Cellular Automata for Image Resizing - Cellular Automata in Image Processing and Geometry - page 29

For a given cell ( x 0 , y 0 ) and range r , the Moore neighborhood can be defined by the

following equation:

M

, ) = { ( , ) | − |≤ ( ) ,| − |≤ ( ) }

(2.3)
In most practical applications, when simulating a CA rule, it is impossible to deal
with an infinite lattice. The system must be finite and have boundaries. Clearly, a
site belonging to the lattice boundary does not have the same neighborhood as other
internal sites. In order to define the behavior of these sites, the neighborhood is
extending for the sites at the boundary. Extending the neighborhood leads to var-
ious types of boundary conditions such as periodic (or cyclic), fixed, adiabatic or
reflection.
The transition rule f determines the way in which each cell of the automaton
is updated. The state of each cell is affected by the cell values in its neighborhood
and its value on the previous time step, according to the transition rule or a set of
rules. The state of every cell in the CA is updated simultaneously by applying the
transition rule f , thus, providing an inherent parallel system.

N ( x 0 y 0 x y : x x 0 r y y 0 r

2.2.2 Canny Edge Detector

The Canny edge detector [4] is an edge detection operator that uses a multi-stage al-
gorithm to detect a wide range of edges in images. The Canny edge detector is actu-
ally an optimal technique of edge detection and creation and its application relies on
the following criteria: correct detection, accurate localization and minimal response.
To satisfy these requirements, a technique which finds the function which optimizes
a given functional was used, namely the calculus of variations. The method produces
binary edge maps by applying sequentially the following processing stages:

• Stage 1: Filtering the image. The image is initially filtered by a 2-D Gaussian fil-

ter of zero mean value and a predefined standard deviation in order to eliminate

possible noise. The result is a slightly blurred version of the original image.

σ

• Stage 2: Defining the intensity gradient of the filtered image. At this stage, ele-

mentary edge detection operators, like Sobel, are used in order to define the first
derivative both in the horizontal and the vertical direction. Thus, the gradient and
direction of each pixel are defined by the following equations:

arctan I Gy
I Gx

I Gx +

=

I Gy , E d =

E g (2.4)

• Stage 3: Non maximum suppression. Given estimates of the image gradients, a

search is applied to determine if the gradient magnitude assumes a local maxi-
mum in the gradient direction. A pixel is defined as an edge pixel if its direction
is larger than the average direction of its area.

• Stage 4: Hysteresis thresholding. The last stage intends to further reduce the

number of edge pixels that resulted during the above stages. For this purpose, two
thresholds are used. The process starts by applying a high threshold and using the