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image processing; partial differential equations; nonlinear advection-diffusion equations; flux-based level set method; subjective surface method; embryogenesis
We develop a method for counting number of cells and extraction of approximate cell centers in 2D and 3D images of early stages of the zebra-fish embryogenesis. The approximate cell centers give us the starting points for the subjective surface based cell segmentation. We move in the inner normal direction all level sets of nuclei and membranes images by a constant speed with slight regularization of this flow by the (mean) curvature. Such multi- scale evolutionary process is represented by a geometrical advection-diffusion equation which gives us at a certain scale the desired information on the number of cells. For solving the problems computationally we use flux-based finite volume level set method developed by Frolkovič and Mikula in [FM1] and semi-implicit co-volume subjective surface method given in [CMSSg, MSSgCVS, MSSgchapter]. Computational experiments on testing and real 2D and 3D embryogenesis images are presented and the results are discussed.
[2] Evans L. C., Spruck J.: Motion of level sets by mean curvature I. J. Differential Geom. 33 (1991), 635–681 MR 1100206 | Zbl 0726.53029
[4] Handlovičová A., Mikula, K., Sgallari F.: Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution. Numer. Math. 93 (2003), 675–695 DOI 10.1007/s002110100374 | MR 1961884 | Zbl 1065.65105
[5] Mikula K., Sarti, A., Sgallari F.: Co-volume method for Riemannian mean curvature flow in subjective surfaces multiscale segmentation. Comput. Visual. Sci. 9 (2006), 23–31 DOI 10.1007/s00791-006-0014-0 | MR 2214835
[6] Mikula K., Sarti, A., Sgallari F.: Semi-implicit co-volume level set method in medical image segmentation. In: Handbook of Biomedical Image Analysis: Segmentation and Registration Models (J. Suri et al., eds.), Springer, New York, 2005, pp. 583–626
[7] Osher S., Sethian J.: Fronts propagating with curvature dependent speed: algorithm based on Hamilton–Jacobi formulation. J. Comput. Phys. 79 (1988), 12–49 DOI 10.1016/0021-9991(88)90002-2 | MR 0965860
[8] Sarti A., Malladi, R., Sethian J. A.: Subjective Surfaces: A Method for Completing Missing Boundaries. Proc. Nat. Acad. Sci. U. S. A. 12 (2000), 97, 6258–6263 DOI 10.1073/pnas.110135797 | MR 1760935 | Zbl 0966.68214
[9] Sarti A., Citti G.: Subjective surfaces and Riemannian mean curvature flow of graphs. Acta Math. Univ. Comenian. 79 (2001), 1, 85–104 MR 1865362 | Zbl 0995.65100
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