Implicit Surface Reconstruction from 3D Scattered Points Based on Variational Level Set Method

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1 Implicit Surface econstruction from D Scattered Points Based on Variational Level Set Method Hanbo Liu Department ofshenzhen graduate school, Harbin Institute oftechnology, Shenzhen, 58055, China liu_hanbo@hit.edu.cn Xin Wang Department ofshenzhen graduate school, Harbin Institute oftechnology, Shenzhen, 58055, China wang_xin@hit.edu.cn Wenyi Qiang Space Control and Inertial Technology esearch Center, Harbin Institute oftechnology, Harbin, 50080, China qiangwy@hit.edu.cn Abstract-In this paper we propose a novel variational form ulation for arbitrary surface reconstruction from D scattered points. An implicit surface is adopted for its more advantages, such as continuity and differentiability, easy estimation of points inside or outside the shape, and convenient implementation for these complicated set operations. The presented new energy functional in this paper considers more factors on properties ofthe scattered points (including distances and normal vectors), smoothness and the constraint of the signed distance function. The gradient flow which minimizes the total energy functional updates and drives the motion ofthe zero level set interface to the desired surface. The derived formulations were applied to certain D surface reconstruction with good results. I. INTODUCTION Constructing a computer model from an existing object is a common problem in everse Engineering. The sampled data might be scanned with a device like the laser range scanner, or be measured on the object with a mechanical probe. Sometimes, not only the spatial location of points, but also normal vectors can be obtained. The problem ofautomatically reconstructing a topologically consistent and geometrically accurate CAD model of complex object from these sampled data widely occurs in manufacturing industry. Obviously solving this problem depends heavily and only on the dense and uniform sampled data itself. And our research aims to provide a compact and topologically consistent representation ofthe complex object and usable and accurate reconstruction method from dense samples. The level set method is a theoretical and numerical method for implicit surfaces. It initially was introduced in the area of fluid dynamics establishing a connection between the family of evolving curves and the family of evolving surfaces []. The level set method was devised by Osher and Sethian as a simple and versatile method for computing and analyzing the motion of an interface under a velocity field in two or three dimensions [][]. The velocity can depend on position, time, the geometry of the interface and the external physics. The level set method is widely used in D image processing and segmentation [4]. The implicit surface methods for reconstruction have received considerable attention in the past few years by researchers, such as Hoppe, Carr, Turk and Ohtake et al. Carr fit a radial basis function (BF) to the signed distance function and used BF to polygonize sampled points to create a triangulated surface [5], but this algorithm is difficult to implement. Subsequently Ohtake proposed MPU method [6]. And Zhao et al. first combined variational and PDE formulation with level set method for D surface reconstruction from unorganized data [7][8]. But they not make us ofthe measurable normal vector. In this paper we mainly propose a novel energy functional additionally involving factors of distance and unit normal vector, constraint ofsigned distance function and smoothness. In section II the implicit surface representation is presented. In section III we introduce our energy functional and the corresponding gradient flow. In section IV concerned details of the numerical implementation are presented. Finally in section V gives the results of reconstructed surface by our variational level set method. II. IMPLICIT SUFACE EPESENTATION We consider the following reconstruction problem: let an unorganized set of points S = {X i }: E associated with normal vectors V = {n i }: E and position vectors P = {Pi = (Xi' Y i, Zi ):} E.And the points S are sampled from a boundary surface r of a three-dimensional object in domain Q in.so we can utilize the level set theory, for a given open region Q with smooth boundary surface r,there exists a level set function (X,Y, z), which at any time is Lipschitz continuous, and do not depend on any underlying parameterization, satisfying /08/$ IEEE

2 φ( xyz,, ) < 0 for( xyz,, ) Ω φ( xyz,, ) = 0 for( xyz,, ) φ( xyz,, ) > 0 for( xyz,, ) Ω. So the coordinates ( xyz,, ) of the points on the surface are defined by the equation φ ( xyz,, ) = 0. The surface remains the zero level set of φ even over time. Moreover, the implicit surface is convenient implementation for complicated set operations. Using convention, the outward unit normal n and the mean curvature k of are given by (). n = and k = () Where φ = φ + φ + φ. x y z Taking the derivatives of φ ( xyz,, ) = 0 with respect to time t gives (), which is a first order Hamilton-Jacobi equation. + = + v n φ = 0 () t t t Where is proportional to the surface normal, x affects t φ only in the direction of the normal, which motion in any direction is merely a change in the parameterization. v n is the velocity normal to the surface. This PDE is solved on a fixed grid in the domain Ω.one of the advantages of implicit surface representation is that the topology of the surface is allowed to change as the surface evolves, thus making it easy to represent complex surfaces that can merge or split and also with holes. III. VAIATIONAL FOMULATION The main idea of the level set method is simple: using energy minimization techniques a gradient flow derived as the steepest descent of the energy functional is obtained and then implemented in the level set space. So the minima correspond to desired surface. In this paper we construct a novel energy functional E( x ). Given a set of scattered D points S = { x} N i i=, for each xi with properties of normal vectors ni and position vectors p i, define the energy functional E( x) to be minimized as Ex ( ) = En + Ed + Esmooth. (4) Furthermore, if f ( x ) is a minimum of the energy functional E( x ), then ' E ( f( x); v ) = 0 for all v. The idea behind (5) is to minimize the difference between the desired normal and the normal of the surface in nature. Normal vectors are coupled with the surface, thus the normal vectors drag the surface while their values are modified during processing. c () En = ( nx ( ) Nx ( )) ds (5) Where N( x) is the extension of the desired normal nx ( i ) at point xi to the whole domain Ω, nx ( ) is the surface normal vector, ds is the surface area. If is the zero level set of φ ( x), then the energy functional corresponding to the normal can be reformulated as En = ( n ( x) N( x)) ds (6) = ( nx ( ) Nx ( )) δφ ( ) dx Where δ ( x) is the one dimensional Delta function and δφ ( ) dx is the surface area element at the zero level set of φ ( x). So (6) can extend the integration surface to a D computation domain containing the zero level set of φ ( x). Let F( x, φ, φ) = ( N( x) ) δ ( φ), then the condition for a minimum given by the Gateaux derivative is: F F F F ( ) ( ) ( ) = 0 (7) x x y y z z So the gradient flow obtained from (7) is φt_ n = ( ( ) N( x)) (8) = ( k N( x)) Similarly equation (9) deals with the position difference between the desired and the reconstructed surface. The energy functional corresponding to distance is defined as: E = d ( x) ds = d ( x) δφ ( ) dx Where d (9) d( x) = dist( x, S) = min dist( x, p ), dist( x, y) = x y. i N So the steepest descent flow is: φt_ d = d ( ) φ + ( d ) = dk φ + ( d). i (0) Es is the smoothing term, and we choose a small positive ω as the smoothing constant. Es = ωds = ωδ( φ( x)) ( x) dx. () Which gets the Well-known mean curvature motion (). φt_ s = ω ( ) φ = ωk () If object with more edges and corners, we change the mean curvature motion to constant speed motion. φt_ s = ω φ () Combing the energy functional (5), (9) and (), the overall PDE motion is expressed as (4). φt = φt_ d + φt_ s + φt_ n = dk φ + ( d) + ωk φ + ( k N( x)) (4) = ( d ) φ + ( d + ω) k N( x) A B C Where ω = ω+. The first term A is corresponding to the

3 advection motion, the middle B is corresponding to motion by mean curvature and the last C corresponding to motion in the normal direction. And this total motion leads the energy functional E( x) decrease until to the equilibrium. PDEs are solved on discrete fixed grids in some domains. IV. NUMEICAL IMPLEMENTATION A. The Signed Distance Function The signed distance of a point x from a closed, oriented manifold S is defined as the Euclidean distance from x to the closest point on S, with a positive sign if x lies outside S and a negative sign otherwise. So computes the distance function to an arbitrary data set on rectangular fixed grids. dist( x, ( t)), x Ω( t) d( x, t) = 0, x ( t) (5) dist( x, ( t)), x \ Ω( t) We use the same numerical approximation to δ ( x) as defined in [0] in this paper. 0, x > α δα ( x) = π x [ + cos( )] x α α α And in (5) N( x ) is defined as nx ( i) if x Bε ( xi) N( x) = nx ( ) otherwise (6) Where Bε ( x i ) a ball with radius ε centered at x i and covers the nearest gird. B. Numerical Scheme This family of PDEs and the numerical scheme for solving them on discrete grids use the traditional level set method [][9]. The spatial derivatives of A and C terms in (4) are approximated by the upwind finite difference scheme, while the spatial derivatives of B term are approximated by the central difference scheme. And the temporal partial derivative φ t is approximated by the forward difference. So the approximation of (4) by the above difference scheme can be expressed as m+ m m φ = φ + τ( φ ) i, j, k i, j, k i, j, k m Where L( φi, j, k) is the approximation of the right hand side in (4) by the above spatial difference schemes. The curvature k usually used is computed by φxφyy φxφyφxy + φyφxx + φxφzz φxφzφxz + φz φxx k = φφ y zz φφφ y z yz + φφ z yy +. In addition, when solving the PDE (4), we must make a point of the spatial and time step. According to the density of D scattered points and K-nearest radius of each point, we decide the spatial and time step. At the same time, the time step and spatial step must satisfy the CFL condition for stable evolution. C. Initialization and e-initialization A good initial surface guess is crucial to the efficiency of the variational level set method. We use the traditional and common method that build (7) an offset of the distance function to the scattered points as the initial surface in initialization process. d( x, t) ε (7) e-initialization has been widely used as a numerical remedy in traditional level set method. In practice, the evolving level set function can deviate greatly from its value as signed distance after several iterations especially when the time step is not chosen small enough for fast evolution. It is well known that a signed distance function must satisfy φ =. Conversely, any function φ satisfying φ = is the signed distance function plus a constant. Illumined by D image re-initialization [], we additionally introduce a penal term for avoiding the deviation of the level set function φ from a signed distance function in D surface reconstruction. F( φ) = ( ) dx (8) Ω Also to minimize the F( φ ), we derive the evolving motion as =Δφ div( ) = div[(- ) ]. (9) t Where Δ is the Laplacian operator. If φ >, this term makes φ more even and therefore reduce the contrary, if φ < will increase φ. φ. On the V. ESULTS Fig. shows the constructed grids for an easy sphere model. According to the red scattered points, we can computer the signed distances and determine the domain of effective grids for initial surface. Fig. illustrates the reconstructed surfaces by the presented variational level set method in this paper. The dots mark the D scattered points all measured directly from real D objects. And the reconstructed surfaces are smooth and express the original shapes well with good precision. esults demonstrate that our method can be applied to arbitrary D scattered points without reference to surface topology and geometrical features.

4 geometrical features. Fig.. Grids and the initial distance to the scattered points EFEENCES [] S. Osher and J.A. Sethian, Fronts Propagating with Curvature Dependent Speed: Algorithm based on the Hamilton-Jacobin Formulation, Computational Physics (79), pp.-49, 988. [] J.A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, New York, 999. [] S. Osher and. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York, 00. [4] S. Osher and. Fedkiw, Level Set Methods: an Overview and Some ecent esults, Journal of Computational Physics (69), pp , 00. [5] J. C. Carr,. K. Beatson, J. B. Cherrie, et al, econstruction and epresentation of D Object with adial Basis Functions, Proceedings of SIGGAPH00, Los Angeles, pp ,00. [6] Y. Ohtake, A. Belyaev and H. Seidel, A Multi-scale Approach to D Scattered Data Interpolation with Compactly Supported Basis Functions, Proceedings of SMI00, Seoul, pp.5-6, 00. [7]. H.K. Zhao, S. Osher, B. Merriman, and M. Kang, Implicit and Non-parametric Shape econstruction from Unorganized Data Using Variational Level Set Method, Computer Vision and Image Understanding, 80(), pp.95-9, 000. [8]. H.K. Zhao, S. Osher and. Fedkiw, Fast Surface econstruction Using the Level Set Method, proceedings of the IEEE Workshop on Variational and Level Set Methods in Computer Vision, Vancouver, pp.94-0, 00. [9] D. Peng, B. Merriman, S. Osher, et al, A PDE-Based Fast Local Level Set Method, Journal of Computational Physics (55), pp.40-48, 999. [0] H.K. Zhao, B. Merriman and S. Osher, A Variational level set approach to multiphase motion, Journal of Computational Physics (7), pp.79-95, 996. [] C.M. Li, C.Y. Xu, C.F. Gui and M.D. Fox, Level Set Evolution Without e-initialization: A New Variational Formulation, proceedings of Computer Vision and Pattern ecognition, pp , 005. Fig.. econstructed surfaces and the initial D scattered points VI. CONCLUSIONS AND FUTUE WOK In this paper we propose a new variational formulation for arbitrary surface reconstruction from D scattered points. The implicit surface representation is adopted because of its advantages, such as continuity and differentiability, easy to identify whether a point lies inside or outside the shape, and convenient implementation for complicated set operations. Furthermore, the novel energy functional in this paper considers more factors on properties of the scattered points (including distances and normal vectors), smoothness and the constraint of the signed distance function. The gradient flow which minimizes the total energy functional updates and drives the motion of the zero level set interface to the desired surface. The derived formulation and method constructs the implicit surface on fixed rectangular grids and has been applied to certain D surface reconstruction with good results. In future work we will pay more attention on efficiency and speed-up of our variational method for D surface reconstruction from large scattered points with more

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