Rapid all-sky star map identification method suitable for star sensor
By combining a data structure of navigation star catalog, star diagonal distance catalog, and index catalog, and using a two-level matching mechanism, the problems of slow all-sky star map recognition speed and insufficient anti-false star capability of star sensors are solved, and fast and accurate star map matching is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- CENT CHINA OPTOELECTRONICS TECH RES INST (CHINA STATE SHIPBUILDING CORP 717TH RES INST)
- Filing Date
- 2025-12-30
- Publication Date
- 2026-05-01
AI Technical Summary
Star sensors involve a large amount of computation when matching star charts across the entire sky. In particular, when the brightest star is a false star, there are too many invalid matching attempts, resulting in slow recognition speed and insufficient resistance to false stars.
Star map matching is performed using a navigation star table, a star-to-diagonal distance table, and a star-to-diagonal distance index table. By quickly calculating the matching results of navigation star pairs, and combining the parallel recording and judgment of bit arrays, a two-level matching mechanism is adopted, which involves local star pair matching followed by overall star-to-diagonal distance consistency verification. This reduces computational complexity and improves anti-pseudo-star capability.
It significantly reduces the search space in the star map matching process, improves the recognition speed and anti-pseudo-star capability, and ensures high matching accuracy and robustness.
Smart Images
Figure CN121962671A_ABST
Abstract
Description
A fast all-sky star map identification method suitable for star sensors Technical Field
[0001] This invention relates to the field of photoelectric imaging technology, and more specifically, to a method for rapid all-sky star map identification suitable for star sensors. Background Technology
[0002] A star sensor is a fixed-focus lens camera, with the photoelectric sensor mounted on the focal plane of the lens. Starlight (parallel light) is focused onto the photoelectric sensor after passing through the lens, where photoelectric conversion is completed to form a digital star map. Stars appear as small bright spots at different positions in the image.
[0003] The star sensor extracts the position coordinates of the star image in the sensor and the position vector in the star sensor coordinate system by extracting the star target; it identifies the star by using a star map recognition algorithm; and it calculates the current attitude of the star sensor based on the position vector of the star point in the star sensor coordinate system and the position vector in the inertial frame.
[0004] When there is no initial attitude information or other attitude information, the star sensor needs to use the all-sky star map identification method to identify stars. Based on information such as star vectors and diagonal distances, it matches the diagonal distances of navigation stars in the navigation star table. When the relevant information of a sufficient number of star vectors and navigation star vectors is within the error range, the star sensor successfully matches the star vectors and can then calculate the attitude. When there are not enough star vectors to match the relevant information of the navigation star vectors, the star sensor fails to identify stars in that frame of the star map.
[0005] Star sensors involve a large amount of computation when matching star charts across the entire sky, especially when the brightest star is a false star, resulting in too many invalid matches. Summary of the Invention
[0006] The purpose of this invention is to propose an innovative star map matching algorithm to improve the running speed, solve the bottleneck of star map recognition speed of star sensors, and improve the anti-spoofing ability of star sensors.
[0007] The technical solution of this invention is: to provide a fast all-sky star map identification method suitable for star sensors. This method uses a navigation star catalog, a star diagonal distance catalog, and a star diagonal distance index table for star map matching, and quickly calculates the matching results of navigation star pairs. Specifically, it includes the following steps:
[0008] S1. The star sensor captures a star image, and the star image is digitized and saved using a photoelectric sensor. Star points are extracted and calibrated from the digitized star image to obtain N star vector information relative to the coordinate system of the star sensor.
[0009] S2. Sort the N star vectors in descending order of brightness and number them from 1 to N. Establish the search boundary parameter M and let M=K, where K is the minimum number of stars required to determine a successful match.
[0010] S3. Enter the loop search phase: Select the combination that has not yet been searched in the K star vectors within the range of 1 to M. If it does not exist, go to S4; if it exists, go to S5.
[0011] S4. Let M = M + 1. If M > N, then the matching is determined to be a complete failure and a matching failure report is output. Otherwise, return to S3.
[0012] S5. Search and determine whether the K star vectors match the information recorded in the navigation star table, star diagonal distance table and star diagonal distance index table. If the match is successful, search for the matching results of other star vectors based on the successful matching information and output a successful matching report. If the match fails, return to S3.
[0013] Step S5 includes the following sub-steps:
[0014] S51. Initialize an unsigned array to 0. The length of the array is the number of navigation stars, and the number of bits in the array elements is greater than or equal to K(K-1) / 2.
[0015] S52. Calculate the diagonal distances between each pair of K star vectors and number the diagonal distances as 1 to K(K-1) / 2.
[0016] S53. Traverse the star diagonal distances. For the star diagonal distance numbered k, multiply the star diagonal distance by a preset specific multiple to obtain a discretized value D. Based on the preset error range d, retrieve all navigation star pair IDs whose discretized values are in the interval [Dd, D+d] through the star diagonal distance index table. For the retrieved and successfully matched navigation star pair, set the Kth bit of the array corresponding to the navigation star ID to 1.
[0017] S54. Search for the candidate star ID of each observed star vector. If the corresponding positions of the K-1 stars diagonal distances to a certain observed star vector are all 1 in the in-position array, then the index number corresponding to the observed star vector is its candidate star ID.
[0018] S55. If the number of candidate star IDs for any observed star vector is 0, then the matching is determined to be a complete failure and the matching failure result is output and the process is exited; otherwise, proceed to S56.
[0019] S56. Traverse the candidate star ID combinations of the observed star vector, recalculate the diagonal distance between candidate navigation stars and compare it with the observed value. If all diagonal distances are within the preset error range, output the matching success result; otherwise, continue searching for the next combination until all combinations fail to match, then output the matching failure result.
[0020] In any of the above technical solutions, the navigation star table further records the attribute information of all navigation stars, including star ID, brightness, right ascension and declination, and inertial frame vector information.
[0021] In any of the above technical solutions, the star-to-diagonal distance table further stores information on all possible navigation star pairs within the maximum star-to-diagonal distance range; each entry in the star-to-diagonal distance table contains a discretized star-to-diagonal distance value and its corresponding two navigation star IDs, and the star-to-diagonal distance table is sorted in ascending order according to the discretized star-to-diagonal distance values.
[0022] In any of the above technical solutions, the discretized value of the star diagonal distance is further calculated by multiplying the angle value of the star diagonal distance by a preset specific multiple and then rounding down.
[0023] In any of the above technical solutions, the length of the star diagonal distance index table is the maximum star diagonal distance discretization value a. Each element of the star diagonal distance index table stores an index information b. The index information b indicates that the discretization value of the b-th and subsequent entries in the star diagonal distance table is greater than or equal to the discretization value a corresponding to the index table, while the discretization values of the entries before the b-th entry are all less than the discretization value a.
[0024] The beneficial effects of this invention are:
[0025] The technical solution in this invention introduces a combined data structure of navigation star table, star diagonal distance table and star diagonal distance index table, and performs discretization and indexing of star diagonal distance, which enables the rapid location of candidate navigation star pairs within a given error range, significantly reducing the search space in the star map matching process and reducing computational complexity.
[0026] In a preferred embodiment of the present invention, a search strategy is adopted that expands gradually from bright stars to dark stars and from small-scale combinations to large-scale combinations. When the brightest star is a false star or an interfering star, the matching failure combination containing the false star can be quickly skipped, and star vector combinations that do not contain the false star are tried first, thereby improving the algorithm's ability to resist false stars and its overall robustness.
[0027] During the matching process, a bit array is used to record and judge the matching relationship of star pairs in parallel. The consistency judgment of the angular distance of multiple star pairs is transformed into bit operation, which improves the efficiency of candidate star ID screening and is conducive to realizing fast star map recognition.
[0028] By employing a two-level matching mechanism that first performs local star pair matching and then verifies the consistency of the overall star pair diagonal distance for candidate star ID combinations, the recognition speed is further improved while ensuring the accuracy of the matching and avoiding high-cost overall calculations for all combinations. Attached Figure Description
[0029] The advantages of the above and additional aspects of the present invention will become apparent and readily understood in the description of the embodiments in conjunction with the following drawings, wherein:
[0030] Figure 1 is an overall flowchart of a fast all-sky star map identification method for star sensors according to an embodiment of the present invention;
[0031] Figure 2 is a flowchart of a single-match method for a fast all-sky star map identification method for star sensors according to an embodiment of the present invention;
[0032] Figure 3 is a diagram showing the relationship between the navigation star table, the star diagonal distance table, and the star diagonal distance index table in a fast all-sky star map identification method for star sensors according to an embodiment of the present invention. Detailed Implementation
[0033] To better understand the above-mentioned objectives, features, and advantages of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments of the present invention and the features thereof can be combined with each other.
[0034] In the following description, many specific details are set forth in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and therefore the scope of protection of the invention is not limited to the specific embodiments disclosed below.
[0035] This embodiment provides a fast all-sky star map identification method suitable for star sensors. The method uses a navigation star catalog data structure, a star diagonal distance table data structure, and a star diagonal distance index table data structure to perform star map matching. Specifically, when searching for all matching navigation star diagonal distances within the error range, the star diagonal distance is first calculated and multiplied by a specific multiple to obtain the star diagonal distance divergence value D. If the matching error range is d, then based on the values of Dd and D+d in the star diagonal distance index table, the information of all star pairs within the error range in the navigation star diagonal distance table is obtained, thereby quickly calculating the matching result of the navigation star pairs.
[0036] As shown in Figure 3, the navigation star table records the attribute information of all navigation stars, including star ID, magnitude, right ascension and declination, and inertial frame vector information.
[0037] The diagonal distance table stores information on all possible navigation star pairs within the maximum diagonal distance range. Each entry contains a discretized diagonal distance value and the IDs of the two corresponding navigation stars. The entire table is sorted in ascending order of the diagonal distance discretized value. The diagonal distance discretized value is calculated by multiplying the angle value of the diagonal distance by a preset specific multiple and then rounding down.
[0038] The length of the star diagonal distance index table is the discretized value of the maximum star diagonal distance, which is a. Each element of the star diagonal distance index table stores an index information b, indicating that in the navigation star diagonal distance table, the discretized value of the b-th and subsequent entries is greater than or equal to the discretized value a corresponding to the index table, while the discretized values of the entries before the b-th entry are all less than a.
[0039] As shown in Figure 1, the overall process of this method includes:
[0040] S1. First, after the star sensor captures the star map by taking a picture, it uses a photoelectric sensor to digitize and save the star map. Then, it extracts and calibrates the star points in the digitized star map to obtain N star vector information of the star points relative to the coordinate system of the star sensor.
[0041] S2. Sort the detected N star vectors from largest to smallest according to their brightness (professional term: magnitude) and number them from 1 to N. Establish the search boundary parameter M and let M=K, where K is the minimum number of stars required to determine a successful match. In this invention, all equal signs indicate that the latter assigns a value to the former, which will not be explained further below.
[0042] S3. Enter the loop search stage. Select the combination that has not been searched in the K star vectors within the range of 1 to M. If it does not exist, jump to step S4. If it exists, jump to step S5.
[0043] S4. Let M = M + 1. When M > N, determine that the match has completely failed, exit the process directly and output a report of match failure. Otherwise, jump to S3 and search again in a new range.
[0044] S5. Search the navigation star table, star diagonal distance table, and star diagonal distance index table to determine whether the K star vectors match the information recorded in the table. If a match is found to be successful, search for the matching results of other star vectors based on the information of the successful match and output them together with the report of successful match. If a match fails, jump to step S3 to search again.
[0045] Through this process, when the brightest star is a false star, the algorithm can quickly skip the failed matching combinations that contain the false star and prioritize trying combinations that do not contain the false star, thereby significantly improving the ability to resist false stars and finding the K stars that are successfully matched as soon as possible.
[0046] As shown in Figure 2, the process of step S5, searching and judging the matching results of K star vectors, includes:
[0047] S51. Initialize an unsigned array to 0. The length of the array is the number of navigation stars, and the number of bits in the array elements is greater than or equal to K(K-1) / 2.
[0048] S52. Calculate the diagonal distances between each pair of K star vectors and number them from 1 to K(K-1) / 2.
[0049] S53. Traverse the star diagonal distances. For the star diagonal distance numbered k, multiply the star diagonal distance by a specific multiple to obtain a discretized value D. According to the preset error range d, use the star diagonal distance index table to retrieve all navigation star pair IDs whose discretized values are in the interval [Dd, D+d]. For a successfully matched navigation star pair, set the Kth bit of the array corresponding to the navigation star ID to 1.
[0050] S54. Search for the candidate star IDs of each star vector. If the corresponding positions of the K-1 stars diagonally distanced to a certain star vector are all 1 in the bit array, then the index number corresponding to that star vector is the candidate star ID of that star vector.
[0051] For example, when K is 4, if the diagonal distances between the star vector of star 1 and stars 2, 3 and 4 are numbered 1, 2 and 3 respectively, then the index number of the element whose first, second and third positions are all 1 is searched in the array. The index number found is the candidate star ID of the star vector of star 1.
[0052] S55. If the number of possible star IDs for a star vector is 0, then the matching is determined to be a complete failure. Output the matching failure result and exit. Otherwise, jump to step S56.
[0053] S56. Traverse the candidate ID combinations of the star vectors, recalculate the diagonal distances between the navigation stars and compare them with the observed values. If all diagonal distances are within the error range, output the matching success result; otherwise, search for the next group until all combinations fail to match, then output the matching failure result.
[0054] By employing the aforementioned method of moving from local to global approaches, combining bitwise operations for acceleration, and employing a specific search order, this invention can significantly reduce computation time while maintaining a high matching accuracy.
[0055] In another embodiment of the present invention, to further illustrate the overall process of the method, taking a star sensor parameter N of 15 and K of 4 as an example, assuming that the brightest star in the single-frame input star vector is a false star, the algorithm sorts the star vectors by magnitude, with star vector 1 being a false star and star vectors 2 to 15 being true stars. First, let M = 4, and select K stars from 1 to M to participate in the matching, i.e., the combination of 1, 2, 3, and 4. When matching the results of star vectors 1, 2, 3, and 4, initialize the array, calculate the diagonal distance between each pair of stars, and update the matching information. Since star number 1 is a fake star, the set of star IDs in the array where all corresponding bits of 1-2, 1-3, and 1-4 are 1 is empty, so this combination fails to match. Since there are no other combinations with M=4, let M=5. Select K star combinations from 1 to 5. Since all combinations that do not contain star number 5 have been searched, prioritize combinations that contain star number 5, such as 1, 2, 3, and 5. Since star number 1 is a fake star, the set of star IDs in the array where all corresponding bits of 1-2, 1-3, and 1-5 are 1 is empty, so this combination fails to match. Similarly, 1, 2, ... Matches 4 and 5 with 1, 3, 4, 5 failed. When 2, 3, 4, 5 were selected for matching, the array was initialized, the diagonal distances between each pair of stars were calculated, and the navigation star IDs were searched based on the diagonal distances and error range, and the array was updated. The set of star IDs where all bits of 2-3, 2-4, and 2-5 were 1 was searched and identified as {id2}, which was used as the possible star IDs for the vector of star 2. Similarly, star 3 was identified as {id3}, star 4 as {id4}, and star 5 as {id5}. The navigation star information for id2, id3, id4, and id5 was read. Calculate the diagonal distance between each pair of stars and compare it with the diagonal distances of stars 2-5. If all are within the error range, the match is successful, and the star IDs corresponding to stars 2-5 are id2-id5. After a successful match, calculate the navigation star information for the remaining star vectors 1 and 6-15 based on the recognition results of stars 2-5. For example, if the diagonal distance between star vector 6 and star vectors 2-5, and the diagonal distance between id6 and star vectors id2-id5 are all within the error range, then star vector 6 is identified as navigation star id6. After all calculations are completed, output the matching results.
[0056] In another embodiment of the present invention, a comparative experiment was designed with several existing algorithms to demonstrate that the present invention has the advantages of faster processing speed and stronger anti-pseudo-star capability compared with other all-sky star map recognition algorithms.
[0057] The algorithms used include: triangular matching algorithm, pyramid algorithm, navigation star field algorithm, and angular distance voting algorithm. All algorithms use the K-vector algorithm to quickly search for the navigation star ID information corresponding to the angular distance between star pairs, and use bitwise operations when they can be used to speed up the matching process.
[0058] The experimental environment parameters are as follows:
[0059] CPU: 13th Gen Intel(R) Core(TM) i7-13700KF, 3.40 GHz.
[0060] Operating system: Windows 10 Professional.
[0061] Compilation and runtime environment: Microsoft Visual Studio Professional 2015 version 14.0.23107.0 D14REL.
[0062] The experimental method was designed as follows: A right ascension and declination were randomly generated within the all-sky attitude framework, with their directions uniformly distributed across a unit sphere. A specific roll value was set. Using the right ascension and declination as the center and the maximum field-of-view radius (15°) of the star sensor in the navigation star catalog as the radius, all navigation star information falling within this radius was searched. Then, based on the navigation star vector table and attitude information such as right ascension, declination, and roll, a star vector table in the star sensor coordinate system was calculated. Fifteen star vectors (the average number of stars detected by the star sensor) were randomly selected for matching. The first K star vectors were randomly varied within the field of view to create dummy stars. Different all-sky star matching algorithms were used to obtain different attitude information. The calculated attitude information was compared with the original attitude information. If the results were within the error range, the match was correct; otherwise, the match was incorrect.
[0063] The experimental results are shown in Tables 1 and 2 below:
[0064] Table 1. Number of matching errors in 3000 attempts.
[0065]
[0066] Table 2, Average time taken for 3000 matches (unit: microseconds).
[0067]
[0068] The experimental results show that, compared with traditional all-sky star map recognition algorithms, the algorithm proposed in this paper has the highest matching accuracy and the highest detection speed, which demonstrates the fast operation speed and strong anti-pseudo-star capability of the proposed algorithm.
[0069] In summary, this invention proposes a fast all-sky star map identification method suitable for star sensors, comprising:
[0070] S1. The star sensor captures a star image, and the star image is digitized and saved using a photoelectric sensor. The star points are extracted and calibrated from the digitized star image to obtain N star vector information relative to the coordinate system of the star sensor.
[0071] S2. Sort the N star vectors in descending order of brightness and number them from 1 to N. Establish the search boundary parameter M and let M=K, where K is the minimum number of stars required to determine a successful match.
[0072] S3. Enter the loop search phase: Select the combination that has not yet been searched in the K star vectors within the range of 1 to M. If it does not exist, go to S4. If it exists, go to S5.
[0073] S4. Let M = M + 1. If M > N, then the matching is completely failed and a matching failure report is output. Otherwise, return to S3.
[0074] S5. Search and determine whether the K star vectors match the information recorded in the navigation star table, star diagonal distance table, and star diagonal distance index table. If the match is successful, search for the matching results of other star vectors based on the successful matching information and output a successful matching report. If the match fails, return to S3.
[0075] Step S5 includes the following sub-steps:
[0076] S51. Initialize an unsigned array to 0. The length of the array is the number of navigation stars, and the number of bits in the array elements is greater than or equal to K(K-1) / 2.
[0077] S52. Calculate the diagonal distances between each pair of K star vectors, and number the diagonal distances as 1 to K(K-1) / 2.
[0078] S53. Traverse the star diagonal distances. For the star diagonal distance numbered k, multiply the star diagonal distance by a preset specific multiple to obtain a discretized value D. Based on the preset error range d, retrieve all navigation star pair IDs whose discretized values are in the interval [Dd, D+d] through the star diagonal distance index table. For the retrieved and successfully matched navigation star pair, set the Kth bit of the array corresponding to the navigation star ID to 1.
[0079] S54. Search for the candidate star IDs of each observed star vector. If the corresponding positions of the K-1 stars diagonally opposite to a certain observed star vector are all 1 in the in-place array, then the index number corresponding to the observed star vector is its candidate star ID.
[0080] S55. If the number of candidate star IDs for any observed star vector is 0, then the matching is determined to be a complete failure and the matching failure result is output and the process is exited; otherwise, proceed to S56.
[0081] S56. Traverse the candidate star ID combinations of the observed star vector, recalculate the diagonal distance between candidate navigation stars and compare it with the observed value. If all diagonal distances are within the preset error range, output the matching success result; otherwise, continue searching for the next combination until all combinations fail to match, then output the matching failure result.
[0082] The steps in this invention can be adjusted, combined, or deleted according to actual needs.
[0083] The units in the device of the present invention can be merged, divided, or reduced according to actual needs.
[0084] In this invention, the terms "installation," "connection," "linking," and "fixing" should be interpreted broadly. For example, "connection" can be a fixed connection, a detachable connection, or an integral connection; "linking" can be a direct connection or an indirect connection through an intermediate medium. Those skilled in the art can understand the specific meaning of these terms in this invention according to the specific circumstances.
[0085] The shapes of the components in the accompanying drawings are schematic and may differ from their actual shapes. The drawings are only used to illustrate the principles of the present invention and are not intended to limit the present invention.
[0086] Although the invention has been disclosed in detail with reference to the accompanying drawings, it should be understood that these descriptions are merely exemplary and not intended to limit the application of the invention. The scope of protection of the invention is defined by the appended claims and may include various modifications, alterations, and equivalents made to the invention without departing from the scope and spirit of the invention.
Claims
1. A method for rapid all-sky star map identification suitable for star sensors, characterized in that, The method employs a navigation star table, a star diagonal distance table, and a star diagonal distance index table for star map matching, quickly calculating the matching results of navigation star pairs. Specifically, it includes the following steps: S1, a star sensor captures a star map, a photoelectric sensor digitizes and saves the star map, and star points are extracted and calibrated from the digitized star map to obtain N star vector information relative to the coordinate system of the star sensor; S2, the N star vectors are sorted from largest to smallest brightness and numbered 1 to N, a search boundary parameter M is established and set M=K, where K is the minimum number of stars required for a successful match; S3, the cyclic search phase begins: within the range of 1 to M, K stars are selected... If no combination of star vectors has been searched, proceed to S4 if it does not exist, and to S5 if it does exist; S4: Let M = M + 1. If M > N, the matching is determined to be a complete failure and a matching failure report is output; otherwise, return to S3; S5: Search and determine whether the K star vectors match the information recorded in the navigation star table, star diagonal distance table, and star diagonal distance index table. If the matching is successful, search for matching results of other star vectors based on the successful matching information and output a matching success report; if the matching fails, return to S3; wherein, step S5 includes the following sub-steps: S51: Initialize an unsigned array to 0. The length of this array is the navigation star S52. Calculate the diagonal distances between each pair of K star vectors and number them from 1 to K(K-1) / 2; S53. Iterate through the diagonal distances. For a diagonal distance numbered k, multiply it by a preset specific multiple to obtain a discretized value D. Based on a preset error range d, retrieve all navigation star pair IDs whose discretized values are in the interval [Dd, D+d] using the diagonal distance index table. For a retrieved and successfully matched navigation star pair, set the Kth bit of the array corresponding to the navigation star ID to 1; S54. Search for each observed star vector... Candidate Star ID: If the corresponding positions of the K-1 diagonal distances of a certain observed star vector are all 1 in the bit array, then the index number corresponding to the observed star vector is its candidate star ID; S55: If the number of candidate star IDs for any observed star vector is 0, then the matching is determined to be completely unsuccessful and the matching failure result is output and the process is exited; otherwise, proceed to S56; S56: Traverse the candidate star ID combinations of the observed star vectors, recalculate the diagonal distances between candidate navigation stars and compare them with the observed values. If all diagonal distances are within the preset error range, the matching success result is output; otherwise, continue searching for the next combination until all combinations fail to match and the matching failure result is output.
2. The rapid all-sky star map identification method for star sensors as described in claim 1, characterized in that, The navigation star table records the attribute information of all navigation stars, including star ID, brightness, right ascension and declination, and inertial frame vector information.
3. The rapid all-sky star map identification method for star sensors as described in claim 1, characterized in that, The diagonal distance table stores information on all possible navigation star pairs within the maximum diagonal distance range; each entry in the diagonal distance table contains a discretized diagonal distance value and its corresponding two navigation star IDs, and the diagonal distance table is sorted in ascending order of the discretized diagonal distance values.
4. The rapid all-sky star map identification method for star sensors as described in claim 3, characterized in that, The method for calculating the discretized value of the star diagonal distance is as follows: multiply the angle value of the star diagonal distance by a preset specific multiple and then round down.
5. The rapid all-sky star map identification method for star sensors as described in claim 1, characterized in that, The length of the star diagonal distance index table is the maximum star diagonal distance discretization value a. Each element of the star diagonal distance index table stores an index information b. The index information b indicates that the discretization value of the b-th and subsequent entries in the star diagonal distance table is greater than or equal to the discretization value a corresponding to the index table, while the discretization values of the entries before the b-th entry are all less than the discretization value a.