In  Marr and Hildreth proposed the use of zero crossings of the Laplacian of a Gaussian (LoG), that is, the use of the rotationally symmetric convolution filter,
where is the convolution filter, A and are constants and determines the spatial scale of the Gaussian. The LoG convolution can be approximated by convolution with the difference of two Gaussians (DoG, sometimes referred to as the ``Mexican hat'') of appropriate scale and magnitude. The zero crossings can be tested for spurious responses by thresholding the magnitude of the gradient of the zero crossing. The synthesis of results found at different scales is suggested, to enable the detection of a larger range of edge ``widths''. (See  for a description of a successful approach to the synthesis of multiple scale zero crossings, and  for a detailed discussion of this subject.)
Contours produced using the LoG filter have the property, convenient for some purposes, of producing closed contours. However, connectivity at junctions is poor, and corners are rounded. Also, the use of non-directional derivatives means that the edge response parallel to an edge is always measured (as well as the expected response perpendicular to it), reducing the signal to noise ratio. The use of directional first and second derivatives improves on this. Finally, the LoG filter gives no indication of edge direction, which may be needed by higher level processes.