#### 3.3.2. Defect Characteristic Exactions

By the procedure given, a local MFL image of the localized defects is presented, and the geometric features and moment invariants of the MFL image can be used to identify defects. The geometric features describe the basic shape of object, and the moment invariant is the average description of area gray distribution, which is calculated by all points in the area and is less susceptible to noise. In total, ten characteristics of the MFL image were selected, including the equivalent area, the slenderness ratio, and the circularity and first- to seven-order moment invariants (Φ_{1}–Φ_{7}).

- 1.
Basic Description of Image Shape

Geometric features were calculated by brief description of features, such as: area (S), perimeter (L), major axis (L_{1}), and minor axis (L_{2}). All these descriptions are as follows.

The area of defect image is:

where

R is the set of points in the defect region;

S is the amount of high-value in binary image; and

I is the binary image.

The defect perimeter is the total length of the outer boundary, which can be expressed by the sum of the distance between adjacent pixels. If the number of pixels of the outer boundary is

n, its chain code,

c_{i} is followed by

c_{1},

c_{2},

c_{3}, …,

c_{n} and the perimeter can be presented as follows:

where

L_{1} is defined as the maximum distance between any two points in the outer boundary, and

L_{2} is defined as the longest straight line which is vertical to

L_{1}, as shown in

Figure 9.

Assuming two random points are present in the outer boundary,

α_{1}(

x_{1},

y_{1}) and

α_{2}(

x_{2},

y_{2}). The endpoints of the vertical line are

v_{1}(

m_{1},

n_{1}) and

v_{2}(

m_{2},

n_{2}) to

L1. The

L_{1} and

L_{2} are calculated as follows:

- 2.
Characteristic Descriptions of Shape

Because of the different sizes of wire rope and lift-off variations, the detected defect area, perimeter, and length-width are not similar in the same case. Therefore the basic description is not taken as the recognition features. Nevertheless the equivalent area, which is the ratio of area and perimeter, represented by

G, is taken as the recognition feature:

where the equivalent area

G reflects the surrounded region of defects by unit perimeter. If the shape of the defect is circular, the ratio of area and perimeter is the minimum.

Slenderness ratio

F is defined as the ratio of the major axis

L_{1} and the minor axis

L_{2}:

The slenderness ratio reflects the shape of the defects. It is a sensitive parameter of the circular boundary. When the defect shape forms a circle, the long and short diameters are relatively close, and the F value is close to 1. The greater the ratio of the long axis to the short axis, the slimmer the shape of the defect.

The circular degree of the image is the complexity degree of the area shape measured on the basic of the area and perimeter. Its mathematical expression is as follows:

where

e is the circularity of the defect;

S is the area; and

C is the perimeter.

When the object region is circular with the radius, r, its area $S=\pi {r}^{2}$ and its perimeter is $C=\text{}2\pi r$. That is, its circularity $e\text{}=\text{}1$. This characteristic reflects the complexity of the shape in the area. If the shape is closed to a circle, e is bigger, and the maximum is value 1. If the shape is more complex, e is closer to 0.

- 3.
Characteristics of Invariant Moment

Invariant moments are established on statistical analyses of the gray distribution of the target area, a sort of statistical description on average. It describes the overall characteristics of an object from a global view, thus, it is less susceptible to noise and would not change with the translation, rotation and scale of the image [

25]. For this paper, we chose to describe the shape characteristics of the defect image.

Given an image

f(

x,y), if it is piecewise continuous, with a limited non-zero number available on the plane, its varied order exists. The two-dimensional (

p + q) order moment of

f(

x,y) is defined as [

25]:

The value of moments,

m_{pq}, will change when (

x,y) is translated. To reduce and eliminate unfavorable effects, the central moments are defined as:

where

$\overline{x}$ and

$\overline{y}$ are the center of gravity, defined as:

On the basis of the defined central moments, the seven invariant moments are defined as follows [

25]:

Equations (10) to (19) are implemented in the defects image. Seven invariant moments are calculated as the characteristic vectors of the image. In this paper, we selected four different wire rope structures as detection examples. They were 6 × 19, 6 × 36, 6 × 37, and 7 × 27. According to the characteristic extraction above, parts of vectors of one wire rope are listed in

Table 1.