Abstract
We assume that edge detection is the task of measuring and localizing changes of light intensity in the image. As discussed by V. Torre and T. Poggio (1984), “On Edge Detection,” AI Memo 768, MIT AI Lab), edge detection, when defined in this way, is λ problem of numerical differentiation, which is ill posed. This paper shows that simple regularization methods lead to filtering the image prior to an appropriate differentiation operation. In particular, we prove (1) that the variational formulation of Tikhonov regularization leads to λ convolution filter, (2) that the form of this filter is similar to the Gaussian filter, and (3) that the regularizing parameter λ in the variational principle effectively controls the scale of the filter.
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