Overview of Image Processing
65
detection of region boundaries, shape and object classification and so on.
The edge related operations might prepare an image for further higher-level
operations such as pattern recognition.
An edge is defined as a significant transition from an image region at one
gray level to another. Clearly, this is a subjective definition. To arrive at an
objective definition one has to quantify the degree of transition. If images are
acquired in continuous space then the sharpness of transition is measured by
the magnitude of the gradient operator. Given an image I (x,y)in continuous
space, the gradient operation is given by
v I (x, y) = - 1 + - J I,
( a. a.)
ax ay
(2.12)
where i and j are the unit vectors along the x and y directions respectively. For
digital images, several approximations to the gradient have been proposed.
The Sobel edge operator is a common and simple approximation to the
gradient. This approximates the horizontal and vertical derivatives using filter
kernels
[
-1
h](k,£) = -2
-1
o 1]
o 2
o 1
and
2
o
-2
~] ,
-1
(2.l3)
respectively. Suppose I] (m, n), and 12(m, n) are the respective outputs of applying these filters to an image I(m, n). The edge intensity is calculated as
jI?(m, n) + Ii(m, n).
An alternative to gradient approximations is provided by the Laplacian of
Gaussian (LoG) approach. For a continuous image I (x,y), the Laplacian is
provided by the operation
L{I}= (a
2
2 + (
2
2 ) I (x,y) .
ax ay
(2.14)
Unlike the gradient, the Laplacian is a scalar operator and is isotropic.
A discrete space approximation is provided by the filter kernel
h(k,e) ~ [! 1 0]
-4 1 .
1 0
(2.15 )
By itself, the Laplacian is sensitive to noise. Therefore, a smoothing filter is
first applied and then the Laplacian is performed. Because of its isotropic nature
and smooth falloff, the Gaussian function is employed as a smoothing kernel.
In continuous-space, the filter kernel for the combined Gaussian filtering and
Laplacian, otherwise known as the LoG operation is given by
1 [X2 + y2 ] x 2 +r
hLoG (x,y) = --4 - - 2 - -14 e- 2a2
na
2a
(2.16)
65
detection of region boundaries, shape and object classification and so on.
The edge related operations might prepare an image for further higher-level
operations such as pattern recognition.
An edge is defined as a significant transition from an image region at one
gray level to another. Clearly, this is a subjective definition. To arrive at an
objective definition one has to quantify the degree of transition. If images are
acquired in continuous space then the sharpness of transition is measured by
the magnitude of the gradient operator. Given an image I (x,y)in continuous
space, the gradient operation is given by
v I (x, y) = - 1 + - J I,
( a. a.)
ax ay
(2.12)
where i and j are the unit vectors along the x and y directions respectively. For
digital images, several approximations to the gradient have been proposed.
The Sobel edge operator is a common and simple approximation to the
gradient. This approximates the horizontal and vertical derivatives using filter
kernels
[
-1
h](k,£) = -2
-1
o 1]
o 2
o 1
and
2
o
-2
~] ,
-1
(2.l3)
respectively. Suppose I] (m, n), and 12(m, n) are the respective outputs of applying these filters to an image I(m, n). The edge intensity is calculated as
jI?(m, n) + Ii(m, n).
An alternative to gradient approximations is provided by the Laplacian of
Gaussian (LoG) approach. For a continuous image I (x,y), the Laplacian is
provided by the operation
L{I}= (a
2
2 + (
2
2 ) I (x,y) .
ax ay
(2.14)
Unlike the gradient, the Laplacian is a scalar operator and is isotropic.
A discrete space approximation is provided by the filter kernel
h(k,e) ~ [! 1 0]
-4 1 .
1 0
(2.15 )
By itself, the Laplacian is sensitive to noise. Therefore, a smoothing filter is
first applied and then the Laplacian is performed. Because of its isotropic nature
and smooth falloff, the Gaussian function is employed as a smoothing kernel.
In continuous-space, the filter kernel for the combined Gaussian filtering and
Laplacian, otherwise known as the LoG operation is given by
1 [X2 + y2 ] x 2 +r
hLoG (x,y) = --4 - - 2 - -14 e- 2a2
na
2a
(2.16)