3.1. Image Preprocessing of Polar Co-Ordinate Sea Surface Static Features
The pixel value distribution is clustered, and the prevalent pixel categories are identified based on the features of the sea surface static feature image. The prevalent pixel categories extracted are enhanced [
36]. When a sea surface static feature image turns to RGB,
is input and the pixel-level cumulative illumination value
can be formulated as:
where
denotes the pixel value of image
at location
in channel m and the channel wise weight parameters
,
, and
is the proportion of the corresponding
pixels in channel to the total pixels of the image, and jointly meet
. By employing different weights on the
channels, high-contrast-rate colors such as yellow and orange will be suppressed and low-contrast-rate colors such as red will be amplified in the image
as
. Then, the log-average cumulative luminance
is given as in [
37]:
where
is the total number of pixels in the image and
is a relatively small quantity to avoid a zero value of
in
. Eventually, the adaptive enhancement factor map
can be obtained as:
where
denotes the maximum value of
. The aim of factor calculation is to further adaptively alter the local color value of three intensity channels at each pixel to realize image enhancement, as follows:
where
denotes the enhanced version of the original
. The area in red that has been enhanced accurately highlights the features of wind field modulation. Through the derivation operator, the local auto-enhancement image
can be constructed as:
According to , the value of local auto-enhancement image can thus vary beside the original image . Thereby, the class of low contrast rate can be indicated as the informative local region. To be specific, the average value and standard deviation of the low contrast rate in are computed. Following a three-sigma criterion in statistics, pixels in the range are considered the informative local region, while others are less informative and should be cast away after the preprocessing step for higher efficiency.
To acquire the pixel where the dominant trend is situated in a locally enhanced marine radar image
, Gaussian differential filtering is applied to the enhancement image [
37]. After Gaussian differential filtering, the image’s two-dimensional Gaussian spectral function
satisfies the following formula [
38]:
where
is the Gaussian differential filtering of the locally enhanced image
and the gray portion of the Gaussian filter can smooth the image and reduce noise.
is the two-dimensional Gaussian spectral function under the initial phase angle.
is the two-dimensional Gaussian spectral function with a phase angle of
, which takes the
, and
.
The range is optimized in depth after the Gaussian differential filtered image is searched in four neighborhoods. To produce the preprocessed marine radar image , two sets of feature groups are extracted. is the final preprocessed radar image; the portion that is black after depth optimization can clearly and precisely extract wind field stripe information.
3.2. Fast-Convergence Gray-Level Co-Occurrence Matrix
The traditional GLCM only focuses on the scenarios of direction extraction under the Cartesian co-ordinate system, which is a malfunction for the polar co-ordinated problems such as wind direction extraction for radar images. On the other hand, the accuracy of the retrieved wind direction information based on marine radar images is restricted due to practical project demands. This paper proposes a FC-GLCM to address the endogenous problem of the GLCM in order to extend the applicability of traditional GLCM to polar co-ordinate systems and improve its convergence competence to meet practical requirements. The visualization of the algorithm is shown in
Figure 5. The overall mathematical model of the FC-GLCM can be formulated as follows:
where
is the optimal value of the matrix
in the FC-GLCM at
k-th iteration,
is the cost function for jointly considering the variable pair
to obtain the minimal
, while
is the GCLM process for
applied to the
k-th iteration and
in the FC-GLCM.
is to obtain the supremum of the specific set. According to the above definitions, the value of
can also represent the optimization degree of
; thus, the supremum
denotes the minimal
among all
iterations and can obtain the corresponding optimal
. Furthermore,
is the threshold for amplitude
, while
and
are the lower bound and upper bound for the argument
, respectively. It should be noted that the world polar co-ordinates pair
can specify a co-ordinate in the polar image
. Similar variable setting rules are employed for the relative polar co-ordinate pair
and the corresponding
. Finally, the relative polar coordinate pair
is oriented at the coordinate designated by the global polar coordinate pair
.
To be specific, the GLCM under the polar co-ordinates system can be formulated according to the following statements.
The gray-level ranging section of an image
at position
varies from 0 to
and the GLCM is a matrix of size
. In the traditional GLCM direction estimation, only the GLCM matrix of a relative position is used, which corresponds to an individual pixel in the image. For a GLCM of the relative position
oriented at position
, its matrix element at
can be calculated by counting the pixel pairs, as follows [
39]:
where
denotes the counting function whose output is the number of elements in a set and
represents a line orientated at the position
with a polar amplitude value
and an angle value
with respect to the horizontal orientation of the position
. Furthermore,
is the pixel set of the image and
is the gray level at position
in the polar co-ordinate system. The normalization factor
is:
where
is the number of pixel pairs that satisfy the relative position
. Then, the gray-level co-occurrence matrix satisfies the following formula:
where
is an
-oriented increasing function, which can be written as:
where
is the distance measure function between
and its orientation point
.
This paper uses the integral of
with respect to
, which can be calculated as:
In general, the discrete form of
can be formulated as:
where
is the integration range for
.
The optimal orientation
, which denotes the minimum value of
, is calculated as the following formula:
As shown in
Figure 6, the FC-GLCM is not meaningless to repeatedly calculate k times of GLCM but uses a method of coarse–fine estimation of cyclic iteration. During the first GLCM, the algorithm finds an optimal solution in the transparent semicircle region of the graph
. At this time, the estimated wind direction is 47°, which is located at the black line of the semicircle. The estimated wind direction for the first time above
is the center and the step size is half of the last time. The GLCM is calculated again and an optimal solution is found in the purple sector area in the figure to obtain the iterative updated estimated wind direction
, which is 48°. Finally, the estimated wind direction updated by the second iteration
is the center and the step size is half of the last time. The GLCM is calculated again and an optimal solution is found in the orange sector area of the figure. Finally, the estimated wind direction after three iterations
is 48.1°. Experiments show that the algorithm converges to the unique optimal direction after three iterations. The mathematical proofs of the uniqueness of optimal direction and the convergence properties of the FC-GLCM are presented in
Appendix A.
3.3. How to Improve the Efficiency of the FC-GLCM
According to Zheng et al. [
33], the GLCM’s wind direction estimation speed is distributed in
, where fps is the frame rate, which represents the number of images the algorithm can process per second. The frame rate of the FC-GLCM algorithm proposed in this paper is distributed in
through the actual computation of the experimental phase. In practical applications, the computation speed of the actual detection data of the direction station is distributed in
. To improve the algorithm’s efficiency and meet real-time requirements, this paper proposes two interpolation algorithms based on the features of the algorithm and the processed data.
According to the algorithm’s properties, the following
Table 1 compares the time complexity of the FC-GLCM with the traditional GLCM.
As shown in
Table 1, the time complexity distribution of the FC-GLCM for each portion and the traditional GLCM is similar, with the primary amount of computation concentrated on the
and
calculations. Due to the accuracy requirements of the algorithm, only the
and
calculation processes with a large amount of calculation are interpolated. In terms of the interpolation implementation method, the sequence of interval calculation first and interpolation later is adopted. The specific implementation steps are as follows:
According to the calculation formula of , each variable’s time complexity can be simplified to , where is the interval order. Similarly, the temporal complexity after the interval calculation is .
In this paper, the mathematical expectation of the probability model distribution of the processed image is variable and there is no fixed distribution mode. Therefore, a robust and accurate interpolation approach must be required.
Approach 1: Kriging interpolation
All functions
should have:
However, due to the complexity of the spatial variables analyzed by Kriging estimation technology changing with different spatial positions and the insufficient information given by the limited amount of observation data, determining the general form of the function clearly and totally is impossible, so we can only estimate
, which is the form of
. When
is considered as a linear function of
, the following results are obtained [
40]:
The limit of this estimation is based on the linear range. It is essential to decide which standard to apply for estimating the function before determining the constant
. The minimum variance is commonly employed as the estimate standard in a Kriging estimation scheme. The formula can be written as:
Kriging technology can be used for linear minimum variance estimation.
The pan-Kriging approach was chosen because the random function involved in this study is neither defined nor constant.
Assume that the draft function
has the following form:
where
is a specific constant,
is a known function with
as its independent variable, and
is the superscript of
,
and is not equal to the product of
.
The mean value of the random function
at a point x is the drift
, as:
If it is a deterministic function that depicts the spatial trend of the random function
, then:
where
is referred to as the residual function of
at
.
If
, then the drift at
can be expressed as:
Eventually, the mutation function
with only the residual function
is listed. The pan-Kriging equations for estimating
in existence are as follows:
Similarly, the random function is also interpolated by the above method.
Approach 2: Natural neighbor interpolation
Spatial interpolation creates continuous surface modeling from discrete sample locations and estimates attribute values. Spatial autocorrelation serves as the foundation for spatial interpolation, which is the closer the distance is, the more similar the objects are in [
41].
Spatial autocorrelation is also used in natural neighbor interpolation. Its primary premise is to generate Tyson polygons for all sample locations. When interpolating unknown points, these Tyson polygons will be updated and a to-be-interpolated Tyson polygon will be constructed for unknown points. The sample points in the Tyson polygons that intersect the interpolated Tyson polygons are utilized in the interpolation. The influence weight of the to-be-interpolated Thyson polygon is determined by the intersecting area between the original Thyson polygon and the to-be-interpolated Thiessen polygon, as shown in
Figure 7. Formula (31) can be used to represent it:
where
is the interpolation result at point
,
is the weight of interpolation sample points
with respect to the interpolation point
, and
is the value at sample point
.
The weight can be written using the following formula:
where
is the area of the Tyson polygon in which the sample point participating in the interpolation is located,
is the area of the Tyson polygon in which the point
to be interpolated is located, and
is the area where the two intersect. Similarly, the random function
is also interpolated by the above method.
3.5. How to Resolve the 180° Ambiguity Problem
It is challenging to resolve the 180° ambiguity problem in spatial image processing when retrieving sea surface wind direction information. The GLCM method applied in this paper uses a counterclockwise semicircle search method to restrict the angle to 180°, but also has the option to select between a tiny angle and a 180° turnover as other algorithms. The following is a theoretical description of the modified GLCM search strategy, which can effectively solve the 180° ambiguity problem and be applied to other wind direction retrieval methods.
Firstly, the 180° search domain of the GLCM is extended to 360°, or from the counterclockwise semicircular search mode to the standard circular search mode. The 360° search range in this study domain extends the angle search range of the GLCM. Because of the mirror relationship between the angles below 180° and the angles reversed by 180°, the calculation accuracy of the GLCM method will be slightly decreased. The results of the GLCM’s wind direction inversion will also be impacted by the unstable calculation accuracy, which makes it difficult to resolve the 180° ambiguity problem.
While Step 1 overcomes the GLCM method’s inability to extract information regarding a wind direction of over 180°, the algorithm’s computation accuracy is decreased. This section utilizes a coupling angle determination strategy to address the mentioned problems and further resolve the problem of 180° ambiguity, while simultaneously increasing the calculation accuracy of the algorithm.
Source identification. The source of the sea surface wind direction can be identified based on the fact that the origins between the large and small angles of the wind are mirrored. According to the definition given in this study, the source of sea surface wind is the point at which the local maximum value is within the predominant trend range of the current wind direction data. The likelihood that the wind direction is a small-angle wind direction increases with the proximity of the wind direction source to the origin of the polar co-ordinates. Otherwise, the wind direction is a large-angle wind direction.
Trend analysis. Based on the conclusions of the source identification, the trend of the wind direction is determined as a straight line with three points: the origin of the polar co-ordinates, the source of the wind direction, and the lower boundary of the data, with the source of the wind direction serving as the initial point. The characteristics of small-angle wind are that it begins at the source of the wind and extends to the downward boundary, and the proximity of the source to the origin is another feature. However, the source of the wind direction is closer to the lower border and the trend of the large-angle wind direction extends from the source of the wind direction to the origin of the polar co-ordinates.
The distribution of the P matrix will alter based on the consequence of various trends within the calculating GLCM, which will have a positive impact on the wind direction retrieval result.
Figure 8a illustrates that the distribution of the small-angle wind direction
in the P matrix is defined by several peaks, which will lead to the general migration tendency. The distribution of
displays a multi-peak and wide-area trend, an overall left tilt, and evenly spaced pixels. As shown in
Figure 8b, the distribution of the large-angle wind direction
in the P matrix has a single peak, the characteristic value is rather considerable, and it has a consistent tendency to be non-tilting. The distribution of
exhibits a single peak, a trend confined to a narrow region, and a right-tilted, steep pixel distribution.
In summary, firstly, the feature distribution direction and position of the wind direction are judged in the process of extracting the dominant features in the preprocessing part for the input radar image. Secondly, the characteristic change trend of the wind direction is evaluated. Finally, utilizing the distinctive distribution direction and location of the wind direction that will cause the distribution of the P matrix to change, the final estimated sea surface wind direction can be accurately determined in GLCM calculations.
Figure 5 is taken as an example to illustrate this principle, where the current reference wind direction is 48°. In terms of characteristic distribution direction and wind direction, the characteristic distribution in
Figure 5 is characterized by a multi-segment distribution, with a general left-leaning dominant trend. This distribution mode will cause the P matrix to have multi-peak characteristics, resulting in an overall migration tendency. The change trend in
Figure 5 is relatively gentle, with an obvious transition trend. As a result, during the feature extraction process, the result frequently has the feature of edge augmentation and there will be a certain number of edge transition zones. Due to the smooth pixel distribution of the wind direction, the P matrix distribution presents a trend of multi-peak and large areas, as shown in
Figure 8a. Due to different characteristic distribution directions and positions, their peak positions will also be different, as shown in the peak position of the curve in
Figure 7. In addition, since the characteristic change trend of the wind direction in
Figure 5 is relatively gentle, the peak value of its P matrix distribution will also produce more secondary peaks, as shown in the pink curve in
Figure 8a.