Application Of RBF Meshless Method Based On Organ Image Reconstruction

In recent years,the research of 3D reconstruction and visualization of medical images is increasing day by day,which has important academic significance and application value.In the course of studying the imaging of human organs,Washington University School of Medicine has used the method of fundamental solutions(MFS)of meshless method,and has achieved remarkable results in clinical experiments.3D heart images have been obtained by sampling cardiac data by distributing electrodes on the surface of human body,and the abnormal heart can be found in time.They are now studying the prevention of preterm labor in pregnant women,and if the same technique is used,3D images of the uterus can be obtained to detect fetal abnormalities,but there is a risk of harm to the fetus.Based on the desire to minimize radiation and reduce cost,and only collect surface data in relevant positions,we are possible to render a complete 3D image.Based on the above question,the feasibility of numerical realization is discussed in this paper.Partial differential equations based algorithm has been proposed for the 3D implicit surface reconstruction from a set of scatter cloud data.In the solution process,the method of approximate particular solutions(MAPS)with radial basis function(RBF)has been employed for solving the modified Helmholtz or Poisson equation with constant Dirichlet boundary condition.We also consider to repair the surfaces when a certain region of cloud data points are missing.The selection of several parameters is studied for the optimal recovery of the surfaces.Finally,we extend the research object to the case of irregular complex surface,and reconstruct the complex surface efficiently by constructing special internal points as auxiliary calculation.The main contents and results of this study are as follows:(1)The closed-type special solutions of Helmholtz-type partial differential equations are given.The MAPS used in image reconstruction requires a particular solution of the correlation operator corresponding to the RBF.We derive closed-form particular solutions for Helmholtz-type partial differential equations.These are derived explicitly using the Matern basis functions.The derivation of such particular solutions is further extended to the cases of products of Helmholtz-type operators in two and three dimensions.The main idea of the paper is to link the derivation of the particular solutions to the known fundamental solutions of certain differential operators.The newly derived particular solutions are used,in the context of the method of particular solutions,to solve boundary value problems governed by a certain class of products of Helmholtz-type equations.The leave-one-out cross validation(LOOCV)algorithm is employed to select an appropriate shape parameter for the Matern basis functions.In order to avoid the ill-conditioned dense matrix when the data of MAPS is large,we use local MAPS(LMAPS)and demonstrate the effectiveness of the proposed method.(2)A 3D implicit surface reconstruction algorithm via PDEs is proposed.We propose to reconstruct the 3D surfaces defined by a set of point cloud data through solving nonhomogeneous modified Helmholtz equation with constant Dirichlet boundary condition( = 1).The MAPS with the use of radial basis functions has been selected as the numerical method due to its effectiveness and simplicity.The PDE model is nonhomogeneous in which interior collocation points are required.In our study,these interior points can be easily obtained.Furthermore,we discover the number and the location of interior points have little effect on the reconstruction of the simple surface.Only a few interior points are sufficient in our model.If the forcing term 1)((3)is properly chosen,the value of will continue to extend outside the domain with >1.Since <1 inside the domain and >1 outside the domain and the solution of the PDE is continuous,there exists solution =1 which is the surface of the given domain.As a result,the identification of the implicit surface can be achieved by finding all the points with =1 in a bounding box containing the domain.(3)Reconstruction and Restoration of Organ Images.The data points of abdominal are easily available through a scanning device.Since the data points of uterus are difficult to obtain,in many occasions we may not be able to obtain the complete data set.A large portion of the data points are not available and the direct reconstruction using the incomplete data has resulted in cut-off surfaces on the top and bottom.After adding two augmented points on the top and bottom of the uterus.We can recover most of the missing surface.On the other hand,we try to remove some data points from the one side of the uterus.A large hole has appeared in one side.To patch up the hole,we choose the isosurface with PDE solution at the level of =0.98 instead of =1.The hole disappeared but the surface has shrunk slightly.The reconstructive experiment of tooth let us know the importance of internal point selection.(4)Reconstruction of irregular complex images.This is a challenging cproblem.We chose images of the stanford Rabbit,Hand,Head,Dragon with concave and convex surfaces and containing sharp vertices as experimental objects.One of the limitations of our proposed method is that the resulting matrix is dense and full and we could only handle a limited number of cloud data points.Some special internal points need to be selected in the reconstruction process: One way is to randomly select boundary points and apply normals inward to generate these internal points;Another way is to estimate the normals of sparse 3D point cloud by using "Find 3D Normals and Curvature ",which we use these given normals to produce the interior points in this paper.The more internal points are extracted in the position of greater concavity and convexity,and the parameter in the PDE is λ set a larger value,which can effectively complete the image reconstruction,otherwise the reconstructed image will appear the spurious surfaces.In conclusion,the implicit surface reconstruction algorithm proposed in this paper can effectively reconstruct all kinds of regions and play an important role in 3D surface reconstruction.