Shape from Shading with a Linear Triangular Element Surface Model

815 of CG g 2 (r , y) can be speeded up by a preprocessing like that for (13). We have decided not to lengthen this correspondence by the tedious derivation of this time-saving technique. We instead simply state our result that g 2 (s. y) can be evaluated in O(1) time for any given G with an O (161) preprocessing and O ([GI) space. time, the worst-case time complexity of the ROFS algorithm-aided by the preprocessing can be bounded by O(Ii(T, + Xy)). If we assume that, on average, each bipartition of G halves G in size and each cutting direction has equal chance to the optimal one (and denote by C (n) the cost of bipartitioning a subimage of size n), then the total cost of the procedure ROFS becomes 2'C(lGl/2'). Under the above assumptions, we have C (n) = O (n o ') since G is not extremely prolonged in any cutting direction. Thus, the expected time complexity of the ROFS algorithm is E:= " , " 2'C([G1/2') = The required preprocessing of computing (13) clearly takes O ([GI) operations and O((G1) space, based on the parameter ranges in (13). Thus, the cost of the preprocessing asymptotically dominates that of the ROFS algorithm in the expected case for Zi << [GI. However, this will only be true for very large 15'1 because there is a considerable constant before the order complexity O((I<lgl)o ') incurred by the overheads of computing the key vertices of newly formed canonical polygons and polygon scan conversion to determine the digital cutting line segment A,B,. The recursive optimal four-way split and the RAG-guided minimum cost merge were proposed to improve the validity of the classic split-and-merge segmentation algorithm. The new optimized split-and-merge algorithm achieves the unification of edge detection and segmentation within a tree hierarchy. As a result of this data structure and a statistic preprocessing, the optimization can be done without increasing the time complexity order of the previous algorithms. The success of the adaptive four-way split naturally leads to the following question: Why should we not consider every orientation when we optimize the split to capture edges of arbitrary orientations? This is a matter of compromise between computational costs and the optimality. Although adaptive four-way cuts greatly improve the segmentation validity over regular rectilinear cuts, they still have relatively simple digital geometry and manageable computational cost. The same cannot be said …