chapter 3 binary image analysis - ulisboa · dilation and erosion – generalization for...

Chapter 3 – Binary Image Analysis

Comunicação Visual Interactiva

Pixels and Neighborhoods

Vizinhança N4

• Most common neighborhoods

• Use of masks

– Example:

Vizinhança N8

origem

input output

CVI - Binary Image Analysis 2

Example

input saída

0 1 0

1 0 1

0 1 0

80 160 200 240 160

120 200 280 320 240

120 200 280 320 240

120 200 280 320 240

80 160 200 240 160

output

0 0 0

0 1 0

0 0 0

and with the mask ?


Counting objects

• Algorithm

– Hypothesis: An object is a set of connected pixels (connectivity 4) and

without inner holes

Outer corners Inner corners


How many objects?

e - 21

i - 5

44

16

4

521#

e

i

e

e e

e

e e

e

e e

e

e

e

e e

e

i

i

e

i

e

e

i

e

e


And now?

e

i

e

e

e

e e

e

e e

e

e

e

e e

e

i

i

e

i

e

e

i

e

e

e

e e

e - 23

i - 4

?4

19

4

423#


Connected Component Analysis – recursive algorithm

input

output


Recursive algorithm - Example

Neighbor 4

Neighbor 8


Union-Find structure


Classic CCA using Union-Find

input

1º step

2º step

equiv. classes


Exercise

• Apply the CCA algorithm considering connectivity 4

• What would be the result considering connectivity 8?

1 2

1 1 1 1

1

0

2

0 0

3

1 1

1 1

3 3

4

1

3

3 3

3

1

5 4

0

4

0

5

4


Morphological operators

• Structuring elements

– We need to define the center (i.e. origin)

• Definition: A dilation of the binary image B by the structuring

element S is defined as

• Definition: A erosion of the binary image B by the structuring

element S is defined as

Bb

bSSB

SsBsbbSB

ones(3,5) disk(5) ring(5)

SsbsSb


Morphological operations - Examples


Dilation and erosion – Generalization for monochromatic images

Operação Rule

Dilation

The output pixel value is the maximum value of all

pixels in the neighborhood of the input pixel. It is

assigned minimum value (0) to the other pixels

Erosion

The output pixel value is the minimum value of all

pixels in the neighborhood of the input pixel. It is

assigned a maximum value (1 or 255) to the other pixels

Dilation of the binary image Dilation of a monochromatic image


Morphological operations

• Definition: The closing of binary imge B by the structuring

element S is defined by

• Definition: A opening of a binary image B by the structuring

element S is defined by

SSBSB

SSBSB

B

S

SB

SB SB

SB


• Medical applications (image resolution of 512x512)

– opening with disk (13) folowed by closing with disk (2)

• Extraction of shape primitives

– Subtraction between the original image and the image obtained with the

opening operator (using a disck as the structuring element)

Examples of morphology to extract shape primitives

original binarized processed


original opening corners

Inspection procedure

input

output

a) b)

c) d)

e) f)


Conditional Dilation

• Definition: Given two binary images, original B, and processed

C, and the structuring element S, and let

The conditional dilation of C by S with respect to B is defined by

BSCCCC nn 10 e

mB CSC

where m is the smallest integer satisfying 1 mm CC

B

V

VBC SC B

S


Region properties

• Area

• Centroid

• Perimeter pixels

– Perimeter length

• Circularity (1)

Rcr

A),(

1

Rcr

Arr

),(

1

Rcr

Acc

),(

1

RcrNRcrP ),(),( 84

RcrNRcrP ),(),( 48

),(),(),(2

),(),(

4811

411

kkkkkk

kkkk

crNcrNcrk

crNcrkP

A

PC

2

1


Properties (cont.)

• Circularity (2)

– Mean radial distance

– Standard deviation of radial distance

R

RC

2

1

0

),(),(1 K

k

kkR crcrK

2

11

0

2),(),(

1

K

k

RkkR crcrK


Properties – lengths and boundaries

• Rectangle (and octagon) bounding box

• Axis length

º45:

º45:)(

)sin(1

)cos(1

Q

QccrrD 2

12

2

1222D

21

2D

º0 º45


Properties– 2nd order moments

• Second order moments

• Relation between moments and elliptic regions

• Axis with least second order moment

– Formulation

– Solution

ccrr

rc

2ˆ2tan

12|, 22 fcercdrcrR

rrrc

rccc

rcccrrfe

ed

24

1

Rcr

Acc cc),(

21

Rcr

Arr rr),(

21

Rcr

Arc ccrr),(

1

RcrA

RcrAcr

V

d

),(

21

),(

21,,

sin,cos

2/


Region Adjacency Graphs

• Problem: regions that have inner holes (in the background)

• Solution: algorithm with 3 steps

– Aplication of the CCA algorithm twice: (1) foreground pixels and (2)

background pixels

– (3) Building of graph relations


Limiar Threshold

• Definition: the histogram h of a monochromatic image I is

defined by mcrIcrmh ),(),()(


CVI - Análise de Imagens Binárias

Automatic computation of the Threshold

• Otsu Method

– Let us assume that we have an image , with and

denotes the gray level intensity in the coordinates .

– Also, assume that represents the gray levels in the

image .

– Defining as the nº of pixels at a given level , the total nº of the

pixels in the image is given by

– We can compute the weights of the pixels at the level as

– We are going to assume that we have two classes of gray levels,

using a threshold

RI :

),( yxI ),( yx

MaxValM ,...,3,2,1

I

2R

in Mi

MaxVal

i

inn1

mC ,...,2,11 MaxValmmC ,...,2,12 m

i

nniP i /)(

25 CVI - Binary Image Analysis

Automatic computation of the Threshold

• Otsu Method

– Ideia: intra-class minimization variance

– The weights allow to compute the mean of the gray levels of the two

classes;

– Now, we can compute the variances, from the above means:

)()()()( 2

22

2

11

2 mmmmW

)()(1

1 iPmwm

i

MaxVal

mi

iPmw1

2 )()(

m

i

mwiiPm1

11 )(/)()(

MaxVal

mi

mwiiPm1

22 )(/)()(

m

ii

mwiPmim )(/)()()( 1

2

1

2

1

MaxVal

mi

mwiPmim1

2

2

2

2

2 )(/)()()(


Cálculo automático do limiar

• Otsu then proposes the following “goodness” of the threshold

where

22111

2 )()()(1)( mmmwmwB

22 / WB

22121 )()()()( mmmwmw

)(1)( 12 mwmw

2

22

2

11

2 wwW


2

W

2

B

- within class (intra-class) variance

- between class (inter-class) variance

Recursive Algorithm

• Synopsis of the Otsu algorithm to find the threshold t

– Initializaion

– For m := 0 to MaxVal

– Limiar computation

)1()()1( 11 mPmwmw

)0()0(1 Pw

)1(

)1()1()()()1(

1

111

mw

mPmmmwm

)1(1

)1()1()1(

1

112

mw

mmwm

0)0(1

22111

2 )()()(1)()( mmmwmwmB

MaxVal

i

iiP0

)(

CRihiP /)()(

)(maxarg 2 mm Bm

t = 93

Original (MaxVal=255)


1 Compute histogram and probabilities of each intensity level

2 Set up initial \omega_i(0) and \mu_i(0)

3 Step through all possible thresholds t = 1 … maximum intensity

3.1 Update \omega_i and \mu_i

3.2 Compute \sigma^2_b(t)

4 Desired threshold corresponds to the maximum \sigma^2_b(t)


chapter 3 binary image analysis - ulisboa · dilation and erosion – generalization for...

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