- Morphological Gradient
In
mathematical morphology anddigital image processing , morphological gradient is the difference between the dilation and the erosion of a given image. It is an image where eachpixel value (typically non-negative) indicates the contrast intensity in the close neighborhood of that pixel. It is useful foredge detection andsegmentation applications.Mathematical definition and types
Let be a grayscale image, mapping points from an Euclidean space or discrete grid "E" (such as "R"2 or "Z"2) into the real line. Let be a grayscale
structuring element . Usually, "b" is symmetric and has short-support, e.g., :.Then, the morphological gradient of "f" is given by:
:,
where and denote the dilation and the erosion, respectively.
An internal gradient is given by:
:,
and an external gradient is given by:
:.
The internal and external gradients are "thinner" than the gradient, but the gradient peaks are located "on" the edges, whereas the internal and external ones are located at each side of the edges. Notice that .
If , then all the three gradients have non-negative values at all pixels.
References
* "Image Analysis and Mathematical Morphology" by Jean Serra, ISBN 0126372403 (1982)
* "Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances" by Jean Serra, ISBN 0-12-637241-1 (1988)
* "An Introduction to Morphological Image Processing" by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)
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