Morphological Image Processing
Basics of Set Theory
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Dilation
Additional Set Operations for Images
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Mathematical morphology – tools based on set theory that are
useful in the representation and description of shape
We will start with binary images; each pixel is either 0 or 1, on or
off, black or white; however, see below...
We will consider sets that are collections of pixels
The concept of “pixel value” is not nearly as important as “set
membership”
These sets will “live” in a “universe” consisting of all the pixels in
an image
Dilation – the set of all pixels z such that the reflection of B
translated by z overlaps with A by at least one pixel
B is called the structuring element
Dilation tends to make sets bigger
Dilation fills in gaps smaller than the structuring element
Application: fill in gaps in characters in a poorly scanned
document
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Erosion
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Erosion – the set of all pixels z such that B translated by z is
contained entirely in A
Erosion tends to make sets smaller
Erosion completely eliminates sets smaller than the structuring
element
Erosion removes protrusions smaller than the structuring element
Duality relationship – dilation and erosion are duals with respect
to complementation and reflection
Application: remove all components smaller than a given size
from a segmented image
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Opening
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Erosion Examples
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Dilation Examples
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“ Rolling ball” analogy; ball rolls on inside of set
Smooths contours
Breaks narrow ithmuses
Eliminates thin protrusions
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Closing
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“ Rolling ball” analogy; ball rolls on outside of set
Fuses narrow breaks and long thin gulfs
Eliminates small holes
Fills gaps
Duality relationship – opening and closing are duals with respect
to complementation and reflection
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Closing Example
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Opening Example
Opening Example
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Hit-or-miss Transformation
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Closing Example
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Hit-or-miss Transformation
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Used to search for match to a particular shape
Also used in thinning and thickening
B actually represents a pair of structuring elements, B1 and B2
B1 and B2 will never be the same set (why?)
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