Edge Detection by Adaptive Splitting

Edge Detection by Adaptive Splitting II. The Three-Dimensional Case

“…The comportment of AI T (f) when / is continuous is given by the following result. If / is smooth the above result can be sharpened, see [24]. From the above result, we can conclude, in the case of continuous functions, that A/r(/) ->• 0 as |T| approaches zero.…”

“…An adaptive splitting approximation algorithm was proposed in [25]. In [24] it was proved (for d = 1 and d = 2) that the average integral used in its splitting criterion is an effective jump detector. These results imply that the algorithm is divergent for functions with jump discontinuities, but they allow us to modify it to obtain an efficient edge detection method.…”

“…In Table 2 we increase the precision of EDAS-3 by decreasing the value of the parameter E 2 -As a result, a large increase of tetrahedra arises around the discontinuity surface; see Fig. 13 in [24]. In Table 3 the increase of a makes ff difficult to approximate by a piecewise affine function away from the discontinuities.…”

A Doubly Adaptive Algorithm for Edge Detection in 3D Images

“…It is proved at [21,22] that the behavior of the next average integral AI ( ) depends on the continuity or lack of continuity of on :…”

“…Then, it considers straight lines perpendicular to the plane 1 2 and computes the edge points lying on these straight lines. This task is accomplished using the 1D edge detector EDAS-1 [22]. If the number of edges detected in the region is high or the distance between them is small, the algorithm concludes that the region under consideration is complex and performs a subdivision process; otherwise, the region is not divided and the algorithm studies a contiguous zone.…”