240-373: chapter 6: morphological image processing 1 montri karnjanadecha [email protected] ....

240-373: Chapter 6: Morphological Image

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Montri [email protected]://fivedots.coe.psu.ac.th/~montri

240-373 Image Processing


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Chapter 6

Morphological Image Processing


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Morphological and Other Area Operations

• Introduction– Morphology is the science of form and structure– It is about regions or shapes, in computer vision

• Morphology can be used for the following tasks:– Smoothing the edges of a region– Forcing shapes onto region edges– Counting regions– Estimating sizes of regions


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Morphological Operations

• Morphological operations are easily seen on bin ary images but they can be extended to work o

- n gray level images• Basic Morphological Operations

– Consider the following image 1 * 1 * 1

* 1 * 1 *

1 * 1 * 1

– This could be represented as a set of those pixels in 5 x 3 image that have value ‘1’, namely

00 02 04 11 13 20{( , ), ( , ), ( , ), ( , ), ( , ), ( , ), 22 24


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• And now it is possible to do set operations on images

1 * 1 * 1 * * * 1 1

A * 1 * 1 *= B 1** * 1 1

1 * 1 * 1 * * * 1 1

Giving

1 * 1 1 1 **** 1

A union B 1 1 *1 * 1 1 A intersection B 1 ***1

*

1 * 1 1 1 **** 1


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• The set on unknown pixels (*) can, theoretically, extend infinitely

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 * 1 * 1 * * * * * * * * * * * 1 * 1 * * * * * *

* * * * * 1 * 1 * 1 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

where the circle around the top left 1 indicates the position of the origin


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Dilation

1Technique : Dilation (Minkowski addition)

USE: Region edge valley filling (for binary image)

OPERATION:– A template (made from 1’s and *’s) is create

d with a known origin– The origin of this template is stepped over ev

ery element in the whole of the image


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Dilation

OPERATION: (cont’d)

– The template is ‘unioned’ where the origin of the template corresponds to a 1 in the image

– - The resulting template sized matrix is then u - nioned with all other template sized results (

using their original position)


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Dilation Example

Image Template

Applying the template to the first line gives

1111111****

1111111****

1*11111****

1111111****

*11*11*****

1***1******

*1**1******

11

*1

11*11******

*1**1******


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Dilation Example

Application across the whole image gives new

row

new column– Continually applying this to the images fills out all the

holes and makes the image grow, one row and one column at a time

11111111****

11111111****

11111111****

11111111****

11111111****

1111111*****

*11*11******

**1**1******


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Erosion

2:Technique Erosion (Minkowski subtraction)

USE: To remove spikes from the edges of regions

OPERATION:– A template is created– The template is stepped over the image but it is not al

lowed to go outside the image– For every position, if the template is the same as the i

mage window, then the element corresponding to the originof t he t empl at e i s set t o a 1


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Erosion Example

1111111****

1111111****

1*11111****

1111111****

*11*11*****

1***1******

*1**1******

11

*1

Image Template


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Erosion Example

• Final result is

unused row and column

***********

**1********

***********

*1**1******

***********

***********

***********


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Gray-level Erosion and Dilation

Technique 3: Gray-level erosion and dilation

USE: Flattening spikes and filling valleys on region edges in gray-level images

OPERATION: – Let I(x,y) be an image of gray levels and

R(x,y) be resulting image after I(x,y) has been dilated/eroded with m x n template T(i,j)

0 <= i <= m-1, 0 <= j <= n-1


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OPERATION: – Gray level dilation is defined as

– and gray level erosion is

),(),(),( max1010

jiTjyixIyxR

njmi

),(),(),( min1010

jiTjyixIyxR

njmi


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NOTES: – For both of the above definitions, I(x,y) = 0

for locations outside of an M x N image– Dilation is precisely the dual of the erosion

operation. A dilation can be performed by reflecting the template and negating the original image and then negating the result.


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Example

Consider the following initial image

Dilating by the template

00000000

00244200

00488400

00244200

00000000

11

11


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Example

gives

or subtracting 1 throughout

11111111

13555311

11599511

11599511

13555311

11111111

00000000

02444200

00488400

00488400

02444200

00000000


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Example

Conversely, erode by the same template gives

11111111

11111111

11113111

11113111

11111111


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Example

or if 1 is subtracted throughout

00000000

00000000

00024200

00024200

00000000


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Example

NOTE:– Dilationgi ves a l ar ger i mage, whi l e er osi on gi ves a smal l

er i mage– Inbot h cases, t he or i gi nal st r uct ur e of t he i mage i

s mai nt ai ned– If the template is not constant, say

10

01


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Example

the results are as follows:

- 1 1Original Dilate Erode +

• D(E(D(I))) = D(I) and E(D(E(I))) = E(I)

00000000

00244200

00488400

00244200

00000000

02443100

04887300

03788400

01344200

00000000

00000000

00000000

00024300

00034200

00000000


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Opening and Closing Operations

• OPENING = dilating after eroding• CLOSING = eroding after dilating

4:Technique Opening and closing operations

USE: As a step towards the skeleton, count - ing, and size estimating operations belo

w


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OPERATION:– Let OPEN(I,T ) be the opening of image I by so

me template T– Then using previous terminology and the te

mplate T throughoutOPEN( I,T) = D(E(I))

This gives


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Original Opened with Opened with

Clearly, this restores the image to its original and avoids the scaling of the result. It introduces the shaping of the template into the original.

00000000

00244200

00488400

00244200

00000000

00000000

00244200

00244200

00244200

00000000

00000000

00244200

00344200

00234200

0000000011

1111

01


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More interesting is its operation on clear edges Original Opened with

Note how the peak (1 1 2 5 5 4) has been smoothed to the shape of the template but that the trough (1 3) has been left alone.

32100000

32100000

31000000

32100000

44210000

44210000

41000000

00000000

32210000

32100000

31000000

32100000

45521100

54210000

41000000

00000000

1

1


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Closing is defined asCLOSE(I,T’) = E(D(I))

with E and D using T’ where T’ is the 180o rotation of T

Original Closed with

Now the trough has been filled but the peak has been left.

32210000

32100000

32100000

32100000

45521100

54210000

41000000

00000000

32210000

32100000

31000000

32100000

45521100

54210000

41000000

00000000

1

1


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Properties of open and close operations

• OPEN -> forces the shape of the template on the convex parts of an edge

• CLOSE -> forces its shape on the concave parts of an edge

• These two operations can be combined to produce a skeleton operation


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Skeleton Operations

5:Technique Skeleton operations

USE: To create a skeleton of a region. This will consist of a set of lines correspondin

g to complete thinning of the region with out losing the essential shape.


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Skeleton Operations

OPERATION:– Let D

1 , D2

, …, Dn be structuring templates t hat are square, with sizes 1x1, 2x2, …, n x n ,

all of them filled with 1’s.

– Wi t h i mage I , using template D1

, D2

, …, Dn , i n turn, evaluate

resulti = - E(I) OPEN(E(I))

– Then determine

)result()(SK max allfor

ii

I


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Skeleton Operations

Example: Original Final

image

using only D1, D2, D3, and D4

• This corresponds to a skeleton of the image, retaining length but reduce thickness

000000000

000000000

002100000

001000000

001000000

001000000

003110000

001000000

000000000

32210000

32100000

31000000

32100000

45521100

54210000

41000000

00000000


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Granule sizing and granule counting

Technique 6: Granule sizing and granule counting

USE: To estimate the number of bright regions in an image. To estimate the size of the regions in an image.


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Granule sizing and granule counting

OPERATION:– Using templates such as a set of vertical or horizontal

1 ’s with increasing length to estimate the size of the area

– Usingver t i cal or hor i zont al t empl at es wi t h bi nar y st r i 2ngshavi ng val ues n +1(i.e. 1, 11, 101, 1001, 10001, etc.) t o count111 111 111 11 11111

– Inbothcases, t he i mage i s er oded by t he st r uct ur i ng t emp lates and a measure of how many elements are now non

- 11 1 111 11 111 111 11 11 111 111111 11 1111111111,– 111 111111 11 1111111 1111111 111 11111111 111111111 111111 11

give a covariance curve whose peaks indicate the size a nd number of granules (areas) in the image.


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Example

• The following image was eroded with the templates as shown below. After each erosion the number of pixels > 0 was counted and the power of the resulting image was summed.

Original image

0000000000

0000040000

1000022000

6500034410

3650021100

4400003200


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Example

Length Type

of string 11….1 10…01

power count powercount

1 63 20 63 202 31 11 31 113 8 4 8 44 1 1 2 25 0 0 3 36 7 77 7 78 4 49 1 110 0

0


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Example

Given a slightly altered image

0000000000

2334222400

1220222000

6500034410

3650021100

4400003200


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Example

which eroded with 1 0 …0 1 pattern gives

length 1 2 3 4 5 6 7 8 9

count 30 21 13 9 10 11 9 5 1power 87 52 25 14 15 15 12 6 1

240-373: chapter 6: morphological image processing 1 montri karnjanadecha [email protected] ....

Documents

createdthe template

image of gray levels

j0 graylevel erosion

gray level dilation

image grow

image window

image processingmontri

columngraylevel erosion