On the discrete representation of the Laplacian of Gaussian

The Laplacian of Gaussian (LoG) is commonly employed as a second-order edge detector in image processing, and it is popular because of its attractive scaling properties. However, its application within a finite sampled domain is non-trivial due to its infinite extent. Heuristics are often employed to determine the required mask size and they may lead to poor edge detection and location. We derive an explicit relationship between the size of the LoG mask and the probability of edge detection error introduced by its approximation, providing a strong basis for its stable implementation. In addition, we demonstrate the need for bias correction, to correct the offset error introduced by truncation, and derive strict bounds on the scales that may be employed by consideration of the aliasing error introduced by sampling. To characterise edges, a zero-crossing detector is proposed which uses a bilinear surface to guarantee detection and closure of edges. These issues are confirmed by experimental results, which particularly emphasise the importance of bias correction. As such, we give a new basis for implementation of the LoG edge detector and show the advantages that such analysis can confer