CN111723335B - Target symmetry axis detection method based on concentric circumference filter - Google Patents
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Abstract
The invention discloses a target symmetry axis detection method based on a concentric circumference filter. And then taking the overlapped part of the concentric circumference and the target edge as a sampling point, carrying out statistics on angle information under a polar coordinate system through coordinate system conversion, and finally carrying out angle judgment to obtain the direction of the symmetry axis.
Description
Technical Field
The invention relates to a target symmetry axis detection method based on a concentric circumference filter, and belongs to the technical field of digital image processing.
Background
In supervised learning, the classification problem can be generally decomposed into two parts, namely, the image extraction features and the classification method. A strong feature can have the ability to well distinguish different attributes of different targets, so the core part of the classification problem is the feature extraction method. However, many features do not directly have rotation invariance, and therefore, data enhancement is required when the sample direction is not sufficient, and a lot of work is added on training samples.
The circular filter is a filter for detecting a target by utilizing Fourier transform of a circular vector, essentially only amplitude information is utilized, and phase information does not play any role, so that information waste is caused.
Aiming at the situation, polar coordinates are introduced into a circular filter, the collinear situation of the symmetrical targets is solved through relative coordinate clustering, and finally a tangent function is calculated to obtain the direction of the symmetrical axis. After the direction of the symmetry axis is found, the target can be rotated to a specific direction no matter what the original direction of the target is, so that certain rotation invariance can be ensured no matter what features are extracted subsequently.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the problems in the prior art, the invention provides a target symmetry axis detection method based on a concentric circumference filter. According to the method, the concentric circumference filter is constructed, collinear pixel points are accumulated, the collinearity is quantized according to the similarity of relative angles, and the direction of a target symmetry axis can be effectively calculated.
The technical scheme is as follows: in order to realize the purpose of the invention, the technical scheme adopted by the invention is as follows: a target symmetry axis detection method based on a concentric circumference filter comprises the following steps:
(1) Constructing a concentric circular filter: the input target image is taken as an original image img, the original image img is filtered through a circumferential filter, corresponding frequency can be extracted through priori knowledge, a frequency response graph can be obtained, a frequency response threshold value is preset, a series of concentric circles with the same circle center and the same radius and gradually increased can be constructed by taking the frequency position larger than the frequency response threshold value as the center, frequency response is extracted from each concentric circle, and the concentric circles smaller than the frequency response threshold value are discarded, so that a concentric circle filter can be obtained;
(2) Angle statistics: firstly, extracting an edge image edge from an original image img, overlapping a concentric circle filter with the edge image edge, wherein the overlapped part is a sampling point, converting the coordinate of the sampling point from a Cartesian coordinate to a polar coordinate, then calculating the relative angle of the sampling point relative to a reference point by taking the alpha-layer circle sampling point as the reference point, measuring the similarity between the relative angles of the sampling points of different layers, and finally performing collinear statistics by taking the similar angle as a collinear angle;
(3) And (3) angle judgment: and converting the angle into a tangent value, and obtaining the angle in the direction of the symmetry axis.
Further, in the step (1), a specific calculation method for constructing the concentric circular filter is as follows:
(1.1) assuming that a target needing to be subjected to symmetry axis detection is an airplane target, the priori knowledge shows that a general airplane is provided with a left wing, a right wing, a front fuselage and a rear fuselage in front of and behind the wings, and 4 components in total, so that a circumference can be constructed on an input optical image img, a circumference vector is taken as a one-dimensional signal to be subjected to Fourier transform, whether the circumference contains the target or not can be judged according to a transform result, and the coordinates of the target on the img can be judged by updating all coordinates on the img to the circle center of the circumference:
where π represents the circumference ratio, sin and cos represent the sine and cosine functions, respectively, and z k Expressing as the kth pixel value on the circumference, length expresses the total number of circumferential pixels, fourier (p, q) expresses the Fourier transform response corresponding to the circumferential vector with the circle center (p, q), and simultaneously expresses that the response value is placed at the position (p, q) of the frequency response graph, and the frequency response graph fourier can be generated by sequentially updating all coordinates on the original graph img as the circle center for response extraction;
(1.2) constructing a concentric circumference filter by taking the frequency response position larger than the frequency response threshold as the center of a circle from the frequency response graph fourier:
circleFilter＝[filter 0 (p,q),...,filter num (p,q)]s.t.fourier(p,q)≥FreqThres
wherein, fourier (p, q) is expressed as frequency response on coordinates (p, q), freqThres is frequency response threshold, filter 0 (p, q) as the layer 0 circumference filter with the circle center (p, q), namely the layer 0 circumference, and the same principle is that the filter num (p, q) is the num layer of circumference filter with the circle center (p, q), namely the outermost layer circumference, num is the total number of the circumferences, and circleFilter represents a concentric circumference filter set;
wherein, the single-layer circumference curve equation with the circle center being (p, q) is as follows:
wherein (x, y) represents the coordinates of a pixel point on the circumference, four k (p, q) and radius k (p, q) are each a filter corresponding to a circular filter k Fourier transform response and radius of (p, q), and filter k (p, q) is the k-th layer circular filter with the center of the circle being (p, q), the frequency response of each circle in the concentric circles is greater than the frequency response threshold value FreqThres, and once the frequency response is less than the threshold value, the circle is determined to be beyond the target boundary.
Further, in the step (2), a specific statistical method of angle statistics is as follows:
(2.1) firstly, extracting an edge image edge from img by using a canny operator, overlapping the edge image edge with a circleFilter, wherein the overlapped part is a sampling point, and a sampling point set
Wherein->
Representing a circular filter j The coordinates of the ith sampling point are (p, q), and edge (p, q) represents the pixel value at the coordinates (p, q) on the edge image edge, and similarly, the filter j (p, q) denotes a circular filter j A pixel value at coordinate (p, q);
(2.2) obtaining the polar coordinate representation of the sampling point by using the following coordinate system conversion formula:
wherein,
is the sampling point->
Relative angle relative to the origin in a polar coordinate system, and
is the sampling point->
Expressed in terms of the distance from the origin in a polar coordinate system>
Then it is the sampling point pick>
Abscissa in the Cartesian coordinate system, for the same reason, in>
Then it is the sampling point pick>
Ordinate in cartesian coordinate system;
(2.3) taking the sampling points on the alpha layer circumference as reference points, namely calculating the relative angles of the rest sampling points relative to the reference points, wherein the following formula gives calculation by taking the gamma layer and the alpha layer as examples
The specific mode of (1):
wherein,
represents the ith sample point of the most gamma layer, and>
represents the jth sampling point of the alpha-most layer, and
is the relative angle between the two sampling points;
(2.4) by measuring the relative angle similarity between the sampling point of the other layer and the sampling point of the beta layer, two sampling points with the difference smaller than a similarity threshold value can be considered to be collinear, and the circumferences of the two collinear layers are considered to be in the same weight
Marking:
wherein,
the method is characterized in that the relative angles of the ith layer q point and the 1 st layer p point are respectively determined, the alpha layer j point is used as a starting point, the similar thresholds are used as similar thresholds, the above formula constraint shows that all sampling points of the ith layer are traversed, and if the sampling points of the ith layer and the/H are available>
Is less than the similarity threshold, then it is assumed that the two layers are collinear and that the similarity weight->
Indicating whether or not the i-th layer contains collinearly>
Is sampled, the similar weights of all layers are summed to obtain->
Number of collinear layers in direction:
Wherein m represents the total number of alpha layer sampling points, sW p Representing the statistical result with the largest number of collinear layers of the p-th sampling point in the beta-th layer under the condition of all different reference circumference starting points, and obtaining the similarWeight = [ sW ] after the statistical results of all the points on the beta-th layer are all calculated 1 ,...,sW n ]N represents the total number of sampling points of the beta layer, namely a statistical result set;
(2.5) reserving the direction in which the number of layers is greater than the threshold value of the collinear number of layers, and regarding sampling points in the direction smaller than the threshold value as interference:
wherein,
represents the statistical result that the p th collinear layer number in the beta layer meets the condition, layerThres is a collinear layer number threshold, pick _ num is the total number of sampling points meeting the condition, and/or is/is selected>
And showing that the statistical result set of which the number of collinear layers is smaller than the collinear threshold layerThres is screened out.
Further, in step (3), a specific determination method of the angle determination is as follows:
(3.1) step (2) to obtain
Is arranged therein>
The corresponding concrete similarity weight is->
Thus, by similar weight, the relative angle set retained by the beta layer is combined into ≦ based on the weight of the beta layer>
Since the target of the airplane is divided into 4 parts such as the front fuselage, the rear fuselage, the left wing, the right wing and the like, the relative angle set can be divided into 4 parts by the KNN method, and the relative angle can be further expressed by 4 angles, namely [ relative 1 ,...,relative 4 ]Substituting the relative angle value into the tangent function, and determining the direction of the symmetry axis by the following formula:
symmetryAngle＝relative j s.t.tan(relative i )-tan(relative j )＝tan(relative j )-tan(relative k )
wherein tan represents a tangent function, relative i 、relative k Respectively showing the relative angles of the left wing and the right wing j I.e. the relative angle of the fuselage, and symmertyangle is the direction of the symmetry axis, since the left wing and the right wing are symmetric to the fuselage, the difference between the relative angles should be consistent, and in the actual operation, to avoid the calculation of the negative angle, all the negative angles are added with pi so as to map all the angles to 0, pi]In order to ensure that mapping does not interfere with judgment, the problem can be solved by utilizing the characteristic that tangent values of angles before and after adding pi are consistent, and if the condition of the formula is met, the symmetry angle of the machine body can be judged to be the direction of the symmetry axis.
Has the advantages that: compared with the prior art, the technical scheme of the invention has the following beneficial technical effects:
(1) A target symmetry axis detection method based on a concentric circumference filter is provided. Aiming at the problem that the circumferential filter does not utilize phase information, the collinear condition of the pixel points is counted from the accumulated angle of the relative angle, the relative angle is further judged, the detection task of the symmetry axis is realized, and the rotation invariance of the subsequent characteristics is positively influenced.
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FIG. 1 is a block diagram of an embodiment of the present invention.
Detailed Description
The present invention is further illustrated by the following examples, which are intended to be purely exemplary and are not intended to limit the scope of the invention, as various equivalent modifications of the invention will occur to those skilled in the art upon reading the present disclosure and fall within the scope of the appended claims.
As shown in fig. 1, a target symmetry axis detection method based on a concentric circular filter includes the following steps:
firstly, an input target image is taken as an original image img, the original image img is filtered through a circumferential filter, corresponding frequencies can be extracted through priori knowledge, a frequency response graph can be obtained, a frequency response threshold value is preset, a series of concentric circles can be constructed by taking the frequency position larger than the frequency response threshold value as the center, frequency response is extracted from each concentric circle, the concentric circles smaller than the frequency response threshold value are discarded, and the concentric circle filter can be obtained.
Firstly, supposing that a target needing to be subjected to symmetry axis detection is an airplane target, the airplane is provided with a left wing, a right wing, a front fuselage and a rear fuselage in front of and behind the wings, and 4 components in total, so that a circumference can be constructed on an input optical image img, the circumference vector is regarded as a one-dimensional signal to be subjected to Fourier transform, whether the circumference contains the target or not can be judged according to a transform result, and all coordinates are updated to the circle center of the circumference on the img so as to judge the coordinates of the target on the img:
where π represents the circumference ratio, sin and cos represent the sine and cosine functions, respectively, and z k The method comprises the steps of representing the k-th pixel value on the circumference, representing the total number of circumferential pixels by length, representing Fourier (p, q) representing Fourier transform response corresponding to a circumferential vector with the circle center being (p, q), simultaneously representing that a response value is placed at the position (p, q) of a frequency response graph, and sequentially updating all coordinates on an original graph img as the circle center to perform response extraction, so that the frequency response graph fourier can be generated.
Then, constructing a concentric circumference filter by taking the frequency response position larger than the frequency response threshold as the center of a circle from the frequency response graph fourier:
circleFilter＝[filter 0 (p,q),...,filter num (p,q)]s.t.fourier(p,q)≥FreqThres
wherein, fourier (p, q) is expressed as frequency response on coordinates (p, q), freqThres is frequency response threshold, filter 0 (p, q) as the layer 0 circumference filter with the circle center (p, q), namely the layer 0 circumference, and the same principle is that the filter num (p, q) is the num layer of circumference filter with the circle center of (p, q), namely the outermost circumference, num is the total number of circumferences, and circleFilter represents the concentric circumference filter set.
The single-layer circumference curve equation with the circle center (p, q) is as follows:
wherein (x, y) represents the coordinates of a pixel point on the circumference, fourier k (p, q) and radius k (p, q) are each a filter corresponding to a circular filter k Fourier transform response and radius of (p, q), and filter k (p, q) is the k-th layer circular filter with the center of the circle being (p, q), the frequency response of each circle in the concentric circles is greater than the frequency response threshold value FreqThres, and once the frequency response is less than the threshold value, the circle is determined to be beyond the target boundary.
And secondly, extracting an edge image edge from the original img by using a canny operator, overlapping the concentric circle filter with the edge image edge, wherein the overlapped part is a sampling point, and converting the coordinate of the sampling point from a Cartesian coordinate to a polar coordinate. Then, taking the 0 th layer of circumferential sampling points as reference points, calculating the relative angles of the sampling points relative to the reference points, measuring the similarity among the relative angles of the sampling points on different layers, and finally, regarding the similar angles as collinear angles to carry out collinear statistics.
Firstly, extracting an edge image edge from img by using a canny operator, overlapping the edge image edge with a circleFilter, wherein the overlapped part is a sampling point, and a sampling point set
Wherein
Representing a circular filter j The coordinates of the ith sampling point are (p, q), and edge (p, q) represents the pixel value at the coordinates (p, q) on the edge image edge, and similarly, the filter j (p, q) denotes a circular filter j Pixel value at coordinate (p, q).
Then, the polar coordinate representation of the sampling point can be obtained by using the following coordinate system conversion formula:
wherein,
is the sampling point->
Relative angle relative to the origin in a polar coordinate system, and
is the sampling point->
Expressed in terms of the distance from the origin in a polar coordinate system>
Then it is the sampling point pick>
Abscissa in the Cartesian coordinate system, for the same reason, in>
Then it is a sampling point>
Ordinate in cartesian coordinate system.
Then, using the sampling points on the circumference of layer 0 as reference points, the relative angles of the rest sampling points to the reference points can be calculated, and the following formula gives the calculation by taking layer 1 and layer 0 as examples
The specific mode of (1): />
Wherein,
represents the jth sampling point of the most 0 layer, <' > or>
Represents the ith sample point of the most 1 layer, and->
Is the relative angle between the two sample points.
Then, by measuring the relative angle similarity between the sampling points of other layers and the sampling point of the layer 1, two sampling points with the difference smaller than a similar threshold value can be determined to be collinear, and the circumferences of the collinear two layers are determined to be collinear by similar weights
Marking:
wherein,
the method is characterized in that the relative angles of the qth point of the ith layer and the pth point of the 1 st layer are respectively determined, the jth point of the 0 th layer is taken as a starting point, the similarThres is taken as a similar threshold, and the above formula constraint shows that all sampling points of the ith layer are traversed, and if the sampling points of the ith layer and the/or the & lt/EN & gt exist>
Is less than the similarity threshold, then it is assumed that the two layers are co-linear. And a similar weight->
Indicating whether or not the i-th layer contains collinearly>
The sampling points of (a). The sum of similar weights for all layers results in->
Number of collinear layers in direction:
Wherein m represents the total number of layer 0 samples, sW p Representing the statistical result with the largest number of collinear layers of the p-th sampling point in the 1 st layer in the situation of all different reference circumference starting points, and obtaining the similarWeight = [ sW ] after the statistical results of all the points on the 1 st layer are all calculated 1 ,...,sW n ]And n represents the total number of layer 1 sample points, i.e., the statistical result set.
And finally, reserving the direction in which the number of layers is greater than the threshold value of the collinear number of layers, and regarding sampling points in the direction smaller than the threshold value as interference:
wherein,
represents the statistical result that the p th collinear layer number in the layer 1 meets the condition, layerThres is a collinear layer number threshold, pick _ num is the total number of sampling points meeting the condition, and->
And showing that the statistical result set of which the number of collinear layers is less than the collinear threshold value layerThres is screened out.
And thirdly, converting the angle into a tangent value, and obtaining the angle in the direction of the symmetry axis.
In the second step canTo obtain
Is arranged therein>
The corresponding specific similarity weight is
Therefore, as can be seen from the similar weights, the set of relative angles retained by layer 1 is
Since the target of the airplane is divided into 4 parts such as the front fuselage, the rear fuselage, the left wing, the right wing and the like, the relative angle set can be divided into 4 parts by the KNN method, and the relative angle can be further expressed by 4 angles, namely [ relative 1 ,...,relative 4 ]Substituting the relative angle value into the tangent function, and determining the direction of the symmetry axis by the following formula:
symmetryAngle＝relative j s.t.tan(relative i )-tan(relative j )＝tan(relative j )-tan(relative k )
wherein tan represents a tangent function, relative i 、relative k Respectively showing the relative angles of the left wing and the right wing j I.e. the relative angle of the fuselage, symmetryAngle is the direction of the symmetry axis, the left wing and the right wing should have the same relative angle difference because they are symmetric to the fuselage, and in the actual operation, to avoid the calculation of the negative number angle, the negative number angle is added by pi so as to map all angles to 0, pi]In order to ensure that mapping does not interfere with judgment, the problem can be solved by utilizing the characteristic that the tangent values of the angle before and after adding pi are consistent, and if the condition of the formula is met, the symmetrical angle of the machine body can be judged to be the direction of the symmetrical axis.
Claims (3)
1. A target symmetry axis detection method based on a concentric circle filter is characterized by comprising the following steps:
(1) Constructing a concentric circular filter: taking an input target image as an original image img, filtering the original image img through a circumferential filter, extracting preset corresponding frequency to obtain a frequency response image, presetting a frequency response threshold, constructing a series of concentric circles with the same circle center and gradually increased radius by taking a frequency position larger than the frequency response threshold as a center, extracting frequency response from each concentric circle, and discarding the concentric circles smaller than the frequency response threshold to obtain a concentric circle filter;
(2) Angle statistics: firstly, extracting an edge image edge from an original image img, overlapping a concentric circle filter with the edge image edge, wherein the overlapped part is a sampling point, converting the coordinate of the sampling point from a Cartesian coordinate to a polar coordinate, then calculating the relative angle of the sampling point relative to a reference point by taking the alpha-layer circle sampling point as the reference point, measuring the similarity between the relative angles of the sampling points of different layers, and finally, regarding the similar angle as a collinear angle to carry out collinear statistics;
(3) And (3) angle judgment: converting the angle into a tangent value to obtain a symmetry axis direction angle;
the angle statistical method in the step (2) is as follows:
(2.1) firstly, extracting an edge image edge from img by using a canny operator, overlapping the edge image edge with a circleFilter, wherein the overlapped part is a sampling point, and the sampling point set comprises:
wherein,
representing a circular filter j The coordinates of the ith sampling point are (p, q), and edge (p, q) represents the pixel value at the coordinates (p, q) on the edge image edge, and similarly, the filter j (p, q) denotes a circular filter j A pixel value at coordinate (p, q);
(2.2) obtaining the polar coordinate representation of the sampling point by using the following coordinate system conversion formula:
wherein,
as a sampling point
Relative angle relative to the origin in a polar coordinate system, and
as a sampling point
The distance from the origin in a polar coordinate system represents,
is the sampling point
The abscissa in the cartesian coordinate system, similarly,
is the sampling point
Ordinate in cartesian coordinate system;
(2.3) taking the sampling points on the alpha layer circumference as reference points, and calculating the relative reference points of the rest sampling pointsThe following formula gives the calculation by taking the gamma layer and the alpha layer as an example
The specific mode of (1):
wherein,
represents the ith sample point of the gamma-most layer,
represents the jth sampling point of the alpha-most layer, and
is the relative angle between the two sampling points;
(2.4) determining two sampling points with the difference smaller than a similarity threshold as collinear by measuring the relative angle similarity of the sampling points of other layers and the sampling point of the beta layer, and determining the two sampling points with the difference smaller than the similarity threshold as collinear two-layer circumferences with similar weights
Marking:
wherein,
the relative angles of the ith layer of the q point and the beta layer of the p point are respectively shown, the alpha layer of the j point is used as a reference point, the similarThres is used as a similar threshold value, the above formula constraint shows that all sampling points of the ith layer are traversed, if the ith layer of the sampling points and the beta layer of the p point exist, the ith layer of the sampling points and the beta layer of the p point are used as the reference points, the similarity between the sampling points is used as the similarity threshold value, and the formula constraint shows that the ith layer of the p point is used as the similarity threshold value
Is less than the similarity threshold, then the two layers are considered co-linear and the similarity weight is determined to be
Indicating whether the ith layer contains collinearity to
The similar weights of all layers are summed to obtain
Number of collinear layers in direction:
Wherein m represents the total number of alpha layer sampling points, sW p Representing the statistical result with the maximum number of collinear layers of the p-th sampling point in the beta-th layer under the condition of all different reference circumference starting points, and obtaining the simiarweight = [ sW ] after the statistical results of all the points on the beta-th layer are all calculated 1 ,...,sW n ]N represents the beta layerThe total number of sampling points is the statistical result set;
(2.5) reserving the direction in which the number of layers is greater than the threshold value of the collinear number of layers, and regarding sampling points in the direction smaller than the threshold value as interference:
wherein,
showing the statistical result that the p-th collinear layer number in the beta layer meets the condition, layerThres is a collinear layer number threshold value, pick _ num is the total number of sampling points meeting the condition,
and showing that the statistical result set of which the number of collinear layers is smaller than the collinear threshold layerThres is screened out.
2. The concentric circular filter-based target symmetry axis detection method according to claim 1, wherein the method for constructing the concentric circular filter in step (1) is as follows:
(1.1) assuming that a target needing to be subjected to symmetry axis detection is an airplane target, the airplane is provided with a left wing, a right wing, a front fuselage and a rear fuselage in front of and behind the wings, and 4 components in total, so that a circumference is constructed on an input optical image img, a circumference vector is regarded as a one-dimensional signal to be subjected to Fourier transform, whether the circumference contains the target or not is judged according to a transform result, all coordinates are updated to the circle center of the circumference on the img, and the coordinates of the target on the img are judged:
where π represents the circumference ratio, sin and cos represent the sine and cosine functions, respectively, and z k Expressed as the kth pixel value on the circumference, and length expresses the circumferential pixel pointThe total number, namely Fourier (p, q) represents Fourier transform response corresponding to a circumferential vector with a circle center (p, q), and simultaneously represents that a response value is placed at the position (p, q) of the frequency response graph Fourier, and all coordinates on the original graph img are sequentially updated as the circle center to perform response extraction to generate the frequency response graph Fourier;
(1.2) constructing a concentric circumference filter by taking the frequency response position larger than the frequency response threshold as the center of a circle from the frequency response graph fourier:
circleFilter＝[filter 0 (p,q),...,filter num (p,q)]s.t.fourier(p,q)≥FreqThres
wherein, fourier (p, q) is expressed as frequency response on coordinates (p, q), freqThres is frequency response threshold, filter 0 (p, q) as the layer 0 circumference filter with the circle center (p, q), namely the layer 0 circumference, and the same principle is that the filter num (p, q) is the num layer of circumference filter with the circle center of (p, q), namely the outermost circumference, num is the total number of circumferences, and circleFilter represents a concentric circumference filter set;
wherein, the single-layer circumference curve equation with the circle center being (p, q) is as follows:
wherein (x, y) represents the coordinates of a pixel point on the circumference, four k (p, q) and radius k (p, q) are each a filter corresponding to a circular filter k Fourier transform response and radius of (p, q), and filter k (p, q) is a k-th layer circle filter with the center of the circle being (p, q), the frequency response of each circle in the concentric circles is larger than a frequency response threshold value FreqThres, and if the frequency response is smaller than the threshold value, the circles are determined to be beyond the target boundary.
3. The concentric circular filter-based target symmetry axis detection method according to claim 1, wherein the angle determination in step (3) is as follows:
(3.1) obtaining the product in the step (2)
Is provided therein
The corresponding specific similarity weight is
Therefore, from similar weights, the set of relative angles retained by the beta-th layer is
And because the plane target is divided into 4 parts of the front plane body, the rear plane body, the left wing and the right wing, the relative angle set is divided into 4 parts by the KNN method, and the 4 parts are expressed by 4 angles, namely the relative 1 ,...,relative 4 ]Substituting the relative angle value into a tangent function, and determining the direction of the symmetry axis according to the following formula:
symmetryAngle＝relative j s.t.tan(relative i )-tan(relative j )＝tan(relative j )-tan(relative k )
wherein tan represents a tangent function, relative i 、relative k Respectively showing the relative angles of the left wing and the right wing j I.e., the relative angle of the fuselage, and symmetryAngle is the direction of the axis of symmetry, since the left wing and the right wing are symmetric to the fuselage, the relative angle difference between them should be consistent, and all negative angles are added by pi so that all angles are mapped to 0, pi]Therefore, if the above formula condition is satisfied:
s.t.tan(relative i )-tan(relative j )＝tan(relative j )-tan(relative k )
therefore, the symmetry angle of the fuselage is determined to be the direction of the symmetry axis.
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