Surface Reconstruction by Structured Light Projection

Transcription

1 J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(5): DOI: /s Surface Reconstruction by Structured Light Projection YANG Rong-qian 1,2 ( ), CHEN Ya-zhu 2 ( ) (1. Department of Biomedical Engineering, South China University of Technology, Guangzhou , China; 2. Biomedical Instrument Institute, Med-X Research Institute, Shanghai Jiaotong University, Shanghai , China) Shanghai Jiaotong University and Springer-Verlag Berlin Heidelberg 2010 Abstract: A rapid and practical method is proposed to reconstruct surface based on the linked structured light stripes which are produced by structured light projection. The subpixel points on a stripe are linked firstly one by one to form a stripe ensemble which is then transformed to a point ensemble in 3D space. The initial mesh with local optimization is generated by triangulating each two adjacent point ensembles. In order to obtain a better mesh, our improved edge flipping algorithm is employed to optimize the initial mesh globally. Because of employing the information of the linked structured stripes, our reconstruction algorithm is performed fastly. Moreover, the subpixel points on each stripe are already linked on the captured images such that they do not require the high sampling density. The experiments show that the proposed method constructs a surface rapidly and effectively. Key words: edge flip, point clouds, structured light, surface reconstruction, triangulation CLC number: TP 38 Document code: A 1 Introduction Surface reconstruction from point-sampled geometry is an important problem in computer graphics, computer aided design (CAD), medical imaging, and solid modeling. Laser range finders record the geometry of real-world 3D objects in the form of point coordinates sampled from their surfaces, plus some auxiliary information. The current methods for surface reconstruction are complex, time-consuming, and high requirement on computer configuration. As a very significant work, surface reconstruction has received considerable attention in the computer graphics community in recent years. Amenta et al [1] proposed the crust algorithm to reconstruct surface from unorganized sample points in 3D space. They also gave a simple combinatorial algorithm that computes a piecewiselinear approximation of a smooth surface from a finite set of sample points and proved it correct by showing that for densely sampled surfaces [2]. All the algorithms based on crust require the point clouds dense enough, the target object with a smooth surface, and the sample points without noise. They also require that the surface be smooth, no holes exists in the surface, and Received date: Foundation item: the Fundamental Research Funds for the Central Universities of SCUT (No. 2009ZM0235) and the National Natural Science Foundation of China (No ) rqyang@scut.edu.cn the sampling points be dense enough. Otherwise the reconstruction performance is undesirable [3]. Because the point clouds contain thousands or even millions of points in 3D space, the complexity of reconstruction algorithm is a very important index. The basic concept of the methods based on spatial subdivision is that the boundary hull (convex hull, box around points, etc.) of the point set is divided to independent areas. Typical example is divided by regular grid, Octree, or irregular tetrahedronization. Xie et al [4] organized the sample points by an Octree and fitted an implicit quadric surface in each Octree cell. Tobor et al [5] presented a method for the multi-scale reconstruction of implicit surfaces with attributes from large unorganized point sets which are arranged over the overlapping local subdivisions using a perfectly balanced binary tree. Furthermore, the moving least squares (MLS) methods [6] were used effectively to smooth, refine, and reconstruct surfaces from possibly noisy point clouds. Cheng et al [7] showed a Delaunay refinement algorithm for meshing a piecewise smooth complex in 3D which protects edges with weight points to avoid the difficulty posed by small angles between adjacent input elements, and then they introduced a novel modification of the algorithm to make it implementable in practice [8]. These methods are required to resample the obtained 3D points, so that they are time-consuming and only obtain the approximation of surface from the points. Some researchers constructed Delaunay triangulations

2 588 J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(5): from a given manifold triangle mesh. Bobenko et al[9] defined the intrinsic Delaunay triangulations (idts) of the vertex set of piecewise flat surfaces. An algorithm proposed by Dyer et al[10] swaps the physical mesh edges based on the locally Delaunay criterion. They proved the surface area of the mesh is reduced, when a physical edge that is not locally Delaunay (NLD) is swapped[11]. Fisher et al[12] used the intrinsic Laplace-Beltrami operator which is based on idts to construct triangulations together with an overlay structure which captures the relationship between the extrinsic and intrinsic triangulations. Afterward, Dyer et al[10] proposed a geometry-preserving algorithm for producing a Delaunay remeshing of a given manifold triangle mesh and proved that it is guaranteed to terminate. In this study, a novel method is proposed to reconstruct surface, which effectively uses the information of the linked structured light stripes. Then all stripes produced by the same structured light are linked to form a stripe ensemble. A stripe ensemble is transformed to the 3D space to form a point ensemble, which retains the same topological structure. An initial triangular mesh is generated by triangulating each two adjacent point ensembles with local optimization. Then an improved Dyer s method[12] is employed to globally optimize the initial mesh to make the reconstructed surface desirable. pixel locations and link them into lines are described in our previous work[13]. Figure 1(a) shows an original image captured by structured light projection. The individual subpixel locations are extracted and linked into lines, which is exhibited in Fig. 1(b). Figure 1(b) shows all subpixel points are linked one by one in a stripe and form a line on the captured image. Let L = {lm m = 1, 2,, M } represents the lines constructed on all captured images. All line points are arranged in the specified order. That is to say, a subpixel point qj and qj+1 in a certain line lm = {qj j = 1, 2,, J} are the adjacent points on an image. The subpixel points on a stripe are linked into a line, which exhibits in Fig. 2(b) as well. qj 1, qj, and qj+1 which are corresponding to the points pj 1, pj, and pj+1 on the real object surface, respectively, are the adjacent subpixel points in a linked stripe. Because of some factors, such as discontinuity of surface and shading, several stripes in some areas are produced by a structured light projecting to the target object such that holes exist in the corresponding areas in the reconstructed surface (e.g., Ref. [3]). In Fig. 2, a structured light projects to the target object to produce two stripes. The stripes produced by the same structured light assign an identical identification code. In order to facilitate generating triangular mesh, the stripes with the same identification code should be linked correctly. 2 Linking Stripes with the Same Identification Code The structured lights projecting to the target object can obtain an image which is shown in Fig. 1(a). Each structured light stripe can be considered as a curvilinear structure. A curvilinear detector which combines the zero-crossing detection algorithm to Steger s detector is employed to detect the subpixel locations of the light stripes. The details of how to obtain the sub- Fig. 2 Fig. 1 Performance of subpixel detection and linking algorithm The relationship between the linked points in 2D and 3D space All subpixel points on a stripe have a unique identification code and are obtained from the same structured light. These subpixel points on a stripe lm are linked into a line and organized in order. So lm has an identification code which can be obtained in the correspondence matching of gray code and line-shifting techniques[14]. As to two stripes lm and ln, they result from the same structured light projection if they have the same identification code. Then the stripes with the same identification code are gathered in a stripe ensemble se = {lk k = 1, 2,, K}. Then all stripes in

3 J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(5): s e should be linked by one line. The criterion of linking the stripes in s e is to make the linked results smooth. That is to say, the linked result has a minimum angle between the line linked to the endpoints in two adjacent stripes and the direction vector of the corresponding endpoints. Moreover, the centroids of all stripes in s e are computed to determine the linking sequence. Then s e can be linked to be a line. As to stripe a, the mean value of all the points is computed as the centroid c a. Assume that r points from the endpoint a 1, as shown in Fig. 3, are considered in determining the direction of a 1, there are r 1 the direction vectors obtained, in which the tth vector v at = (a t a t+1 )/ a t a t+1. Each vector v at is put a weight λ t in order to better represent the direction the endpoints of a. Then the direction of a 1 is computed by ( r 1 ) / r 1 n a1 = λ t v at λ t v at. (1) t=1 t=1 Likewise, another endpoint a u is computed as well. All direction vectors of the stripes endpoints in s e can be obtained by this way. Fig. 3 a 3 a 2 a 1 n a1 c a a a u 2 a u 1 a u 1 n b1 n au c b 2 b 1 b 2 b n bv b v b v 1 b v 2 Demonstration of linking stripes In Fig. 3, another stripe b which has a shorter distance from c a to its centroid c b than others. a and b with four endpoints should be linked into a line. Therefore, four candidate choices are provided, one of which is shown in Fig. 3. The endpoint a u in a is linked to b 1 in b. Two angles ω 1 and ω 2 are formed between the linked line and the direction vectors of the corresponding endpoints respectively. The sum value ω 1 +ω 2 characterizes the level of smoothness. The less the sum value is, the smoother the linked line is. Meanwhile, the other angles are also obtained. The choice with less sum value is considered as the optimal one. In Fig. 3, linking a u and b 1 is the best choice. Then all subpixel points in b are combined into a to form a new a, in which b 1 is the neighbor of a u. The centroid of a is replaced by c a = (uc a + vc b )/(u + v), (2) where u and v are the total number of subpixel points in a and b respectively before combination. The direction vectors n a1 and n bv are also those in the new a. b is removed from s e and the K stripes in s e become K 1 ones. This linking process is repeated until K = 1. Finally, all stripes in s e are linked into one line and all subpixel points in s e are organized in order. By handling all captured images, all the stripe ensembles S E = {s eh h = 1, 2,, H} are obtained. The pseudocode of stripes linking procedure is provided as follows. Input: s e = {l k k = 1, 2,, K} Output: s e = {l k k = 1} (1) Compute the centroids and direction vectors of endpoints for each stripe l k, where k = 1, 2,, K. (2) Choose a stripe a arbitrarily and compute the distances from its centroid to others. Another stripe b with less distance is specified. (3) Choose the optimal one in four of the linking solutions and combine a with b. (4) The centroid of the new a is computed by Eq. (2) and direction vectors of endpoints are still those of the retained endpoints. (5) K K 1. (6) Return, if K = 1; otherwise, go to (2). 3 Mesh Generation 3.1 Edge Flipping Algorithm Dyer et al [12] proposed a geometry-preserving algorithm is for producing a Delaunay remeshing of a given manifold triangle mesh and proved that the algorithm was guaranteed to terminate. The edge flipping process is applied in Delaunay triangulation for a given mesh. As shown in Fig. 4(a), two adjacent triangles T 1 = [p q f] and T 2 = [p q w] share the same edge e which links two vertices p and q. Let the Euclidean line segment e = [f w]. e is the opposing edge to e. The edges of T 1 and T 2 together with e form a flip tetrahedron. Performing an edge flip on e involves replacing e with the new edge e and faces T 1 and T 2 with faces T 1 = [p f w] and T 2 = [q f w]. p w e f e' (a) Fig. 4 q p e (b) Illustration of flipping an edge For remeshing a triangle T 1, an adjacent triangle T 2 can be found if the edge e is not a boundary one. If e is already in the mesh, flipping e would result in a non-manifold edge. Otherwise, the edge e is flippable and flips to e if the sum of the two angles opposite to an edge e is greater than that for the opposing edge e (i. e., pfq+ pwq > fpv+ fqw). Dyer et al proved e' f w q

4 590 J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(5): that the combined area of the two triangles adjacent to e is less than that of the two triangles adjacent to e if the edge flip happens. Hence, the mesh surface area is reduced after each Delaunay flip. However, if the point clouds to construct the initial mesh are undersampled, as shown in Fig. 4(b), the edge flipping may be inappropriate and the change in geometry may be too large. Dyer et al added some new vertices to produce a geometry-preserving Delaunay mesh. The refinement algorithm inserts a new vertex along an original edge of the input mesh and connects the newly inserted vertex with two vertices opposite to the current mesh edge being split such that each added vertex is on the mesh. Actually, the given mesh is not the optimal representation of the real surface. So these added vertices are not on the real surface. The new mesh after edge flipping cannot perfectly represent the real surface. If the point clouds are not dense enough and the initial mesh is not optimal, the geometry should be changed to make the representation of the surface better. So the geometry-preserving is not necessary and the Delaunay triangulation is not the best way to represent the real surface. In Fig. 4(b), the triangle T 2 = [q f w] is too sharp. So a constraint is added to the edge flipping algorithm exploited by Ref. [12]. As shown in Fig. 4(a), let θ pq be the maximum of the opposing angles fpw and fqw, and θ fw the maximum of the opposing angles pf q and pwq. Assume that the edge e can be flipped. If θ pq θ fw > γ where γ is a constant, the edge e cannot flip to the opposing one e. In contrast, the edge e must flip if θ fw θ pq > γ. Otherwise, if θ fw θ pq γ, the aforementioned Delaunay edge flipping algorithm is employed. In the implementation, γ = π/6 is used. The performance is shown in Fig. 5. The six points are sampled by a cylinder surface. The Delaunay triangles are obtained using Dyer s edge flipping algorithm in Fig. 5(a). This mesh not only contains some sharp triangles, but exits a sharp edge on the reconstructed surface. The constraint is employed in edge flip and then the result is produced in Fig. 5(b). Clearly, it is an optimal representation of the cylinder surface. 3.2 Constructing Initial Triangular Mesh with Local Optimization The gray code and line-shifting techniques [14] are employed to obtain the 3D points. After the linked stripes on the captured images are obtained, all the subpixel points can be transformed to 3D space to describe the object surface [15]. Therefore, the 3D point ensembles PE = {pe h h = 1, 2,, H} are computed from the stripe ensembles SE = {se h h = 1, 2,, H}. Figure 2 shows that each point in 3D space has a corresponding coordinate in 2D image. If there are a point q j and its neighbor point q j+1 in se h, the corresponding points p j and p j+1 are adjacent with each other in pe h. So an edge is constructed by linking p j and p j+1 in the process of constructing the initial triangular mesh. All the points in pe h remain the same adjacent relationship with the corresponding points in se h and thus can be linked one by one to form a curve in 3D space. The two adjacent structured lights projecting to the target object can produce two 3D point ensembles pe h = {p i i = 1, 2,, I} and pe h+1 = {s j j = 1, 2,, J}. pe h and pe h+1 are adjacent on the object surface, which is demonstrated in Fig. 2. Triangulating the points of pe h and pe h+1 by the specified rules, we can obtain a triangle set ts which can cover the space between pe h and pe h+1 on the object surface. Linking the points one by one in pe h produces I 1 edges of the triangles in ts. Likewise, J 1 edges from pe h+1 are provided to the triangles in ts. In each triangle of ts, one of three edges is in these I +J 2 edges and the other two edges are obtained by linking the points from pe h to pe h+1 such that ts can cover and only cover the space between pe h and pe h+1. To construct the triangles ts, the edge flipping algorithm mentioned above is employed to perform the local optimization. It would speed up the following global optimization procedure. As shown in Fig. 6, start with the linked endpoints p 1 and s 1 which belong to pe h and pe h+1 respectively, the adjacent two points p 2 and s 2 are chosen. So a tetrahedron is constructed by these four points, from which the edge flipping algorithm is applied to obtain two triangles. The one with the vertices p 1 and s 1 is retained. In Fig. 6, assume the triangle with vertices p 1, s 1 and s 2 is kept and appended to ts. The edge which is chosen to construct the next triangle should satisfy two conditions: the two vertexes belong to two point ensembles pe h and pe h+1 p 2 pe h p 1 p t pt+1 p I pe h+1s1 s2 s 3 s J 1 s J Fig. 5 Comparative performance of edge flip Fig. 6 Illustration of meshing two adjacent lines

5 J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(5): respectively; ② it is an edge of the current constructed triangle. Therefore, the edge p1 s2 is chosen to construct the next triangle. This procedure continues until the current constructed triangle contains an endpoint in peh or peh+1. In Fig. 6, sj is an endpoint of peh+1 and the edge pt sj is chosen to construct the next triangle. However, only one candidate point Fig. 7 pt+1 can be used and hence is chosen to construct the triangle. Likewise, all the retained points in peh and peh+1 are triangulated. Hereto, there are I + J 2 triangles are obtained and assembled in ts = {Tg g = 1, 2,, I + J 2}. In this way, all the point ensembles PE are triangulated to provide an initial mesh TS = {tsh h = 1, 2,, H 1}, which is shown in Fig. 7(b). The procedure of mesh generation In constructing one initial triangle with the local optimization, only one of three interior angles is considered, which does not guarantee that the constructed triangle is better than other choices. When another endpoint of a point ensemble is met, there is only one choice to construct the next triangle which sometimes is also not optimal. Furthermore, the reconstructing process does not employ the information of other point ensembles. Each triangle produced by the two adjacent point ensembles is the neighbor of another one which does not belong to the same ts. Even if the two adjacent triangles are all optimal, it is possible to form a sharp fringe between the two adjacent triangles for the mesh is generated in 3D space, which is undesirable in surface reconstruction. The initial triangular mesh in Fig. 7(b) is not optimal. Consequently, the generated mesh is not optimal to represent the object surface and thus is called initial triangular mesh which can be further optimized in a global view. However, this way to generate the initial mesh uses the information of the linked and organized stripe ensembles such that only two candidate points need to consider in reconstructing a triangle, which makes triangulation very fast. In addition, the generated mesh is also optimized to some extent, and makes the most triangles satisfying the global optimization criterion and thus costs less time to optimize the mesh Global Optimization In initial triangle generation, only two adjacent triangles are considered to produce a triangle. The relationships between a triangle and all its adjacent ones in TS are not paid attention comprehensively. In order to obtain a better mesh, the edge flipping algorithm is used to optimize the initial mesh in a global view. Each triangle in TS is tested with the edge flipping algorithm. All triangles in TS are arranged in a priority queue TQ according to the harmonic index which is explained in Ref. [9]. The harmonic index of a triangle can be expressed as the sum of the cotangents of its interior angles. hrm(t) = 4(cot α + cot β + cot η), (3) where α, β and η are the interior angles of the triangle. Start from the first triangle in TQ, one or more triangles adjacent to the chosen triangle can be found. Then this triangle and one of its adjacent ones compose a tetrahedron. The edge flipping algorithm is introduced to judge whether the shared edge flips or not. If it happens, the two new triangles are generated to replace the current triangle and its adjacent one. Moreover, these two new triangles and their adjacent ones are labeled to test again in the next turn. After the tetrahedrons composed by a triangle and its adjacent ones are tested and no edges flip, this triangle is optimal and labeled to no longer test in the next turn. In the course of global optimization, the harmonic index is employed to estimate the mesh s quality. The harmonic index of the mesh T is the mean value of the

6 592 J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(5): harmonic indices of all triangles and hence expressed as hrm(t ) = 1 hrm(t), (4) N t T p 1 n i p 2 n' i n i b i 1 where N is the number of total triangles. In each turn of edge flip, hrm(t ) decreases. If hrm(t ) do not decrease any more, the global optimization terminates. In addition, if all triangles are labeled to no longer test in the next turn, this global optimization also terminates. Therefore, it is guaranteed to terminate. 3.4 Boundary Determination In the point clouds acquisition using the gray code and line-shifting techniques, the boundary, as shown in Figs. 7(b) 7(c), is usually not smooth because of some factors such as uneven object surface. Therefore, smoothing the boundary can make the reconstructed surface better. The linking procedure links the broken segments produced by individual structured light beams, which can fill holes induced by the uneven surface and shading on the reconstructed surface. However, the broken segments are not necessary to link if they are on the boundary in the single view. Furthermore, the endpoints of each curve are not trim with its neighbors. Consequently, the procedure of boundary determination is divided into two steps: shrink and smoothness. If an edge on the boundary, this edge just belongs to a unique triangle. According to this property, the boundary edges are obtained. Let the vertexes related to these boundary edges be B = {b i i = 1, 2,, M}, where M represents the number of boundary points. In the shrinking process, if the length of a boundary edge which links two boundary vertex b i and b i+1 is greater than a threshold λ bound, this edge is pruned and the other vertex p of the triangle which b i and b i+1 belong to are regarded as boundary vertex and insert into b i and b i+1. So the boundary vertexes become B = {b 1, b 2,, b i, p, b i+1,, b M } and hence M + 1 vertices are on the boundary. The boundary shrinks by repeating this process. Thus Fig. 7(c) becomes Fig. 7(d). After the boundary shrinks, the updated boundary vertices are represented with B = {b i i = 1, 2,, N}. As to a boundary vertexes b i, the triangles attached to b i and their normal vectors can be obtained. The mean value of the normal vectors of all these triangles is regarded as the normal vector n i of b i. There are two possibility of boundary needed to smooth, which is shown in Fig. 8. The vertices b i 1, b i, and b i+1 are obtained in B and then the angle b i 1 b i b i+1 is computed. If b i 1 b i b i+1 < ψ where ψ is constant, this boundary vertex b i should be handled. Then The vector n i obtained by computing the cross product of the vectors b i b i+1 and b i b i 1. (i.e., n i = b i b i+1 b i b i 1 ). If b i 1 Fig. 8 b i b i+1 b i+1 (a) p 3 n' i b i (b) Boundary smoothness (a) and two possibility of boundary (b) n i b i n i > π/2, which is shown in Fig. 8(a), the triangle T with the vertices b i 1, b i, and b i+1 are regarded as a triangle on the surface and added to the mesh. Then b i is removed from B and the edge b i 1 b i+1 is a boundary one. Otherwise, b i and its connecting triangles are removed in Fig. 8(b). p 1 becomes a boundary point and the edges b i 1 p 1 and p 1 b i+1 are on the boundary. Fig. 7(e) shows the smoothness result of Fig. 7(d). However, the added triangles, as shown in Fig. 7(e), are not optimized. So these added triangles are optimized by the edge flipping algorithm, which is shown in Fig. 7(f). 4 Experimental Results We obtained 271 stripe ensembles by our own designed equipment. All the stripe ensembles SE are mapped to 3D space to obtain 271 point ensembles containing points which is exhibited in Fig. 9(a). All these points are used to produce the initial mesh with triangles. The edge flipping algorithm is used in generating the initial mesh in Fig. 9(b). Then the initial mesh is optimized globally in Fig. 9(d). The global optimization terminated after 68 times iteration and the result is shown in Fig. 9(d). The changes of the total tested triangles, the total edge flipped triangles, and the harmonic indexes are shown in Fig. 10. The optimizing procedure converges very fast. The performance of optimization is shown by comparing Figs. 9(c) and 9(e). The triangles with the red edge are optimized with better representation. By observing Figs. 9(c), 9(e), and Fig. 10(b), most of triangles do not change, which indicates the local optimization in generating the initial mesh is very effective. Set λ bound = 5 and ψ = π/2, the boundary of the mesh is smoothed and the final result is shown in Fig. 9(f). The changes of triangles and their processing times are listed in Table 1. The edge flipping algorithm introduced by Dyer et al [12] is employed to optimize the initial mesh and the result is exhibited in Fig. 11(a). Figure 11(b) is a part of mesh on the bottom of nose in which some sharp triangles exist. From the render view in Fig. 11(c), p 1

7 J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(5): Iteration times (a) The total tested triangles versus iteration times Fig. 10 Fig. 11 The process of mesh generation 8.4 Harmonic index 106 Edge filped triangles Tested triangles Fig Iteration times (b) The edge flipped triangles versus iteration time Iteration times (c) The harmonic index versus iteration times Characteristics of the parameters in the global optimization Comparison between our improved edge flipping algorithm and Dyer s one

8 594 J. Shanghai Jiaotong Univ. (Sci.), 2010, 15(5): Table 1 Output statistics for our surface reconstruction algorithm Procedures Total triangles Boundary points Time/s Initial meshing Global optimization Shrinking boundary Smoothing boundary Optimizing boundary the reconstructed surface is not smooth. Therefore, a constraint is added to the edge flipping algorithm introduced by Dyer et al. Figure 11(d) is the reconstructed result. Compared with Figs. 11(b) and 11(c), Figs. 11(e) and 11(f) are smoother and the reconstructed result are more perfect and desirable. The improved crust algorithm[2] and our method are running in a notebook computer (CPU: INTEL Celeron M 1.5 GHz, Memory: 768 MB). As shown in Fig. 12(a), it costs s to reconstruct the surface using the improved crust algorithm. Since the sampling density of point cloud does not satisfy the improved crust algorithm, there are some holes on the reconstructed surface. The boundary shown in Fig. 12(b) is not smooth. Meanwhile, our method just spends s to reconstruct the surface which is shown in Fig. 12(c) under the same platform. The holes are filled and the boundary is smooth. Hence, compared with the improved crust algorithm, our algorithm is faster and more effective. 5 Conclusion This study proposes a method for surface reconstruction according to the information of the linked structured light stripes. In the procedure of linking stripes, most of stripe ensembles only contain one stripe and need not to link. The two adjacent subpixel points on a structured light image are transformed to 3D space to obtain two points which are still adjacent topologically. Two adjacent subpixel points are on the two adjacent pixels on an image and the corresponding 3D points maybe far away from each other, which is shown in Figs. 2(b) and 2(c). These points can also be linked in 3D space since each point ensemble is on a structured light sheet, but it is time-consuming and the sampling density directly determines the linking performance. Linking the stripes on the captured images facilitates to generate the initial mesh and further can fill the holes on the reconstructed surface. If expecting to keep some of holes, the triangles with an edge which links two adjacent stripes, as shown in the red line in Fig. 9(c), can be removed after generating the initial mesh. In the initial mesh generation, only two candidate points need to be determined. Thus, the generation procedure is performed very fast. Most of time is spent on globally optimizing the mesh, but the global optimization algorithm terminates very fast. Moreover, just a few edges flip in the globally optimization since the edge flip criterion is also employed in generating the initial mesh. In some applications, the initial mesh can perfectly represent the surface and satisfy the user s requirement such that the global optimization can be avoided and most of time is saved. Although many factors, such as shading, the topological structures of points, affect the speed of reconstruction, the experiments show our algorithm can reconstruct surface fast and effectively. In conclusion, the proposed method for surface reconstruction consists of four steps: linking stripes; initial mesh generation with local optimization; global optimization and boundary processing. The methods for linking stripes and mesh generation using the linking information can be extented to the other surface measurement system based on the structured lights. The low requirement of the computing configuration and low cost of computing time show its potential benefit in the industry applications. Although the perfect mesh is constructed, this proposed method just applied in triangulating the point clouds which are obtained from structured light projection in a single view. References Fig. 12 Comparison between the improved crust algorithm and our method [1] Amenta N, Bern M, Kamvysselis M. A new Voronoi-based surface reconstruction algorithm [C] // International Conference on Computer Graphics and

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