An adaptive track smoothing method against motion direction abrupt change delay

Abstract

The application relates to track smoothing, in particular to an adaptive track smoothing method resisting motion direction mutation delay, obtaining a track of a previous frame and a related point track, and updating state information of a track of a current frame; calculating a direction change factor of the related point track of the current frame; constructing an adaptive weighted least square smoothing window to perform smoothing processing on the related point track of the current frame; and outputting smoothed coordinates; when a new track is updated, the above process is repeated to calculate smoothed coordinates of the new track; the technical scheme provided by the application can effectively overcome the defects of the prior art that it is difficult to quickly and accurately perform track smoothing under motion direction mutation.

Description

Technical Field

[0001] This invention relates to track smoothing, and more specifically to an adaptive track smoothing method that resists abrupt delays in motion direction. Background Technology

[0002] With the continuous development of millimeter-wave radar technology, its target detection and tracking functions have been widely applied in security, transportation, and water surveillance. Millimeter-wave radar calculates the path time difference between the transmitted and received signals, and through a series of signal processing steps, obtains information such as the target's distance, elevation angle, and signal strength. Radar data processing takes the target point data output from the signal processing as input, and through a series of processing steps such as track correlation, generates the target's track information.

[0003] In the above processing, due to the influence of atmospheric disturbances, electromagnetic interference, and other factors, the measured target position always carries random errors. Least squares is the most commonly used method for radar track smoothing in engineering. However, traditional least squares smoothing has an inherent flaw: when the target's true direction of motion changes significantly (such as a vehicle turning or a pedestrian turning back), the smoothed track exhibits a noticeable lag. This is because least squares fitting performs an overall fit between historical and current points within a fixed or sliding time window, attempting to minimize the sum of squared fitting errors. At points of abrupt change in direction, the window simultaneously includes points before the change (old direction) and points after the change (new direction). The fitting result inevitably compromises between the old and new directions, causing the smoothed trajectory to "tail" into the new direction, deviating from the true path.

[0004] Existing technologies address the delay caused by sudden changes in motion direction, including: reducing the window length, which lowers the delay but drastically reduces noise smoothing capabilities and makes the trajectory prone to jitter; and methods such as detecting the turn first and then smoothing in segments. For example, the aircraft track data smoothing method and system disclosed in patent application CN121366510A requires setting a fixed threshold, which is prone to false positives or false negatives when noise is high, and the segments are discontinuous. Therefore, there is an urgent need for a track smoothing method that maintains computational convenience while adaptively responding to sudden changes in direction and eliminating the delay caused by these changes. Summary of the Invention

[0005] (a) Technical problems to be solved In view of the above-mentioned shortcomings of the existing technology, the present invention provides an adaptive track smoothing method to resist the delay caused by sudden changes in motion direction, which can effectively overcome the shortcomings of the existing technology in that it is difficult to perform track smoothing quickly and accurately in the case of sudden changes in motion direction.

[0006] (II) Technical Solution To achieve the above objectives, the present invention provides the following technical solution: An adaptive trajectory smoothing method to resist abrupt changes in motion direction delay includes the following steps: S1. Obtain the track and its associated points from the previous frame, and update the status information of the track in the current frame. S2. Calculate the orientation change factor of the associated points in the current frame, including: By calculating the velocity angle difference of multiple frames of track, and statistically analyzing its mean and standard deviation, the velocity angle change factor and heading change feature factor are generated respectively. The change in direction is determined by comparison, and the direction change factor of the current frame associated point track is calculated by combining the acceleration change feature factor, so as to realize the quantitative characterization of the sudden change in track direction. S3. Construct an adaptive weighted least squares smoothing window to smooth the associated points in the current frame, including: An adaptive weighted least squares smoothing window centered on the associated points of the current frame is constructed. The forgetting rate and weight are dynamically adjusted based on the direction change factor. The associated points of the current frame are smoothed by weighted least squares fitting to obtain smooth coordinates. S4. Output smooth coordinates. When updating the track of a new frame, repeat S1~S3 to calculate the smooth coordinates of the new frame.

[0007] Preferably, in S1, the track and its associated points of the previous frame are obtained, and the status information of the track in the current frame is updated, including: S11, Obtain the track T of the previous frame. i-1 ={P i-1 ,X i-1 ,Y i-1 V i-1 ,θ i-1 A i-1}, where P i-1 For the associated point trace of the previous frame, P i-1 ={r i-1 ,α i-1 ,d i-1 ,f i-1}, r i-1 α i-1 d i-1 f i-1 These are the distance, azimuth, Doppler channel number, and amplitude information of the associated point trace in the previous frame, respectively. i-1 Y i-1 These are the X-axis and Y-axis distances of the trajectory in the previous frame, respectively, V i-1 θ i-1 A i-1 These are the velocity, velocity angle, and acceleration of the trajectory in the previous frame, respectively. S12, associate the point P of the previous framei-1 ={r i-1 ,α i-1 ,d i-1 ,f i-1 Transform to Cartesian coordinates to obtain the transformation-related point P from the previous frame. i-1 '={r i-1 ,α i-1 ,x i-1 ,y i-1 ,v i-1}, where x i-1 y i-1 These are the x and y coordinates of the associated point in the previous frame in the Cartesian coordinate system, respectively. , v i-1 The velocity of the associated point trace in the previous frame; S13. Calculate the X-axis distance X of the current frame's trajectory using the CA model. i Y-axis distance Y i X-axis velocity V X,i Y-axis velocity V Y,i X-axis acceleration A X,i Y-axis acceleration A Y,i : ; Among them, V X,i-1 V Y,i-1 These are the X-axis velocity and Y-axis velocity of the trajectory in the previous frame, respectively. X,i-1 A Y,i-1 These are the X-axis acceleration and Y-axis acceleration of the trajectory in the previous frame, respectively. , These are the X-axis and Y-axis velocity increments of the trajectory in the current frame, respectively, where T is the frame duration, and all velocities are positive with the direction closer to the radar as the positive value; S14. Calculate the acceleration A of the trajectory in the current frame. i Velocity angle θ i : ; .

[0008] Preferably, in S2, the direction change factor of the associated point trace in the current frame is calculated, including: S21. Obtain the velocity angle θ of the trajectory in the previous k frames. i-k θ i-k+1 、…、θ i Calculate the velocity angle difference Δθ between adjacent frames. i-k+1 , △θ i-k+2 ..., △θ i and constrained to Between, the velocity angle θ of the trajectory in the current framei Velocity angle θ of the trajectory compared to the previous frame i-1 The velocity angle difference Δθ between them i Calculate using the following formula: ; S22. Calculate the mean velocity angle difference between adjacent frames. Standard deviation s θ 2 : ; ; in, The velocity angle θ of the trajectory in the nth frame n The velocity angle θ of the trajectory in the (n-1)th frame n-1 The speed angle difference between them S23, Calculate the change factor of velocity angle : ; Among them, s θ The standard deviation of the velocity angle difference between adjacent frames. The angle noise threshold, The standard deviation of the reference angle; S24. Calculate the heading change characteristic factor. : ; in, This is the turning sensitivity adjustment coefficient; S25, Calculate the change in direction I θ : ; S26. Obtain the acceleration A of the trajectory in the previous k frames. i-k A i-k+1 A i Calculate the mean acceleration : ; Among them, A m Let be the acceleration of the trajectory in the m-th frame; S27. Calculate the characteristic factor I of acceleration change. A : ; in, The threshold value is the acceleration noise threshold, and β is the sensitivity adjustment coefficient. The larger β is, the more gradual the acceleration increases when the acceleration is small, and the faster the acceleration saturates when the acceleration is large. S28. Calculate the associated point P of the current frame. iDirection change factor I i : ; Here, w1 and w2 are both weighting coefficients, and w1+w2=1.

[0009] Preferably, in S3, an adaptive weighted least squares smoothing window is constructed to smooth the associated points in the current frame, including: S31. Construct an adaptive weighted least squares smoothing window for the current frame's associated point P. i Select the sliding window centered on it. , where L i R represents the left length of the window, i.e., the number of history points. i This represents the right length of the window, i.e., the number of future points. S32, Calculate the sliding window W i The j-th point P j Adaptive weights w j : ; Among them, t i t j The corresponding point P in the current frame i Sliding window W i The j-th point P j timestamp, To adapt to the forgetting rate, , Based on the forgetting rate, it is used to control the smoothing intensity of the trajectory during linear motion. This is an adaptive gain coefficient used to determine the amplification effect of directional changes on the forgetting rate. When the directional change factor I... i When the adaptive forgetting rate increases, The synchronous increase leads to adaptive weight w j With time interval The decay rate increases and then decreases more rapidly, thus enabling a rapid response to maneuvering targets; S33. Associat the current frame's associated point P with adaptive weights. i Smoothing is performed to obtain smoothed coordinates (x). i ,y i ): ; .

[0010] (III) Beneficial Effects Compared with the prior art, the adaptive trajectory smoothing method for resisting sudden changes in motion direction delay provided by the present invention has the following beneficial effects: 1) This invention, through adaptive forgetting rate, automatically accelerates the "forgetting" of old trajectories at turning points, effectively shortening the lag time; 2) In straight or gently curved areas, when the direction change factor is close to 0, the algorithm degenerates into the traditional exponential weighted least squares method, maintaining a stable trajectory smoothing effect. 3) This invention is based solely on real-time radar data, requiring no additional hardware devices, and has the advantages of low data computation, strong real-time performance, and high accuracy. Attached Figure Description

[0011] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are merely some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without any creative effort.

[0012] Figure 1 This is a schematic diagram of the process of the present invention; Figure 2 This is a comparison diagram of the effects of the present invention and the existing fixed-window trajectory smoothing method. Detailed Implementation

[0013] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0014] In this embodiment, information such as the target's distance and azimuth relative to the radar is obtained based on the echo signal received by the millimeter-wave radar. The 24GHz millimeter-wave radar used in this embodiment has a range resolution of 1.34m and a velocity resolution of 0.138m / s. Because of the high resolution of this radar and the multiple radar reflection points of the target, the DBSCAN algorithm is used for point clustering, and the clustered points are used in subsequent association.

[0015] The core of this invention lies in dynamically calculating the direction change factor of each point and adjusting the weight allocation (including forgetting rate and window shape) in the adaptive weighted least squares smoothing window in real time based on the factor. This allows noise to be fully suppressed when the target motion direction is stable and the old direction to be quickly "forgotten" and followed when the target motion direction changes abruptly, thereby simultaneously satisfying the requirements of smoothness and low latency.

[0016] The following describes the specific process of the adaptive trajectory smoothing method for resisting sudden changes in motion direction provided by this invention, using a concrete example (e.g.) Figure 1 (as shown) and technical effects.

[0017] S1. Obtain the track and its associated points from the previous frame, and update the status information of the track in the current frame, including: S11, Obtain the track T of the previous frame. i-1 ={P i-1 ,X i-1 ,Y i-1 V i-1 ,θ i-1 A i-1}, where P i-1 For the associated point trace of the previous frame, P i-1 ={r i-1 ,α i-1 ,d i-1 ,f i-1}, r i-1 α i-1 d i-1 f i-1 These are the distance, azimuth, Doppler channel number, and amplitude information of the associated point trace in the previous frame, respectively. i-1 Y i-1 These are the X-axis and Y-axis distances of the trajectory in the previous frame, respectively, V i-1 θ i-1 A i-1 These are the velocity, velocity angle, and acceleration of the trajectory in the previous frame, respectively. S12, associate the point P of the previous frame i-1 ={r i-1 ,α i-1 ,d i-1 ,f i-1 Transform to Cartesian coordinates to obtain the transformation-related point P from the previous frame. i-1 '={r i-1 ,α i-1 ,x i-1 ,y i-1 ,v i-1}, where x i-1 y i-1 These are the x and y coordinates of the associated point in the previous frame in the Cartesian coordinate system, respectively. , v i-1 The velocity of the associated point trace in the previous frame; S13. Calculate the X-axis distance X of the current frame's trajectory using the CA model. i Y-axis distance Y i X-axis velocity V X,i Y-axis velocity V Y,iX-axis acceleration A X,i Y-axis acceleration A Y,i : ; Among them, V X,i-1 V Y,i-1 These are the X-axis velocity and Y-axis velocity of the trajectory in the previous frame, respectively. X,i-1 A Y,i-1 These are the X-axis acceleration and Y-axis acceleration of the trajectory in the previous frame, respectively. , These are the X-axis and Y-axis velocity increments of the trajectory in the current frame, respectively, where T is the frame duration (T=1s). All velocities are positive with the direction closer to the radar as the positive value. S14. Calculate the acceleration A of the trajectory in the current frame. i Velocity angle θ i : ; .

[0018] S2. Calculate the orientation change factor of the associated points in the current frame, including: By calculating the velocity angle difference of multiple frames of track, and statistically analyzing its mean and standard deviation, a velocity angle change factor and a heading change feature factor are generated respectively. The change in direction is determined by comparison, and the direction change factor of the current frame associated point track is calculated by combining the acceleration change feature factor, thereby realizing the quantitative characterization of the sudden change in track direction.

[0019] S2 calculates the orientation change factor of the associated point trace in the current frame, specifically including: S21. Obtain the velocity angle θ of the trajectory in the previous k frames. i-k θ i-k+1 、…、θ i Calculate the velocity angle difference Δθ between adjacent frames. i-k+1 , △θ i-k+2 ..., △θ i and constrained to Between, the velocity angle θ of the trajectory in the current frame i Velocity angle θ of the trajectory compared to the previous frame i-1 The velocity angle difference Δθ between them i Calculate using the following formula: ; In this embodiment, the information of the points associated with multiple flight paths is shown in Table 1: Table 1. Multi-frame track association point information table

[0020] S22. Calculate the mean velocity angle difference between adjacent frames. Standard deviation s θ 2 : ; ; in, The velocity angle θ of the trajectory in the nth frame n The velocity angle θ of the trajectory in the (n-1)th frame n-1 The speed angle difference between them S23, Calculate the change factor of velocity angle : ; Among them, s θ The standard deviation of the velocity angle difference between adjacent frames. The angle noise threshold, As a reference angle standard deviation, when the velocity is less than 2 m / s, Take 3°, Take 1°; when the velocity is not less than 2 m / s, Take 6°, Take 4°; S24. Calculate the heading change characteristic factor. : ; in, This is the turning sensitivity adjustment coefficient; when the speed is less than 2 m / s, Take 1; when the speed is not less than 2 m / s, Take 2; S25, Calculate the change in direction I θ : ; S26. Obtain the acceleration A of the trajectory in the previous k frames. i-k A i-k+1 A i Calculate the mean acceleration : ; Among them, A m Let be the acceleration of the trajectory in the m-th frame; S27. Calculate the characteristic factor I of acceleration change. A : ; in, The threshold value is β, which is the sensitivity adjustment coefficient. A larger β indicates a more gradual increase in acceleration at lower speeds and rapid saturation at higher speeds. When the velocity is less than 2 m / s... Take 0.2 m / s 2 β is set to 1; when the velocity is not less than 2 m / s, Take 1.5 m / s 2 β is 2; S28. Calculate the associated point P of the current frame. i Direction change factor I i : ; Where w1 and w2 are both weighting coefficients, and w1 + w2 = 1; In this embodiment, w1=w2=0.5, and at 64 frames, the direction change factor I i =0.105; at 65 frames, the orientation change factor I i =0.955, which shows that the trajectory has obvious signs of turning at frame 65.

[0021] S3. Construct an adaptive weighted least squares smoothing window to smooth the associated points in the current frame, including: An adaptive weighted least squares smoothing window centered on the associated points of the current frame is constructed. The forgetting rate and weights are dynamically adjusted based on the direction change factor. The associated points of the current frame are smoothed by weighted least squares fitting to obtain smooth coordinates.

[0022] In S3, an adaptive weighted least squares smoothing window is constructed to smooth the associated points in the current frame, specifically including: S31. Construct an adaptive weighted least squares smoothing window for the current frame's associated point P. i Select the sliding window centered on it. , where L i R represents the left length of the window, i.e., the number of history points. i This represents the right length of the window, i.e., the number of future points. In this embodiment, for real-time smoothing, R i Set to 0; for offline post-processing, R can be set to... i =L i When the current frame is associated with the point P i Direction change factor I i When the window size is large, it is necessary to reduce the smoothing window size and perform real-time smoothing. In this case, L i Take 5, R i Take 0, when I i When L > 0.9, i Reduced to 3; S32, Calculate the sliding window W i The j-th point P j Adaptive weights w j : ; Among them, t i t j The corresponding point P in the current frame i Sliding window W i The j-th point P j timestamp, To adapt to the forgetting rate, , Based on the forgetting rate, it is used to control the smoothing intensity of the trajectory during linear motion. This is an adaptive gain coefficient used to determine the amplification effect of directional changes on the forgetting rate. When the directional change factor I... i When the adaptive forgetting rate increases, The synchronous increase leads to adaptive weight w j With time interval The decay rate increases and then decreases more rapidly, thus enabling a rapid response to maneuvering targets; In this embodiment, =0.3, =3, therefore =1.458, the adaptive weight for frame 65 is 1, the adaptive weight for frame 64 is 0.32, and the adaptive weight for frame 63 is 0.11; S33. Associat the current frame's associated point P with adaptive weights. i Smoothing is performed to obtain smoothed coordinates (x). i ,y i ): ; ;

[0023] In this embodiment, based on the adaptive weights calculated in S32, the smoothed coordinates of the associated points in the current frame are obtained as (12.06, 342.17). Figure 2 This is a comparison diagram of the effects of the present invention and the existing fixed-window track smoothing method. The red line represents the smoothed track obtained by using the technical solution of the present invention, the green line represents the smoothed track obtained by using the existing fixed-window track smoothing method, and the blue dots represent radar measurement points.

[0024] S4. Output smooth coordinates. When updating the track of a new frame, repeat S1~S3 to calculate the smooth coordinates of the new frame.

[0025] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions will not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. An adaptive trajectory smoothing method to resist sudden changes in motion direction delay, characterized in that: Includes the following steps: S1. Obtain the track and its associated points from the previous frame, and update the status information of the track in the current frame. S2. Calculate the orientation change factor of the associated points in the current frame, including: By calculating the velocity angle difference of multiple frames of track, and statistically analyzing its mean and standard deviation, the velocity angle change factor and heading change feature factor are generated respectively. The change in direction is determined by comparison, and the direction change factor of the current frame associated point track is calculated by combining the acceleration change feature factor, so as to realize the quantitative characterization of the sudden change in track direction. S3. Construct an adaptive weighted least squares smoothing window to smooth the associated points in the current frame, including: An adaptive weighted least squares smoothing window centered on the associated points of the current frame is constructed. The forgetting rate and weight are dynamically adjusted based on the direction change factor. The associated points of the current frame are smoothed by weighted least squares fitting to obtain smooth coordinates. S4. Output smooth coordinates. When updating the track of a new frame, repeat S1~S3 to calculate the smooth coordinates of the new frame.

2. The adaptive trajectory smoothing method for resisting sudden changes in motion direction delay according to claim 1, characterized in that: S1 retrieves the track and its associated points from the previous frame and updates the status information of the track in the current frame, including: S11, Obtain the track T of the previous frame. i-1 ={P i-1 ,X i-1 ,Y i-1 V i-1 ,θ i-1 A i-1 }, where P i-1 For the associated point trace of the previous frame, P i-1 ={r i-1 ,α i-1 ,d i-1 ,f i-1 }, r i-1 α i-1 d i-1 f i-1 These are the distance, azimuth, Doppler channel number, and amplitude information of the associated point trace in the previous frame, respectively. i-1 Y i-1 These are the X-axis and Y-axis distances of the trajectory in the previous frame, respectively, V i-1 θ i-1 A i-1 These are the velocity, velocity angle, and acceleration of the trajectory in the previous frame, respectively. S12, associate the point P of the previous frame i-1 ={r i-1 ,α i-1 ,d i-1 ,f i-1 Transform to Cartesian coordinates to obtain the transformation-related point P from the previous frame. i-1 '={r i-1 ,α i-1 ,x i-1 ,y i-1 ,v i-1 }, where x i-1 y i-1 These are the x and y coordinates of the associated point in the previous frame in the Cartesian coordinate system, respectively. , v i-1 The velocity of the associated point trace in the previous frame; S13. Calculate the X-axis distance X of the current frame's trajectory using the CA model. i Y-axis distance Y i X-axis velocity V X,i Y-axis velocity V Y,i X-axis acceleration A X,i Y-axis acceleration A Y,i : ； Among them, V X,i-1 V Y,i-1 These are the X-axis velocity and Y-axis velocity of the trajectory in the previous frame, respectively. X,i-1 A Y,i-1 These are the X-axis acceleration and Y-axis acceleration of the trajectory in the previous frame, respectively. , These are the X-axis and Y-axis velocity increments of the trajectory in the current frame, respectively, where T is the frame duration, and all velocities are positive with the direction closer to the radar as the positive value; S14. Calculate the acceleration A of the trajectory in the current frame. i Velocity angle θ i : ； 。 3. The adaptive trajectory smoothing method for resisting sudden changes in motion direction delay according to claim 2, characterized in that: S2 calculates the orientation change factor of the associated point trace in the current frame, including: S21. Obtain the velocity angle θ of the trajectory in the previous k frames. i-k θ i-k+1 、…、θ i Calculate the velocity angle difference Δθ between adjacent frames. i-k+1 , △θ i-k+2 ..., △θ i and constrained to Between, the velocity angle θ of the trajectory in the current frame i Velocity angle θ of the trajectory compared to the previous frame i-1 The velocity angle difference Δθ between them i Calculate using the following formula: ； S22. Calculate the mean velocity angle difference between adjacent frames. Standard deviation s θ 2 : ； ； in, The velocity angle θ of the trajectory in the nth frame n The velocity angle θ of the trajectory in the (n-1)th frame n-1 The speed angle difference between them S23, Calculate the change factor of velocity angle : ； Among them, s θ The standard deviation of the velocity angle difference between adjacent frames. The angle noise threshold, The standard deviation of the reference angle; S24. Calculate the heading change characteristic factor. : ； in, This is the turning sensitivity adjustment coefficient; S25, Calculate the change in direction I θ : ； S26. Obtain the acceleration A of the trajectory in the previous k frames. i-k A i-k+1 A i Calculate the mean acceleration : ； Among them, A m Let be the acceleration of the trajectory in the m-th frame; S27. Calculate the characteristic factor I of acceleration change. A : ； in, The threshold value is the acceleration noise threshold, and β is the sensitivity adjustment coefficient. The larger β is, the more gradual the acceleration increases when the acceleration is small, and the faster the acceleration saturates when the acceleration is large. S28. Calculate the associated point P of the current frame. i Direction change factor I i : ； Here, w1 and w2 are both weighting coefficients, and w1+w2=1.

4. The adaptive trajectory smoothing method for resisting sudden changes in motion direction delay according to claim 3, characterized in that: In S3, an adaptive weighted least squares smoothing window is constructed to smooth the associated points in the current frame, including: S31. Construct an adaptive weighted least squares smoothing window for the current frame's associated point P. i Select the sliding window centered on it. , where L i R represents the left length of the window, i.e., the number of history points. i This represents the right length of the window, i.e., the number of future points. S32, Calculate the sliding window W i The j-th point P j Adaptive weights w j : ； Among them, t i t j The corresponding point P in the current frame i Sliding window W i The j-th point P j timestamp, To adapt to the forgetting rate, , Based on the forgetting rate, it is used to control the smoothing intensity of the trajectory during linear motion. This is an adaptive gain coefficient used to determine the amplification effect of directional changes on the forgetting rate. When the directional change factor I... i When the adaptive forgetting rate increases, The synchronous increase leads to adaptive weight w j With time interval The decay rate increases and then decreases more rapidly, thus enabling a rapid response to maneuvering targets; S33. Associat the current frame's associated point P with adaptive weights. i Smoothing is performed to obtain smoothed coordinates (x). i ,y i ): ； 。

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