Multiscale filtering method for derivative computation
Bingcheng Li, Songde Ma · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1994
In this paper, we propose a multiscale filtering method to compute derivatives with any orders. As a special case, we consider the computation of the second derivatives, and show that the difference of two smoothers with the same kernel, but different scales constructs a Laplacian operator and has a zero crossing at a step edge. Selecting a Gaussian function as the smoother, we show the DOG (difference of Gaussian) itself is a zero crossing edge extractor, and it needn't approximate to LoG (Laplacian of Gaussian). At the same time, we show that even though DOG for bandwidth ratio 0.625 (1:1.6) is the optimal approximation to LoG, it is not optimal for edge detection. Finally, selecting an exponential function as the smoothing kernel, we obtain a Laplacian of exponential (LoE) operator, and it is shown theoretically and experimentally that the LoE has a high edge detection performance, furthermore its computation is efficient and its computational complexity is independent of the filter kernel bandwidths.