
-
Previous Article
Computational optimization in solving the geodetic boundary value problems
- DCDS-S Home
- This Issue
-
Next Article
Two notes on the O'Hara energies
3D image segmentation supported by a point cloud
1. | Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05 Bratislava, Slovakia |
2. | Department of Mathematics University of Nigeria 410001 Nsukka, Nigeria |
3. | Mammalian Embryo and Stem Cell Group, University of Cambridge, Department of Physiology, Development and Neuroscience, Downing Street, Cambridge CB2 3EG, UK |
4. | Department of Biological Sciences University of Cyprus University Avenue 1, Nicosia 2109, Cyprus |
Here, we report a novel method of 3D image segmentation, using surface reconstruction from 3D point cloud data and 3D digital image information. For this task, we apply a mathematical model and numerical method based on the level set algorithm. This method solves surface reconstruction by the application of advection equation with a curvature term, which gives the evolution of an initial condition to the final state. This is done by defining the advective velocity in the level set equation as the weighted sum of distance function and edge detector function gradients. The distance function to the shape, represented by the point cloud, is computed using the fast sweeping method. The edge detector function is applied to the presmoothed 3D image. A crucial point for efficiency is the construction of an initial condition by a simple tagging algorithm, which allows us also to highly speed up the numerical scheme when solving PDEs. For the numerical discretization, we use a semi-implicit co-volume scheme in the curvature part and implicit upwind scheme in the advective part. The method was tested on representative examples and applied to real data representing 3D biological microscopic images of developing mammalian embryo.
References:
[1] |
N. Christodoulou, C. Kyprianou, A. Weberling, R. Wang and G. Cui,
Sequential formation and resolution of multiple rosettes drive embryo remodeling after implantation, Nature Cell Biology, 20 (2018), 1278-1289.
doi: 10.1038/s41556-018-0211-3. |
[2] |
S. Corsaro, K. Mikula, A. Sarti and F. Sgallari,
Semi-implicit covolume method in 3D image segmentation, SIAM J. Sci. Comput., 28 (2006), 2248-2265.
doi: 10.1137/060651203. |
[3] |
S. Dyballa, T. Savy, P. Germann, K. Mikula and M.Remešíková, et al., Distribution of neurosensory progenitor pools during inner ear morphogenesis unveiled by cell lineage reconstruction, 6 (2017).
doi: 10.7554/eLife.22268.001. |
[4] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature. Ⅰ, J. Differential Geom., 33 (1991), 635-681.
doi: 10.4310/jdg/1214446559. |
[5] |
R. Eymard, A. Handlovičvá and K. Mikula,
Study of a finite volume scheme for the regularised mean curvature flow level set equation, IMA J. Numer. Anal., 31 (2011), 813-846.
doi: 10.1093/imanum/drq025. |
[6] |
E. Faure, et al., A workflow to process 3D+time microscopy images of developing organisms and reconstruct their cell lineage, Nature Communications, 7 (2016).
doi: 10.1038/ncomms9674. |
[7] |
B. Kósa, J. Haličková-Brehovská and K. Mikula, New efficient numerical method for 3D point cloud surface reconstruction by using level set methods, Proceedings of Equadiff 2017 Conference, 2017,387–396. Google Scholar |
[8] |
K. Mikula, N. Peyriéras, M. Remešíková and A. Sarti, 3D embryogenesis image segmentation by the generalized subjective surface method using the finite volume technique, in Finite Volumes for Complex Applications V, ISTE, London, 2008,585-592. |
[9] |
K. Mikula and M. Remešíková,
Finite volume schemes for the generalized subjective surface equation in image segmentation, Kybernetika, 45 (2009), 646-656.
|
[10] |
K. Mikula and A. Sarti, Parallel co-volume subjective surface method for 3D medical image segmentation, in Deformable Models, Topics in Biomedical Engineering. International Book Series, Springer, NY, 2007,123–160.
doi: 10.1007/978-0-387-68343-0_5. |
[11] |
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, 153, Springer-Verlag, New York, 2003.
doi: 10.1007/b98879. |
[12] |
P. Perona and J. Malik,
Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intelligence, 12 (1990), 629-639.
doi: 10.1109/34.56205. |
[13] |
A. Sarti, R. Malladi and J. A. Sethian,
Subjective surfaces: A method for completing missing boundaries, Proc. Natl. Acad. Sci. USA, 97 (2000), 6258-6263.
doi: 10.1073/pnas.110135797. |
[14] |
J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science, Cambridge Monographs on Applied and Computational Mathematics, 3, Cambridge University Press, Cambridge, 1999.
![]() |
[15] |
M. N. Shahbazi, A. Scialdone, N. Skorupska, A. Weberling and G. Recher,
Pluripotent state transitions coordinate morphogenesis in mouse and human embryos, Nature, 552 (2017), 239-243.
doi: 10.1038/nature24675. |
[16] |
M. N. Shahbazi and M. Zernicka-Goetz,
Deconstructing and reconstructing the mouse and human early embryo, Nature Cell Biology, 20 (2018), 878-887.
doi: 10.1038/s41556-018-0144-x. |
[17] |
C. Zanella, M. Campana, B. Rizzi, C. Melani and G. Sanguinetti,
Cells segmentation from 3-D confocal images of early zebrafish embryogenesis, IEEE Trans. Image Process., 19 (2010), 770-781.
doi: 10.1109/TIP.2009.2033629. |
[18] |
H. Zhao,
A fast sweeping method for Eikonal equations, Math. Comp., 74 (2005), 603-627.
doi: 10.1090/S0025-5718-04-01678-3. |
[19] |
H. Zhao, S. Osher, B. Merriman and M. Kang,
Implicit and nonparametric shape reconstruction from unorganized data using a variational level set method, Comput. Vision Image Understanding, 80 (2000), 295-319.
doi: 10.1006/cviu.2000.0875. |
show all references
References:
[1] |
N. Christodoulou, C. Kyprianou, A. Weberling, R. Wang and G. Cui,
Sequential formation and resolution of multiple rosettes drive embryo remodeling after implantation, Nature Cell Biology, 20 (2018), 1278-1289.
doi: 10.1038/s41556-018-0211-3. |
[2] |
S. Corsaro, K. Mikula, A. Sarti and F. Sgallari,
Semi-implicit covolume method in 3D image segmentation, SIAM J. Sci. Comput., 28 (2006), 2248-2265.
doi: 10.1137/060651203. |
[3] |
S. Dyballa, T. Savy, P. Germann, K. Mikula and M.Remešíková, et al., Distribution of neurosensory progenitor pools during inner ear morphogenesis unveiled by cell lineage reconstruction, 6 (2017).
doi: 10.7554/eLife.22268.001. |
[4] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature. Ⅰ, J. Differential Geom., 33 (1991), 635-681.
doi: 10.4310/jdg/1214446559. |
[5] |
R. Eymard, A. Handlovičvá and K. Mikula,
Study of a finite volume scheme for the regularised mean curvature flow level set equation, IMA J. Numer. Anal., 31 (2011), 813-846.
doi: 10.1093/imanum/drq025. |
[6] |
E. Faure, et al., A workflow to process 3D+time microscopy images of developing organisms and reconstruct their cell lineage, Nature Communications, 7 (2016).
doi: 10.1038/ncomms9674. |
[7] |
B. Kósa, J. Haličková-Brehovská and K. Mikula, New efficient numerical method for 3D point cloud surface reconstruction by using level set methods, Proceedings of Equadiff 2017 Conference, 2017,387–396. Google Scholar |
[8] |
K. Mikula, N. Peyriéras, M. Remešíková and A. Sarti, 3D embryogenesis image segmentation by the generalized subjective surface method using the finite volume technique, in Finite Volumes for Complex Applications V, ISTE, London, 2008,585-592. |
[9] |
K. Mikula and M. Remešíková,
Finite volume schemes for the generalized subjective surface equation in image segmentation, Kybernetika, 45 (2009), 646-656.
|
[10] |
K. Mikula and A. Sarti, Parallel co-volume subjective surface method for 3D medical image segmentation, in Deformable Models, Topics in Biomedical Engineering. International Book Series, Springer, NY, 2007,123–160.
doi: 10.1007/978-0-387-68343-0_5. |
[11] |
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, 153, Springer-Verlag, New York, 2003.
doi: 10.1007/b98879. |
[12] |
P. Perona and J. Malik,
Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intelligence, 12 (1990), 629-639.
doi: 10.1109/34.56205. |
[13] |
A. Sarti, R. Malladi and J. A. Sethian,
Subjective surfaces: A method for completing missing boundaries, Proc. Natl. Acad. Sci. USA, 97 (2000), 6258-6263.
doi: 10.1073/pnas.110135797. |
[14] |
J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science, Cambridge Monographs on Applied and Computational Mathematics, 3, Cambridge University Press, Cambridge, 1999.
![]() |
[15] |
M. N. Shahbazi, A. Scialdone, N. Skorupska, A. Weberling and G. Recher,
Pluripotent state transitions coordinate morphogenesis in mouse and human embryos, Nature, 552 (2017), 239-243.
doi: 10.1038/nature24675. |
[16] |
M. N. Shahbazi and M. Zernicka-Goetz,
Deconstructing and reconstructing the mouse and human early embryo, Nature Cell Biology, 20 (2018), 878-887.
doi: 10.1038/s41556-018-0144-x. |
[17] |
C. Zanella, M. Campana, B. Rizzi, C. Melani and G. Sanguinetti,
Cells segmentation from 3-D confocal images of early zebrafish embryogenesis, IEEE Trans. Image Process., 19 (2010), 770-781.
doi: 10.1109/TIP.2009.2033629. |
[18] |
H. Zhao,
A fast sweeping method for Eikonal equations, Math. Comp., 74 (2005), 603-627.
doi: 10.1090/S0025-5718-04-01678-3. |
[19] |
H. Zhao, S. Osher, B. Merriman and M. Kang,
Implicit and nonparametric shape reconstruction from unorganized data using a variational level set method, Comput. Vision Image Understanding, 80 (2000), 295-319.
doi: 10.1006/cviu.2000.0875. |







[1] |
Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073 |
[2] |
Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390 |
[3] |
Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233 |
[4] |
Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048 |
[5] |
Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 |
[6] |
Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347 |
[7] |
S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020435 |
[8] |
Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020355 |
[9] |
Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 |
[10] |
Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 |
[11] |
Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078 |
[12] |
Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219 |
[13] |
Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 |
[14] |
Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020404 |
[15] |
Philippe G. Ciarlet, Liliana Gratie, Cristinel Mardare. Intrinsic methods in elasticity: a mathematical survey. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 133-164. doi: 10.3934/dcds.2009.23.133 |
[16] |
Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020176 |
[17] |
Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389 |
[18] |
Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386 |
[19] |
Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020320 |
[20] |
Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 |
2019 Impact Factor: 1.233
Tools
Metrics
Other articles
by authors
[Back to Top]