In general surfaces can be expressed in an explicit or an implicit form. An explicit form of a surface is the graph of a function of two variables. In the context of a computer vision system, the depth maps produced by a rangefinder can be considered as sample data sets from an explicit surface,

here, z is the distance from the camera origin to the surface, and are the image plane coordinates. For example, the equation of a planar surface in explicit form is

The implicit form of a surface in is expressed in the form

where are the Cartesian points on the surface. Thus, f is a function which maps a point from the space of possible surfaces to a real number in the space , which defines the coordinates of a surface point. For a plane,

The implicit form represents a surface in 3D space; in general the explicit form is a special case of the implicit form. As an example, the equation of an ellipsoid is

This can be re-arranged in the explicit form,

This describes the position of all points of the surface; there are two solutions for z for a given value of . However, if we view an ellipsoid we cannot see the points on the rear of the surface. A graph surface is the single root of equation similar to Equation 6. For example, if we viewed the ellipsoid from a position at , shown in Figure 1, below, then we would take the positive square root. This graph surface corresponds to the orthogonal depth data we would obtain from a rangefinder.

The less general form which is equivalent to the graph surface ( or Monge patch) is more simply,

Grey level surfaces in intensity images, and depth surfaces in range images conform to this common representation and can be so analysed. We consider only smooth surfaces, where all three parametric functions have continuous second partial derivatives. In general, a range image may have several smooth surfaces separated by points of discontinuity, i.e. the geometric edges we considered earlier. In this special case, we may rewrite Equation 8 as

or

Figure 2:   The differential geometry of a surface patch

[ Contents | Surface description and differential geometry ]

Comments to: Sarah Price at ICBL.