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Biomedical Signal and Image Processing
Compactness: While both area and perimeter are important features in image processing, it is often desirable to combine these two measures to create a rather unified
size measure. This measure is called compactness and is defined as follows:
Perimeter
2
Compactness =
(7.1)
Area
By dividing these two measures, compactness provides a feature that identifies the
size of the perimeter for a given unit of area. The reason why perimeter appears in the
equation as a squared order term is rather straightforward; the resulting measure is not
supposed to have a unit. Compactness can easily distinguish between long and narrow
oval-shaped objects that have large compactness values and circular objects that have
small compactness values. Since many objects in biomedical images have oval and
circular shapes (e.g., cells, nuclei, and tumors), features such as compactness are often
considered as the main geometric measures during the classification process.
Major and minor axes: In order to express the dimension of the objects, it is a common
practice to calculate the major axis of the object. The major axis is defined as the line
connecting a pair of points located on the contour of the object whose distance from
each other is maximal. In other words, in order to find the major axis, a pair of points
on the contour is found whose distance from each other is more than any other pair
of points on the contour. The line connecting these two points is the major axis. It is
straightforward to see that the major axis of an oval is the line passing through both
focal points. The length of the major axis is the largest dimension of the object that
has physical and biological significance. The axis perpendicular to the major axis is
called the minor axis. The minor axis of an oval is the smallest line connecting a pair
of points on the contour. In a circle, any line passing though the center is both a major
and a minor axis. The major and minor axes are important diagnostics features in cell
image classification.
Eccentricity: An important feature called eccentricity is defined to evaluate the
deviation of the object’s shape from a symmetric circular shape. Eccentricity is
defined as follows:
Length of Major Axis
Eccentricity =
(7.2)
Length of Minor Axis
As can be seen, while the eccentricity of a circle is one, the eccentricity of an oval is
always more than one. In addition, the larger this measure is, the less circular shape
(and more linear) the object must be.
Fourier descriptors: An important feature of tumors and cells is the smoothness of
the contour. This feature can be captured using measures called Fourier descriptors
(FDs). The FDs are essentially the discrete Fourier transform (DFT) of the points on
the contour of the object. Assume that the points (x 0 , y 0 ), (x 1 , y 1 ),…, (x K−1 , y K−1 ) are
consecutive points forming the contour of the object. In order to form FDs, first, a
sequence of complex numbers is formed as z i = x i + y i , i = 0, 1,…, K − 1. Then, the
Biomedical Signal and Image Processing
Compactness: While both area and perimeter are important features in image processing, it is often desirable to combine these two measures to create a rather unified
size measure. This measure is called compactness and is defined as follows:
Perimeter
2
Compactness =
(7.1)
Area
By dividing these two measures, compactness provides a feature that identifies the
size of the perimeter for a given unit of area. The reason why perimeter appears in the
equation as a squared order term is rather straightforward; the resulting measure is not
supposed to have a unit. Compactness can easily distinguish between long and narrow
oval-shaped objects that have large compactness values and circular objects that have
small compactness values. Since many objects in biomedical images have oval and
circular shapes (e.g., cells, nuclei, and tumors), features such as compactness are often
considered as the main geometric measures during the classification process.
Major and minor axes: In order to express the dimension of the objects, it is a common
practice to calculate the major axis of the object. The major axis is defined as the line
connecting a pair of points located on the contour of the object whose distance from
each other is maximal. In other words, in order to find the major axis, a pair of points
on the contour is found whose distance from each other is more than any other pair
of points on the contour. The line connecting these two points is the major axis. It is
straightforward to see that the major axis of an oval is the line passing through both
focal points. The length of the major axis is the largest dimension of the object that
has physical and biological significance. The axis perpendicular to the major axis is
called the minor axis. The minor axis of an oval is the smallest line connecting a pair
of points on the contour. In a circle, any line passing though the center is both a major
and a minor axis. The major and minor axes are important diagnostics features in cell
image classification.
Eccentricity: An important feature called eccentricity is defined to evaluate the
deviation of the object’s shape from a symmetric circular shape. Eccentricity is
defined as follows:
Length of Major Axis
Eccentricity =
(7.2)
Length of Minor Axis
As can be seen, while the eccentricity of a circle is one, the eccentricity of an oval is
always more than one. In addition, the larger this measure is, the less circular shape
(and more linear) the object must be.
Fourier descriptors: An important feature of tumors and cells is the smoothness of
the contour. This feature can be captured using measures called Fourier descriptors
(FDs). The FDs are essentially the discrete Fourier transform (DFT) of the points on
the contour of the object. Assume that the points (x 0 , y 0 ), (x 1 , y 1 ),…, (x K−1 , y K−1 ) are
consecutive points forming the contour of the object. In order to form FDs, first, a
sequence of complex numbers is formed as z i = x i + y i , i = 0, 1,…, K − 1. Then, the