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Chapter 9: Morphological Image Processing

Digital Image Processing

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Mathematic Morphology

used to extract image components that are useful in the representation and description of region shape, such as

boundaries extraction
skeletons
convex hull
morphological filtering
thinning
pruning

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Mathematic Morphology

mathematical framework used for:

pre-processing

noise filtering, shape simplification, ...

enhancing object structure

skeletonization, convex hull...

Segmentation

watershed,…

quantitative description

area, perimeter, ...

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Z² and Z³

set in mathematic morphology represent objects in an image

binary image (0 = white, 1 = black) : the element of the set is the coordinates (x,y) of pixel belong to the object 🢧 Z²

gray-scaled image : the element of the set is the coordinates (x,y) of pixel belong to the object and the gray levels 🢧 Z³

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Basic Set Theory

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Reflection and Translation

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Logic Operations

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Example

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Structuring element (SE)

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small set to probe the image under study
for each SE, define origo
shape and size must be adapted to geometric

properties for the objects

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Basic idea

in parallel for each pixel in binary image:

check if SE is ”satisfied”
output pixel is set to 0 or 1 depending on used operation

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How to describe SE

many different ways!
information needed:

position of origo for SE
positions of elements belonging to SE

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Basic morphological operations

Erosion

Dilation

combine to

Opening object
Closening background

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keep general shape but smooth with respect to

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Erosion

Does the structuring element fit the set?

erosion of a set A by structuring element B: all z in A such that B is in A when origin of B=z

shrink the object

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Erosion

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Erosion

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Erosion

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Erosion

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Dilation

Does the structuring element hit the set?
dilation of a set A by structuring element B: all z in A such that B hits A when origin of B=z

grow the object

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Dilation

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Dilation

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Dilation

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Dilation

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B = structuring element

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Dilation : Bridging gaps

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useful

erosion

removal of structures of certain shape and size, given by SE

Dilation

filling of holes of certain shape and size, given by SE

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Combining erosion and dilation

WANTED:

remove structures / fill holes
without affecting remaining parts

SOLUTION:
combine erosion and dilation
(using same SE)

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Erosion : eliminating irrelevant detail

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structuring element B = 13x13 pixels of gray level 1

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Opening

erosion followed by dilation, denoted ∘

eliminates protrusions
breaks necks
smoothes contour

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Opening

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Opening

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Opening

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Closing

dilation followed by erosion, denoted •

smooth contour
fuse narrow breaks and long thin gulfs
eliminate small holes
fill gaps in the contour

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Closing

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Closing

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Closing

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Properties

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Opening

A°B is a subset (subimage) of A
If C is a subset of D, then C °B is a subset of D °B
(A °B) °B = A °B

Closing

A is a subset (subimage) of A∙B
If C is a subset of D, then C ∙B is a subset of D ∙B
(A ∙B) ∙B = A ∙B

Note: repeated openings/closings has no effect!

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Duality

Opening and closing are dual with respect to complementation and reflection

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Useful: open & close

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Application: filtering

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Hit-or-Miss Transformation ⊛ (HMT)

find location of one shape among a set of shapes ”template matching

composite SE: object part (B1) and background part (B2)
does B1 fits the object while, simultaneously, B2 misses the object, i.e., fits the background?

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Hit-or-Miss Transformation

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Boundary Extraction

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Example

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Region Filling

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Example

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Extraction of connected components

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Example

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Convex hull

A set A is is said to be convex if the straight line segment joining any two points in A lies entirely within A.

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Thinning

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Thickening

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Skeletons

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Pruning

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H = 3x3 structuring element of 1’s

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5 basic structuring elements

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