# Analysis of Fine Crack Images Using Image Processing Technique and High-Resolution Camera
## 2 Geometric Relationships of Image Acquisition  
Measurement methods using imaging equipment measure the sizes of objects in an image coordinate system by analyzing the correlation between the size of the real object and the image. Therefore, every measurement unit has the characteristic of having pixel units constituting an image. Measurement methods using acquired images require a process of converting the measured pixel value into an engineering unit. As can be seen in Figure 1, image acquisition applies the general lens principle. A digital camera uses an image sensor such as a charge-coupled device (CCD) or a complementary metal-oxide-semiconductor (CMOS). It shows the relationship between the object plane that shows the object and the image plane that shows the image based on the image sensor in the camera. Here, the relationship between the captured image and the real object is determined using the ratio of the distance (\(O\)) from the measured object to the center point of the lens to the distance (\(i\)) from the camera lens to the surface of the image sensor, or the measured value in pixel units can be converted into actual length units by using the number of pixels (\(l\)) of the image that corresponds to the actual size (\(L\)) of the object. Ground sample distance (\(GSD\)) or ground resolution refers to the size of one pixel corresponding to the actual size of the object. The unit used for the foregoing is mm/pixel, and  
Figure 1: The principle of a thin lens. it can be obtained using the size and resolution of the camera sensor, the focal length of the lens, and the working distance. The distance conversion coefficient \(GSD\) can be generalized through the lens principle from previous researchers and the Gaussian imaging equation. The Gaussian imaging equation is as shown in Equation (1), and Equation (2) below can be obtained through the geometric relationship between the size of the object, the number of pixels in the image, the distance from the lens to the object, and the distance from the lens to the image. \[\frac{1}{O}+\frac{1}{i}=\frac{1}{f} \tag{1}\]  
\[L=\frac{O-f}{f}\times l \tag{2}\]  
where \(O\) is the distance (mm) from the lens to the object, which is the working distance; \(i\) is the distance (mm) from the lens to the image sensor; \(f\) is the length (mm); \(L\) is the size (mm) of the object; and \(l\) is the number of pixels in the image corresponding to the object. When an object is photographed using a camera, an image of the object is formed on the image sensor. In this case, the size of the object varies according to the size and resolution of the image sensor. This can be defined as the image-forming principle and is illustrated in Figure 2. The size of the image sensor is shown as \(w\times h\) (mm), and the resolution is expressed as \(W_{re}\times H_{re}\) (pixels). In this case, when the number of measured pixels is expressed as \(p_{1}\), and the corresponding image length is denoted as \(l_{1}\), the following Equation (3) can be obtained from the correlation between the horizontal and vertical sizes of the image sensor and the resolution. \[\frac{W}{w}=\frac{p_{1}}{l_{1}}\text{ or }\frac{H}{h}=\frac{p_{2}}{l_{2}} \tag{3}\]  
where \(W\) and \(H\) are the horizontal and vertical resolution values (pixels) of the sensor and \(w\) and \(h\) are the horizontal and vertical dimensions (mm) of the sensor. If Equation (3) is substituted into Equation (2), vertical or horizontal length is defined as Equation (4) below. \[L=\frac{O-f}{f}\times\frac{w\times p_{1}}{W} \tag{4}\]  
Basically, since one pixel is a square, the value of the size does not change according to the direction. Therefore, there is no need to use the values for different directions separately. Equation (4) is an equation for crack measurement and can be changed and expressed as shown by the following Equation (5). \[GSD=\frac{L-f}{f}\times\frac{S_{s}}{S_{R}}\text{, \ }CW(crack\ width)= GSD\times N_{m} \tag{5}\]  
Figure 2: Image-forming principle of the sensor.