Reconstructing surfaces form sparse depth information

This paper is concerned with interpolating and displaying the surface of an object from depth information obtained at its edge. Linear interpolation is discussed and an approximation with a least error matrix is given. An algorithm is derived which will determine the location of a point by accumulating the angular increment along the boundary of the surface. When the boundary of a surface is composed of a polygon, we count the winding number instead of calculating the angular increment. By applying the Jordan Curve Theorem in interpolation, a great amount of calculation can be avoided. Introduction Any process which transforms images into representations of surfaces has two stages. The first stage consists of computing explicit surface values (depth or surface orientation) at a particular set of points in the image. The second stage consists of interpolating between these known points to obtain a complete specification of the visible surfaces. The depth information obtained from stereoscopic images is sparse, available mainly at the edges of objects where there are significant intensity changes. This paper is concerned with reconstruction and display of the surface of such objects and in particular how distance information obtained at the edge is used to recover every point on its surface(s). We present a method to represent a surface using an implicit polynomial equation f(x, y, z) = O. Planar surfaces are emphasized. There are several reasons to take this approach. The first is that there is a mechanism of interpolation in the human visual system. The second is that surface is an important aspect in the perception of three-dimensional shapes [11]. Representations below the level of surfaces are generally too unstructured to contribute to any substantial vision tasks and in most cases we have only edge information and do not have texture Permission to copy without fee all or part of thie matarial ie granted provided that the copies ara not made or distributed for diract commercial advantage, the ACM copyright notica and the title of tha publication and ite date appear, and notice ie given thet copying ie by permission of the Association for Computing Machinary. To copy otherwise, or to rapublish, requires a fee andlor spacific permission. 01992 ACM 0-89791 -!502-W921000210450...$ 1.50 information. The third is that without texture information the human visual system perceives urdmown surfaces by planar interpolation. We refine this discussion in the following section. Then we give a representation of the surface of an object using sparse depth information from its boundaries. We present an algorithm using the Argument Principle and the Jordan Curve Theorem for interpolation in the fourth section, followed by a section detailing implementation. Finally, there is a conclusion and dkcussion. The Psychophysical and Computational Foundation There is an interpolating mechanism in the human visual system, which can be shown in the following figures. In Figure 1 (a), the perception associated with the figure is of a pair of equilateral triangles occluding a set of circles, as shown in Figure 1 (b). Afthough the sides of the triangle are not explicit, the visual system fills in the gaps to form subjective contours. Figure 2 gives an example of three-dimensional interpolation based on texture information. In Figure 2 (a), the density of the dots represents the depth of a half cylinder, lying on a reference plane. A gap is created in the cylinder. The human visual system fills in the gap and perceives the object as a half cylinder as shown in outline form in Figure 2 (b), where the viewpoint has been altered to give a clearer representation. (a) b) Figure 1. Interpolating Mechanism of the Human Visual System