Definitions

Let P be a point on a smooth curved surface ${\cal S}$. We assume that the imaging system is based on the pinhole model (i.e. perspective projection). Therefore, the vector position r of P can be written  

$$r=C + \lambda T,$$ (1)

where C is the camera centre position, T the unit viewing direction and $\lambda$ the depth of the point P along the viewing direction. For a given camera position there is a locus of points on the surface ${\cal S}$ where the normal N is perpendicular to the viewing direction. This set of points is called the rim (also known as the limb) and its projection onto the image plane is called the occluding contour (also known as the extremal boundary or the profile), as shown in figure 2.

  

Figure 2: Rim and occluding contour under perspective projection.

As the camera moves around ${\cal S}$ the occluding contours sweep a surface in the space of parameters (u, v, t) which is called the spatio-temporal surface [Fau 93,Gib 95].
Since the camera centre position is a function of time t, this surface can be parameterised by (s,t), where the parameter s describes the position on the occluding contours and the parameter t corresponds to time. However, such a parametrisation is not uniquely defined [Cip 90]: curves of constant t are the occluding contours but curves of constant s have no physical interpretation. Until now, the most generally accepted parametrisation of the spatio-temporal surface is the epipolar parametrisation which yields a mapping between successive occluding contours called the epipolar correspondence. The advantage of this parametrisation is that it leads to a local parametrisation of the surface ${\cal S}$ which can always be used except when the occluding contour is singular or when the camera motion is in the plane tangent to the surface [Gib 95]. Furthermore, the epipolar correspondence between points on successive occluding contours constrains the reconstruction problem which becomes a linear estimation, as shall be seen later.