Computing force field-based directional maps in subquadratic time

Spatial knowledge, i.e., knowledge about configurations among distinct spatial entities, plays a key role in everyday life and has wide applications in numerous application domains. In particular, computer vision systems exploit spatial knowledge to enhance their object recognition capabilities. The focus of this article is on the representation of spatial knowledge. Specifically, we concentrate on the representation of spatial relationships among objects in an image, a fundamental problem in pattern recognition and computer vision. In the study of spatial relationships, significant progress has been made by using directional maps. However, the time complexity of map generation is O(N2) and thus not practical. We present an improvement of a state-of-the-art approximation method that is based on an analogy with the force exerted by an object on another object. The improvement is based on the use of a family of forces whose dependence on the pixel–pixel distance r   is of the type 1/rl+11/rl+1 with l > 1. Analysis of time complexity shows that the improved method is O(Nx  ), with x=1+3/(l+2)x=1+3/(l+2). Application of this method to both artificial and real images validates the time complexity analysis and shows that the proposed method is much faster than the original method. Furthermore, for the special case of small objects relative to the image size, the proposed method generates more accurate maps and, in other cases, the difference in accuracy is very small.