Formal Description

The following contour representation has been motivated by the observation, that for active contours

1. crossing in the contour might occur, because the contour elements move in the 2D image plane (see 1, left)
2. no unique ordering is given in the image plane. Only an ordering is defined along the contour, i.e. a neighborhood is defined between contour points. A relationship between contour point and geometric features of the object cannot be found (see 1, right).
3. the definition of region energies, which are capable for the extraction of textured regions, are computational expensive. At present, no real--time implementation is known, which can be used for object tracking.

  
Figure 1: Two main problems of active contours during tracking resulting from the missing ordering in the 2D image plane: Crossings in the 2D contour may occur (top). The contour elements are not fixed at logical features on the object's contour, but they may move around the contour as they like (bottom).

For the representation, we define a reference point , which has to lie within the image contour. Then, a so called active ray is defined on the image plane as a 1D function depending on those gray values of the image, which are on a straight line from the image point in direction

with where is given by the image size. This principle is illustrated in 2. In the following we only look at convex contours. Some concave contours can also be represented; for most concave contour one needs to modify the energy, to allow more than one contour point on each ray. This is described elsewhere.

  
Figure 2: Representation of a contour point by active rays