Recovering three-dimensional shapes from visual data is an important intermediate goal of many vision systems used in robotics, surveillance, guidance and modelling. In such applications, surface reconstruction is often used as an intermediate step in the more important task of 3D representation and recognition.
For non-polyhedral objects, rich and robust information on the shape
are provided by the occluding contours. In fact, if some strong a priori
knowledge on the object is available such as parametric descriptions,
then a single view allows shape recovery [Pon 89,Zer 93], object
pose estimation and object recognition [Kri 90,Gla 92,For 92]. Otherwise, for any smooth object, a sequence of
occluding contours must be considered to recover the shape.
Different approaches
The problem of reconstructing surfaces from occluding contours was
first considered by Koenderink [Koe 84], who
attempted to infer 3D curvature properties from occluding contours.
Later, Giblin and Weiss [Gib 87] proved that surface
reconstruction can be performed for planar motion under the
assumption of orthographic projection. The problem of reconstructing
under perspective projection for general known motions has been
tackled by Cipolla and Blake [Cip 90], and Vaillant [Vai 92]. Both approaches lead to depth computation under
the assumption of continuous observations of the occluding contours
and are based on differential analysis.
Unfortunately, the use of
differential computations with discrete observations requires
numerical approximations and produces
numerical instabilities. With the aim of further constraining the
problem, the occluding contour is supposed to be locally part of a
circle. In addition, the plane containing the circle is
supposed to be known: in [Cip 90], the epipolar plane is used
whereas Vaillant [Vai 92] used the radial plane. Such methods are called osculating circle methods and have been widely used in the literature.
While keeping the circle model, Szeliski [Sze 93] proposed to improve the reconstruction process by computing the epipolar curves on the whole surface together with an estimate of uncertainty. The use of estimation theory leads to a more robust shape recovery using a linear smoother but the basic reconstruction method remains unchanged.
In [Boy 96], we considered the reconstruction problem directly from a discrete point of view. A correct depth formulation was established for occluding contours observed at discrete times. The main interest of such a formulation is to avoid reconstruction constraints (unlike [Cip 90,Vai 92,Sze 93,Jos 95] no reconstruction plane is locally imposed), so that non-linear motion can be easily taken into account. Moreover, it gives a general solution to the reconstruction problem which is always defined except when the camera motion is in the viewing direction. Our approach is based on a local approximation of the surface up to second order, allowing a linear estimation of depth to be derived, given any set of three occluding contours.
The approaches mentioned above only allow a 3D depth
map of points or curves belonging to the surface to be obtained.
Using a different strategy, Zhao [Zha 94] attempts to recover the
global 3D surface description from the occluding contours in a single stage by use of B-spline
patches. This
approach introduces a direct regularisation of the reconstructed
surface, but it requires a complete a priori parametrisation of the surface
which is usually not available. Therefore it is preferable to first recover a more
accurate 3D depth map before computing a 3D surface description.
The approach described in this document
The surface reconstruction algorithm has three main parts (as shown in figure 1) :
This document is organized as follows : elementary notions are first introduced. In section 2 the depth formulation is established, enabling the discrete observations of occluding contours to be handled. The local reconstruction algorithm is then compared to previous methods. In section 3 the triangulation method is presented. Reconstruction examples are shown in section 4.
