A multi-source trajectory information fusion and compression method
By using a multi-source trajectory adaptive fusion and compression method, the problems of inconsistent accuracy of multi-source positioning terminals and transmission delay caused by large data volume were solved, achieving efficient and accurate trajectory data processing and improving the ability to monitor the situation on site in real time.
Patent Information
- Application Number
- CN202311318567.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-10-12
- Publication Date
- 2025-12-26
- Estimated Expiration
- 2043-10-12
AI Technical Summary
In information-based emergency response, the inconsistent positioning accuracy of various positioning terminals makes it impossible to determine the exact location of personnel on site. Furthermore, the large amount of data and poor transmission timeliness mean that existing trajectory fusion technologies suffer from low accuracy or poor compression efficiency.
A multi-source trajectory adaptive fusion method with comprehensive data quality is adopted. Noise points are removed by setting denoising distance and angle thresholds, trajectory roughness is calculated and weighting weights are adjusted, trajectory information is fused by combining an improved exponential weighted average method, and the GeoHash algorithm sub-block size is dynamically adjusted for compression.
It achieves efficient and real-time trajectory fusion and compression with high compression accuracy, improving positioning accuracy and transmission efficiency, and reducing rendering latency.
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Figure CN117313033B_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the field of trajectory data processing, in particular to a multi-source trajectory information fusion and compression method. BACKGROUND
[0002] Under the background of information-based emergency disposal, a person on the scene may carry multiple terminal devices carrying positioning information such as mobile phones, PDT terminals, law enforcement recorders, and special positioning terminals. The positioning accuracy of different terminals is different, and the collected longitude and latitude information at the same time is often different due to the influence of complex geographical environment. At the same time, the adaptability of different terminals to different environmental information is different, and there is no terminal device that has absolute positioning accuracy advantage in all scenarios, which makes it impossible to determine the accurate position of the on-site personnel, so it is necessary to fuse multi-source inconsistent trajectories.
[0003] In addition, the positioning information is collected in real time and continuously, which has the characteristics of large data volume, multiple positioning sources, and rapid growth. Due to the limitations of network bandwidth in remote areas and unstable network environment, the on-site trajectory data may need a lot of time to be transmitted back to the command center from the scene, and the timeliness of data transmission cannot be guaranteed, which greatly affects the real-time grasp and judgment of the situation of the command center. Therefore, it is necessary to compress the trajectory under the premise of meeting the positioning accuracy of the trajectory, and to reduce the data volume as much as possible.
[0004] The traditional multi-source trajectory fusion technology directly calculates the average value of the trajectory information of multiple positioning terminals. This method has the disadvantage of "lowering" the positioning accuracy of the fused trajectory information due to the low positioning accuracy of the trajectory information. Chinese patent document No. CN115793007A discloses a multi-source fusion positioning method and device applied to navigation services. The method is based on the access satellite quantity, satellite signal strength, and access UWB base station quantity, UWB base station signal strength, and other environmental parameters obtained by the positioning terminal. According to the preset rules, the BDS positioning method, inertial navigation positioning method, and UWB positioning method are comprehensively used. Since the positioning terminal generally only provides positioning information such as time, longitude and latitude, and does not provide environmental parameters, this method is not suitable for the multi-source trajectory fusion scene of the upper application system. Chinese patent document No. CN115465296A discloses a long-short-time motion trajectory fusion method based on a spline function. The constant term of the spline function needs to be pre-set according to the experience of road conditions. This method needs to exhaust all road conditions, but this is often impossible in reality, which limits its practical application.
[0005] The existing trajectory compression method usually cannot achieve the balance between compression accuracy and compression efficiency. The GeoHash algorithm is a kind of spatial index method, and the basic principle is to understand the earth as a two-dimensional plane, to divide the plane into smaller sub-blocks by recursive average, and to have all the trajectory points in each sub-block to have the same latitude and longitude data coding under a certain latitude and longitude accuracy. The trajectory compression based on GeoHash only retains one trajectory point in each sub-block, and compared with the compression methods based on key feature points, road network and semantics, the method has the characteristics of simplicity and efficiency, and the time efficiency performance meets the requirements of high-concurrency real-time trajectory compression under resource-limited conditions. However, the sub-block size of the trajectory compression based on GeoHash is related to the positioning accuracy, the smaller the size is, the higher the accuracy is, and the lower the compression rate is, and since which point is retained in each sub-block is not analyzed in detail, the retention of trajectory details is insufficient in some scenarios. Therefore, considering the compression efficiency, compression accuracy and other factors, how to adaptively and dynamically adjust the sub-block size according to different trajectory characteristics is a problem to be solved. SUMMARY
[0006] Therefore, the present application provides a multi-source trajectory information fusion and compression method, which can adaptively fuse multi-source trajectories generated by multiple positioning terminals into one trajectory, and adaptively and dynamically adjust the sub-block size of the GeoHash algorithm according to the trajectory roughness, so as to realize efficient compression under the premise of high compression accuracy.
[0007] In order to achieve the above purpose, the technical scheme adopted by the present application is as follows:
[0008] A multi-source trajectory information fusion and compression method comprises the following steps:
[0009] (1) Set a sliding time window, and in the sliding time window, set a denoising distance threshold and a denoising angle threshold to perform real-time denoising on the trajectory information reported by each positioning terminal, and simultaneously obtain the number of noise points, the number of missing points, the number of stay points and the number of inflection point covers of the current trajectory, wherein the trajectory information comprises the latitude and longitude of each trajectory point;
[0010] (2) Calculate the slope of the line connecting two adjacent trajectory points in the trajectory information, and take the standard deviation of all the slopes in the trajectory information as the trajectory roughness; take the reciprocal of the sum of the number of noise points, the number of missing points, the number of stay points and the number of inflection point covers in the trajectory information and the trajectory roughness as the data quality of the current trajectory information;
[0011] (3) Taking the data quality of each piece of trajectory information as the weighted weight of each trajectory, all trajectories are sorted in the order of data quality from small to large, starting from the trajectory corresponding to the smallest data quality, and according to each piece of trajectory information after real-time denoising, the improved exponentially weighted average method is used to sequentially fuse backward to obtain the final fused multi-source trajectory information;
[0012] (4) For the fused multi-source trajectory information, starting from the initial trajectory segment, a trajectory segment is sequentially added backward and the trajectory roughness of the current multi-source trajectory information is calculated, the trajectory roughness calculated at present is recorded to the set as a new element, and the trajectory roughness calculated at present is compared with other elements in the set, the size order relationship is used to dynamically adjust the GeoHash algorithm sub-block size in real time, the fused multi-source trajectory information is compressed according to the GeoHash algorithm, and the fusion and compression of multi-source trajectories are realized.
[0013] Further, the method for real-time denoising of trajectory information by setting the denoising distance threshold and the denoising angle threshold in step (1) is as follows:
[0014] Firstly, according to the longitude and latitude information of each trajectory point in the trajectory information, the denoising distance threshold l mean in the sliding time window is calculated
[0015]
[0016] wherein p0 is the starting trajectory point, n represents the id of the last trajectory point p n in the sliding time window, l i,j represents the distance from the trajectory point p j to the straight line p0p i , i is the id of a trajectory point p n in p2 to p i , and j is the id of a trajectory point p i-1 in p1 to p j .
[0017] Secondly, the denoising angle threshold θ mean in the sliding time window is calculated
[0018]
[0019] wherein θ i,j represents the value of ∠p j p0p i .
[0020] Starting from the second trajectory point p1, the distance l of the current trajectory point p k from the straight line p0p k+1 and the angle ∠p kp0p k+1 The sine value, where k = 1, 2, 3…n-1, if the distance l is greater than the noise reduction distance threshold l mean or ∠p k p0p k+1 The sine value is greater than the noise reduction angle threshold θ mean Then determine the current trajectory point p k For noise points, delete the current trajectory point p. k , set the current trajectory point p k If the previous trajectory point is connected to the next trajectory point, the calculation and judgment of the next trajectory point will continue; otherwise, the trajectory point will be retained and the calculation and judgment of the next trajectory point will continue.
[0021] Furthermore, step (2) specifically includes the following steps:
[0022] (201) For each trajectory information, starting from the starting point p0, calculate the slope of the line connecting two adjacent trajectory points in sequence. k :
[0023]
[0024] in, Let p be the current trajectory point k latitude coordinates Let p be the current trajectory point k Longitude coordinates;
[0025] (202) For the current trajectory information, the standard deviation of all slopes in the trajectory information is taken as the trajectory roughness.
[0026] (203) Calculate the trajectory data quality Q for each trajectory information:
[0027]
[0028] Among them, C noise The number of noise points in the current trajectory information, C miss The number of missing points in the current trajectory information, C feature This represents the number of stopping points and inflection points covered by the current trajectory information.
[0029] Furthermore, step (3) includes the following steps:
[0030] (301) For the m trajectory information after real-time denoising, sort each trajectory in ascending order of data quality Q, and calculate the data quality Q' of the m normalized trajectory in sequence. a :
[0031] Q' a =Q aQ1+Q2+...+Qm m ), a=1, 2...m
[0032] Q' a=Q a / (Q1+Q2+...+Qm) a is the trajectory data quality of the a-th trajectory information, and Q1<Q2...<Qm m Q' a is the normalized trajectory data quality of the a-th trajectory information. a
[0033] (302) calculating the multi-source trajectory information V after the final fusion of the m pieces of trajectory information m :
[0034] When m=1,
[0035] V1=Q'1×θ1
[0036] When m≥2,
[0037] V m =(1-Q' m )×V m-1 +Q' m ×θ m
[0038] wherein θ m represents the m-th trajectory information.
[0039] Further, the method for dynamically adjusting the GeoHash algorithm sub-block size according to the roughness of the fused multi-source trajectory information in step (4) is specifically as follows:
[0040] For the fused multi-source trajectory information, starting from the initial trajectory segment, a trajectory segment is sequentially added backward and the trajectory roughness of the current multi-source trajectory information is calculated, the trajectory roughness currently calculated is recorded to the set as a new element, if the trajectory roughness currently calculated is greater than the trajectory roughness of c% in the set and less than the trajectory roughness of (100-c)%, the default GeoHash string encoding length is set for the current trajectory segment, corresponding to the default GeoHash algorithm sub-block size; if the trajectory roughness currently calculated is greater than the trajectory roughness of (100-c)% in the set, the encoding length of the default GeoHash string is increased, corresponding to the decrease of the GeoHash algorithm sub-block size; if the trajectory roughness currently calculated is less than the trajectory roughness of c% in the set, the encoding length of the default GeoHash string is decreased, corresponding to the increase of the GeoHash algorithm sub-block size; wherein 0
[0041] Due to the adoption of the above technical solutions, the present application has the following beneficial effects compared with the background art:
[0042] 1, The application sets up a track quality evaluation function Q(C noise ,C miss ,C feature ,Rough) by comprehensively considering the number of noise points, the number of missing points, the number of turning points and inflection points covered, roughness and other characteristic factors, and a real-time, efficient and accurate track quality evaluation method is given.
[0043] 2, The application can realize efficient and real-time compression under the premise of high compression accuracy by comprehensively adjusting the sub-block size of the GeoHash algorithm according to the track roughness.
[0044] 3, The application adopts a track denoising algorithm for adaptively and dynamically adjusting the denoising distance threshold and the denoising angle threshold, thereby avoiding the problems of tedious manual setting of the threshold and large errors. DETAILED DESCRIPTION
[0045] Figure 1 is a track information schematic diagram in the embodiment of the application.
[0046] Figure 2 is a track compression schematic diagram corresponding to the adaptive adjustment of the sub-block size of the GeoHash algorithm according to the track roughness in the embodiment of the application.
[0047] Figure 3 is a flowchart of the multi-source track adaptive fusion and compression method of the application.
[0048] Figure 4 is a global track fusion effect diagram of the multi-source track adaptive fusion method in an outdoor environment, wherein the abscissa represents the longitude of the track point, and the ordinate represents the latitude.
[0049] Figure 5 is a local track fusion effect diagram of the multi-source track adaptive fusion method in an outdoor environment, wherein the abscissa represents the longitude of the track point, and the ordinate represents the latitude.
[0050] Figure 6 is an original positioning point effect diagram in the adaptive compression method of the fused track data.
[0051] Figure 7 is a positioning point effect diagram with a compression rate of 40% in the adaptive compression method of the fused track data.
[0052] Figure 8 is a positioning point effect diagram with a compression rate of 90% in the adaptive compression method of the fused track data. DETAILED DESCRIPTION
[0053] The application will be further described in connection with the accompanying drawings Figures 1-8 The application will be further described in connection with the accompanying drawings
[0054] The multi-source trajectory fusion and compression method flow chart of the application is shown in the figure Figure 3 The multi-source trajectory fusion and compression method specifically includes the following steps:
[0055] A multi-source trajectory information fusion and compression method includes the following steps:
[0056] (1) Set a sliding time window, in the sliding time window, for each positioning terminal reported trajectory information, through the method of setting the denoising distance threshold and denoising angle threshold for real-time denoising, while obtaining the number of noise points, the number of missing points, the number of stay points and inflection point coverage of the current trajectory; wherein the trajectory information includes the latitude and longitude of each trajectory point;
[0057] Specifically, as shown in the figure Figure 1 First, according to the latitude and longitude information of each trajectory point in the trajectory information, the denoising distance threshold l mean in the sliding time window is calculated
[0058]
[0059] Wherein, the starting trajectory point is p0, n represents the id of the last trajectory point p n in the sliding time window, l i,j represents the distance from the trajectory point p j to the straight line p0p i , i is the id of a trajectory point p i in p2 to p n , j is the id of a trajectory point p j in p1 to p i-1 ;
[0060] Secondly, the denoising angle threshold θ mean in the sliding time window is calculated
[0061]
[0062] Wherein, θ i,j represents the value of ∠p j p0p i ;
[0063] From the second trajectory point p1, the distance l of the current trajectory point p k from the straight line p0p k+1 and ∠p k p0p k+1sin (k * l), where k = 1, 2, 3…n-1, if the distance l is greater than the denoising distance threshold l mean or ∠p k p0p k+1 sin (k * l) is greater than the denoising angle threshold θ mean , it is determined that the current trajectory point p k is a noise point, the current trajectory point p k is deleted, the previous trajectory point of the current trajectory point p k is connected with the next trajectory point, and the next trajectory point is continuously calculated and determined; otherwise, the trajectory point is retained, and the next trajectory point is continuously calculated and determined.
[0064] (2) The slope of the connecting line between two adjacent trajectory points in the trajectory information is calculated, the standard deviation of all slopes in the trajectory information is taken as the trajectory roughness, and the reciprocal of the sum of the number of noise points, the number of missing points, the number of stay points and inflection point covers, and the trajectory roughness in the trajectory information is taken as the data quality of the current trajectory information.
[0065] Specifically, for each trajectory information, the slope Slope k between two adjacent trajectory points is calculated in sequence from the starting point p0:
[0066]
[0067] wherein, is the latitude coordinate of the current trajectory point p k , is the longitude coordinate of the current trajectory point p k .
[0068] For the current trajectory information, the standard deviation of all slopes in the trajectory information is taken as the trajectory roughness R;
[0069] The trajectory data quality Q of each trajectory information is calculated:
[0070]
[0071] wherein, C noise is the number of noise points of the current trajectory information, C miss is the number of missing points of the current trajectory information, and C feature is the number of stay points and inflection point covers of the current trajectory information.
[0072] (3) The data quality of each trajectory information is taken as the weighted weight of each trajectory, all trajectories are sorted in the order of data quality from small to large, and starting from the trajectory corresponding to the smallest data quality, each trajectory information after real-time denoising is fused in sequence according to the improved exponential weighted average method to obtain the final fused multi-source trajectory information.
[0073] Specifically, for m pieces of trajectory information after real-time denoising, each trajectory is sorted in order of data quality Q from small to large, and m normalized trajectory data qualities Q' are calculated in turn. a :
[0074] Q' a =Q a / (Q1+Q2+..+Q m ),a=1,2...m
[0075] Wherein, Q a is the trajectory data quality of the a-th trajectory information, and Q1<Q2...<Q m , Q' a is the normalized trajectory data quality of the a-th trajectory information.
[0076] As shown in Figure 4 , Figure 5 , the final fused multi-source trajectory information V m :
[0077] When m = 1,
[0078] V1 = Q'1 x θ1
[0079] When m ≥ 2,
[0080] V m = (1-Q' m ) x V m-1 + Q' m x θ m
[0081] Wherein, θ m represents the m-th trajectory information.
[0082] Suppose there are three devices that collect three trajectories respectively, and the trajectory quality Q1<Q2<Q3. First, normalize the trajectory quality to get Q' a , according to the trajectory data quality, the calculation process of the final fused multi-source trajectory information V3 of 3 pieces of trajectory information is as follows:
[0083] V1 = Q'1 x θ1
[0084] V2 = (1-Q'2) x V1 + Q'2 x θ2
[0085] V3 = (1-Q'3) x V2 + Q'3 x θ3
[0086] Therefore, the fused multi-source trajectory information V3 is:
[0087] V'3 = (1-Q'3)(1-Q'2)Q'1xθ1 + (1-Q'3)Q'2xθ2 + Q'3xθ3
[0088] (4) for the fused multi-source trajectory information, from the starting trajectory segment, sequentially add a trajectory segment and calculate the trajectory roughness of the current multi-source trajectory information, record the currently calculated trajectory roughness to the set as a new element, and compare the currently calculated trajectory roughness with other elements in the set, and dynamically adjust the GeoHash algorithm sub-block size according to the size order relationship, and compress the fused multi-source trajectory information according to the GeoHash algorithm, realize the fusion and compression of multi-source trajectory.
[0089] Specifically, as shown in Figure 2 、 Figure 6 、 Figure 7 、 Figure 8 In the embodiment, from the starting trajectory point p0 of the fused trajectory information V m , a trajectory segment is sequentially added and the trajectory roughness of the current multi-source trajectory information is calculated, the currently calculated trajectory roughness is recorded to the set as a new element, if the currently calculated trajectory roughness is greater than 20% of the trajectory roughness in the set and less than 80% of the trajectory roughness in the set, the default GeoHash string encoding length of the current trajectory segment is set to 9, and the corresponding default GeoHash algorithm sub-block size is 4.7m*4.7m; if the currently calculated trajectory roughness is greater than 80% of the trajectory roughness in the set, the Geohash string encoding length is set to 10, and the corresponding sub-block size is 1.2m*0.6m; if the currently calculated trajectory roughness is less than 20% of the trajectory roughness in the set, the Geohash string encoding length is set to 8, and the corresponding sub-block size is 38.2m*19m.
[0090] In summary, the present application proposes a multi-source trajectory fusion and compression method, which is aimed at the problem of inconsistent data caused by positioning accuracy difference in multi-source terminal historical trajectory query and real-time trajectory mapping, and the problem of high transmission and rendering delay caused by large data volume, adopts a multi-source trajectory adaptive fusion and compression method based on comprehensive data quality, and adaptively fuses the multi-source trajectory generated by multiple positioning terminals into one trajectory; and adaptively dynamically adjusts the GeoHash algorithm sub-block size according to the trajectory roughness, realizes efficient compression under the premise of high compression accuracy, and has the advantages of reasonable fusion weight, high real-time compression efficiency.
[0091] Finally, it should be noted that: the above description is only the preferred embodiment of the present application, and does not limit the present application, any modification, equivalent replacement, improvement, etc. made without departing from the spirit and scope of the present application defined by the appended claims, should be included in the protection scope of the present application.
Claims
1. A fusion and compression method of multi-source trajectory information, characterized in that, The method comprises the following steps: (1) setting a sliding time window, in the sliding time window, for each positioning terminal reported trajectory information, real-time denoising is performed by setting a denoising distance threshold and a denoising angle threshold, and the number of noise points, the number of missing points, the number of stay points and the number of inflection point coverages of the current trajectory are obtained; wherein the trajectory information comprises the latitude and longitude of each trajectory point; (2) calculating the slope of the line between two adjacent trajectory points in the trajectory information, and taking the standard deviation of all slopes in the trajectory information as the trajectory roughness; taking the reciprocal of the sum of the number of noise points, the number of missing points, the number of stay points and the number of inflection point coverages and the trajectory roughness in the trajectory information as the data quality of the current trajectory information; (3) taking the data quality of each trajectory information as the weighted weight of each trajectory, and sorting all trajectories in order of data quality from small to large, starting from the trajectory corresponding to the smallest data quality, and according to the improved exponential weighted average method, each trajectory information is fused in turn to obtain the final fused multi-source trajectory information; (4) for the fused multi-source trajectory information, starting from the starting trajectory segment, a trajectory segment is added in turn and the trajectory roughness of the current multi-source trajectory information is calculated, the recorded trajectory roughness is added to the set as a new element, and the current calculated trajectory roughness is compared with other elements in the set, and the GeoHash algorithm sub-block size is dynamically adjusted in real time according to the size order relationship, and the fused multi-source trajectory information is compressed according to the GeoHash algorithm, realizing the fusion and compression of multi-source trajectories.
2. The method of claim 1, wherein, The method for real-time denoising of trajectory information by setting a denoising distance threshold and a denoising angle threshold in step (1) is as follows: Firstly, according to the longitude and latitude information of each trajectory point in the trajectory information, the denoising distance threshold l in the sliding time window is calculated mean : wherein p0 is the start trajectory point, n represents the id of the last trajectory point p in the sliding time window n l i,j represents the distance from the trajectory point p j to the straight line p0p i , i is the id of the trajectory point p n in p2 to p i , and j is the id of the trajectory point p i-1 in p1 to p j . Secondly, the denoising angle threshold θ in the sliding time window is calculated mean : where θ i,j represents ∠p j p0p i the value of; From the second track point p1, the current track point p is calculated in turn k The distance l of the distance straight line p0p k+1 and the sine value of ∠p k p0p k+1 , wherein k=1, 2, 3…n-1, if the distance l is greater than the denoising distance threshold l mean or the sine value of ∠p k p0p k+1 is greater than the denoising angle threshold θ mean , it is determined that the current track point p k is a noise point, the current track point p k is deleted, the previous track point and the next track point of the current track point p k are connected, and the next track point is continuously calculated and determined; otherwise, the track point is retained and the next track point is continuously calculated and determined.
3. The method of claim 2, wherein, Step (2) specifically comprises the following steps: (201) For each piece of trajectory information, starting from the starting point p0, the slope Slope of the line segment between the two adjacent trajectory points is calculated in turn k : wherein is the latitude coordinate of the current trajectory point p k is the longitude coordinate of the current trajectory point p is the latitude coordinate of the current trajectory point p k is the longitude coordinate of the current trajectory point p (202) for the current trajectory information, the standard deviation of all slopes in the trajectory information is taken as the trajectory roughness R; (203) calculating the trajectory data quality Q of each trajectory information: where C noise is the number of noise points of the current trajectory information, C miss is the number of missing points of the current trajectory information, C feature is the number of stay points and inflection point coverage of the current trajectory information.
4. The method of claim 3, wherein, Step (3) comprises the following steps: (301)For m pieces of trajectory information after real-time denoising, each piece of trajectory is sorted in order of data quality Q from small to large, and m pieces of normalized trajectory data quality Q' are calculated in turn a : Q' a =Q a / (Q1+Q2+..+Q m ),a=1,2...m wherein Q a is the trajectory data quality of the a-th trajectory information, and Q1 m , Q' a is the normalized trajectory data quality of the a-th trajectory information; (302)calculating the multi-source trajectory information V after final fusion of the m pieces of trajectory information m : When m = 1, V1 = Q'1 x θ1 When m ≥ 2, V m = (1 - Q m ) x V m-1 + Q m x Θ m where θ m represents the mth track information.
5. The method of claim 4, wherein, Step (4) in the method for real-time dynamic adjustment of GeoHash algorithm sub-block size according to the roughness of the fused multi-source trajectory information is as follows: For the fused multi-source trajectory information, from the starting trajectory segment, a trajectory segment is sequentially added backward and the trajectory roughness of the current multi-source trajectory information is calculated, the trajectory roughness currently calculated is recorded to the set as a new element, if the trajectory roughness currently calculated is greater than the trajectory roughness of c% in the set and less than the trajectory roughness of (100-c)%, a default GeoHash string encoding length is set for the current trajectory segment, corresponding to a default GeoHash algorithm sub-block size; if the trajectory roughness currently calculated is greater than the trajectory roughness of (100-c)% in the set, the encoding length of the default GeoHash string is increased, corresponding to the decrease of the GeoHash algorithm sub-block size; if the trajectory roughness currently calculated is less than the trajectory roughness of c% in the set, the encoding length of the default GeoHash string is decreased, corresponding to the increase of the GeoHash algorithm sub-block size; wherein 0
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