Introduction

For smooth curved surfaces, their apparent contours are dominant in images, and are rich source of geometric information about the surfaces and motions [ 8 , 5 , 16 ]. These are the projection of the locus of points on the surface which separate the visible and occluded parts on the surface as shown in Fig.  1 . Under perspective projection this locus, the contour generator, can be constructed as the set of points on the surface which are touched by lays through the projection center.

The fundamental difficulty of recovering structure from apparent contours lies in the fact that the apparent contours are not fixed features. That is the contour generator slips over the surface under viewer motion, and the apparent contours observed from the different viewpoints do not have any correspondence in general. Giblin and Weiss [ 8 ] showed that if the camera motion is known and is coplanar, the curved surfaces can be recovered uniquely from their apparent contours. This result has been extended for general camera motion by Cipolla and Blake [ 5 ], and Vaillant and Faugeras [ 16 ]. Unfortunately, all these methods assume that the cameras are calibrated and their motions are known or controlled for a specific position.

Now, the question is that if the camera motion is unknown and if the camera is uncalibrated , is it still possible to recover curved surfaces just from their apparent contours? It has been shown recently that there exist some points on contour generators which are visible on both apparent contours before and after the camera motion. Such points are called frontier points  [ 9 , 14 ] (see Fig.  1 ), and are visible from both viewpoints. The epipolar plane is tangent to the curved surface at the frontier point in 3D space, and the epipolar line is tangent to the apparent contour at the projection of the frontier point in images, i.e epipolar tangency  [ 13 ]. Recent research [ 3 , 1 ] showed that by iteratively searching for the frontier points we can recover the epipolar geometry just from the apparent contours of curved surfaces. Although these works showed the possibility of recovering the epipolar geometry from apparent contours viewed under unknown arbitrary motions of a camera, the methods require non-linear minimisation processes, which are sometimes unstable and fall into local minima. Furthermore, the reconstruction of curved surfaces from uncalibrated cameras has not been addressed.

   
Figure 1: Epipolar geometry and frontier points. The contour generators separate the visible and occluded parts on a surface. The projections of the contour generators in images are apparent contours. The epipolar plane defined by the two projection centers touches to the surface at a frontier point. At this point, the contour generators from two viewpoints intersect each other. In the images, the epipolar lines are tangent to the apparent contours at the projections of the frontier point.

Recent progress [ 6 , 10 ] in non-metric reconstruction showed that if we have fixed features with known correspondences in two views from uncalibrated cameras, 3D structure can be recovered up to a 3D projective ambiguity. Furthermore, it has been shown [ 12 , 11 ] that if the camera motion is pure translation  , the structure can be recovered up to a 3D affine ambiguity.

In this paper, we assume that the cameras are uncalibrated and their motions are unknown but are limited to pure translations. We show that, under this condition, the epipolar geometry can be obtained without using any optimisation method, and the curved surfaces are reconstructed up to an affine ambiguity just from the changes in apparent contours, as is the case of fixed features. The result is applied for labeling image curves as belonging to the projection of curved surfaces or fixed features. We also show that the partial reconstruction of curved surfaces allow us to extract an important cue for visual navigation, i.e. time-to-contact to the curved surfaces, just from the changes in apparent contours viewed from an uncalibrated camera.

In section 2, we define a camera model considered in this paper. In section 3, a method for extracting epipolar geometry from apparent contours with uncalibrated cameras will be shown. In section 4, the computed epipolar geometry is used for recovering the structure of curved surfaces from uncalibrated views. It is also shown that the time-to-contact to the curved surfaces can be computed from the changes in apparent contours. The results of some experiments are shown in section 5.