2.1. Modified MF-FDOG Algorithm for Ground Fissure Extraction
The MF-FDOG algorithm comprises the MF algorithm and the FDOG algorithm. The MF [
26] algorithm was initially applied in the digital signal processing field to detect whether a complex signal contains a simple known signal [
27,
28]. As a digital image is one type of digital signal; thus, methods used in the digital signal field can be used for digital image processing.
In terms of the structure (e.g., width, length, shape) of ground fissures in different images, we can expand a one-dimensional template to two-dimensional space. Then, using the two-dimensional template. The function of the template is as follows:
where x and y denote variables along the horizontal and vertical directions of the two-dimensional template, respectively; and
denotes the standard deviation of the inverted Gaussian distribution. In a Gaussian distribution, the probability that variable x lies in the range of
is up to 99.7%, and the vertical profile of ground fissures is similar to an inverted Gaussian distribution (
Figure 1). Therefore, it is reasonable to select this interval as the range of variable x. In other words, the width of the ground fissure is
. Moreover, L denotes the minimum length of the ground fissure and m denotes the mean template. In computer memory, continuous digital image signals have a regular digital matrix format. Therefore, m is represented by
where
represents the response after MF processing at position
and N represents the size of the two-dimensional template.
FDOG [
24] refers to the first-order derivative function of the Gaussian function, then expands it to a two-dimensional filter template. The template is established in the same way as that of the MF template. The FDOG function is represented by
where all parameters are the same as those in Equation (1).
Figure 2a,b show examples of the three-dimensional diagram about the MF template and FDOG template, respectively. The parameters of both templates are set to
= 1.5,
= 9.0, and
= 90° (i.e., the template is along the vertical direction).
The orientation and distribution of ground fissures are arbitrary. Thus, we must process images with the optimal direction number of templates to ensure that all ground fissures are detected. In experiments, the template direction number will have a strong effect on the sensitivity and accuracy of the modified MF-FDOG algorithm; however, ground fissures are a linear target, which means that they can be regarded as a straight line with an anti-parallel characteristic in a local area [
29]. Therefore, we only need to create a template with directions of 0°–180° in two-dimensional space. To create these templates with different directions, the central pixel of the templates is set as the origin (0, 0) of the coordinate system. We must decide how many directions, N, we want in the 180° space. Then, the angular interval of direction
is calculated using Equation (4). The rotation angle,
, between the direction of the template and the x axis (see
Figure 3) and the rotation matrix
are represented by Equations (5) and (6), respectively:
Through rotation, we can obtain a group of MF and FDOG templates. The template direction corresponding to the maximum response is taken as the direction of ground fissures in the processed images. The derived maximum response represents the result of the MF operation at the current pixel location. The angle corresponding to this response represents the orientation angle of the ground fissures. The function for selecting the template angle is defined as
where
denotes the angle corresponding to the maximum response obtained by convolving an image with n MF templates processing different directions;
represents the UAV image;
denotes the template with the rotation angle
; and
denotes convolution operation.
For ground fissure extraction, the difference between the MF response matrix and the FDOG response matrix must be calculated. Before this step, to reduce the probability of false ground fissure extraction, [
25] suggested implementing some special procedures: (1) after convolution by the MF template, all negative responses should be set to zero; (2) for the FDOG response matrix, a mean filter should be performed before calculating the absolute value for all responses. The two steps are represented by
where
and
denote the response matrices obtained by convolution of the MF and FDOG templates, respectively;
and
denote the template corresponding to the maximum response of the convolution operation in n different directions, and M denotes the template of the mean filter, whose value is set to 6
[
24]. Additionally, the mean filter not only reduces noise and smooths the image, but also expands the width of
. To do this, the differential signals are enhanced.
The results indicate that it is difficult to directly differentiate between
and
matrices because their ranges are substantially different. Therefore, a sensitivity correction parameter should be used to correct their ranges. In [
25], the empirical ranges of the parameter are between 3 and 4. However, the ranges of
and
, do not satisfy the linear relationship shown by ground fissure extraction experiments using different images. Therefore, there is a limitation of using the sensitivity correction parameter to correct their ranges. In this study, we propose a solution to normalize the range of
and
by a linear stretching method. Their difference is then calculated as
where
and
indicate the response matrices of
and
stretched into [0,1], respectively.
Figure 4 is the frequency histogram of the
matrix, which exhibits a Gaussian distribution. Generally, ground fissures are small targets in UAV images, whose area is far less than that of the loess, vegetation, and other ground targets. Ground fissure signals show a strong response after the modified MF-FDOG convolution operation. It is reasonable to choose a threshold
T, according to the mean and standard deviation of
to segment the image using (11) and (12). Then, pixels with values greater than
T can be considered ground fissure locations. Finally, ground fissure candidates can be extracted from the images; i.e., locations with
:
In Equations (11) and (12), denotes the mean of ; denotes the standard deviation of ; T denotes the threshold selected for image segmentation; and denotes the final result the ground fissure extraction.
For a better description of the modified MF-FDOG algorithm, we present a simulation experiment. The simulated signals of the ground fissure and edges, as well as the results of the modified MF-FDOG algorithm, are shown in
Figure 5. Both the ground fissure and edge signals are dramatically enhanced by the MF convolution operation. However, the ground fissure information obtained by FDOG convolution shows the exact opposite response from that of MF convolution. As the response signals of edges are stronger than those of ground fissures, these two targets can be clearly distinguished by mean filter processing.
High-resolution UAV images contain copious features, representing complex and diverse textures; this increases the difficulty of ground fissure identification, as these textures can be mistaken for ground fissures. Therefore, it is inevitable that multiple false ground fissure results will result from modified MF-FDOG processing. These false results must be removed. In this study, a “non-ground fissure removal” method based on random forest (RF) classification was introduced to improve the ground fissure results obtained using the modified MF-FDOG algorithm.