An Optimal Scale for Edge Detection

Many problems in early vision are ill posed 1. Edge detection is a typical example. This paper applies regular-ization techniques to the problem of edge detection. We derive an optimal filter for edge detection with a size controlled by the regularization parameter and compare it to the Gaussian filter. A formula relating the signal-to-noise ratio to the parameter is derived from regularization analysis, showing that the scale of the filter is a function of the signal-to-noise ratio. We also discuss the method of Generalized Cross Validation for obtaining the optimal filter scale. Finally, we use our framework to explain two perceptual phenomena: coarsely quantized images becoming recognizable by either blurring or adding noise.