Chapter 9. Mathematical morphology: ◦ A useful tool for extracting image components in the representation of region shape. Boundaries, skeletons,

Chapter 9

Mathematical morphology:◦ A useful tool for extracting image components in

the representation of region shape. Boundaries, skeletons, and convex hull.

Set theory is usually used to describe mathematical morphology.◦ Sets represent objects in a binary image.

Black: representing object, denoted by 1. White: representing background , denoted by 0.

9.1 Preliminaries9.1 Preliminaries

• Our interest in this chapter is sets in Z2, where each element denotes the coordinates of an object pixel.– If a=(a1, a2), we write if a is an element in A.

– if a is not an element in A.

• The null or empty set is denoted by .

• We use braces, {·}, to specify the content of a set. For example, C={w|w=-d, for }.

Aa

Aa

Dd

Operations of SetsOperations of Sets

c

c

BABwAwwBA

AwwA

BABA

BwAwwBA

BwAwwBA

} ,|{

} |{

) if exclusivemutually or disjoint be tosaid are and (

} and |{

}or |{

Additional DefinitionsAdditional Definitions

• Translation: –

• Reflection: –

}for ,|{)( AazaccA Z

}for ,|{ˆ BbbwwB

9.1.2 Logic Operations Involving Binary Images9.1.2 Logic Operations Involving Binary Images

• The logic operations discussed in this section involve binary images.– Black pixel: 1.– White pixel: 0.

Note: logic operations are restricted to binary variables, which is not the case in general for set operations.

Logic Operations Involving Binary ImagesLogic Operations Involving Binary Images

9.2 Dilation and Erosion9.2 Dilation and Erosion

• These two operations are fundamental to morphological processing.

– Dilation: to enlarge an object along its boundary.

– Erosion: to shrink an object into a smaller size.

9.2.1 Dilation9.2.1 Dilation

• With A and B are sets in Z2, the dilation of A by B, denoted A B, is defined asA B = – Other interpretation: A B =

• B is commonly referred to as the structuring element.

• The dilation of A by B is the set of all displacements, z, such that the reflection of B and A overlap by at least one element.

})ˆ(|{ ABz z

})ˆ(|{ AABz z

The Illustration of DilationThe Illustration of Dilation

The Implementation of DilationThe Implementation of Dilation

• Given a binary image f and the structuring element s, construct a duplicate of f, denoted by g.

• For each pixel p = f(x, y), do the following:– If p is black:

• If p is at the boundary (any of the 4-adjacent neighbors is white) of the object, center the origin of s at (x, y) in g, and fill the pixels black on which s covers.

• Return g.

Application of DilationApplication of Dilation

• One of the simplest applications of dilation is for bridging gaps.

9.2.2 Erosion9.2.2 Erosion

• With A and B are sets in Z2, the dilation of A by B, denoted A B, is defined as

A B = – The erosion of A by

B is the set of all points z such that B, translated by z, is contained in A.

})(|{ ABz z

The Implementation of DilationThe Implementation of Dilation

• Given a binary image f and the structuring element s, construct a duplicate of f, denoted by g.

• For each pixel p = f(x, y), do the following:– If p is white:

• If p is adjacent to the boundary of the object, center the origin of s at (x, y) in g, and fill the pixels white on which s covers.

• Return g.

Application of DilationApplication of Dilation• One of the simplest uses of erosion is for eliminating

irrelevant detail (in terms of size) from a binary image.

Note that objects are represented by white pixels, rather than by black pixels.

9.3 Opening and Closing9.3 Opening and Closing

• Opening: to break narrow isthmuses and to eliminate thin protrusions.

• Closing: to fuse narrow breaks and long thin gulfs, to eliminate small holes, and to fill gaps in the contour.

BBABA ) (

BBABA ) (

Illustration of Opening and ClosingIllustration of Opening and Closing

Example 9.4: Application of Opening and Closing

Example 9.4: Application of Opening and Closing

9.4 The Hit-or-Miss Transformation9.4 The Hit-or-Miss Transformation

• The morphological hit-or-miss transform is a basic tool for shape detection or pattern matching.

• Let B denote the set composed of X and its background.

– B = (B1, B2), where B1=X, B2=W-X.

The match of B in A, denoted by A B, is*

])( [) ( XWAXABA c *

To find objects that may contain X To find objects that may be contained in X

The Hit-or-Miss TransformationThe Hit-or-Miss Transformation

• Other interpretation:

• If B is 3x3, the matching can be done directly rather than computing the background image.

] [) ( 21 BABABA c*

)ˆ () ( 21 BABABA *

9.5 Some Basic Morphological Algorithms9.5 Some Basic Morphological Algorithms

• Boundary extraction

• Region filling

• Extraction of connected components

• Convex Hull

• Thinning

• Skeletons

• Pruning

9.5.1 Boundary Extraction9.5.1 Boundary Extraction

• The boundary of a set A, denoted by β(A), can be obtained by first eroding A by B and then performing the set difference between A and its erosion.

Example 9.5: Boundary ExtractionExample 9.5: Boundary Extraction

• Binary 1’s are shown in white and 0’s in black.• Using 5x5 structuring element would result in a

boundary between 2 and 3 pixels thick.

The structuring element in this example is 3x3; therefore, the boundary is one pixel thick.

9.5.2 Region Filling9.5.2 Region Filling• Goal: given a point p inside the boundary (Fig. (a)), fill the

entire region with 1’s.

• Let X0 = p. The filled set Xk can be obtained by

...3,2,1 ) ( 1 kABXX ckk

The Procedure of Region FillingThe Procedure of Region Filling• The algorithm terminates at iteration step k if Xk=Xk-1.

• The result is obtained from the union of Xk and the boundary in A.

9.5.3 Extraction of Connected Components9.5.3 Extraction of Connected Components

• Goal: given a point p, find the component that connects to p.

• Let X0 = p. The set Xk can be obtained by

• The algorithm terminates at iteration step k if Xk=Xk-1.

• The result Y is obtained from Xk.

...3,2,1 ) ( 1 kABXX kk

The Procedure of Finding Connected Components

The Procedure of Finding Connected Components

Example 9.7Example 9.7

9.5.4 Convex Hull9.5.4 Convex Hull

• A set A is said to be convex.– If the straight line joining any two points in A lies entirely

within A.

• The convex hull H of a set S is the smallest convex set containing S.

– The set H-S is called the convex difference, which is useful for object description.

• The procedure is to implement the equation:

– With Xi0=A. Let Di=Xi

conv, where “conv” indicates that Xi

k=Xik-1. The convex hull of A is

1,2,3... and 4,3,2,1 ) ( 1 kiABXX ik

ik *

4

1

)(

i

iDAC

Convex HullConvex Hull

Limiting Growth of Convex HullLimiting Growth of Convex Hull

9.5.5 Thinning9.5.5 Thinning

• The thinning of a set A by a structuring element B, denoted A B, is defined by

• Each B is usually a sequence of structuring elements:

– B1, B2,…are different rotated versions of B.

• The result of thinning A by one pass is the union of the results obtained by thinning by Bi by one pass.

x

* ) ( BAABA x

},...,,,{}{ 321 nBBBBB

Thinning ProcedureThinning Procedure

9.5.6 Thickening9.5.6 Thickening

• The thickening of a set A by a structuring element B, denoted A B, is defined by

• A more efficient scheme is to obtain the complement of A, say Ac, and then to compute Cc, where C is the thinned result of Ac and Cc is its complement.

·

) ( BAABA · *

9.5.7 Skeletons9.5.7 Skeletons• The dot line : the skeleton of A, S(A).

}) (|max{

) ()( with )()(0

kBAkK

BkBAASASAS k

K

kk

The Procedure of SkeletonizationThe Procedure of Skeletonization

9.5.8 Pruning9.5.8 Pruning

spur

The Procedure of PruningThe Procedure of Pruning

• Thinning an input set A to eliminate the short line segment by

• To restore the character to its original form:– Find the set containing all the end points by

– Dilate the end points and find the intersection with A:

– The union of X3 and X1 yields the desired result:

}{1 BAX

8

112 ) (

k

kBXX *

AHXX ) ( 23

314 XXX

9.6 Extensions to Gray-Scale Images9.6 Extensions to Gray-Scale Images

• Dilation– Let Df and Db be the domains of f and b, where b is the

structuring element.

• The dilated image tends to be brighter.

• The dark details either reduce or eliminated, depending on their values and shapes relate to the structuring element.

• Erosion

• The eroded image tends to be darker.

• The bright details either reduce or eliminated.

}),(;)(),(|),(),(max{),)(( bf DyxDytxsyxbytxsftsbf

}),(;)(),(|),(),(min{),)( ( bf DyxDytxsyxbytxsftsbf

Example 9.9Example 9.9

9.6.3 Opening and Closing9.6.3 Opening and Closing

Example 9.10 Example 9.10

• In (a), the decreased sizes of the small, bright details, with no appreciable effect on the darker gray levels.

• In (b), the decreased sizes of the small, dark details, with relatively little effect on the bright features.

9.6.4 Applications of Gray-Scale Morphology9.6.4 Applications of Gray-Scale Morphology

• Morphological smoothing– i.e. performing opening followed by a closing.

• Morphological gradient– Let g denote the operation, and then

– Depending less on edge directionality.

) ()( bfbfg


• Top-hat transformation

)( bffh


• Textural segmentation– Use closing operation to eliminate the left half.

– Apply opening to restore and join the right half.

– Threshold the result to draw the boundary.


• Granulometry (粒度測量 )– Apply opening with different sizes of structuring elements.

– Calculate image difference.

– Draw the histogram to evaluate the difference with respect to various sizes of structuring elements.

– For some x, particles with similar size of x have higher responses in the histogram.

Chapter 9. Mathematical morphology: ◦ A useful tool for extracting image components in the representation of region shape. Boundaries, skeletons,

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