TABLE OF CONTENTS Objects in images differ from each other via their boundaries or edges, which has made edge detection or extraction one of the oldest yet still the most fundamental tasks in vision and image analysis.
Though primarily arising in image analysis and processing, we must point out that edge detection and adaptivity have also been pivotal in many other areas such as numerical analysis of spectral data (see, e.g., Tadmor and Tanner [295, 296]).
Like any general pattern recognition problem, edge detection crucially depends on
a sound definition of the edge feature or pattern; and
a good algorithm for extracting such edges.
The two are intimately correlated, however.
Edges are traditionally recognized as the collection of pixels where the gradients are noticeably large. Thus, for example, a simple edge detector can be defined by
where 
denotes an image domain, u a given image on ?, and p some suitable threshold.
This naive edge detector can be criticized from two aspects. First, it is oversensitive to noise. Suppose u = ? + n is the superposition of a clean image ? with some additive noise n. Then the detector (7.9) fails to distinguish genuine edge pixels from noisy pixels. That is, almost surely (from the probability point of view), the detected edge set ? u( p) would be the entire domain ?, regardless...
