morphology gonzalez woods

8/12/2019 Morphology Gonzalez Woods

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Lecture 5. Morphological Image

Processing


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Introduction

Morphology : a branch of biology that deals with the formand 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 convex hull


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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


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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 forproperties of interest


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Examples: Structuring Elements (1)

origin


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Examples: Structuring Elements (2) Accommodate theentire structuringelements when itsorigin is on theborder of theoriginal set A

Origin of B visitsevery element of A

At each location ofthe origin of B, if Bis completelycontained in A,then the location isa member of the

new set, otherwiseit is not a memberof 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 Z A B z B A


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Exampleof

Erosion(1)


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Exampleof

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

The set of all displacements , the translated and

overlap by at least one element.

z B A

| z A B z B A A


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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 toset complementation and reflection

c c

c c

A B A B

and

A B A B


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Duality


| |

|

cc

Z

cc Z

c

Z

c

A B z B A z B A

z B A A B


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Duality


|

|

cc

Z

c

Z

c

A B z B A

z B A

A 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 itgenerates 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 ,


A B

A B

A B A B B

The closing of set by structuring element ,


A B

A B A B A B B


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Opening

The opening of set by structuring element ,


| 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 respectto set complementation and reflection

( )

c c A B A B

( )c c A 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 BC D C B D B

A B B A B


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The Hit-or-MissTransformation

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

( ) A A A B


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Example 1


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Example 2


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Hole Filling A hole may be defined as a background region surroundedby a connected border of foreground pixels.

Let A denote a set whose elements are 8-connectedboundaries, each boundary enclosing a background region(i.e., a hole). Given a point in each hole, the objective is tofill all the holes with 1s.


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Hole Filling1. Forming an array X 0 of 0s (the same size as the arraycontaining A), except the locations in X 0 corresponding tothe given point in each hole, which we set to 1.

2. Xk = (X k-1 + B) A c k=1,2,3,

Stop the iteration if X k = X k-1


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Example


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Extraction of Connected ComponentsCentral to many automated image analysis applications.

Let A be a set containing one or more connectedcomponents, and form an array X 0 (of the same size as thearray containing A) whose elements are 0s, except at eachlocation known to correspond to a point in each connectedcomponent in A, which is set to 1.


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Extraction of Connected ComponentsCentral 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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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 smallestconvex set containing S.


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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 ik k

B i

X X B Ai k

0

1

4

1

2,3,...

with .

When the procedure converges, or , let ,

the convex hull of A is

( )

i

i i i ik k k

i

i

X A

X X D X

C A D


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

( * )

( * ) c A B A A B

A A B


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A more useful expression for thinning A symmetrically isbased 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 2The thinning of by a sequence of structuring element { } { } ((...(( ) )...) )n

A B A B A B B B


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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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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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0

The skeleton of A can be expressed in terms oferosion and openings.

( ) ( )

with max{ | };

( ) ( ) ( )

where is a structuring element, and

K

k k

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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0

A can be reconstructed from the subsets by using

( ( ) )

where ( ) denotes successive dilations of A. ( ( ) ) ((...(( ( ) ) )... )

K

k k

k

k k

A S A kB

S A kB k S A kB S A B B B


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Pruning:Example

1 { } X A B


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Pruning:Example

8

2 11* k

k X 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 3 X X X


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Pruning:Example


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Morphological Reconstruction

It involves two images and a structuring element

a. One image contains the starting points for the

transformation (The image is called marker)

b. Another image (mask) constrains the transformation

c. The structuring element is used to define connectivity

M h l i l R i G d i


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Morphological Reconstruction: GeodesicDilation

(1)

(1)

Let denote the marker image and the mask image,

. The geodesic dilation of size 1 of the marker image

with respect to the mask, denoted by ( ), is defined as ( ) G

G

G

F G

F G

D F D F F B

( )

( ) (1) ( 1)

(0)

The geodesic dilation of size of the marker image

with respect to , denoted by ( ), is defined as ( ) ( ) ( )

with ( ) .

nG

n nG G G

G

n F

G D F D F D F D F

D F F


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M h l i l R t ti G d i


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Morphological Reconstruction: GeodesicErosion

(1)

(1)

Let denote the marker image and the mask image.

The geodesic erosion of size 1 of the marker image with

respect to the mask, denoted by ( ), is defined as ( ) G

G

G

F G

E F E F F B

( )

( ) (1) ( 1)

(0)

The geodesic erosion of size of the marker image

with respect to , denoted by ( ), is defined as ( ) ( ) ( )

with ( ) .

nG

n nG G G

G

n F

G E F E F E F E F

E F F


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Morphological Reconstruction by Dilation

Morphological reconstruction by dialtion of a mask image

from a marker image , denoted ( ), is defined as

the geodesic dilation of with respect to , iterated until

stability is achieved; that

DGG F R F

F G

( )

( ) ( 1)

is,

( ) ( )

with such that ( ) ( ).

D k G G

k k G G

R F D F

k D F D F


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Morphological Reconstruction by Erosion

Morphological reconstruction by erosion of a mask image

from a marker image , denoted ( ), is defined as

the geodesic erosion of with respect to , iterated until

stability is achieved; that i

E GG F R F

F G

( )

( ) ( 1)

s,

( ) ( )

with such that ( ) ( ).

E k G G

k k G G

R F E F

k E F E F


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Opening by Reconstruction

( )

The opening by reconstruction of size of an image is

defined as the reconstruction by dilation of from the

erosion of size of ; that is

( )

where

n D R F

n F

F

n F

O F R F nB

F nB

indicates erosions of by .n F B


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Filling Holes

Let ( , ) denote a binary image and suppose that weform a marker image that is 0 everywhere, except at

the image border, where it is set to 1- ; that is

1 ( , ) if ( , ) is on the bor ( , )

I x y F

I

I x y x y F x y

der of

0 otherwise

then

( )cc D

I

I

H R F


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SE : 3 3 1s.


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( )cc D

I H R F


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Border ClearingIt can be used to screen images so that only complete objects

remain for further processing; it can be used as a singal that

partial objects are present in the field of view.

The original image is used as the mask and the followingmarker image:

( , ) if ( , ) is on the border of( , )

0 otherwise

I x y x y I F x y

X I R

( ) D I F


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Summary (1)


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Summary (2)


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Gray-Scale Morphology

( , ) : gray-scale image( , ): structuring element

f x yb x y

Gray-Scale Morphology: Erosion and Dilation


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Gray Scale Morphology: Erosion and Dilationby 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


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Gray-Scale Morphology: Erosion and Dilation


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Gray Scale Morphology: Erosion and Dilationby Nonflat Structuring

( , )

( , )

( , ) min ( , ) ( , )

( , ) max ( , ) ( , )

N N s t b

N N s 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

l d l


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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 c f 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


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i f G l O i


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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

P i f G l Cl i


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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 bc f b b f b


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M h l i l S hi


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Morphological Smoothing

Opening suppresses bright details smaller than thespecified SE, and closing suppresses dark details.

Opening and closing are used often in combination asmorphological filters for image smoothing and noiseremoval.


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T h t d B tt h t T f ti


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Top-hat and Bottom-hat Transformations

The top-hat transformation of a grayscale image f isdefined as f minus its opening:

The bottom-hat transformation of a grayscale image f isdefined as its closing minus f:

( ) ( )hat

T f f f b

( ) ( )hat B f f b f

T h t d B tt h t T f ti


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Top-hat and Bottom-hat Transformations

One of the principal applications of these transformationsis in removing objects from an image by using structuringelement in the opening or closing operation


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Granulometry


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Granulometry

Granulometry deals with determining the size ofdistribution of particles in an image

Opening operations of a particular size should have themost effect on regions of the input image that containparticles of similar size

For each opening, the sum ( surface area ) of the pixelvalues in the opening is computed

Example


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p


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Textual Segmentation


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Segmentation: the process of subdividing an image intoregions.



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Gray Scale Morphological Reconstruction (2)


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Gray-Scale Morphological Reconstruction (2)

The geodesic erosion of size 1 of f with respect to g isdefined as

The geodesic erosion of size n of f with respect to g isdefined as

(1) ( )

where denotes the point-wise maximum operator.

g E f f g g

( ) (1) ( 1) (0)( ) ( ) with ( )n n g g g g E f E E f E f f


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The opening by reconstruction of size n of an image f isdefined as the reconstruction by dilation of f from theerosion of size n of f; that is,

The closing by reconstruction of size n of an image f isdefined as the reconstruction by erosion of f from thedilation of size n of f; that is,

( ) ( )n D R f O f R f nb

( ) ( )n E R f C f R f nb


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Steps in the Example


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1. Opening by reconstruction of the original image using a horizontalline of size 1x71 pixels in the erosion operation

2. Subtract the opening by reconstruction from original image

3. Opening by reconstruction of the f using a vertical line of size 11x1pixels

4. Dilate f1 with a line SE of size 1x21, get f2.

( ) ( )n D R f O f R f nb

( )' ( )n R f f O f

( )1 ( ') ' 'n D R f f O f R f nb



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5. Calculate the minimum between the dilated image f2and and f, get f3.

6. By using f3 as a marker and the dilated image f2 asthe mask,

( )2 2

( ) ( 1)2 2

( 3) 3

with k such that 3 3

D k f f

k k f f

R f D f

D f D f

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