Computer Aided Drafting, Design and Manufacturing Volume 26, Number 2, June 2016, Page 13

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1 Coputer Aided Drafting, Design and Manufacturing Volue 26, uber 2, June 2016, Page 13 CADDM 3D reconstruction of coplex curved objects fro line drawings Sun Yanling, Dong Lijun Institute of Mechanical Engineering, Taiyuan University of Technology, Taiyuan , China. Abstract: An active research topic in coputer vision and graphics is developing algoriths that can reconstruct the 3D surface of curved objects fro line drawings. There are a nuber of algoriths have been dedicated to solve this proble, but they can t solve this proble when the geoetric structure of a curved object becoes coplex. This paper proposes a novel approach to reconstructing a coplex curved 3D object fro single 2D line drawings. Our approach has three steps: (1) decoposing a coplex line drawing into several sipler line drawings and transforing the into polyhedron; (2) reconstructing the 3D wirefrae of curved object fro these sipler line drawings and generating the curved faces; (3) cobining the 3D objects into the coplete objects. A nuber of exaples are given to deonstrate the ability of our approach to successfully perfor reconstruction of curved objects which are ore coplex than previous ethods. Key words: 3D reconstruction; line drawings; coplex curved objects 1 Introduction Line labeling [1-2] is one of the earliest ethod which is used to reconstruct 3D object fro single line drawing, it focuses on seeking a series of labels fro a line drawing without hidden lines. A nuber of ethods that discusses line labeling have been proposed. Methods of 3D reconstruction fro wirefrae odels try to reconstruct a 3D object fro its ultiple orthographic projections [3], there are ore inforation in three orthographic views for the reconstruction of objects than in a single projected view. There have been any papers for reconstructing polyhedral objects fro single-view line drawings [4-6]. However, the ethods they use are all for reconstruction of line drawings of planar objects. Although the research of 3D reconstruction of curved objects is very few, soe works [7-9] do try to cope with this proble. In reference [7] and [8],their works concentrate on architectural odeling where the lines are ostly straight, and they can only deal with siple curved objects. Without the help of user, both can not deterine caera paraeters and for a odel. In reference [9],the work propose a progressive ethod for reconstructing objects with siple curved faces. It's liitation is that it needs the pronounced angular trends of the edges of objects. Copared with these approaches, our ethod can autoatically reconstruct coplex curved solid objects. 2 Assuptions Line drawings is in a generic view where all the edges and vertices of the object are visible, and it is the parallel projection of a wirefrae anifold. All the faces and internal faces of the anifold that a line drawing represents are available. A curved edge in a line drawing is the projection of a 3D planar curve. 3 Decoposition of the coplex line drawing and transforation This section discusses how to decopose a coplex line drawing along the internal faces and transfor the into polyhedrons. The algorith is as follows: Algorith 1. Partition along an internal face. Let f be an internal face, F(f )={f i } i=1 be the set of all the real faces connected to f, and E(f )={e i } i=1 be the set of all the n edges connected to f. We can verify that all separations satisfy conditions as follows:1 Each face connected to f is contained either by F 0 (f ) or F 1 (f ). Two faces that share an edge connected to f always appear in the sae face set F 0 (f ) or F 1 (f ). Above all, we set F 0 (f ), F 1 (f ), Corresponding author: Dong Lijun, Feale, Ph.D, Professor, E-ail: dlj_lq@163.co.

14 Coputer Aided Drafting, Design and Manufacturing (CADDM), Vol.26, o.2, Jun.

2 If one real face in F 1 (f ) and another real face in F 0 (f ) share an edge connected to f,we reove the face in F 1 (f ), put it into F 0 (f ), and repeat the above steps until all the faces in F

Fro Fig.1, we give an exaple of the separation of the line drawing. (a) Fig.

2 14 Coputer Aided Drafting, Design and Manufacturing (CADDM), Vol.26, o.2, Jun E 0 (f ) and E 1 (f ) to be epty sets, and put all the real faces connected to f into F 1 (f ), then reove any one face fro F 1 (f ) and put it into F 0 (f ). 2 If one real face in F 1 (f ) and another real face in F 0 (f ) share an edge connected to f,we reove the face in F 1 (f ), put it into F 0 (f ), and repeat the above steps until all the faces in F 1 (f ) be tested. 3 We put all the edges that are connected to f and in the faces in F 0 (f ) into E 0 (f ), and put all the edges that are connected to f and in the faces in F 1 (f ) into E 1 (f ). Fro Fig.1, we give an exaple of the separation of the line drawing. (a) Fig. 1 (a) f 1, f 2 and f 3 are three internal faces which we used to separate a line drawing; (b) Decoposition along f 1, f 2 and f 3 in (a). (b) There are two propositions [10] show that the decoposition along an internal face is unique, and after the decoposition, the generating line drawing still represent a anifold. Proposition 1. The separation along an internal face of a line drawing representing a anifold exists and is unique. Proposition 2. After the decoposition along an internal face, the line drawing still represents a anifold. Algorith 1 actually achieved the purpose of the partition along an internal face f. The algorith is run repeatedly until all the internal faces have been separated. An iportant step is to transfor a line drawing with curved edges into a polyhedron. When there is only one curve between two vertices, the curve is transfored into a straight line that passes through the two vertices. When there is ore than one curved edge between two vertices, in this case, we first find a singular point in a curve and then replace the curve with two straight lines (Fig.2). Fig. 2 Transforing a line drawing into a polyhedron. 4 3D reconstruction and generating the curved faces As we assue a line drawing to be a parallel or near parallel projection of a 3D object. The x-coordinate and y-coordinate of each vertex have already been derived, and only the z-coordinate has to be obtained. Let C(k) be a 3D curve described in the paraeterized for:

3 Sun Yanling et al., 3D Reconstruction of Coplex Curved Objects fro Line Drawings 15 C(k)=(x(k), y(k), z(k)) T the 2D parallel projection of C(k) is siply C (k) = (x(k), y(k)) T. 2D curve C (k)=(x(k), y(k)) T and the coordinate of endpoint P 0 =(x 0, y 0, z 0 ) T in 3D curve C(k) are given, As long as we find the unit noral vector n=(n x, n y, n z ) T of the plane the 3D curve C(k) lies on, we can get the z-coordinates of the curve by: z(k)=z 0 (n x (x(k) x 0 )+n y (y(k) y 0 ))/n z We use an optiization-based approach to reconstruct the 3D wirefrae of the curved object. There are eight regularities are applied to the reconstruction: iniizing the standard deviation of angles in the reconstructed object (α 1 ), face planarity (α 2 ), corner orthogonality (α 3 ), line parallelis (α 4 ), face perpendicularity (α 5 ), curve concavity (α 6 ), curve parallelis (α 7 ) and isoetry (α 8 ). The objective function to be optiized is defined as: n 1 2 M i i i=1 f( z, z,, z, n, n,, n )= where λ 1 8 are weighting factors, z 1 are the depths of all the vertices of a line drawing, and n 1 M are the unit noral vectors of the M planes. We iniize f using the ethod presented in reference [11]. After reconstructing the 3D wirefrae fro sipler line drawings, we use Bezier and Coons patches [12] to generate curved faces with three or four edges. Let P(a, b), 0 a, b 1, be a curved patch, P(0, b), P(1, b), P(a, 0), and P(a, 1) are its four curving boundary. We can obtain the approxiation of P(a, b) by: P ( a, b)=(1 a) P( a, b) ap(1, b) (1 b) P( a,0) bp( a,1) (1 a) bp(0, 1) a(1 b) P(1, 0) (1 a)(1 b) P(0, 0) abp(1,1) where P (a, b) is a Coons patch, which pass through four boundary curves and four corners of P(a, b) exactly once. After obtaining the Coons patch, we use a sae ethod to get the control points D i,j of the Bezier n patch P(a, b)= Di, jbi, n( a) Bj, ( b) by: i 0 j 0 i i j j D = 1 D D 1 D D n n i j i j 1 1 D0,0 1 D0, n n i j i j 1 Dn,0 Dn, n n i, j 0, j n, j i,0 i, where D 0,j, D n,j, D i, 0, and D i,, 0 i n, 0 j, are four known control points of the boundaries. 5 Cobining 3D objects We cobine all the 3D objects along the internal faces of the original line drawing. Let f be an internal face shared by two 3D object O x and O y, and z x1, z x2,, z x and z y1, z y2,, z y are the depths of vertices on the internal face f in O x and O y, respectively. μ i which eans the depth and standard deviations σ i of f are defined as: 1 i = zij, i=x, y j=1 1 =, i=x, y 2 z i ij i j=1 We odify the depths of all the vertices of O x by: y z xi = y zxi x, i =1,2,, Finally, we cobine O x and O y by: x zxi zbi z xi = z yi =, i=1, 2,, 2 where zx 1, z x2,, z x and z y1, z y2,, z y are odifiedvertex depths of f in O x and O y, respectively. We erge O x and O y by forcing the to be sae. 6 Conclusion In this paper, we have proposed a novel ethod to reconstructing a coplex curved 3D object fro single 2D line drawings. Our strategy is to : 1 divide a coplex line drawing into sipler ones along the internal faces and transfor the into the line drawings of polyhedrons; 2 reconstruct the 3D wirefrae fro these sipler line drawings and generate the curved faces with Bezier patches and triangular eshes; 3 erge the 3D objects into a coplete object. A nuber of exaples indicate the success of our approach (Fig.3). Fig.3 displays soe of results. we give each input line drawing separation result and their transfored line drawings and the final 3D reconstruction result displayed in two views. We use different colors to distinguish different faces. Fro Fig.3, we can find that our approach successfully separates the line drawings and handle 3D reconstruction of coplex curved object. It is worth entioning that soe of the results in Fig.3 are ore coplex than the

16 Coputer Aided Drafting, Design and Manufacturing (CADDM), Vol.26, o.2, Jun.

depending on the coplexity of the. For those in Fig.3, it ranges fro 0.

The previous approaches of 3D reconstruction of curved objects fro single

objects. In contrast, our ethod can handle coplex curved objects autoatically.

ore regularities for curved object reconstruction. Fig.

and the final 3D object by reconstruction.

A review on interpretation of line drawings [J].

The Interpretation of line drawings with contrast failure and shadows [J].

without hidden lines Pattern Analysis and Machine Intelligence, 2008, 30(2):

A divide-and-conquer approach to 3D object reconstruction fro line drawings

3D reconstruction of polyhedral objects fro single parallel projections using

ACM Transactions on Graphics, 2008, 27(2): 1-15. [8] Lee S, Feng D, Gooch B.

Syposiu on Interactive 3D Graphics, Redwood City.

4 16 Coputer Aided Drafting, Design and Manufacturing (CADDM), Vol.26, o.2, Jun objects given in the previous papers for 3D reconstruction of curved objects fro single line drawings. The coputational tie of our algorith varies with different line drawings, depending on the coplexity of the. For those in Fig.3, it ranges fro 0.8 s to 78 s. Aong the three steps, Step 3 take alost all the coputational tie. The previous approaches of 3D reconstruction of curved objects fro single line drawings [7-9] need huan interaction and can only perfor siple 3D curved objects. In contrast, our ethod can handle coplex curved objects autoatically. Our future work includes finding a better fine-tuning schee and developing ore regularities for curved object reconstruction. Fig. 3 Several line drawings and their separation, their transfored line drawings and the final 3D object by reconstruction. References [1] [2] [3] [4] [5] [6] Gao M T, Chu J, Dong L J. A review on interpretation of line drawings [J]. Journal of Engineering Graphics, 2006, 27(5): Cooper M. The Interpretation of line drawings with contrast failure and shadows [J]. International Journal of Coputer Vision, 2001, 43(2): Liu J, Cao L, Li Z, et al. Plane-based optiization for 3d object reconstruction fro single line drawings [J]. IEEE Trans. Pattern Analysis and Machine Intelligence, 2008, 30(2): Cao L, Liu J, Tang X. What the back of the object looks like: 3D reconstruction fro line drawings without hidden lines [J]. IEEE Trans. Pattern Analysis and Machine Intelligence, 2008, 30(2): Chen Y, Liu J, Tang X. A divide-and-conquer approach to 3D object reconstruction fro line drawings [J]. Coputer Vision, 2007, 14(21): 1-8. Lee Y T, Fang F. 3D reconstruction of polyhedral objects fro single parallel projections using cubic corner [J]. Coputer Aided Design, 2011, 43(8): [7] Chen X, Kang S, Xu Y, et al. Sketching reality: Realistic interpretation of architectural designs [J]. ACM Transactions on Graphics, 2008, 27(2): [8] Lee S, Feng D, Gooch B. Autoatic construction of 3D odels fro architectural line drawings [C]//. Syposiu on Interactive 3D Graphics, Redwood City. ew York: ACM Press, 2008: [9] Masry M, Kang D, Lipson H. A freehand sketching interface for progressive construction of 3d objects [J]. Coputers and Graphics, 2005, 29(4): [10] Chen Y, Liu J. Reconstructing 3D anifold object fro line drawings based on a divide-and-conquer approach [D]. Hong Kong: The Chinese University of Hong Kong, [11] Chen Y, Liu J Z. A divide-and-conquer approach to 3D object reconstruction fro line drawings [J]. Coputer Vision, 2007, 14(21): 1-8. [12] Wang Y Z, Chen Y, Liu J Z, et al. 3D reconstruction of curved objects fro single 2d line drawings [C]//

His research fields are engineering graphics and coputer-aided design. Dong Lijun is currently a professor of College of Mechanical Engineering, University.

5 Sun Yanling et al., 3D Reconstruction of Coplex Curved Objects fro Line Drawings 17 Coputer Vision and Pattern Recognition, IEEE Conference on. ew York: IEEE Press, 2009: Sun Yanling currently a graduate student at Taiyuan University of Technology and ajoring in Mechanical Engineering. His research fields are engineering graphics and coputer-aided design. Dong Lijun is currently a professor of College of Mechanical Engineering, University. Taiyuan University of Technology. Her research interests include Coputer Graphics and Coputer Vision. She received the Ph.D ajoring in ME degrees fro orthwestern Polytechnical University in 2005.

TensorFlow and Keras-based Convolutional Neural Network in CAT Image Recognition Ang LI 1,*, Yi-xiang LI 2 and Xue-hui LI 3

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