Edge Detection - Important Concepts in Signal Processing, Image Processing and Data Compression - page 28

where L x , L y ... L yyy denote partial derivatives computed from a scale-space representation
L obtained by smoothing the original image with a Gaussian kernel. In this way, the
edges will be automatically obtained as continuous curves with subpixel accuracy.
Hysteresis thresholding can also be applied to these differential and subpixel edge
segments.
In practice, first-order derivative approximations can be computed by central differences
as described above, while second-order derivatives can be computed from the scale-space
representation L according to:
corresponding to the following filter masks:
Higher-order derivatives for the third-order sign condition can be obtained in an
analogous fashion.
Phase congruency based edge detection
A recent development in edge detection techniques takes a frequency domain approach to
finding edge locations. Phase congruency (also known as phase coherence) methods
attempt to find locations in an image where all sinusoids in the frequency domain are in
phase. These locations will generally correspond to the location of a perceived edge,
regardless of whether the edge is represented by a large change in intensity in the spatial
domain. A key benefit of this technique is that it responds strongly to Mach bands, and
avoids false positives typically found around roof edges. A roof edge, is a discontinuity in
the first order derivative of a grey-level profile.