Implementation Of Calculating
the RTS Invariant Moments Of Area
Why Using Moments of Area?
It is seen that the usage of
moments of area highly supports for the systems using Artificial intelligence.
The reason behind the fact is that these moments tend to be more specific about
the original shapes of the Ligatures (blobs). The main property of moments is
that they are Rotation, translation and Scaling independent and that's why they
are highly used for the recognition process. These moments along with other
features are then usually passed to Neural Nets to get the result.....
It is highly recommended to study
the last article before proceeding next...
http://www.codersource.net/csharp_labelling_connected_components.aspx
Moments of Area
Consider now a two dimensional light intensity function g(x,y)
(the pattern or visual input)
normalized so that the volume under the function is equal to one. For our
purposes it then behaves just like a bi-variate probability density function.
For such a function of two variables the moments
are defined as follows:
mjk= ? ?
x jyk f(x, y)dxdy
for j, k = 0,1,2,....
In this way we now are able to obtain a
countably infinite number of shape measurements for describing g(x,y).
As in the one dimensional case, an infinite number of measurements yields
an impractical system. Luckily, in practice, even if the shape of g(x,y)
is quite complicated not
many moments are needed to adequately describe it.
Affine Transformations
Consider designing a character recognition system
for machine printed characters. Two characters such as A and A
are still considered A?s and therefore quite
similar even though one is bigger than the other. Such patterns would still be
considered A?s even if they differed in location, stretching, squeezing, and
shearing. Rotation is a rather special transformation and depending on the
context it may or may not be desirable to treat patterns that are invariant
under rotation as belonging to the same pattern class. This may result, for
example, in the system?s inability to discriminate between M and W or
p and d.
Transformations such as these are special cases of affine transformations
and are treated in detail in [Ry86]. Let R2 denote the two-dimensional Euclidean
plane.
Definition:
A mapping T of R2
onto R2 is called an affine
transformation if there exists an invert as a pattern P and three of its
collapsing projections on the x, y and x+y axes.
How this method of Moments came into existence was in fact from Sigfried. He opened the book and quickly noticed that different shapes had
different formulas and gave different values even if the shapes had the same
height, width or area. He was now getting excited. He turned to the next page
and was electrified. Here was a page full of letter shapes, exactly his field of
interest. In the book they were called I-beams, X-beams and L-beams and what
have you, but to him they were just letters of the alphabet and they all had
their formulas for the moments. Sigfried realized he had stumbled on a new and
different method for extracting shape features from patterns. He put the
dynamics book back on the shelf, kissed his worn-out paperback copy of de Bono?s
Lateral Thinking and rushed back to his office to tell his room mate he
had found a topic for his thesis, feature extraction for character recognition
by the Method of Moments.
Guidelines for Use
To illustrate RTS invariant moments, we
start with a simple image containing some distinct artificial objects(
specifically text)
Now we apply Grayscale conversion to the image
to convert it to Grayscale image.
Now, we will convert the image into binary so
only two intensities will be present in the image..... Black and white or in
other words 0 and 1.
After scanning this image and labeling the
distinct pixels classes with a different grayvalue, we obtain the labeled output
image.
C# Sample Program:
The algorithm is coded in C# using
unsafe so the quality and speed of the program may not be affected. The class
BitmapData is used to read and process the pixels in the image. This is the specialty
of C# to provide such a speed even on image processing applications. There
is a set of modules that are designed to implement the
algorithm. The program scans the image to convert the image into grayscale
Levels. Then it converts the image into binary. This binary image will be
subjected to splitting it into components. Here we can us 4-Connected or
8-Connected Algorithm but we used 8-Connected Algorithm for finding the Blobs in
the image as described above.
In order to understand the basic
functionality of the program, you must consider my previous articles
<give reference here>. The program consists of
a set of classes that illustrate the basic functionality of the Moments of area.
Let us see how these classes work together to get the values of the invariant
moments. The diagram below explains the hierarchy of the classes.
There are three classes here that
can be seen in the image above. Please refer to the article "Connected
Components" to learn about the functionality of the class ConnectedComponents.
Here we will discuss how does the classes Object and Feature works at all.
Class Object:
It is a
collection of objects having the following information about each object
obtained during the process of finding the connected components.
private
int
startX=0,startY=0;
private int
finalX=0,finalY=0;
public
image.Features feature;
(startX,startY): the top left position of the
object.
(finalX,finalY): the bottom right position of
the object
feature: an object of type Features.
Process:
In the previous article, we have
seen that we get an array of objects when we pass an image to the instance of
the class ConnectedComponents. Now, we want to get some important information
from these objects. When recognition of objects is made, we segment the objects
from the image first and then get Features from these objects. Some of the
features that we calculate are RTS invariant Moments.
Class Features:
public
Features (int
startX,int startY,int
width,int height,int
l)
The constructor takes the
top left position of the object, the width and height and the label i.e.,
intensity value of the object so the object can be reached easily.
private
void calcAreaCenter()
-
It calculates the area of the
object.
-
It calculates the central position
of the object along horizontal axis.
-
It calculates the central position
of the object along vertical axis.
public
double
normCentralMoment (int
p, int q,int
objectId)
p: which moment to be
calculated as directed on the vertical axis
q: which moment to be calculated
as directed on the horizontal axis
object id: The intensity value of
the object.
The function
calculates the normalized central moment with respect to the center.
public
double centralMoment (int
p, int q,
int objectId)
p: which moment to be
calculated as directed on the vertical axis
q: which moment to be calculated
as directed on the horizontal axis
object id: The intensity value of
the object.
The function
calculates the central moment of each of the pixel value.
The final output is a colored image showing separate colors for every blob....
An array named objects of Type Class Objects contains the sizes of each blob or
object being segmented and the object also contains the features of the objects
yet obtained...
Output: A word file containing the feature list for the
RTS invariant Moments.
Attachments:
Project Files: RTS_Invariant.zip