Representation and Description

Digital Image Processing

Lecture 11,12 – Representation and

Description

Lecturer: Ha Dai Duong

Faculty of Information Technology

I. Introduction

After an image has been segmented into regions by

methods such as those discussed in previous lectures,

segmented pixels usually is represented àn described in

a form suiable for further computer processing.

A segmented region can be represented by: external

(boundary) pixels or internal pixels

When shape is important, a boundary representation is used

When colour or texture is important, an internal representation is

used

Digital Image Processing

2

1

Representation and Description

I. Introduction

The description of a region is based on its representation,

for example a boundary can be described by its length

The features selected as descriptors are usually required

to be as insensitive as possible to variations in:

Scale;

Translation;

Rotation;

=> That is the features should be scale, translation and

rotation invariant

Digital Image Processing

3

I. Introduction

Contents:

Representation

Boundary Descriptors

Region Descriptors

Digital Image Processing

4

2

Representation and Description

II.1. Chain code

Chain codes are used to represent a boundary by a connected

sequence of straight-line segments of specified length and direction.

Typically, this presentation is based on 4- or 8-connectivity of the

segments, the direction of each segment is coded by using a

numbering scheme such as the ones shown in Fig 11.1

Digital Image Processing

5

II.1. Chain code

Generation of chain code: follow the boundary in

an clockwise direction and assign a direction to

the segment between successive pixels

Difficulties:

Code generally very long

Noise changes the code

Solution: Resample the boundary using a larger

grid spacing

Digital Image Processing

6

3

Representation and Description

II.1. Chain code

Example

Digital Image Processing

7

II.1 . Chain code

Chain code depends on starting point

Normalization: consider the code to be circular

and choose the starting point in such a way that

the sequence represents the smallest integer

Example:

Digital Image Processing

8

4

Representation and Description

II.1. Chain code

Rotation invariance: consider the first difference

in the code: This code is obtained by counting the

number of direction changes (go counter clockwise)

Example:

4-directional code: 1 0 1 0 3 3 2 2

first difference:

33133030

0

1

3

0

1

3

Scale invariance: chance grid size

Note: when objects differ in scale and orientation

(rotation), they will be sampled differently

2

2

9

Digital Image Processing

II.2. Polygonal Approximations

A digital boundary can be approximated by a pologon.

For closed curve, the approximation is exact when the number of

segments in polygon is equal to the number of points in the boundary

so that each pair of adjacent points defines a segment in the polygon.

In practice, the goal of polygonal approximation is to

capture the “essence” of boundary shape with the fewest

possible polygonal segments

Several polygonal approximation techniques of modest

complexity and processing requirements are well suited

for image processing applications.

Digital Image Processing

10

5

Representation and Description

II.2. Polygonal Approximations

Minimum perimeter polygons

Imagine the boundary as a “rubber band” and let it shrink...

The maximum error per grid cell is √2d, where d is the dimension

of a grid cell

Digital Image Processing

11

II.2. Polygonal Approximations

Merging techniques

1.

2.

3.

4.

5.

6.

7.

Consider an arbitrary point on the boundary

Consider the next point and fit a line through these two points:

E = 0 (least squares error is zero)

Now consider the next point as well, and fit a line through all

three these points using a least squares approximation.

Calculate E

Repeat until E > T

Store a and b of y = ax + b, and set E = 0

Find the following line and repeat until all the edge pixels were

considered

Calculate the vertices of the polygon, that is where the lines

intersect

Digital Image Processing

12

6

Representation and Description

II.2. Polygonal Approximations

Merging techniques

Least Squares Error

Calculate E as:

where

Digital Image Processing

13

II.2 . Polygonal Approximations

Splitting techniques

Joint the two furthest points on the boundary → line

ab

Obtain a point on the upper segment, that is c and a

point on the lower segment, that is d, such that the

perpendicular distance from these points to ab is as

large as possible

Now obtain a polygon by joining c and d with a and b

Repeat until the perpendicular distance is less than

some predefined fraction of ab

Digital Image Processing

14

7

Representation and Description

II.2. Polygonal Approximations

Splitting techniques - example

Digital Image Processing

15

II.3. Signatures

A signature is a 1-D representation of boundary. It might be

generated by various ways

Simplest approach: plot r(θ)

r: distance from centroid of boundary to boundary point

θ: angle with the positive x-axis

Digital Image Processing

16

8

Representation and Description

II.3. Signatures

Translation invariant, but not rotation or scale invariant

Normalization for rotation:

(1) Choose the starting point as the furthest point from the

centroid OR

(2) Choose the starting point as the point on the major axis that

is the furthest from the centroid

Normalization for scale:

Note: ↑ scale => ↑ amplitude of signature

(1) Scale signature between 0 and 1

(2) Divide each sample by its variance - assuming it is not zero

Problem: sensitive to noise

Digital Image Processing

17

II.3. Signatures

Alternative approach: plot Φ(θ)

Φ: angle between the line tangent to the boundary and a

reference line

θ: angle with the positive x-axis

Φ(θ) carry information about basic shape characteristics

Alternative approach: use the so-called slope density function

as a signature, that is a histogram of the tangent-angle values

Respond strongly to sections of the boundary with constant

tangent angles (straight or nearly straight segments)

Deep valleys in sections producing rapidly varying angles

(corners or other sharp inflections)

Digital Image Processing

18

9

Representation and Description

II.4. Boundary segments

Boundary segments are usually easier to

describe than the boundary as a whole

We need a robust decomposition: convex hull

A convex set (region) is a set (region) in which

any two elements (points) A and B in the set

(region) can be joined by a line AB, so that each

point on AB is part of the set (region)

The convex hull H of an arbitrary set (region) S

is the smallest convex set (region) containing S

Digital Image Processing

19

II.4. Boundary segments

Convex deficiency: D = H − S

The region boundary is partitioned by following the

contour of S and marking the points at which a

transition is made into or out of a component of the

convex deficiency

Digital Image Processing

20

10

Representation and Description

II.5. Skeletons

An important approach to representing the

structural shape of a plane region is to reduce it

to a graph.

This reduction may be ascomplished by

obtaining the skeleton of the region via a

thinning algorithm

The skeleton of a region may be defined via the

medial axis transformation proposed by Blum in

1967

Digital Image Processing

21

II.5. Skeletons

Brute force method: Medial axis transformation (MAT - 1967)

Consider a region R with boundary B...

For each point p in R find its closest neighbour in B

If p has more than one closest neighbour, then p is part of the medial axis

Example

Problem: we have to calculate the distance between every internal point

and every point on the edge of the boundary!

Digital Image Processing

22

11

Representation and Description

II.5. Skeletons

Thinning algorithm:

Edge points are deleted in an iterative way so that

1. end points are not removed,

2. connectivity is not broken, and

3. no excessive erosion is caused to the region

This algorithm thins a binary region, where an

edge point = 1 and a background point = 0

Contour point: Edge point (= 1) with at least one

neighbour with a value of 0

Digital Image Processing

23

II.5. Skeletons

Step 1: A contour point is flagged for deletion if

a) 2 < N(p1) < 6

b) T(p1) = 1

c) p2 · p4 · p6 = 0

d) p4 · p6 · p8 = 0

Where:

N(p1) ≡ number of non-zero neighbours of p1

T(p1) ≡ number of 0 − 1 transitions in the sequence

{p2, p3, p4, p5, p6, p7, p8, p9, p2}

Digital Image Processing

24

12

Representation and Description

II.5. Skeletons

Example

- N(p1) ≡ number of non-zero neighbours of p1

- T(p1) ≡ number of 0 − 1 transitions in the sequence

{p2, p3, p4, p5, p6, p7, p8, p9, p2}

Digital Image Processing

25

II.5. Skeletons

Now delete all the flagged contour points and consider

the remaining contour points...

Step 2: A contour point is flagged for deletion if

2 < N(p1) < 6

b) T(p1) = 1

c) p2 · p4 · p8 = 0

d) p2 · p6 · p8 = 0

a)

Delete all the flagged contour points

Repeat steps 1 and step 2 until no contour point is

deleted during an iteration

Digital Image Processing

26

13

Representation and Description

II.5. Skeletons

Reasons for each of these conditions...

Digital Image Processing

27

II.5. Skeletons

Example

Digital Image Processing

28

14

Representation and Description

III.1. Simple boundary descriptors

Length: The number of pixels along a boundary

Diameter: Diameter of B is defined as

Where D is a distance measure

Major Axis

Minor Axis

Basic rectangle

Eccentricity: the ratio of the major to the minor axis.

Digital Image Processing

29

III.2. Shape Numbers

Shape number: Difference of chain code that

represents smallest integer

Order n of shape number: number of digits in its

representation

Suppose that we use a 4-directional chain code,

then...

Closed boundary => n even

n limits the number of possible shapes

Digital Image Processing

30

15

Representation and Description

III.2. Shape Numbers

Example

Digital Image Processing

31

III.2. Shape Numbers

Note: the shape number is rotation invariant, but the

coded boundary depends on the orientation of the grid!

One way to normalize the grid orientation is by aliging

the chain code grid with the sides of the basic ractangle

defined in the previous section

Example: Suppose that n=18 is specified for the

boundary shown in Fig 11.12(a)

1.

2.

3.

Find the basic rectangle - Fig 11.12(b)

Subdivision of the basic rectangle with grid 3x6 - Fig 11.12(c)

Chain code directions are aligned with the resulting grid, find

the shape number Fig 11.12 (d)

Digital Image Processing

32

16

Representation and Description

III.2. Shape Numbers

Example

Digital Image Processing

33

III.3. Fourier Discriptors

Suppose that a boundary is represented by K coordinate pairs in the

xy-plane, (x0, y0), (x1, y1), (x2, y2), . . . , (xK−1, yK−1)

Digital Image Processing

34

17

Representation and Description

III.3. Fourier Discriptors

When we traverse this boundary in an anticlockwise direction the boundary can be

represented as the sequence of coordinates sk

= [xk, yk ] for k = 0, 1, 2, . . . , K − 1

Each coordinate pair can be treated as a

complex number so that sk = xk + i*yk => 2-D

problem → 1-D problem

Digital Image Processing

35

III.3. Fourier Discriptors

Sn (n=0…K-1) are called Fourier descritors of the boundary

Digital Image Processing

36

18

Representation and Description

III.3. Fourier Discriptors

Note that the same number of points exist in the approximate

boundary ˜sk. Also note that the smaller P becomes, the more detail

in the boundary is lost...

Digital Image Processing

37

III.3. Fourier Discriptors

Some basic properties of Fourier descriptors

Digital Image Processing

38

19

Representation and Description

III.4. Statistical moments

Statistical moments can be used to describe the

shape of a boundary segment

A boundary segment can be represented by a 1-D

discrete function g(r) ...

Digital Image Processing

39

III.4. Statistical moments

Digital Image Processing

40

20

Representation and Description

IV.1. Simple Regional Descriptors

Area: Number of pixels in region

Perimeter: Length of boundary

Compactness: Perimeter2/Area

Mean and median gray levels

Min and max gray level values

Number of pixels with values above or below

mean

Digital Image Processing

41

IV.2. Topological Descriptors

Topological properties are useful for global

descriptions of regions in image plane.

Topology is the study of properties of a fingure

that are unaffected by any deformation.

That properties may be:

Number of holes

Number of connected conponents

Digital Image Processing

42

21

Representation and Description

IV.2. Topological Descriptors

Example

Digital Image Processing

43

IV.2. Topological Descriptors

Euler number:

E denotes Euler number, it is denined as:

E=C–H

Where

C: The number of connected components

H: The number of holes

Euler number is also topological property.

Digital Image Processing

44

22

Representation and Description

IV.2. Topological Descriptors

Example

Digital Image Processing

45

IV.3. Texture

An important approach to region description is

to quantify its texture content.

Although no formal definition of texture exist,

intuitively this description provides measures of

properties such as

Smoothness

Coarseness and

Regularity

Digital Image Processing

46

23

Representation and Description

IV.3. Texture

Digital Image Processing

47

IV.3. Texture

Statistical approaches

Using the statistical moments of gray-level histogram of an

image or region

Let z be a random variable denoting gray levels and let p(zi),

i=0,1,.., L-1, be the corresponding histogram, the nth moment

of z about mean is

Digital Image Processing

48

24

Representation and Description

IV.3. Texture

Statistical approaches

Digital Image Processing

49

IV.3. Texture

Statistical

approaches

Digital Image Processing

50

25

Digital Image Processing

Lecture 11,12 – Representation and

Description

Lecturer: Ha Dai Duong

Faculty of Information Technology

I. Introduction

After an image has been segmented into regions by

methods such as those discussed in previous lectures,

segmented pixels usually is represented àn described in

a form suiable for further computer processing.

A segmented region can be represented by: external

(boundary) pixels or internal pixels

When shape is important, a boundary representation is used

When colour or texture is important, an internal representation is

used

Digital Image Processing

2

1

Representation and Description

I. Introduction

The description of a region is based on its representation,

for example a boundary can be described by its length

The features selected as descriptors are usually required

to be as insensitive as possible to variations in:

Scale;

Translation;

Rotation;

=> That is the features should be scale, translation and

rotation invariant

Digital Image Processing

3

I. Introduction

Contents:

Representation

Boundary Descriptors

Region Descriptors

Digital Image Processing

4

2

Representation and Description

II.1. Chain code

Chain codes are used to represent a boundary by a connected

sequence of straight-line segments of specified length and direction.

Typically, this presentation is based on 4- or 8-connectivity of the

segments, the direction of each segment is coded by using a

numbering scheme such as the ones shown in Fig 11.1

Digital Image Processing

5

II.1. Chain code

Generation of chain code: follow the boundary in

an clockwise direction and assign a direction to

the segment between successive pixels

Difficulties:

Code generally very long

Noise changes the code

Solution: Resample the boundary using a larger

grid spacing

Digital Image Processing

6

3

Representation and Description

II.1. Chain code

Example

Digital Image Processing

7

II.1 . Chain code

Chain code depends on starting point

Normalization: consider the code to be circular

and choose the starting point in such a way that

the sequence represents the smallest integer

Example:

Digital Image Processing

8

4

Representation and Description

II.1. Chain code

Rotation invariance: consider the first difference

in the code: This code is obtained by counting the

number of direction changes (go counter clockwise)

Example:

4-directional code: 1 0 1 0 3 3 2 2

first difference:

33133030

0

1

3

0

1

3

Scale invariance: chance grid size

Note: when objects differ in scale and orientation

(rotation), they will be sampled differently

2

2

9

Digital Image Processing

II.2. Polygonal Approximations

A digital boundary can be approximated by a pologon.

For closed curve, the approximation is exact when the number of

segments in polygon is equal to the number of points in the boundary

so that each pair of adjacent points defines a segment in the polygon.

In practice, the goal of polygonal approximation is to

capture the “essence” of boundary shape with the fewest

possible polygonal segments

Several polygonal approximation techniques of modest

complexity and processing requirements are well suited

for image processing applications.

Digital Image Processing

10

5

Representation and Description

II.2. Polygonal Approximations

Minimum perimeter polygons

Imagine the boundary as a “rubber band” and let it shrink...

The maximum error per grid cell is √2d, where d is the dimension

of a grid cell

Digital Image Processing

11

II.2. Polygonal Approximations

Merging techniques

1.

2.

3.

4.

5.

6.

7.

Consider an arbitrary point on the boundary

Consider the next point and fit a line through these two points:

E = 0 (least squares error is zero)

Now consider the next point as well, and fit a line through all

three these points using a least squares approximation.

Calculate E

Repeat until E > T

Store a and b of y = ax + b, and set E = 0

Find the following line and repeat until all the edge pixels were

considered

Calculate the vertices of the polygon, that is where the lines

intersect

Digital Image Processing

12

6

Representation and Description

II.2. Polygonal Approximations

Merging techniques

Least Squares Error

Calculate E as:

where

Digital Image Processing

13

II.2 . Polygonal Approximations

Splitting techniques

Joint the two furthest points on the boundary → line

ab

Obtain a point on the upper segment, that is c and a

point on the lower segment, that is d, such that the

perpendicular distance from these points to ab is as

large as possible

Now obtain a polygon by joining c and d with a and b

Repeat until the perpendicular distance is less than

some predefined fraction of ab

Digital Image Processing

14

7

Representation and Description

II.2. Polygonal Approximations

Splitting techniques - example

Digital Image Processing

15

II.3. Signatures

A signature is a 1-D representation of boundary. It might be

generated by various ways

Simplest approach: plot r(θ)

r: distance from centroid of boundary to boundary point

θ: angle with the positive x-axis

Digital Image Processing

16

8

Representation and Description

II.3. Signatures

Translation invariant, but not rotation or scale invariant

Normalization for rotation:

(1) Choose the starting point as the furthest point from the

centroid OR

(2) Choose the starting point as the point on the major axis that

is the furthest from the centroid

Normalization for scale:

Note: ↑ scale => ↑ amplitude of signature

(1) Scale signature between 0 and 1

(2) Divide each sample by its variance - assuming it is not zero

Problem: sensitive to noise

Digital Image Processing

17

II.3. Signatures

Alternative approach: plot Φ(θ)

Φ: angle between the line tangent to the boundary and a

reference line

θ: angle with the positive x-axis

Φ(θ) carry information about basic shape characteristics

Alternative approach: use the so-called slope density function

as a signature, that is a histogram of the tangent-angle values

Respond strongly to sections of the boundary with constant

tangent angles (straight or nearly straight segments)

Deep valleys in sections producing rapidly varying angles

(corners or other sharp inflections)

Digital Image Processing

18

9

Representation and Description

II.4. Boundary segments

Boundary segments are usually easier to

describe than the boundary as a whole

We need a robust decomposition: convex hull

A convex set (region) is a set (region) in which

any two elements (points) A and B in the set

(region) can be joined by a line AB, so that each

point on AB is part of the set (region)

The convex hull H of an arbitrary set (region) S

is the smallest convex set (region) containing S

Digital Image Processing

19

II.4. Boundary segments

Convex deficiency: D = H − S

The region boundary is partitioned by following the

contour of S and marking the points at which a

transition is made into or out of a component of the

convex deficiency

Digital Image Processing

20

10

Representation and Description

II.5. Skeletons

An important approach to representing the

structural shape of a plane region is to reduce it

to a graph.

This reduction may be ascomplished by

obtaining the skeleton of the region via a

thinning algorithm

The skeleton of a region may be defined via the

medial axis transformation proposed by Blum in

1967

Digital Image Processing

21

II.5. Skeletons

Brute force method: Medial axis transformation (MAT - 1967)

Consider a region R with boundary B...

For each point p in R find its closest neighbour in B

If p has more than one closest neighbour, then p is part of the medial axis

Example

Problem: we have to calculate the distance between every internal point

and every point on the edge of the boundary!

Digital Image Processing

22

11

Representation and Description

II.5. Skeletons

Thinning algorithm:

Edge points are deleted in an iterative way so that

1. end points are not removed,

2. connectivity is not broken, and

3. no excessive erosion is caused to the region

This algorithm thins a binary region, where an

edge point = 1 and a background point = 0

Contour point: Edge point (= 1) with at least one

neighbour with a value of 0

Digital Image Processing

23

II.5. Skeletons

Step 1: A contour point is flagged for deletion if

a) 2 < N(p1) < 6

b) T(p1) = 1

c) p2 · p4 · p6 = 0

d) p4 · p6 · p8 = 0

Where:

N(p1) ≡ number of non-zero neighbours of p1

T(p1) ≡ number of 0 − 1 transitions in the sequence

{p2, p3, p4, p5, p6, p7, p8, p9, p2}

Digital Image Processing

24

12

Representation and Description

II.5. Skeletons

Example

- N(p1) ≡ number of non-zero neighbours of p1

- T(p1) ≡ number of 0 − 1 transitions in the sequence

{p2, p3, p4, p5, p6, p7, p8, p9, p2}

Digital Image Processing

25

II.5. Skeletons

Now delete all the flagged contour points and consider

the remaining contour points...

Step 2: A contour point is flagged for deletion if

2 < N(p1) < 6

b) T(p1) = 1

c) p2 · p4 · p8 = 0

d) p2 · p6 · p8 = 0

a)

Delete all the flagged contour points

Repeat steps 1 and step 2 until no contour point is

deleted during an iteration

Digital Image Processing

26

13

Representation and Description

II.5. Skeletons

Reasons for each of these conditions...

Digital Image Processing

27

II.5. Skeletons

Example

Digital Image Processing

28

14

Representation and Description

III.1. Simple boundary descriptors

Length: The number of pixels along a boundary

Diameter: Diameter of B is defined as

Where D is a distance measure

Major Axis

Minor Axis

Basic rectangle

Eccentricity: the ratio of the major to the minor axis.

Digital Image Processing

29

III.2. Shape Numbers

Shape number: Difference of chain code that

represents smallest integer

Order n of shape number: number of digits in its

representation

Suppose that we use a 4-directional chain code,

then...

Closed boundary => n even

n limits the number of possible shapes

Digital Image Processing

30

15

Representation and Description

III.2. Shape Numbers

Example

Digital Image Processing

31

III.2. Shape Numbers

Note: the shape number is rotation invariant, but the

coded boundary depends on the orientation of the grid!

One way to normalize the grid orientation is by aliging

the chain code grid with the sides of the basic ractangle

defined in the previous section

Example: Suppose that n=18 is specified for the

boundary shown in Fig 11.12(a)

1.

2.

3.

Find the basic rectangle - Fig 11.12(b)

Subdivision of the basic rectangle with grid 3x6 - Fig 11.12(c)

Chain code directions are aligned with the resulting grid, find

the shape number Fig 11.12 (d)

Digital Image Processing

32

16

Representation and Description

III.2. Shape Numbers

Example

Digital Image Processing

33

III.3. Fourier Discriptors

Suppose that a boundary is represented by K coordinate pairs in the

xy-plane, (x0, y0), (x1, y1), (x2, y2), . . . , (xK−1, yK−1)

Digital Image Processing

34

17

Representation and Description

III.3. Fourier Discriptors

When we traverse this boundary in an anticlockwise direction the boundary can be

represented as the sequence of coordinates sk

= [xk, yk ] for k = 0, 1, 2, . . . , K − 1

Each coordinate pair can be treated as a

complex number so that sk = xk + i*yk => 2-D

problem → 1-D problem

Digital Image Processing

35

III.3. Fourier Discriptors

Sn (n=0…K-1) are called Fourier descritors of the boundary

Digital Image Processing

36

18

Representation and Description

III.3. Fourier Discriptors

Note that the same number of points exist in the approximate

boundary ˜sk. Also note that the smaller P becomes, the more detail

in the boundary is lost...

Digital Image Processing

37

III.3. Fourier Discriptors

Some basic properties of Fourier descriptors

Digital Image Processing

38

19

Representation and Description

III.4. Statistical moments

Statistical moments can be used to describe the

shape of a boundary segment

A boundary segment can be represented by a 1-D

discrete function g(r) ...

Digital Image Processing

39

III.4. Statistical moments

Digital Image Processing

40

20

Representation and Description

IV.1. Simple Regional Descriptors

Area: Number of pixels in region

Perimeter: Length of boundary

Compactness: Perimeter2/Area

Mean and median gray levels

Min and max gray level values

Number of pixels with values above or below

mean

Digital Image Processing

41

IV.2. Topological Descriptors

Topological properties are useful for global

descriptions of regions in image plane.

Topology is the study of properties of a fingure

that are unaffected by any deformation.

That properties may be:

Number of holes

Number of connected conponents

Digital Image Processing

42

21

Representation and Description

IV.2. Topological Descriptors

Example

Digital Image Processing

43

IV.2. Topological Descriptors

Euler number:

E denotes Euler number, it is denined as:

E=C–H

Where

C: The number of connected components

H: The number of holes

Euler number is also topological property.

Digital Image Processing

44

22

Representation and Description

IV.2. Topological Descriptors

Example

Digital Image Processing

45

IV.3. Texture

An important approach to region description is

to quantify its texture content.

Although no formal definition of texture exist,

intuitively this description provides measures of

properties such as

Smoothness

Coarseness and

Regularity

Digital Image Processing

46

23

Representation and Description

IV.3. Texture

Digital Image Processing

47

IV.3. Texture

Statistical approaches

Using the statistical moments of gray-level histogram of an

image or region

Let z be a random variable denoting gray levels and let p(zi),

i=0,1,.., L-1, be the corresponding histogram, the nth moment

of z about mean is

Digital Image Processing

48

24

Representation and Description

IV.3. Texture

Statistical approaches

Digital Image Processing

49

IV.3. Texture

Statistical

approaches

Digital Image Processing

50

25

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## Báo cáo khoa học: "The detection and representation of ambiguities of intension and description" pptx

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