Corsaro, K. Mikula, A. Sarti, F. Sgallari, Semi-imlicit co-volume method in 3D image segmentation, SIAM Journal on Scientific Comuting, Vol. 28, No ) [7] L. C. Evans, J. Sruck, Motion of level sets by mean curvature, J. Diff. Geom ) [8] R. Eymard, A. Handlovic ova, K. Mikula, Study of a finite volume scheme for the regularised mean curvature flow level set equation, IMA Journal on Numerical Analysis, Vol ) [9] P. Frolkovic, K. Mikula, Flux-based level set method: a finite volume method for evolving interfaces, Alied Numerical Mathematics, Vol. 57, No ) [10] P. Frolkovic, K. Mikula, N. Peyrie ras, A. Sarti, A counting number of cells and cell segmentation using advection-diffusion equations, Kybernetika, Vol. 43, No ) [11] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, A. Yezzi, Conformal curvature ows: from hase transitions to active vision, Arch. Rational Mech. Anal ), [12] K. Mikula, N. Peyrie ras, M. Remes ı kova, M. Smı s ek, 4D numerical schemes for cell image

11 426 KAROL MIKULA, MICHAL SMÍŠEK AND RÓBERT ŠPIR segmentation and tracking, in Finite Volumes in Comlex Alications VI, Problems & Persectives, Eds. J. Fořt et al. Proceedings of the Sixth International Conference on Finite Volumes in Comlex Alications, Prague, June 6-10, 2011), Sringer Verlag, 2011, [13] K. Mikula, N. Peyriéras, M. Remešíková, O. Stašová, Segmentation of 3D cell membrane images by PDE methods and its alications, Comuters in Biology and Medicine, Vol. 41, No ) [14] E. Rouy, A. Tourin, Viscosity solutions aroach to shae-from-shading, SIAM Journal on Numerical Analysis 29 No ), [15] A. Sarti, R. Malladi, J. A. Sethian, Subjective Surfaces: A Method for Comleting Missing Boundaries, Proceedings of the National Academy of Sciences of the United States of America 12 97) 2000),

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