Multiple points

In the general case, epipolar correspondents p₁ and p₂ are image projections of two different points P₁ and P₂ on ${\cal S}$. This non-stationarity property of occluding contours can be used to discriminate such contours from others [Vai 92]. However, there is a circumstance where the property is not verified. This occurs when the camera motion (C₁ - C₂) is in the tangent plane of ${\cal S}$ at P₁ or P₂. In this situation, points p₁ and p₂ are image projections of the same fixed point P (P=P₁=P₂), and rims of ${\cal S}$ for camera positions C₁ and C₂ intersect at P. We call such a point P a multiple point of the sequence considered.

Definition A multiple point of an image sequence of ${\cal S}$is a point P where two or more consecutives rims of the sequence intersect.

  

Figure 4: A multiple point.

Consequently, if P is a multiple point of an image sequences of ${\cal S}$ then the epipolar plane at P is tangent to ${\cal S}$. In addition, if the n camera centre positions of the image sequences are aligned, then P is of multiplicity n.

Remark In the literature, the locus of rim points where the epipolar plane is tangent to the surface is called the frontier [Gib 95,Cip 95]. For a linear camera motion, the frontier, if it exists, is restricted to isolated points. Thus, multiple points represent the frontier for linear camera motions going through successive camera centre positions of the sequence. Since these rim points are fixed features on the surface, they can be used to derive constraint on the camera motion (see [Rie 86,Por 91,Cip 95,Jos 95] for more information on this subject).