CN112782666A - Rapid detection method for marine radar target - Google Patents
Rapid detection method for marine radar target Download PDFInfo
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Abstract
The invention discloses a method for quickly detecting a marine radar target, which comprises the steps of firstly carrying out an off-line field observation test, dividing an acquired radar image into block sub-images, selecting two types of image samples as reference units, including a target radar echo sample and a pure clutter radar echo sample, respectively extracting HOG characteristic vectors from the two types of image samples, and training by using a machine learning classification method to obtain a training model; then, dividing the original radar image to be detected into blocky sub-images, and extracting the HOG characteristic vector K of each sub-image; and finally, inputting the HOG characteristic vector K into a training model, and judging whether a target exists in the image according to a classification function output result. The invention adopts a block detection strategy, when no target exists in the image, the target can be detected accurately in the follow-up process, the detection speed and the detection precision are greatly improved, the target in the navigation process can be judged more accurately and rapidly, and the influence of false alarm and false alarm omission on navigation is reduced.
Description
Technical Field
The invention relates to a method for quickly detecting a marine radar target, in particular to a method for quickly detecting a marine radar target based on HOG characteristics, and belongs to the technical field of ship target detection under marine environmental conditions.
Background
Through decades of development, modern radars are highly diversified and widely applied to various fields of modern scientific and technical research, social life, military and the like. One of the marine radars has high resolution, and is widely used in the aspects of target detection, ship safety guarantee and the like in the ship sailing process. In the field of ship navigation, the problem of target detection under the background of sea clutter is always a hotspot in research, and plays a vital role in ensuring the safety of ships.
In the aspect of target detection under the background of sea clutter, in published literature, many methods have been proposed by scholars at home and abroad, wherein point-by-point detection by using a Constant False Alarm Rate (CFAR) detection technology is the most deep and extensive method in the field of image target detection of marine radar. According to different detection types, the detection method can be generally divided into an average value type CFAR detection and an ordered statistics type CFAR detection. In 1968, Finn et al first proposed a CA-CFAR detector based on unit averaging. After this, there are mainly: the method comprises the following steps of providing a GO-CFAR detector [1] selected to the maximum, a SO-CFAR detector [2] selected to the minimum, a CFAR detector [3] of an ordered statistic class, and providing a double-threshold CFAR algorithm based on a TM-CFAR detector [4] with an average rejection, 2015, Tanjie and the like according to characteristics and a processing mode of a civil navigation radar (JRC) video image, SO that the detection accuracy is improved while the detection of the marine target under different backgrounds is realized [5 ]. In 2016, ruyi bin et al further proposed a multi-strategy CFAR detector based thereon, which proved the improvement of detection performance through simulation experiments [6 ]. In 2019, there is an article in Science Letter which studies the CFAR detection problem of slow Targets in synthetic aperture radar images [7] see references [1-7] (Amoozegar F, Sundareshan M K. constant face Target detection in filters: and processing algorithms [ C ]. Applications of Artificial Neural Networks V. International resources for Optics and Photonics,1994.Trunk, G.V. Range Resolution of Targets Using Audio Detectors [ J. IEEE Transactions on Optics and electronics, 1978, AES-14(5) 750-755, routing H. radio coding in CFAR, filtration channels and analysis [ 19, 1988. gateway J. (1988, IEEE-427, CFAR) detection in CFAR, filtration and filtration Systems [ 19. 1988. gateway J. (1988, IEEE-3. gateway J.),427, CFAR, sample detection in A. 1984. gateway J. (1988, 1984. gateway J.),445. noise, CFAR, sample detection in A. 3. gateway, 1984. gateway, bear wei, zhonwei, a double-threshold detection method [ J ] based on constant false alarm rate, Radar science and technology, 2015,13(02): 154-; new Data from Naval University of Engineering illuminants In radio and Source Research (Constant fat Alarm Rate Detection of Slow Targets In polar Along-track Interactive Synthetic Aperture radio) [ J ] Science Letter,2019 ]
According to the detection method of the CFAR detector, in terms of detection efficiency, the detection efficiency of the detector is greatly reduced along with the improvement of the radar resolution by adopting a point-by-point scanning method.
Disclosure of Invention
Aiming at the prior art, the technical problem to be solved by the invention is to provide a method for quickly detecting a marine radar target based on HOG characteristics, solve the problem of low detection efficiency caused by point-by-point line-by-line scanning of the traditional marine radar target detection, extract characteristics from the angle of image characteristic detection, quickly screen a sample with the target and improve the target detection precision.
In order to solve the technical problem, the invention provides a method for quickly detecting a marine radar target, which comprises the following steps:
step 1: carrying out an off-line field observation test, dividing the acquired radar image into block sub-images, selecting two types of image samples as reference units, including a target radar echo sample and a pure clutter radar echo sample, respectively extracting HOG (histogram of oriented gradient) characteristic vectors from the two types of image samples, using the extracted HOG characteristic vectors as input, and training by using a machine learning classification method to obtain a training MODEL MODEL;
step 2: dividing an original radar image to be detected into blocky sub-images, and extracting an HOG characteristic vector K of each sub-image according to the method for extracting the HOG characteristic vector in the step 1;
and step 3: and inputting the obtained HOG characteristic vector K of each sub-image into a training MODEL MODEL, and judging whether a target exists in the image according to a classification function output result.
The invention also includes:
1. the step 1 of extracting the HOG feature vectors from the two types of image samples respectively specifically comprises the following steps:
step 1.1: carrying out gray level processing on the two types of image samples to obtain a gray level image, and then carrying out Gamma correction, wherein the Gamma correction formula is as follows:
I(x,y)＝I(x,y)gamma
wherein x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, I (x, y) is the pixel of the selected reference unit image, gamma is any normal number less than 1;
step 1.2: the gradient operator is adopted to process the image in the horizontal direction and the vertical direction simultaneously to obtain a gradient image, the gradient image comprises the gradient size and the gradient direction, and the gradient size and the gradient direction of the pixel point I (x, y) are specifically as follows:
the gradient of pixel point I (x, y) is:
Gx(x,y)＝I(x+1,y)-I(x-1,y)
Gy(x,y)＝I(x,y+1)-I(x,y-1)
wherein x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, Gx(x,y)，Gy(x, y), I (x, y) respectively representing the horizontal direction gradient, the vertical direction gradient and the pixel value at the pixel point (x, y) in the input image;
the gradient value and gradient direction at pixel point (x, y) are:
in the formula: x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, Gx(x,y)，Gy(x, y), G (x, y), which respectively represent the horizontal direction gradient, the vertical direction gradient and the gradient value at the pixel point (x, y) in the image, and α (x, y) represents the gradient direction at the pixel point (x, y) in the image;
step 1.3: extracting the HOG characteristics of each image, specifically: dividing an image into cell cells with the size of i x j, dividing the gradient direction into k direction blocks, counting the gradient information of each cell by adopting k histograms to obtain a k-dimensional feature vector, forming each m cell into a block, scanning a sample image by using the block, wherein the scanning step length is one cell, and finally, connecting the features of all the blocks in series to obtain the HOG feature of each image.
2. In step 1, the extracted HOG feature vector is used as input, and is trained by using a machine learning classification method, and the obtained training MODEL MODEL specifically comprises:
defining a classification function f (x), and recording the classification mark of a radar echo sample containing a target as 1 and the classification mark of a pure clutter sample without the target as 0;
the classification function is calculated as:
f(x)＝wTx+b
wherein w represents a weight vector, x represents an n-dimensional vector, and b represents a bias;
randomly dividing the HOG characteristic vectors of the two types of image samples obtained in the step 1 into a training set and a sample set, taking the training set as input, selecting a classification method in machine learning to train to obtain a training MODEL MODEL, taking the test set as test input, and obtaining MODEL parameters w and x when the test result is the best.
The invention has the beneficial effects that: according to the method, after the HOG characteristics of the radar data image are obtained through an off-line field test, the detection model is determined, the HOG characteristics of the actually-measured radar data image can be input into the detection model for comparison, the radar image area without the target can be quickly eliminated, the radar image without the target is not required to be detected again after the accurate detection is carried out, the detection precision is improved, the detection time is reduced, the requirements of the radar system on the real-time performance and the accuracy of the target detection are met, and the method is used as the rough detection of the target quick detection. Compared with the single traditional CFAR method, the method adopts a block detection strategy, and when no target exists in the image, the target does not need to be accurately detected subsequently, so that the detection speed and the detection precision are greatly improved. Therefore, the method can more accurately and quickly judge the target in the navigation process, and further reduce the influence of false alarm and missed alarm on navigation.
The rapid detection technology for the marine radar target based on the HOG characteristics, which is provided by the invention, is effective in actual measurement, has higher goodness of fit with the actual measurement result, is superior to the traditional mean variance detection technology in terms of false alarm rate and total detection time consumption, and can be widely popularized and applied in marine observation equipment.
Drawings
FIG. 1 is a block sub-image containing a target radar echo;
FIG. 2 is a block sub-image of a radar echo without a target
FIG. 3 is a HOG feature diagram containing target radar returns
FIG. 4 is a HOG feature diagram of a sea clutter radar echo without a target
FIG. 5 is a flow chart of an embodiment of the present invention
Detailed Description
The following further describes the embodiments of the present invention with reference to the drawings.
The implementation steps of the invention are as follows:
step 1, extracting echo sample HOG characteristic vectors and determining a detection MODEL MODEL. And (3) carrying out an on-site observation test off line, and segmenting the acquired radar image into two types of blocky subimages. Selecting a certain amount of two types of image samples as reference units, wherein one type of image samples comprises target radar echo samples, and the other type of image samples comprises pure clutter radar echo samples, and extracting HOG characteristic vectors from the two types of samples respectively. And selecting a certain amount of HOG characteristics of the two types of samples as input, and training by utilizing a machine learning classification method to obtain a training MODEL MODEL.
And 2, extracting the echo characteristic vector K of the radar image to be detected. Dividing all radar original images to be detected into block-shaped sub-images, and extracting HOG characteristic vectors K one by one.
And 3, judging whether the target exists or not. Inputting the obtained feature vector K of each sub-image into a training MODEL MODEL, and judging whether a target exists in the image according to an output result;
the step 1 comprises the following steps:
step 1.1, carrying out an off-line field observation test, dividing the obtained radar image into blocky sub-images, selecting a certain amount of two types of image samples, and dividing the two types of image samples into a target radar echo sample and a pure clutter radar echo sample, wherein the selection of the samples meets the following three requirements:
the sub-images of the division must be equal in size.
And secondly, the sub-image containing the target needs to contain complete ship targets with various sizes, and the pure clutter radial echo needs to contain clutter areas under different conditions.
③ the sample must be representative.
Step 1.2, carrying out gray processing on the obtained complete ship target and pure clutter target images to obtain a gray image, then carrying out Gamma correction, and giving a correction formula:
the formula for Gamma correction is:
I(x,y)＝I(x,y)gamma
in the formula: x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, I (x, y) is the pixel size of the selected reference unit image coordinate point, and gamma is any normal number less than 1.
Step 1.3, processing the image in the horizontal direction and the vertical direction by adopting a gradient operator to obtain a gradient image, wherein the gradient image comprises the gradient size and the gradient direction, and the gradient size and the gradient direction of a pixel point (x, y) in the image are given:
the gradient of pixel point (x, y) is:
Gx(x,y)＝I(x+1,y)-I(x-1,y)
Gy(x,y)＝I(x,y+1)-I(x,y-1)
in the formula: x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, Gx(x,y)，Gy(x, y), I (x, y) respectively representing the horizontal direction gradient, the vertical direction gradient and the pixel value at the pixel point (x, y) in the input image;
the gradient value and gradient direction at pixel point (x, y) are:
in the formula: x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, Gx(x,y)，Gy(x, y), G (x, y), which respectively represent the horizontal gradient, the vertical gradient and the gradient value at the pixel point (x, y) in the image, and α (x, y) represents the gradient direction at the pixel point (x, y) in the image.
Step 1.4, extracting the HOG characteristics of each image, wherein the extraction process comprises the following steps: the method comprises the steps of dividing an image into small cells, namely cell cells with the size of i x j, dividing a gradient direction into k direction blocks, counting gradient information of each cell by adopting k histograms to obtain a k-dimensional feature vector, forming each m cell into one block, wherein each block has the m x k-dimensional feature vector, and scanning a sample image by using the block, wherein the scanning step length is one cell. And finally, connecting the features of all the blocks in series to obtain HOG features of each image, wherein for M × N of each image, each M units form one block, and the scanning windows in the horizontal direction and the vertical direction are (M/i-1) × (N/j-1) in total, so that each image has M × k (M/i-1) × (N/j-1) features in total.
And step 1.5, selecting a classification method in machine learning to obtain a training MODEL MODEL of the HOG feature vector through the two HOG feature vectors with a certain amount obtained in the step 1.4.
The step 2 comprises the following steps:
and 2.1, dividing the radar image to be detected into block-shaped sub-images.
And 2.2, extracting the HOG characteristic vector corresponding to each image from each sub-image according to the steps 1.2, 1.3 and 1.4 in the step 1.
The step 3 comprises the following steps:
and 3.1, when the HOG feature vector K of the block-shaped sub-image of the radar is input into a training MODEL MODEL, judging that the target exists if the output result of the classification function f (x) is 1.
And 3.2, when the HOG feature vector K of the block-shaped sub-image of the radar is input into a training MODEL MODEL, judging that no target exists if the output result of the classification function f (x) is 0.
Step 1.5 comprises the following steps:
step 1.5.1, a classification function f (x) is defined, the classification mark of the radar echo sample containing the target is 1, and the classification mark of the pure clutter sample without the target is 0.
The classification function is calculated as:
f(x)＝wTx+b
in the formula: w denotes a weight vector, x denotes an n-dimensional vector, and b denotes an offset.
Step 1.5.2, randomly dividing the two sample HOG characteristic vectors obtained in step 1.4 into p training sets and q sample sets, taking the training sets as input, selecting a classification method in machine learning to train to obtain a training MODEL MODEL, taking the rest test sets as tests, and obtaining MODEL parameters w and x when the test result is the best as optimal parameters.
Examples are given below with specific parameters.
With reference to fig. 5, the method may specifically include the following steps, in the first step, extracting an echo sample HOG feature vector to determine a detection MODEL mode, in the second step, extracting an echo feature vector K of a radar image to be detected, and in the third step, determining whether a target exists.
The X-band navigation radar for the experimental ship equipment is used in the embodiment of the invention, the experimental radar takes the rotation periodic scanning observation as an observation mode, the time resolution of a radar image is 2.7s, the radial resolution is 7.5m, and the monitoring range of the radar is a circle with the radius of 0.5-4.3 Km.
The main technical parameters of the above-mentioned X-band navigation radar are shown in table one:
table one: technical parameters of X-band navigation radar
With reference to the attached drawings 1-5, the method comprises the following specific implementation steps:
firstly, extracting an echo sample HOG characteristic vector and determining a detection MODEL MODEL. The method comprises the following steps:
step 1.1, carrying out an off-line field observation test, dividing the obtained radar image into blocky sub-images, selecting 455 groups of radar echo samples containing targets and 504 groups of pure clutter radar echo samples, wherein the selected radar echo samples containing targets are shown in fig. 1, and the selected pure clutter radar echo samples are shown in fig. 2, wherein the samples meet the following three requirements:
the sub-images of the division must be equal in size.
And secondly, the sub-image containing the target needs to contain complete ship targets with various sizes, and the pure clutter radial echo needs to contain clutter areas under different conditions.
Thirdly, the sample must be representative;
step 1.2, carrying out gray processing on the obtained complete ship target and pure clutter target images to obtain a gray image, then carrying out Gamma correction, and giving a correction formula:
the formula for Gamma correction is:
I(x,y)＝I(x,y)gamma
in the formula: x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, and I (x, y) is the pixel size at the coordinate of the selected reference unit image, in this embodiment, the gamma value is 1/2.
Step 1.3, in this embodiment, the [ -1,0,1] operator is adopted to process the image in the horizontal and vertical directions respectively to obtain a gradient image, including the gradient size and the gradient direction, and the gradient size and the gradient direction of the pixel point (x, y) in the image are given:
the gradient of pixel point (x, y) is:
Gx(x,y)＝I(x+1,y)-I(x-1,y)
Gy(x,y)＝I(x,y+1)-I(x,y-1)
in the formula: x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, Gx(x,y)，Gy(x, y), I (x, y) respectively representing the horizontal direction gradient, the vertical direction gradient and the pixel value at the pixel point (x, y) in the input image;
the gradient value and gradient direction at pixel point (x, y) are:
in the formula: x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, Gx(x,y)，Gy(x, y), G (x, y), which respectively represent the horizontal gradient, the vertical gradient and the gradient value at the pixel point (x, y) in the image, and α (x, y) represents the gradient direction at the pixel point (x, y) in the image.
Step 1.4, extracting the HOG characteristics of each image, wherein the extraction process comprises the following steps: the image is divided into small cells, in this embodiment, cells with the size of 16 × 16, the gradient direction is divided into 9 direction blocks, the gradient information of each cell is counted by using 9 histograms to obtain a 9-dimensional feature vector, each four cells are combined into one block, and then one block has a 36-dimensional feature vector, the sample image is scanned by using the block, and the scanning step length is one cell. Finally, the features of all the blocks are connected in series to obtain the HOG features of each image, for each image of 128 × 128 in the invention, each 2 × 2 unit (32 × 32) forms one block, and the scanning windows in the horizontal direction and the vertical direction are 7, so that each image has 36 × 7 × 1764 features in total. Fig. 3 is a schematic diagram of the selected any radar echo HOG characteristic containing the target, and fig. 4 is a schematic diagram of the selected any pure clutter radar echo HOG characteristic.
Step 1.5, in the embodiment, a classification method of a Support Vector Machine (SVM) in machine learning is selected for classification, the HOG features are used as input, and an SVM MODEL is used for learning to obtain a training MODEL mode.
The model of the SVM is:
f(x)＝wTx+b
in the formula: w represents the weight vector, x represents the 1764-dimensional HOG feature vector of each acquired image, b represents the bias, and f (x) is a classification function.
And (3) calculating the classification mark of the radar echo sample containing the target as 1 and the classification mark of the pure clutter sample without the target as 0, randomly extracting 959 positive and negative samples according to the HOG characteristics of the two types of echo samples obtained in the step 1.4, and selecting 850 training set samples and 109 residual testing set samples. And taking the training set as input, training by using a self-contained SVM classification tool in MATLAB, detecting the trained classification function f (x) through the test set, and continuously adjusting parameters to obtain the optimal classification parameters w and b with the best classification effect of the test set.
And secondly, extracting the echo characteristic vector K of the radar image to be detected. The method comprises the following steps:
and 2.1, dividing the radar image to be detected into block-shaped sub-images.
And 2.2, extracting the HOG characteristics corresponding to each image from the rest images to be detected according to the steps 1.2, 1.3 and 1.4 in the step 1.
And the third step is to judge whether the target exists or not. The method comprises the following steps:
step 3.1, when the HOG characteristic K of the blocky sub-image of the radar is input into a training MODEL MODEL, judging that a target exists if the output result of a classification function f (x) is 1;
and 3.2, when the HOG characteristic K of the block-shaped sub-image of the radar is input into a training MODEL MODEL, judging that no target exists if the output result of the classification function f (x) is 0.
In 2016-2018, a large amount of radar data and sea state information of relevant time intervals are acquired during the course of navigation of the experimental ship in the east sea area. The rapid marine radar target detection technology based on the HOG characteristics provided by the invention is applied to experimental analysis of a large amount of acquired radar data, and a single traditional CFAR detection algorithm and the method provided by the invention are used for carrying out comparison experiments.
The detection performances of the two detection methods are compared, the false alarm detection is used, the advantages and the disadvantages of the text method and the single traditional CFAR detection method are measured by calculating the false alarm rate, the smaller the value of the false alarm rate is, the more accurate the detection target of the method is, the performance of the detector of the invention is compared only by comparing the false alarm rates of the two methods, and the evaluation function is as follows:
in the formula: TF represents the number of negative samples judged to be positive samples, and TT represents the number of positive samples judged to be positive samples.
The experimental data were counted according to the determined detection threshold, and the results are shown in table two.
Comparison of the two detection methods
The experimental result shows that the false alarm rate obtained by target detection by the traditional CFAR method is far greater than that obtained by target detection by the method. This fully demonstrates that the method proposed herein can be used to achieve better detection results than the conventional CFAR method.
The rapid detection technology for the marine radar target based on the HOG characteristics, which is provided by the invention, is effective in actual measurement, has higher goodness of fit with the actual measurement result, is superior to the traditional mean variance detection technology in terms of false alarm rate and total detection time consumption, and can be widely popularized and applied in marine observation equipment.
Claims (3)
1. A method for quickly detecting a marine radar target is characterized by comprising the following steps:
step 1: carrying out an off-line field observation test, dividing the acquired radar image into block sub-images, selecting two types of image samples as reference units, including a target radar echo sample and a pure clutter radar echo sample, respectively extracting HOG (histogram of oriented gradient) characteristic vectors from the two types of image samples, using the extracted HOG characteristic vectors as input, and training by using a machine learning classification method to obtain a training MODEL MODEL;
step 2: dividing an original radar image to be detected into blocky sub-images, and extracting an HOG characteristic vector K of each sub-image according to the method for extracting the HOG characteristic vector in the step 1;
and step 3: and inputting the obtained HOG characteristic vector K of each sub-image into a training MODEL MODEL, and judging whether a target exists in the image according to a classification function output result.
2. The method for rapidly detecting a marine radar target according to claim 1, wherein: the step 1 of respectively extracting the HOG feature vectors from the two types of image samples specifically comprises the following steps:
step 1.1: carrying out gray level processing on the two types of image samples to obtain a gray level image, and then carrying out Gamma correction, wherein the Gamma correction formula is as follows:
I(x,y)＝I(x,y)gamma
wherein x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, I (x, y) is the pixel of the selected reference unit image, gamma is any normal number less than 1;
step 1.2: the gradient operator is adopted to process the image in the horizontal direction and the vertical direction simultaneously to obtain a gradient image, the gradient image comprises the gradient size and the gradient direction, and the gradient size and the gradient direction of the pixel point I (x, y) are specifically as follows:
the gradient of pixel point I (x, y) is:
Gx(x,y)＝I(x+1,y)-I(x-1,y)
Gy(x,y)＝I(x,y+1)-I(x,y-1)
wherein x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, Gx(x,y)，Gy(x, y), I (x, y) respectively representing the horizontal direction gradient, the vertical direction gradient and the pixel value at the pixel point (x, y) in the input image;
the gradient value and gradient direction at pixel point (x, y) are:
in the formula: x is the abscissa of the pixel point in the image, y is the abscissa of the pixel point in the image, Gx(x,y)，Gy(x, y), G (x, y), which respectively represent the horizontal direction gradient, the vertical direction gradient and the gradient value at the pixel point (x, y) in the image, and α (x, y) represents the gradient direction at the pixel point (x, y) in the image;
step 1.3: extracting the HOG characteristics of each image, specifically: dividing an image into cell cells with the size of i x j, dividing the gradient direction into k direction blocks, counting the gradient information of each cell by adopting k histograms to obtain a k-dimensional feature vector, forming each m cell into a block, scanning a sample image by using the block, wherein the scanning step length is one cell, and finally, connecting the features of all the blocks in series to obtain the HOG feature of each image.
3. A method for the rapid detection of a marine radar target according to claim 1 or 2, characterized in that: step 1, using the extracted HOG feature vector as input, training by using a machine learning classification method, and obtaining a training MODEL specifically:
defining a classification function f (x), and recording the classification mark of a radar echo sample containing a target as 1 and the classification mark of a pure clutter sample without the target as 0;
the classification function is calculated as:
f(x)＝wTx+b
wherein w represents a weight vector, x represents an n-dimensional vector, and b represents a bias;
randomly dividing the HOG characteristic vectors of the two types of image samples obtained in the step 1 into a training set and a sample set, taking the training set as input, selecting a classification method in machine learning to train to obtain a training MODEL MODEL, taking the test set as test input, and obtaining MODEL parameters w and x when the test result is the best.
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