Boundary based segmentation (edge detection)

Changes (or discontinuous) in an image amplitude are important primitive characteristics of an image that carry information about object borders.

Detection methods of image discontinuities are principal approaches to image segmentation and identification of objets in a scene.

Local discontinuities in image intensity fall into three categories: points, lines, and edges.

Segmentation by edge detection

0 10 20 30 40 50 60 70 80 90 10060

80

100

120

140

160

180

200

220

240

Distance along profile

0 10 20 30 40 50 60 70 80 90 10060

80

100

120

140

160

180

200

220

240

Distance along profile

Point and line detection

The most common way to look for an arbitrary image pattern (e.g., point, or edge) is to convolve the image with a mask of size N1×N2 (e.g., 3×3, 5×5).The size of the mask and its content depends on the type of the detected object.

W1

W4

W2

W5 W6

W3

W8 W9 W7

Detection masks

W1

W4

W2

W5 W6

W3

W8 W9 W7

Vector inner product of the mask coefficients with image gray levels covered by the mask :

zwzwzwzwzwD TN

iiiNN ==+++= ∑

=12211 L

A 3×3 detection mask

-1

-1

-1

8 -1

-1

-1 -1-1

Point detection mask:

Point detection mask

The point is rendered if:

where D is a similarity measure between the image and the template, and T is a non-negative threshold.

TD>

Line detection masks

-1

-1

2

2 -1

-1

2 -1-1

-1

-1

-1

2 -1

2

-1 -12

2

-1

-1

2 -1

-1

-1 2-1

Horizontal Vertical

+45° -45°

-1

2

-1

-1 -1

-1 -1

2 2

Line detectors

Edge detection

An edge is the boundary between two regions with distinct gray-level properties.

Edges characterise the physical extent of objects thus their accurate detection plays a key role in image analysis and pattern recognition problems.

The main idea underlying most edge-detection techniques is the computation of a local derivative of an image.

Edge detection

“Ideal” and real edge profiles

0 10 20 30 40 50 60 70 80 90 10060

80

100

120

140

160

180

200

Distance in pixel units

?

How gradient operators work?

v

Image

Firstderivative

Secondderivative

Brightnessprofile

Edgedetection by means ofgradiantoperators

Edgedetection by means ofgradiantoperators

Edge detection

Note the following points about image derivative operators:

• the magnitude of the first derivative can be used to detect the presence of an edge in an image,

• the sign of the second derivative can be used to determine whether an edge pixel is on the dark or light side of an edge,

the second derivative has z zero crossing at the midpoint of a gray-level transition.

Gradient operators

=

=∇

y

fx

f

G

Gf

y

x

∂∂∂∂

The gradient of an image f(x,y) at location (x,y) is defined as the vector:

The gradient vector points in the direction of maximum rate of changeof at (x,y). Magnitude of this vector, called simply the gradient, is computedfrom:

( ) [ ] yx/

yx GGGGfmagf +≈+=∇=∇2122

Gradient of an image

5 10 15 20

2

4

6

8

10

12

14

16

18

20

Gradient operators

Z1 Z2

Z3 Z4

X

Y

3121 zzzzf −+−≈∇

3241 zzzzf −+−≈∇

Gradient for a discrete image:

or for diagonal directions:

Examples of gradiant masks

Roberts

1 0

-10

h1

0 1

0-1

h2

g(x,y) = |h1**f(x,y)| + |h2**f(x,y)|

Examples of gradiant masks

-1

0

-1

0 0

-1

1 11

-1

-1

0

0 1

1

0 1-1

Prewitt

h1 h2

g(x,y) = |h1**f(x,y)| + |h2**f(x,y)|

g(x,y) = ([h1**f(x,y)]2 + [h2**f(x,y)]2)1/2

%Matlabh=fspecial(‘prewitt’)

Examples of gradiant masks

Sobel

1

0

2

0 0

1

-2 -1-1 h1

-1

-2

0

0 2

1

0 1-1h2

g(x,y) = |h1**f(x,y)| + |h2**f(x,y)|

g(x,y) = ([h1**f(x,y)]2 + [h2**f(x,y)]2)1/2

%Matlabh=fspecial(‘sobel’)

Sobel

Prewitt

Roberts

Original

Gradient images

Original

Gradient images

Roberts

CannySobel

%MATLABI = imread(‘cameraman.tif');BW1 = edge(I,‘roberts');BW2 = edge(I,'sobel');BW3 = edge(I,'canny');figure, imshow(BW1); figure, imshow(BW2);figure, imshow(BW3);

Edge detection procedure

The pixel location is considered as an edge location if ∇f(x,y) exceeds some threshold T. Typically, T may be selected using the cumulative histogram of the gradient image.

The edge map (a binary image) gives the necessary data for tracing the object boundaries in an image.

Gx

Gy

G=IGxI+IGyI f(x,y) Edge

map T

Thresholding

Gradient image

Edge detection

Horizontal

Vertical

Edge detection

MATLAB

Demo

Laplacian

2

2

2

22

y

f

x

ff

∂∂

∂∂ +=∇

( )864252 4 zzzzzf +++−≈∇

Laplacian of a 2-D function f(x,y) is a second derivative defined as:

The Laplacian of a discrete image can be approximated by a difference equation:

Laplacian

-1

-1

-1

8 -1

-1

-1 -1-1

0

-1

-1

4 -1

0

-1 00

(a) (b)

Two possible versions of the Laplacian masks.

%Matlabh=fspecial(‘laplacian’)

Gradient operators at work

-1-1-1

-18-1

-1-1-1

%MATLABoutput=filter2(h,input);

help fspecial-1-1-1

-19-1

-1-1-1

Laplacian in frequency domain

Fourier transform of a derivative of a function

( ){ } ( ) ( )ωω Fjxf nn =ℑ

For Laplacian one gets:

higher frequency components are „amplified”.

{ } ( ) ( )yxyx ,Ff ωωωω 222 +−=∇ℑ

Laplacian

The Laplacian plays a secondary role in edge

detection due to the following shortcomings:

• is unacceptably sensitive to noise (second

derivative),

• produces double edges,

• unable to detect edge direction.

2

2

2

22

y

f

x

ff

∂∂

∂∂ +=∇

Laplacian

A more suitable use of the Laplacian is in finding the location of edges using its zero-crossingproperty. This concept is based on convolving an image with the Laplacian of a 2-D Gaussian function of the form:

+−= 2

22

2σyx

exp)y,x(h

where σ is the standard deviation. Assume r2=x2+y2. Then, the Laplacian of h with respect to r is:

−

−=∇ 2

2

4

222

2σσσ r

expr

h

-4-2

02

4

-4-2

02

4

0

0.5

1

XY

Z

Laplacian of a 2-D Gaussian function for σ =1 (also called the Mexican hat function).

Laplacian

zero crossings

Operator LoG (Laplacian of Gaussian)

%MATLABI = imread('saturn.tif');h = fspecial('log',[5 5], 0.2);I2 = filter2(h,I)/255;imshow(I), figure, imshow(I2)

Lapalcian of an image

An image and its zero-crossings.