EEM 463 Introduction to Image Processing

Chapter 9Morphological Image

Processing

Fall 2017

Assist. Prof. Cihan Topal

Anadolu University, Dept. EEE

Slides Credit: Frank (Qingzhong) Liu

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Introduction

► Morphology: a branch of biology that deals with theFORM and STRUCTURE of animals and plants

► Morphological image processing is used to extract imagecomponents for representation and description of regionshape, such as boundaries, skeletons, and the convexhull, connectivity analysis, blob analysis, ...

► Develop methods (region filling, thinning, thickening, andpruning) that are frequently used in conjunction withthese algorithms as pre-or post-processing steps

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Preliminaries (1)

► Reflection

► Translation

The reflection of a set , denoted , is defined as

{ | , for }

B B

B w w b b B

1 2The translation of a set by point ( , ), denoted ( ) ,

is defined as

( ) { | , for }

Z

Z

B z z z B

B c c b z b B

𝐵

𝐵

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Example: Reflection and Translation

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Preliminaries (2)

► Structure elements (SE)

Small sets or sub-images used to probe an image under study for properties of interest

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Examples: Structuring Elements (1)

origin

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Examples: Structuring Elements (2)Accommodate the entire structuring elements when its origin is on the border of the original set A

Origin of B visits every element of A

At each location of the origin of B, if B is completely contained in A, then the location is a member of the new set, otherwise it is not a member of the new set.

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Erosion

2With and as sets in , the erosion of by , denoted ,

defined

| ( )Z

A B Z A B A B

A B z B A

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

| ( ) c

ZA B z B A

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Example of

Erosion (1)

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Example of

Erosion (2)

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Dilation

2With and as sets in , the dilation of by ,

denoted , is defined as

A B= |z

A B Z A B

A B

z B A

• Does the structuring element hit the set?

𝐵

|z

A B z B A A 𝐵

The set of all displacements , the translated and

overlap by at least one element.

z B A𝐵

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Examples of Dilation (1)

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Examples of Dilation (2)

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Duality

► Erosion and dilation are duals of each other with respect to set complementation and reflection

c c

c c

A B A B

and

A B A B

𝐵

𝐵

The duality property is useful particularly when the structuring element is symmetric with respect to its origin (as often is the case)

𝐵 = 𝐵

|

|

|

cc

Z

cc

Z

c

Z

c

A B z B A

z B A

z B A

A B

𝐵

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Duality

► Erosion and dilation are duals of each other with respect to set complementation and reflection

|

|

cc

Z

c

Z

c

A B z B A

z B A

A B

𝐵

𝐵

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Duality

► Erosion and dilation are duals of each other with respect to set complementation and reflection

𝐵

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Opening and Closing

► Opening generally smoothes the contour of an object,

breaks narrow isthmuses, and eliminates thin

protrusions.

► Closing tends to smooth sections of contours but it

generates fuses narrow breaks and long thin gulfs,

eliminates small holes, and fills gaps in the contour.

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Opening and Closing

The opening of set by structuring element ,

denoted , is defined as

A B

A B

A B A B B

The closing of set by structuring element ,

denoted , is defined as

A B

A B

A B A B B

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Opening

The opening of set by structuring element ,

denoted , is defined as

|Z Z

A B

A B

A B B B A

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Example: Opening

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Example: Closing

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Duality of Opening and Closing

► Opening and closing are duals of each other with respect to set complementation and reflection

( )c cA B A B

( )c cA B A B

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The Properties of Opening and Closing

► Properties of Opening

► Properties of Closing

(a) is a subset (subimage) of

(b) if is a subset of , then is a subset of

(c) ( )

A B A

C D C B D B

A B B A B

(a) is subset (subimage) of

(b) If is a subset of , then is a subset of

(c) ( )

A A B

C D C B D B

A B B A B

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The Hit-or-Miss Transformation

if denotes the set composed of

and its background,the match

(or set of matches) of in ,

denoted ,

* c

B

D

B A

A B

A B A D A W D

1 2

1

2

1 2

,

: object

: background

( )c

B B B

B

B

A B A B A B

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Some Basic Morphological Algorithms (1)

► Boundary Extraction

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

( )A A A B

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Example 1

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Example 2

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Some Basic Morphological Algorithms (2)

► Hole Filling

A hole may be defined as a background regionsurrounded by a connected border of foregroundpixels.

Let A denote a set whose elements are 8-connectedboundaries, each boundary enclosing a backgroundregion (i.e., a hole). Given a point in each hole, theobjective is to fill all the holes with 1s.

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Some Basic Morphological Algorithms (2)

► Hole Filling

1. Forming an array X0 of 0s (the same size as the array containing A), except the locations in X0 corresponding to the given point in each hole, which we set to 1.

2. Xk = (Xk-1 + B) Ac k=1,2,3,…

Stop the iteration if Xk = Xk-1

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Example

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Some Basic Morphological Algorithms (3)

► Extraction of Connected Components

Central to many automated image analysis applications.

Let A be a set containing one or more connectedcomponents, and form an array X0 (of the same size asthe array containing A) whose elements are 0s, exceptat each location known to correspond to a point in eachconnected component in A, which is set to 1.

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Some Basic Morphological Algorithms (3)

► Extraction of Connected Components

Central to many automated image analysis applications.

1

-1

( )

: structuring element

until

k k

k k

X X B A

B

X X

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Some Basic Morphological Algorithms (4)

► Convex Hull

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

The convex hull H or of an arbitrary set S is the smallest convex set containing S.

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Some Basic Morphological Algorithms (4)

► Convex Hull

1

Let , 1, 2, 3, 4, represent the four structuring elements.

The procedure consists of implementing the equation:

( * )

1, 2,3,4 and 1,

i

i i

k k

B i

X X B A

i k

0

1

4

1

2,3,...

with .

When the procedure converges, or , let ,

the convex hull of A is

( )

i

i i i i

k k k

i

i

X A

X X D X

C A D

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Some Basic Morphological Algorithms (5)

► Thinning

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

( * )

( * )c

A B A A B

A A B

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Some Basic Morphological Algorithms (5)

► A more useful expression for thinning A symmetrically is based on a sequence of structuring elements:

1 2 3

-1

, , ,...,

where is a rotated version of

n

i i

B B B B B

B B

1 2

The thinning of by a sequence of structuring element { }

{ } ((...(( ) )...) )n

A B

A B A B B B

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Some Basic Morphological Algorithms (6)

► Thickening:

The thickening is defined by the expression

*A B A A B

1 2

The thickening of by a sequence of structuring element { }

{ } ((...(( ) )...) )n

A B

A B A B B B

In practice, the usual procedure is to thin the background of the set

and then complement the result.

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Some Basic Morphological Algorithms (6)

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Some Basic Morphological Algorithms (7)

► Skeletons

A skeleton, ( ) of a set has the following properties

a. if is a point of ( ) and ( ) is the largest disk

centered at and contained in , one cannot find a

larger disk containing ( )

z

z

S A A

z S A D

z A

D and included in .

The disk ( ) is called a maximum disk.

b. The disk ( ) touches the boundary of at two or

more different places.

z

z

A

D

D A

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Some Basic Morphological Algorithms (7)

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Some Basic Morphological Algorithms (7)

0

The skeleton of A can be expressed in terms of

erosion and openings.

( ) ( )

with max{ | };

( ) ( ) ( )

where is a structuring element, and

K

kk

k

S A S A

K k A kB

S A A kB A kB B

B

( ) ((..(( ) ) ...) )

successive erosions of A.

A kB A B B B

k

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Some Basic Morphological Algorithms (7)

0

A can be reconstructed from the subsets by using

( ( ) )

where ( ) denotes successive dilations of A.

( ( ) ) ((...(( ( ) ) )... )

K

kk

k

k k

A S A kB

S A kB k

S A kB S A B B B

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Some Basic Morphological Algorithms (8)

► Pruning

a. Thinning and skeletonizing tend to leave parasitic components

b. Pruning methods are essential complement to thinning and

skeletonizing procedures

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Pruning: Example

1 { }X A B

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Pruning: Example

8

2 11

* k

kX X B

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Pruning: Example

3 2

: 3 3 structuring element

X X H A

H

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Pruning: Example

4 1 3X X X

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Pruning: Example

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Summary (1)

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Summary (2)

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Gray-Scale Morphology

►Unlike binary images we deal with digital imagefunctions of the form f(x,y) as an input imageand b(x,y) as a structuring element.

► f(x,y) and b(x,y) are functions that assign graylevel value to each distinct pair of coordinates.

►Remember that, the domain of gray values canbe 0-255, whereas 0 is black, 255 is white.

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Gray-Scale Morphology

( , ) : gray-scale image

( , ): structuring element

f x y

b x y

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Gray-Scale Morphology: Erosion and Dilation by Flat Structuring

( , )

( , )

( , ) min ( , )

( , ) max ( , )

s t b

s t b

f b x y f x s y t

f b x y f x s y t

► The condition that (x-s),(y-t) need to be in the domain off and x,y in the domain of b,

It is completely analogous to the condition in the binary definition oferosion, where the structuring element has to be completely combinedby the set being eroded.

It is analogous to the condition in the binary definition of dilation, wherethe two sets need to overlap by at least one element.

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Effects of Grayscale Erosion

► General effect of performing an erosion in grayscaleimages:

If all elements of the structuring element are positive, the outputimage tends to be darker than the input image.

The effect of bright details in the input image that are smaller inarea than the structuring element is reduced, with the degree ofreduction being determined by the grayscale values surrounding bythe bright detail and by shape and amplitude values of thestructuring element itself.

► Similar to binary image grayscale erosion and dilation areduals with respect to function complementation andreflection.

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Effects of Greyscale Dilation

►The general effects of performing dilation on a gray scale image is two fold:

If all the values of the structuring elements are positive than the output image tends to be brighter than the input.

Dark details either are reduced or eliminated, depending on how their values and shape relate to the structuring element used for dilation.

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Gray-Scale Morphology: Erosion and Dilation by Nonflat Structuring

( , )

( , )

( , ) min ( , ) ( , )

( , ) max ( , ) ( , )

N Ns t b

N Ns t b

f b x y f x s y t b s t

f b x y f x s y t b s t

► The resulting image will not be bounded by the values of f, which can present problem in interpreting it.

► Also, it is difficult to select meaningful SEs and it increases computational burden.

► Gray-scale SEs are seldom used in practice because of this.

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Duality: Erosion and Dilation

( , ) ( , )

where ( , ) and ( , )

c c

c

c c

f b x y f b x y

f f x y b b x y

f b f b

( )c cf b f b

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Opening and Closing

f b f b b

f b f b b

c c

c c

f b f b f b

f b f b f b

► In the opening of a gray-scale image, we remove smalllight details, while relatively undisturbed overall gray levelsand larger bright features

► In the closing of a gray-scale image, we remove small darkdetails, while relatively undisturbed overall gray levels andlarger dark features

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Properties of Gray-scale Opening

1 2 1 2

( )

( ) if , then

( )

where denotes is a subset of and also

( , ) ( , ).

a f b f

b f f f b f b

c f b b f b

e r e r

e x y r x y

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Properties of Gray-scale Closing

1 2 1 2

( )

( ) if , then

( )

a f f b

b f f f b f b

c f b b f b

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Morphological Smoothing

► Opening suppresses bright details smaller than the specified SE, and closing suppresses dark details.

► Opening and closing are used often in combination as morphological filters for image smoothing and noise removal.

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Morphological Smoothing

Dilation & Erosion & Smoothing

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Morphological Gradient

► Dilation and erosion can be used in combination with image subtraction to obtain the morphological gradient of an image, denoted by g,

► The edges are enhanced and the contribution of the homogeneous areas are suppressed, thus producing a “derivative-like” (gradient) effect.

( ) ( )g f b f b

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Morphological Gradient

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Top-hat and Bottom-hat Transformations

► The top-hat transformation of a grayscale image f is defined as f minus its opening:

► The bottom-hat transformation of a grayscale image f is defined as its closing minus f:

( ) ( )hatT f f f b

( ) ( )hatB f f b f

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Top-hat and Bottom-hat Transformations

► One of the principal applications of these transformations is in removing objects from an image by using structuring element in the opening or closing operation

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Example of Using Top-hat Transformation in Segmentation

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Granulometry

► Granulometry deals with determining the size distribution of particles in an image

► Opening operations of a particular size should have the most effect on regions of the input image that contain particles of similar size

► For each opening, the sum (surface area) of the pixel values in the opening is computed

Granulometry

► In practice, particles seldom are neatly separated, which makes particle counting by identifying individual particles a difficult task.

►Morphology can be used to estimate particle size distribution indirectly, without having to identify and measure every particle in the image.

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Example

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Textual Segmentation

► Segmentation: the process of subdividing an image into regions.

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Textual Segmentation