In this paper we propose a new method for extending 1-D step edge detection filters to two dimensions via complex-valued filtering. Complex-valued filtering allows us to obtain edge magnitude and direction simultaneously. Our method can be viewed either as an extension of n-directional complex filtering of Paplinski to infinite directions or as a variant of Canny’s gradient-based approach. In the second view, the real part of our filter computes the gradient in the x direction and the imaginary part computes the gradient in the y direction. Paplinski claimed that n-directional filtering is an improvement over the gradient-based method, which computes gradient only in two directions. We show that our omnidirectional and Canny’s gradient-based extensions of the 1-D DoG coincide. In contrast to Paplinski’s claim, this coincidence shows that both approaches suffer from being confined to the subspace of two 2-D filters, even though n-directional filtering hides these filters in a single complex-valued filter. Aside from these theoretical results, the omnidirectional method has practical advantages over both n-directional and gradient-based approaches. Our experiments on synthetic and real-world images show the superiority of omnidirectional and gradient-based methods over n-directional approach. In comparison with the gradient-based method, the advantage of omnidirectional method lies mostly in freeing the user from specifying the smoothing window and its parameter. Since the omnidirectional and Canny’s gradient-based extensions of the 1-D DoG coincide, we have based our experiments on extending the 1-D Demigny filter. This filter has been proposed by Demigny as the optimal edge detection filter in sampled images.

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