# American Institute of Mathematical Sciences

March  2021, 14(3): 971-985. doi: 10.3934/dcdss.2020351

## 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

Received  January 2019 Revised  October 2019 Published  March 2021 Early access  May 2020

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.

Citation: Balázs Kósa, Karol Mikula, Markjoe Olunna Uba, Antonia Weberling, Neophytos Christodoulou, Magdalena Zernicka-Goetz. 3D image segmentation supported by a point cloud. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 971-985. doi: 10.3934/dcdss.2020351
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##### References:
The voxel grid cell with a tetrahedral finite element
Notation for the additional points of a grid cell used for calculation of $Gu_{i,j,k}^{n,l},\;l = 1,...,24$
Tetrahedral finite element with marked edges for approximation of partial derivatives
In the first column there are 3D images of a sphere with 6 symmetrically placed holes. The experiment was executed on spheres with holes of different shapes and sizes. The second column shows the result of the mathematical model (1) with $\theta = 0$ and $\rho = 1$ for the advective velocity (3). In the third column we show the result after we added points to the missing parts of the sphere and set $\theta = 1$, $\rho = 1$ in (3)
Sections of the embryo together with the marked points for visceral endoderm (VE)
3D image of embryo structure, visualized with the reconstructed ExE part of the embryo in the upper picture and with VE part in the lower one
Reconstructed surface of extraembryonic ectoderm (ExE) and visceral endoderm (VE), visualized with point cloud
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