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3D Reconstruction of Curved Objects

from Single 2D Line Drawings

CVPR'09

Reporter: PoHan 05/04/2010

Outline

Assumptions

Reconstruction of Curved Manifolds

Experimental Results

Introduction

Conclusions

Introduction

propose an approach to the 3D reconstruction

from line drawings of solids with not only planar but also curved faces

Introduction

a line drawing LDa is transformed into another line drawing LDb

with only straight lines.

Assumptions

(a) A line drawing, represented by a single edge-vertex graph, is the parallel or near-parallel projection of a wireframe manifold object in a generic view where all the vertices and edges of the object are visible.

Manifold: a solid ，在 2D Euclidean space 下，其表面可以被攤 開成一片連續的 2D 平面 property :each edge is shared exactly by two faces

(b) Every curved edge of a line drawing is the projection of a 3D planar curve.

(c) All the faces of a manifold that a line drawing represents are available.

Reconstruction of Curved Manifolds

Distinguishing between curved & planar faces

Transformation of Line Drawings

Regularities

3D Wireframe Reconstruction

Generating Curved Faces

The Complete 3D Reconstruction Algorithm

Reconstruction of Curved Manifolds

Distinguishing between curved & planar faces

Reconstruction of Curved Manifolds

Property 1 Two faces that share a straight edge can be either planar or curved.

Property 2 At least one of the two faces that share a curved edge is curved.

Property 3 Both faces that share a silhouette are curved.

Property 4 Two or more co-surface faces indicated by artificial lines are all planar or all curved faces.

Reconstruction of Curved Manifolds

In some cases, multiple solutions occur

Reconstruction of Curved Manifolds

Transformation of Line Drawings

Reconstruction of Curved Manifolds

Singular point. the points having the maximal distance to the line passing through the two endpoints of the curve.

Reconstruction of Curved Manifolds

Regularities

Reconstruction of Curved Manifolds

based on the transformed line drawing and the original line drawing, to recover the 3D wireframe of the curved object

Reconstruction of Curved Manifolds

Curve Parallelism.

G(s) :the normalized arc-length parametrization of the curve C(t)

Reconstruction of Curved Manifolds

Generalized Face Perpendicularity. [19]

[19] H. Lipson andM. Shpitalni. Optimization-based reconstructionof a 3d object from a single freehand line drawing. Computer-

Aided Design, 28(8):651–663, 1996.

(a, e, f, g, c, b, a) perpendicular to(a, d, c, b, a) and (e, h, g, f, e).

K is the number of the combinations

Reconstruction of Curved Manifolds

Curve Concurvity. [19] e1 and e2 are curved,

p12, p21, and v are collinear

e1 is curved and e2 is straight, p12 and v and v2 are collinear

N is the number of vertices of the line drawing

ε(i) is the set of all the edges ending at vertex i

When Pi, Pj , and Pk arenearly collinear, w close to 1;

Reconstruction of Curved Manifolds

3D Wireframe Reconstruction

Reconstruction of Curved Manifolds

(1)minimizing the standard deviation of the angles (MSDA) in the reconstructed object

[25] T. Marill. Emulating the human interpretation of line-drawings as three-dimensional objects. IJCV, 6(2):147–161, 1991.

(2)face planarity [15] Y. Leclerc and M. Fischler. An optimization-based approach to the interpretation of

single line drawings as 3D wire frames. IJCV, 9(2):113–136, 1992.

(3)line parallelism [19]

(4)corner orthogonality [19]

three new regularities:(5)(6)(7) (8)regularity isometry [19]

Reconstruction of Curved Manifolds

Not only the depths but also the 3D curves are required to compute all the regularity terms

3D curve C(t) = (x(t), y(t), z(t))T

parallel projection↓ C’(t) = (x(t), y(t))T

Reconstruction of Curved Manifolds

the 3D curve is planar (assumptions(b))

unit normal vector n = (nx, ny, nz)T

endpoint P0 = (x0, y0, z0)T

z1−N are the depths of all the N vertices of the line drawing,

n1−M are the unit normal vectors of the M planes on which each of the M curved edges is the intersection of two curved faces

兩曲面交集的 m 個曲邊 , 這 m 個曲邊所在的 m 個平面的 unit normal vectors

hill-climbing method presented in [15] to minimize F

Reconstruction of Curved Manifolds

Generating Curved Faces

A Bezier patch is generated for a curved face with three or four edges

a triangle mesh is used to create a curved face with more than four edges

Reconstruction of Curved Manifolds

Bezier and Coons patches [12] [12] A. Davies and P. Samuels. An introduction to computational

geometry for curves and surfaces. New York: Oxford University Press Inc., 1996.

Reconstruction of Curved Manifolds

triangle mesh

N(i) is the set of mesh points connected to the ith point in the mesh

S is the set of mesh points located on the 3D wireframe

1.the first term enforces the smoothness on the mesh2.the second term is used to maintain the continuity of the curvature in the mesh3.the last term is the fitting constraint that requires the mesh to fit the points on the wireframe well

[34] G. Taubin. A signal processing approach to fair surface design.Proc. SIGGRAPH, 7:351–358, 1995.

Reconstruction of Curved Manifolds

The Complete 3D Reconstruction Algorithm

Experimental Results

Conclusions

proposed a novel approach to 3D curved manifold object reconstruction from single 2D line drawings

In contrast, our approach can reconstruct complex curved objects automatically.

Our future work includes fine-tuning the results and developing more regularities for curved object reconstruction

Thanks you !