電腦視覺 Computer and Robot Vision I Chapter2: Binary Machine Vision: Thresholding and Segmentation Instructor: Shih-Shinh Huang 1

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電腦視覺Computer and Robot Vision

I

Chapter2: Binary Machine Vision:

Thresholding and Segmentation

Instructor: Shih-Shinh Huang

2

Contents

Introduction

Thresholding

Connected Components Labeling

Signature Segmentation and Analysis

3

Computer and Robot Vision I

2.1 Introduction

Introduction

4

Binary Machine Vision

Binary Image Binary Value 1: Part of Object

Binary Value 0: Background Pixel

Definition of Binary Machine Vision Generation and analysis of such a binary image

2.1 Introduction

5

Binary Machine Vision

Thresholding It is the first step of binary machine vision

It is a labeling operation

Connected Components / Signature Analysis They are multilevel vision grouping techniques.

They make a transformation from image pixels to

more complex units.

• Regions

• Segments

2.1 Introduction

6


2.2 Thresholding

Introduction

7

Introduction

What is Thresholding ? It is a labeling operation.

It assigns a binary value to each pixel.

• Binary Value 1: pixels have higher intensity values

• Binary Value 0: pixels have higher intensity values

2.2 Thresholding

128T

8

Introduction

Mathematical Formulation

: row and column

: gray-level intensity image

: intensity threshold

: binary intensity image

2.2 Thresholding

TcrIif

TcrIifcrB

),(0

),(1),(

),( cr

(.,.)I

(.,.)B

T

9

Introduction

2.2 Thresholding

(.,.)I

5T

(.,.)BHow to select an appropriate threshold ?

10

Introduction

Approaches

Global Thresholding: use a global value to make the pixel

distinction in the image.

Local Thresholding: use spatial varying threshold to label

the local pixels.

2.2 Thresholding

T

Image Image

1T 2T

3T 4T

11

Histogram

Definition of Histogram

Histogram Probability

2.2 Thresholding

}),(|),{(#)( mcrIcrmh

(.)h

number of elements m spans each gray level value e.g. 0 - 255

255,...,1,0:)( IIP

CR

IhIP

)()(

12

Histogram

Examples

2.2 Thresholding

)(mh

)(mh

13

Histogram

2.2 Thresholding

)(mh

Histogram

14

2.2 Thresholding

T=110 T=130 T=150 T=170

15

Within-Group Variance

Observations A group is a set of pixels with intensity homogeneity.

Homogeneity is measured by the use of variance

• High homogeneity group has low variance

• Low homogeneity group has high variance

Objective Select a dividing score such that the weighted sum

of the within-group variances is minimized.

2.2 Thresholding

16


Definition: weighted sum of group variances

: probability for the group with values

: probability for the group with values

: variance for the group with values

: variance for the group with values

)()()()()( 222

211

2 ttqttqtW

)(22 t

)(21 t

)(1 tq

)(2 tq

t

t

t

t

2.2 Thresholding

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Objective Formulation

Find a threshold which minimizes

2.2 Thresholding

*t )(2 tW

)(minarg 2* tt W

t

i

iPtq1

1 )()(

t

i

tqiiPt1

11 )(/)()(

I

ti

iPtq1

2 )()(

I

ti

tqiiPt1

22 )(/)()(

)(/)()]([)( 12

11

21 tqiPtit

t

i

)(/)()]([)( 22

222 tqiPtit

I

iti

18


Implementation Issue

Step1: For t=0,…,255

Step2: Compute , , , and

Step3: Compute

Step4: If is less than the value in the

previous iteration

2.2 Thresholding

)(22 t)(21 t)(1 tq )(2 tq

)()()()()( 222

211

2 ttqttqtW

)(2 tW

)(2 tW

tt *

All variables should be re-compute at each iteration.

19


Implementation Issue Speed-Up Formulation

2.2 Thresholding

I

i

iPi1

22 )()(

T

i

iiP1

)(

T

ti

t

i

iPttiiPtti1

222

1

211

2 )())()(()())()((

.......)())(())())(((2))((1

2111

21

2

t

i

iPtttiti

20



2.2 Thresholding

.......)())(())())(((2))((1

2111

21

2

t

i

iPtttiti

t

i

iPtti1

11 0)())())(((

t

i

iPtti1

22

22 )())(())((

t

i

iPtti1

21

21

2 )())(())((

21



2.2 Thresholding

)(])([)())(( 12

11

21

2 tqtiPtit

i

)(])([)())(( 22

21

22 tqtiPti

I

ti

)()()()( 222

211

2 ttqttq

2222

11 ])()[(])()[( ttqttq

22



2.2 Thresholding

)()()()( 222

211

2 ttqttq

2222

11 ])()[(])()[( ttqttq

2122

1122 ])()[(])()[( ttqttqw

)]()()][(1)[( 211122 tttqtqw

)()()()( 2211 ttqttq

23



2.2 Thresholding

)]()()][(1)[( 211122 tttqtqw

constant minimize maximize

)]()()][(1)[(maxarg 2111* tttqtqt

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• We have recursive form to compute optimal threshold.

2.2 Thresholding

)]()()][(1)[(maxarg 2111* tttqtqt

)1()()1( 11 tPtqtq

)1(

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

1

111

tq

tPtttqt

25


Example

2.2 Thresholding

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Kullback Information Distance

Assumption The observations come from a weighted mixture of

two Gaussians distributions.

• Gaussian Distribution of Background

• Gaussian Distribution of Object

2.2 Thresholding

),( 211 u

),( 222 u

2

2

22

1

1 )(2

1

2

2)(

2

1

1

1

22)(

ii

eq

eq

if

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2.2 Thresholding

),( 211 u ),( 2

22 u

Background Gaussian Distribution

ObjectGaussian Distribution

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Objective Formulation Determine a threshold T that results in two Gaussian

distributions which minimize Kullback divergence

• P(I) : observed histogram distribution

• f(I) : a mixture of Gaussian distributions determined by T

2.2 Thresholding

])(

)(log[)(

1 if

iPiPJ

I

i

2

2

22

1

1 )(2

1

2

2)(

2

1

1

1

22)(

ii

eq

eq

if

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Objective Formulation Known Parameter: Observed Histogram

Unknown Parameter: Two Gaussian Distributions

2.2 Thresholding

)(),...2(),1( IPPP

(.)P

2

2

22

1

1 )(2

1

2

2)(

2

1

1

1

22)(

ii

eq

eq

if

),,(),,,( 222111 qq

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Solution Derivation

2.2 Thresholding

])(

)(log[)(

1 if

iPiPJ

I

i

)(log)(log)(1

ifiPiPI

i

)(log)()(log)(11

ifiPiPiPI

i

I

i

Constant

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Solution Derivation

Assumption: The modes are well separated.

2.2 Thresholding

)(log)(minarg])(

)(log[)(minarg

11

ifiPif

iPiP

I

i

I

i

tieq

tieq

if i

i

2

2

2

2

1

1

)(2

1

2

2

)(2

1

1

1

2

2)(

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Solution Derivation

2.2 Thresholding

)(log)()(1

ifiPtHI

i

2

2

22

1

1 )(2

1

2

2

1

)(2

1

1

1

1 2)(

2)()(

iI

ti

it

i

eq

iPeq

iPtH

2211 loglog2

2log1)( qqqqtH

2211 loglog2

1 qq

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Implementation Issue

Step1: For t=0,…,255

Step2: Compute , , , and

Step3: Compute

Step4: If is less than the value in the

previous iteration

2.2 Thresholding

)(22 t)(21 t)(1 tq )(2 tq

)(tH

tt *

)(log)()(1

ifiPtHI

i

)(tH

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Example

2.2 Thresholding

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2.2 Thresholding

Within Group Variance (Otsu)

Kullback Information (Kittler-Illingworth)

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2.3 Connected Component Labeling

Introduction

37

Introduction

Description

Connected Components labeling is a grouping operation.

It performs the unit change from pixel to region or segment.

All pixels are given the same identifier

• Have value binary 1

• Connect to each other


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Introduction

Terminology label: unique name or index of the region

connected components labeling: a grouping

operation

pixel property: position, gray level or brightness

level

region property: shape, bounding box, position,

intensity statistics


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Connected Component Operators

Definition of Connected Component

Two pixels and belong to the same

connected component if there is a sequence of

1-pixels , where

•

•

• are neighbor


qpC),...,( 10 nppp

pp 0qpn

nipp ii ,...,1:,1

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Connected Component Operators

Neighborhood Types


4-connected

8-connected

Original Image Connected Components

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Connected Component Algorithms

Common Features Process a row of image at a time

Assign a new labels to the first pixel of each component.

Propagate the label of a pixel to its neighbors to the right or below it.


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Connected Component Algorithms

Common Features


• What label should be assigned to A

• How does the algorithm keep track of the equivalence of two labels

• How does the algorithm use the equivalence information to complete the processing

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Algorithm1: Iterative Algorithm

Algorithm Steps

Step1 (Initialization): Assign an unique label to each pixel.

Step2 (Iteration) : Perform a sequence of top-down and bottom-up label propagation.


• Use no auxiliary storage• Computational Expensive

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Algorithm2: Classic Algorithm

Two-Pass Algorithm Pass 1:

• Perform label assignment and label propagation

• Construct the equivalence relations between labels

when two different labels propagate to the same pixel.

• Apply resolve function to find the transitive closure of

all equivalence relations.

Pass 2:

• Perform label translation.


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Example:


{2=4}

{3=5}

{1=5}

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Example: Resolve Function


{2=4}{3=5}{1=5} {2=4}{1=3=5}

• Computational Efficiency • Need a lot of space to store equivalence

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2.4 Signature Segmentation and

Analysis

Introduction

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Introduction

Description Signature analysis perform unit

change from the pixel to the segment.

It was firstly used in character

recognition Definition of Signature

The signature, which is a projection,

is the histogram of the non-zero pixels

of the masked image.

2.4 Signature Segmentation and Analysis

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Introduction

General Signatures Vertical Projection

Horizontal Projection

Diagonal Projection


},1),(|),{(#)( rxyxyxrh

},1),(|),{(#)( cyyxyxrv

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Signature Segmentation

Steps

Thresholding: generate the binary image.

Projection Computation: compute the vertical, horizontal, or diagonal projections.

Projection Segmentation: divide the image into several segments or regions according to the signatures.


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OCR: Optical Character Recognition

MICR: Magnetic Ink Character Recognition

The End


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電腦視覺 Computer and Robot Vision I Chapter2: Binary Machine Vision: Thresholding and Segmentation Instructor: Shih-Shinh Huang 1