Elsevier

Image and Vision Computing

Volume 22, Issue 10, 1 September 2004, Pages 829-836
Image and Vision Computing

Reconstruction of surfaces of revolution from single uncalibrated views

https://doi.org/10.1016/j.imavis.2004.02.003Get rights and content

Abstract

This paper addresses the problem of recovering the 3D shape of a surface of revolution from a single uncalibrated perspective view. The algorithm introduced here makes use of the invariant properties of a surface of revolution and its silhouette to locate the image of the revolution axis, and to calibrate the focal length of the camera. The image is then normalized and rectified such that the resulting silhouette exhibits bilateral symmetry. Such a rectification leads to a simpler differential analysis of the silhouette, and yields a simple equation for depth recovery. It is shown that under a general camera configuration, there will be a 2-parameter family of solutions for the reconstruction. The first parameter corresponds to an unknown scale, whereas the second one corresponds to an unknown attitude of the object. By identifying the image of a latitude circle, the ambiguity due to the unknown attitude can be resolved. Experimental results on real images are presented, which demonstrate the quality of the reconstruction.

Introduction

Two-dimensional images contain cues to surface shape and orientation. However, their interpretation is inherently ambiguous because depth information is lost during the image formation process when 3D structures in the world are projected onto 2D images. Multiple images from different viewpoints can be used to resolve these ambiguities, and this results in techniques like stereo vision and structure from motion. On the other hand, under certain appropriate assumptions, it is also possible to infer scene structure, like surface orientation and curvature, from a single image. In this paper, a simple technique for recovering the 3D shape of a surface of revolution (SOR) from a single view is introduced. The symmetry properties exhibited in the image of a SOR are exploited to calibrate the focal length of the camera, and to rectify the image so that the resulting silhouette exhibits bilateral symmetry. Surface normals along the contour generator are then determined from the rectified silhouette, and depth information can then be recovered using a coplanarity constraint between the surface normal and the revolution axis.

This paper is organized as follows. Section 2 briefly reviews existing techniques in the literature for shape from contour using single view. Section 3 gives the theoretical background necessary for the development of the algorithm presented in this paper. A parameterization for surfaces of revolution is presented and the symmetry properties exhibited in the silhouettes are summarized. In particular, the surface normal and the revolution axis are shown to be coplanar. This coplanarity constraint is exploited in Section 4 to derive a simple technique for reconstructing a SOR from its silhouette in a single view. It is shown that under a general camera configuration, there will be a 2-parameter family of solutions for the reconstruction. The first parameter corresponds to an unknown scale in the reconstruction resulting from the unknown distance of the surface from the camera. The second parameter corresponds to the ambiguity in the orientation of the revolution axis on the y–z-plane of the camera coordinate system.1 The algorithm and implementation are described in Section 5 and results of real data experiments are presented in Section 6. Finally, conclusions are given in Section 7.

Section snippets

Previous works

The earliest study of silhouettes in single views dates back to 1978, when Barrow and Tenenbaum [1] showed that surface orientation along the silhouette can be computed directly from image data. In Ref. [2], Koenderink showed that the sign of the Gaussian curvature is equal to the sign of the curvature of the silhouette, and convexities, concavities and inflections of the silhouette indicate convex, hyperbolic and parabolic surface points, respectively. In Ref. [3], Cipolla and Blake showed

Properties of surfaces of revolution

Let C̃r(s)=[X(s)Y(s)0]T be a regular and differentiable planar curve on the x–y-plane where X(s)>0 for all s. A SOR can be generated by rotating C̃r about the y-axis, and is given byS̃r(s,θ)=X(s)cosθY(s)X(s)sinθ,where θ is the angle parameter for a complete circle. The tangent plane basis vectors∂S̃r∂s=Ẋ(s)cosθẎ(s)Ẋ(s)sinθand∂S̃r∂θ=−X(s)sinθ0X(s)cosθ,are independent since Ẋ(s) and Ẏ(s) are never simultaneously zero and X(s)>0 for all s. Hence, S̃r is immersed and has a well-defined tangent

Reconstruction from a single view

Consider a SOR S̃r whose revolution axis coincides with the y-axis, and a pin-hole camera P̂=[I3t] where t=[00dz]T and dz>0. Let the contour generator be parameterized by s asΓ̃(s)=c̃+λ(s)p(s),wherep(s)·n(s)=0.

In Eq. (9), c̃ indicates the camera center at [00−dz]T, p(s) is the viewing vector from c̃ to the focal plane at unit distance for the point Γ̃(s), and λ(s) is the depth of the point Γ̃(s) from c̃ along the z-direction. Note that p(s) has the form [x(s)y(s)1]T, where (x(s),y(s)) is a

Estimation of the harmonic homology

The silhouette ρ of a SOR is first extracted from the image by applying a Canny edge detector [21], and the harmonic homology W that maps each side of ρ to its symmetric counterpart is then estimated by minimizing the geometric distances between the original silhouette ρ and its transformed version ρ′=Wρ (Fig. 4). This can be done by sampling N evenly spaced points xi along ρ and optimizing the cost functionCost(vx,ls)=∑i=1Ndist(W(vx,ls)xi,ρ)2,where dist(W(vx,ls)xi,ρ) is the orthogonal distance

Experiments and results

Fig. 6 shows the reconstruction of a candle holder. The rectification of the silhouette (Fig. 6(b)) was done using the algorithm described in Section 5. An ellipse was fitted to the bottom of the rectified silhouette for computing the orientation of the revolution axis. The radius of the topmost circle and the height of the candle holder, measured manually using a ruler with a resolution of 1 mm, were 5.7 and 17.1 cm, respectively. The ratio of the radius of the topmost circle to the height of

Conclusions

By exploiting the coplanarity constraint between the revolution axis and the surface normal, a simple technique for recovering the 3D shape of a SOR from a single view has been developed. The technique presented here assumes perspective projection and uses information from the silhouette only. The invariant properties of the SOR and its silhouette have been used to calibrate the focal length of the camera, and to rectify the image so that the silhouette becomes bilaterally symmetric about the y

Acknowledgements

The work described in this paper was partially supported by a research grant from The University of Hong Kong.

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