2.3
Edges correspond to abrupt changes or discontinuities in certain image
properties between neighboring areas. The image properties may be of
non-texture or of texture. In this section, we are interested in
non-texture edges due to changes in image intensity, such as jump edges
and roof edges. Jump edges correspond to the discontinuities in the
underlying surface 
, or to the maxima and
minima (More exactly, positive and negative impulses) of its
first (directional) derivative, 
. Roof edges correspond to the
discontinuities in the first derivative, 
, or to the maxima and
minima of its second derivative, 
. However, pixel sites are
spatially quantized (in the x-y plane) the pixel values are subject
to noise. In this case, discontinuities, maxima and minima are not well
defined (their definitions becomes part of the solution to the edge
detection problem). These cause problems for edge detection.
The first part in edge detection [Torre and Poggio 1986 ; Canny 1986] is the estimating the derivatives from noisy and spatially quantized data. The second is to detect the zeros and extrema of the estimated derivative function. Another issue is to link edges, which are detected based on local changes, to form boundaries which are coherent in a more global sense. The latter part also relates to perceptual organization [Lowe 1985]. This section concerns only the first two parts by which pixel locations where sharp changes occur are marked.
Edge detection is closely related to those for image restoration and surface reconstruction involving discontinuities. There, we are mainly interested in removing noise and getting (piecewise) smooth surfaces; although discontinuities are also taken care of to avoid oversmoothing, they are not required to be marked explicitly. In this subsection, we modify the piecewise continuous restoration model to obtain explicit labeling of edges.