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We consider edge detection as the problem of measuring and localizing changes of light intensity in the image. As discussed by Torre and Poggio (1984), edge detection, when defined in this way, is an ill-posed problem in the sense of Hadamard. The regularized solution that arises is then the solution to a variational principle. In the case of exact data, one of the standard regularization methods (see Poggio and Torre, 1984) leads to cubic spline interpolation before differentiation. We show that in the case of regularly-spaced data this solution corresponds to a convolution filter---to be applied to the signal before differentiation -- which is a cubic spline. In the case of non-exact data, we use another regularization method that leads to a different variational principle. We prove (1) that this variational principle leads to a convolution filter for the problem of one-dimensional edge detection, (2) that the form of this filter is very similar to the Gaussian filter, and (3) that the regularizing parameter $lambda$ in the variational principle effectively controls the scale of the filter. |
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