Retrieval system and method using distance index6845377Abstract Similarity retrieval is performed through the use of a distance index which contains a concatenated key and pointer information and retains a dictionary type order in the concatenated key. The concatenated key is produced by discretizing the distance from each of a plurality of reference points to an object point representing information to be retrieved in a multidimensional space consisting of a plurality of dimensions corresponding to the plurality of feature parameters and concatenating the resulting discretized distances. The use of the concatenated key allows candidate points in the d-distance neighborhood to be narrowed down. Claims What is claimed is: Description BACKGROUND OF THE INVENTION
Original distance Discrete distance
0.0-0.1 0
0.1-0.2 1
0.2-0.3 2
. . .
0.9-1.0 9
In EXAMPLE 1, the continuous distance in the range of 0.0 to 1.0 is divided into 10 groups each having an equal width of 0.1. In this case, the original distance, whether it be 0.11 or 0.15, is represented as a discretized distance of 1. In this manner, distance is divided into a plurality of groups by discretization. We shall refer to each group as a band. Also, we shall refer to the difference between maximum and minimum values of distance in each band as the width of it. In EXAMPLE 1, the width of each band is 0.1. Each band is allowed to have a different width. Representing a function for converting original distance into discretized distance by .gamma. yields discretized distance=.gamma. (original distance) We shall refer to this function .gamma. as the discretization function. Using this representation in EXAMPLE 1 yields, for example .gamma.(0.15)=1 The discretization function must satisfy the following condition: If d1<=d2, then .gamma.(d1)<=.gamma.(d2) That is, the discretization function must be monotone increasing. Further, we shall refer to a function for mapping points v in multidimensional space into discretized distances as a discretized distance function. Representing the reference point as s, the discretized distance function .delta.(v) is given by .delta.(v)=.gamma.(d(s, v)) EXAMPLE 2
Original distance Discretized distance
0.0-1.0 1
1.0-10.0 2
10.0-100.0 3
. . .
In this example, the bands have varying widths: 1.0, 10.0, and 90.0. Such discretization is utilized in a case where one wants to make the number of object points in each band uniform. For example, this corresponds to a case where the density of object points becomes reduced as the distance increases. EXAMPLE 3
Original distance Discretized distance
5,000,000.0-5,100,000.0 1
5,100,000.0-5,200,000.0 2
5,200,000.0-5,300,000.0 3
5,900,000.0-6,000,000.0 10
In this example, very great values are simplified. Even if a reference point is set at a long distance, the discretization allows it to have a simple value. The discretized distances, which are ordered quantities, can be represented as integers, offering advantages with respect to space efficiency. In general, the discretized distance does not always satisfy the properties of distance. In the three examples shown above, the discrete distances might be called the numbers allocated to regions divided on the basis of the distance from a reference point in the order of distance beginning with the region nearest to the reference point. The objectives of the discretization are to perform information retrieval at high speed and increase the space efficiency. The discretization is also involved in clustering to be described later. Next, a concatenated key is created by concatenating a plurality of discretized distances determined from a plurality of reference points. Let the reference points be s1, s2, . . . , sm and the original distances from the respective reference points be d1, d2, . . . , dm. The discretization function may generally vary from reference point to reference point. Let the discretization functions for the reference points be .gamma.1, .gamma.2, . . . , .gamma.m. These functions may, of course, be the same function. The discrete distances d1', d2', . . . , dm' then become d1'=.gamma.1(d1) d2'=.gamma.2(d2) dm'=.gamma.m(dm) The concatenated key is logically the same as the following arrangement of those discretized distances: (d1', d2', . . . , dm') For example, when each discretized distance is represented by one byte, the concatenated key is composed of m bytes. An index is created based on this concatenated key. The index is a set of two values: concatenated discretized distance and pointer. The dictionary type order is naturally defined for the concatenated discretized distance, which is retained by the index. The methods for implementing such an index includes a B-tree. Information concerning each object point is generally stored in the secondary storage unit 113 as a record. Information that allows access to the record is a pointer. The objective of the index is to allow fast access to information of the reference point on the basis of the concatenated discretized distances. The advantage of concatenating distances is that the index is allowed to be single in number but not plural. A method might be considered by which an index is created for each distance. However, from the standpoint of not only performance but also space efficiency, it is greatly desirable to use the concatenated distances. The index thus created should accurately be referred to as the discretized concatenated distance index; however, in the description that follows, it is called the discretized distance index for simplicity. For description's sake, an original distance-based index is used which is formed in a similar way to the discretized distance index and referred hereinafter to as the original distance index. Next, a method for seeking the distance neighborhood will be described with reference to FIGS. 3 through 9. The distance neighborhood is calculated by the similarity retrieval section 121. First, a description is given of a procedure of seeking the d-distance neighborhood associated with a certain specified point p on the basis of the original distance from a reference point s. The procedure is 1. Determine D=d(s, p) 2. Using the original distance index, determine from a set T of object points a set T' of object points t that satisfy the condition D-d<=d(s, t)<=D+d (A) The condition (A) means that the distance between each object point t and the reference point lies in the range of D-d to D+d. 3. Calculate d(p, t) for each point t in T' and determine a set of points that satisfy d(p, t)<=d. This set is the d-distance neighborhood to be determined. We shall refer to the set T' in the above procedure as the set of candidate points in a sense that it contains candidates for the d-distance neighborhood. Also, the points in the set T' are referred to as candidate points. FIG. 3 shows the relationship between the d-distance neighborhood and the set of candidate points in a two-dimensional space. A set of object points existing between two circles 131 and 132 is the set T' of candidate points. A set of object points within a circle 133 is the d-distance neighborhood. Next, a description is given of a procedure of determining the d-distance neighborhood of a certain specified point p on the basis of its original distance from two or more reference points s1, s2, . . . , sm. The procedure is 1. As in the case of a single reference point, determine D1=d(s1, p) D2=d(s2, p) Dm=d(sm, p) 2. Using the original distance index, determine from a set T of object points a set T' of candidate points as sets of points that respectively satisfy the conditions D1-d<=d(s1, t)<=D1+d (B1) D2-d<=d(s2, t)<=D2+d (B2) Dm-d<=d(sm, t)<=Dm+d (Bm) Assuming the candidate point sets that respectively satisfy the conditions (B1), (B2), . . . , (Bm) to be T1', T2', . . . , Tm' yields T'=T1'.andgate.T2' . . . .andgate.Tm' 3. Calculate d(p, t) for each point t in T' and determine a set of points that satisfy d(p, t)<=d. This set is the d-distance neighborhood to be determined. FIG. 4 shows the relationship between the d-distance neighborhood and the sets of candidate points based on the original distances from two reference points in a two-dimensional space. Two doughnut-shaped regions 141 and 142 around the reference points s1 and s2 form sets T1' and T2', respectively. The region where the two regions intersect with each other forms the set T' of candidate points. A set of object points within a circle 143 is the d-distance neighborhood to be obtained. Next, a description is given of a procedure of determining the d-distance neighborhood of a certain specified point p on the basis of its discretized distance from a reference point s. The procedure is 1. Determine D=d(s, p) Note here that d(s, p) represents the original distance. 2. Subject the upper and lower limits D+d and D-d in the condition (A) to discretization, i.e., determine Dlower=.gamma.(D-d) Dupper=.gamma.(D+d) where .gamma. is the aforementioned discretization function. Next, using the discretized distance index, determine a set T' of candidate points whose discretized distances satisfy Dlower<=.gamma.(d(s, t))<=Dupper Using discretized distance function .delta.(v)=.gamma.(d(s, v)), the condition is rewritten as .gamma.(D-d)<=.delta.(t)<=.gamma.(D+d) (C) where D=.delta.(p). 3. Calculate d(p, t) for each point t in T' and determine a set of points that satisfy d(p, t)<=d. This set is the d-distance neighborhood to be obtained. Next, a description is given of a procedure of determining the d-distance neighborhood of a certain specified point p on the basis of its discretized distances from a plurality of reference points s1, s2, . . . , sm. FIG. 5 is a flowchart for the determination of the distance neighborhood. Let the discretized distance corresponding to each reference point si (i=1, 2, . . . , m) be .gamma.i 0 and the discretized distance function be .gamma.i 0. The procedure is 1. As in the case of a single reference point, using the discretized distance function, in step S1, determine D1=d(s1, p)=.delta.1(p) D2=d(s2, p)=.delta.2(p) Dm=d(sm, p)=.delta.m(p) 2. As in the case of a single reference point, determine D1lower=.gamma.1(D1-d), D1upper=.gamma.1(D1+d) D2lower=.gamma.2(D2-d), D2upper=.gamma.2(D2+d) Dmlower=.gamma.m(Dm-d), Dmupper=.gamma.m(Dm+d) Next, as in the case where original distances from two or more reference points are used, using the discretized distance index, in step S2 determine from a set T of object points a set T' of candidate points that simultaneously satisfy D1lower<=.gamma.1(d(s1, t))<=D1upper D2lower<=.gamma.2(d(s2, t))<=D2upper Dmlower<=.gamma.m(d(sm, t))<=Dmupper These conditions can be rewritten in the same manner as when the condition (C) is determined as .gamma.1(D1-d)<=.delta.1(t)<=.gamma.1(D1+d) (C1) .gamma.2(D2-d)<=.delta.2(t)<=.gamma.2(D2+d) (C2) .gamma.m(Dm-d)<=.delta.m(t)<=.gamma.m(Dm+d) (Cm) 3. Calculate d(p, t) for each point t in T' and determine a set of points that satisfy d(p, t)<=d (step S3) This set is the d-distance neighborhood to be obtained. FIG. 6 is a diagram illustrating the discretization of two-dimensional space of a square shape according to discretized distances from two reference points s1 and s2. Points t1, t2 and t3 are object points. A sphere q represents a region corresponding to the d-distance neighborhood in the similarity retrieval. Strictly speaking, a sphere in a two-dimensional space is a circle; however, it is referred herein to as the sphere. The discretized distances of the object points and the sphere are as follows:
Object point Discretized Discretized
(sphere) distance from s1 distance from s2
t1 4 3
t2 3 5
t3 3 5
q 3, 4 4, 5
As for the sphere q, as shown above, its discretized distance from s1 is contained in the bands of 3 and 4 and its discretized distance from s2 is included in the bands of 4 and 5. In this case, the conditions representing the set T' of candidate points are 3<=.delta.1(t)<=4 (D1) 4<=.delta.2(t)<=5 (D2) The object point t satisfies the condition (D1) but not the condition (D2); thus, it is not contained in the sphere q. In contrast, the object points t2 and t3 both satisfy the conditions (D1) and (D2); thus, it is possible that they are contained in the sphere q. However, as can be seen from FIG. 6, it is only the object point t3 that is contained in the sphere q. This is the reason why the decision condition, d(p, t)<=d, is set up in the calculation of the d-distance neighborhood. That is, T' is merely a set of candidates to which many object points are narrowed down through filtering. That T' has only to be examined in order to determine the d-distance neighborhood holds for all distances that satisfy the properties of distance. That is, this principle can be applied to all distances that generally satisfy the properties of distance, including not only Euclidean distances which are distances in a normal sense but also Manhattan distances to be described later. This is greatly significant in practice because various distances are used with multidimensional spaces. However, this is difficult to understand intuitively and hence needs to be illustrated logically. To this end, it is simply required to show that a point t contained in the d-distance neighborhood of p satisfies the conditions (C), (C1), (C2), . . . , (Cm). A lemma must be to prove is as follows. Lemma: Let .gamma.(d) be a discretization function that satisfies if d1<=d2, then .gamma.(d1)<=.gamma.(d2) (1) Let an arbitrary point s on a multidimensional space be a reference point and an arbitrary point on the multidimensional space be p. Let the discretization distance function be .delta.(t)=.gamma.(d(s, t)). At this point, an arbitrary point v within the d-distance neighborhood of p, i.e., a point v that satisfies d(p, v)<=d (2) will satisfy D-d<=d(s, v)<=D+d (3) where D=d(s, p) (4). Also, it satisfies .gamma.(D-d)<=.delta.(v)<=.gamma.(D+d) (5) The proof of the above lemma is demonstrated as follows. Proof: From the aforementioned property 3 of distance, we obtain d(s, v)<=d(s, p)+d(p, v) Combining this expression with expressions (2) and (4) yields d(s, v)<=D+d (6) On the other hand, from the property 3 of distance, we obtain d(s, p)<=d(s, v)+d(v, p) (7) From the property 2 of distance, we obtain d(v, p)=d(p, v) Combining this expression with expressions (4) and (7) yields D<=d(s, v)+d(p, v) Substituting expression (2) into this expression and rearranging yields D-d<=d(s, v) (8) Combining expressions (6) and (8) yields expression (3). Expression (1) indicates that the function .gamma. satisfies the property of monotonous increasing. Hence, from expression (3) we obtain .gamma.(D-d)<=.gamma.(d(s, v))<=.gamma.(D+d) i.e., .gamma.(D-d)<=.delta.(v)<=.gamma.(D+d) Hence, expression (5) holds. (The end of proof) Next, the method of determining a set of candidate points using a discretized distance index will be described in more detail. First, the reason for discretization will be described using such an exemplary index as shown in FIG. 7. In FIG. 7, id corresponds to an identifier of a record that forms an object point and d1 and d2 represent distances from different reference points. For comparison with discretized distance, original distance is used here. The index corresponds to the results of sorting object points in a dictionary type order using d1 and d2. The typical example of index is a B-tree. The index, which is generally a tree structure, is described logically in FIG. 7. The pointer section is omitted. In actual similarity retrieval, access is made to a candidate record using a pointer. It is then required to check whether the similarity conditions are met using the feature parameter of the record. Although d1 and d2 are described separately in FIG. 7, they form a concatenated key. Assume the conditions for similarity retrieval to be 0.4<-d1<-0.6 (E1) 0.1<=d2<=0.3 (E2) Then, the use of the index allows direct access to the first record to satisfy the conditions (E1) and (E2). In this example, the first record is the id=4 record (d1=0.4 and d2=0.2). Following records beginning with the id=4 record, scanning is made until it is found that the conditions (E1) and (E2) are no longer satisfied, i.e., until the id=7 record (d1=0.7 and d2=0.4) is reached. In this case, therefore, four records will be followed. One record (id=4) that satisfies the conditions (E1) and (E2) can then be found from among the four records. Here, consider the following discretization in this example: 0.0<=.times.<0.4 1 0.4<=.times.<0.7 2 0.7<=.times.<1.0 3 In this case, the index becomes as depicted in FIG. 8. Since the object points are rearranged in the dictionary type order of discretized distances, the index of FIG. 8 differs from that of FIG. 7 in the id order. The conditions for similarity retrieval become d1=2 (E3) d2=1 (E4) The use of the index allows direct access to the record (id=4) that satisfies the conditions (E3) and (E4). In order to make sure that there is no other record that satisfies the conditions (E1) and (E2), it is required to check the next id=6 record. Thus, a total of two records will be followed. In this example, the index using non-discretized distances requires to follow four records, whereas the index using discretized distances allows a record that satisfies the conditions to be found by following only two records. Thus, by the discretization of distances, records can be clustered to reduce the number of records to be followed. The clustering refers to collecting data into storage areas close to one another. This advantage is a first reason for the discretization of distances. Although, in this example, this advantage might be difficult to understand because of few records and dimensions, it will become more remarkable as the records increase in number. The reason is that, considering the conditions (E3) and (E4) as a sphere of radius d with center at a specified point representing the conditions for similarity retrieval, the possibility of following records which are not ones to be sought increases with increasing number of records contained in a band that contains that sphere. The above example is an ideal one of discretization. That is, the case where both the conditions (E3) and (E4) are represented by equations rather than by inequalities is ideal. In contrast, when the conditions for similarity retrieval are represented by inequalities 2<=d1<=3 (E5) 1<=d2<=2 (E6) records that satisfy the conditions are ones of id=4, 6, 7, and 9; however, the intervention of records of id=5 and 8 that do not satisfy the conditions is involved. This will never occur in the case of equations alone. Considering this situation in terms of a sphere representing the conditions for similarity retrieval, that the conditions contain signs of inequality means that the sphere spans two or more bands. It is therefore desirable to set the sphere not to span bands. To this end, it is basically required to set the bandwidth large to some degree. Further, the following measures will be considered. In a multidimensional space, a threshold radius is expected to be frequently set up on similarity conditions. For example, this corresponds to a case where, outside the threshold radius, similarity is not considered to exist. In this case, since searching is performed within the threshold radius, a distance two or more times the threshold radius is used as the unit of discretization. The distance two times the threshold radius represents the minimum condition for allowing the sphere to be contained in a band. In general, therefore, a longer distance is used as the unit of discretization. In practice, however, even the band width thus increased will not perfectly prevent the sphere from spanning bands. It is thus required to consider some processing to be performed in that event. In the case of similarity retrieval to satisfy the conditions (E5) and (E6), two basic processes are considered: sequential retrieval and divided condition retrieval. First, the sequential retrieval is a method which, after access has been made to the first record that satisfies the conditions, searches through records sequentially until it is known that the conditions will be satisfied no more. In the above example, direct access is made to the first record (id=4, d1-2, d2=1) that satisfies the conditions (E5) and (E6) and then records are followed in sequence for which a decision is made as to whether the conditions are met. The processing is stopped when the id=7 record has been checked because the condition (E5) is not met for the subsequent records. The reason is that the d1 values are sorted in the ascending order of magnitude and, since the condition (E5) is not met at that point, the subsequent records will also not meet the condition (E5). The divided condition retrieval is a method which divides conditions containing signs of inequality into conditions consisting of equations and retrieves records based on individual divided conditions. In the above example, the conditions (E5) and (E6) are divided into four conditions d1=2 and d2=1 (E7) d1=2 and d2=2 (E8) d1=3 and d2=1 (E9) d1=3 and d2=2 (E10) Direct access is made to each of records which satisfies a respective one of the conditions using the index and then records are traced until a record that satisfies none of the conditions is reached. The advantage of this method is that unnecessary records which are traced in the sequential retrieval method can be skipped over. This advantage will become more remarkable as the number of records increases. However, the sequential retrieval also has a merit. In the similarity retrieval in a multidimensional space, the range of conditions and the number of records increase and hence, in general, the divided condition retrieval is effective. However, since the index is a tree structure as exemplified by a B-tree, direct access requires to follow the tree structure from its root. Thus, if the range of retrieval is narrow, the sequential retrieval may be more effective. With these matters taken into consideration, the two methods can be combined into one which involves the divided condition retrieval using some preceding conditions of given conditions and the sequential retrieval using the subsequent conditions. Specifically, retrieval is switched from the divided condition retrieval to the sequential retrieval when the range of the index for retrieval becomes narrowed down. To explain this method, such an index as shown in FIG. 9 is presented. In FIG. 9, d1, d2 and d3 represent discretized distances. Similarity conditions are set up such that 2<-d1<=3 (E11) d2=2 (E12) 1<=d3<=2 (E13) Only the two preceding conditions (E11) and (E12) are divided into d1=2 and d2=2 (E14) d1=3 and d2=2 (E15) Direct access is made to the id=14 record using the condition (E14) and then sequential access is made to records until a record is reached which does no meet the conditions (E11), (E12) and (E13), i.e., until the id=16 record is reached. In the same manner, direct access is made to the id=22 record using the condition (E15) and then sequential access is made to records until a record is reached which does no meet the conditions (E11), (E12) and (E13), i.e., until the id=25 record is reached. In this example, even with sequential retrieval, at most three records are merely followed in each sequential process. It is apparent that this method is faster than the divided condition retrieval using all the conditions. Here, the problem is when to switch from the divided condition retrieval to the sequential retrieval. This cannot be determined unconditionally because the amount and distribution of data are involved. With discrete distances of d1, d2, . . . , dn, it is appropriate that the divided condition retrieval is performed on the basis of conditions except a condition associated with dn and the sequential retrieval is performed on the basis of the condition associated with dn. The alternative is that the divided condition retrieval is performed on the basis of conditions except conditions associated with dn-1 and dn and the sequential retrieval is performed on the basis of the conditions associated with dn-1 and dn. In general it appears appropriate to perform the sequential retrieval using conditions associated with the last some discretized distances. Next, a description is given of a method for determining the order neighborhood, which is computed by the similarity retrieval section 121. We shall refer to a set of k object points collected in the order of increasing distance from a specific point p as the k-order neighborhood. FIG. 10 is a flowchart for the order neighborhood calculation procedure of obtaining the k-order neighborhood of the specified point p. The procedure is 1. Determine a certain distance d appropriately (step S11). 2. Obtain d-distance neighborhood N of the specified point p in accordance with the aforementioned method (step S12). 3. According to the number, n, of points contained in the resultant d-distance neighborhood N, perform the following processes (step S13). 3.1 When n>k+1, sort the points contained in N in the order of increasing distance from p and obtain a set of k points closest to p as the k-order neighborhood. 3.2 When n=k, take N as the k-order neighborhood. 3.3 When n<k, determine a new distance greater than distance d and take it as the distance d (step S15). A return is made to step S12 and the procedure is repeated. The flowchart of FIG. 10 is depicted in the form of a simple algorithm in order to make the process flow readable. However, this procedure includes a fruitless process of retrieving the already determined d-distance neighborhood again. In order to avoid such fruitlessness, after the determination of d-distance neighborhood with d set at da, with d newly set at db (db>da), a differential set of db-distance neighborhood and da-distance neighborhood is simply determined as opposed to determining the db-distance neighborhood. In this case, assuming the number of points in the da-distance neighborhood to be na (na<k), (k-na) points have only to be extracted from the differential set; nevertheless, when k points are not still determined, the same process is repeated. Thus, the determination of the k-order neighborhood generally involves determining the d-distance neighborhood repeatedly, which results in a string of distances d: d(1), d(2), . . . , d(i), . . . Thus, how to determine the string of distances becomes a problem. Although various methods therefor are considered, a method based on arithmetical progression and a method based on geometrical progression are described here. First, with the arithmetical progression-based method, the i-th distances d(i) are given by d(i)=d(1)*i where d(1) is the initial value of the distance d. This method, while being natural, has the possibility that the process is repeated many times when there is no object point near by. In contrast, with the geometrical progression-based method, the distances d(i) are given by d(i)=d(1)*a^(i-1) If a=2, the distances d will be increased progressively by a factor of two (d(1), 2d(1), 4d(1), 8d(1), . . . ). With this method, since the distances increase exponentially, the number of times the process is repeated is significantly reduced even in the case where there is no object point near by. However, there is the possibility that the process for a certain distance may involve more points than is necessary. Next, how to determine a reference point in a multidimensional space will be described. Where to take the reference point significantly affects the performance of similarity retrieval. As an extreme example, consider the case where all the object points in a two-dimensional space are uniformly distributed on a circle 51 of radius r with center at the origin O as shown in FIG. 11. First, when the reference point s is taken at the origin O, all the elements in the distance index are collected at distance r, in which case the object points cannot be narrowed down. Thus, the distance from specified point p has to be calculated for all the points. Next, when the reference point s is set at the intersection P of the x-axis and the circle 151, the distance values range from 0 to 2r. When the reference point s is taken at a point distance d from the origin O between the point P and the origin O, the distance values lie in the range of r-d to r+d. The smaller the distanced from the origin O, the narrower the range becomes. The range becoming narrower is not preferable as suggested by the case where the reference point s is taken at the origin O. The reason is that a set T' of candidate points, when determined to create a distance index, comes to contain too many points for the index to attain its function of narrowing down the retrieval range. In order to make the distance values extend over a wide range, it is preferable to take the reference point s at the point p or far away from it on the x-axis. Incidentally, even if the reference point s is taken at a point far away from p, the distance values will still range from 0 to 2r as in the case where the reference point is taken at P. As another example, consider the case where object points are uniformly distributed on a line connecting points P and Q as shown in FIG. 12. Let the distance between O and p and the distance between O and Q be equal to r. In this case as well, the performance of similarity retrieval depends greatly on the way to select the reference point. First, an undesirable example is adduced. When a point far away from the origin O on the y-axis is taken as the reference point s, all points on the line PQ will be at substantially the same distance from the reference point. Conversely, a desirable example is to take the reference point at the point P or Q or at a point on an extension of the line PQ. In this case, the distance values are uniformly distributed over the range of 0 to 2r. Taking a point between P and Q as the reference point and letting its coordinates be (d, 0) where 0<=.vertline.d.vertline.<r, the distance values will be distributed over a range the width of which is r+d. This width is less than 2r. Even when the reference point is set at a point that does not lie on the x-axis, the width of the range of distance values will still become less than 2r. Thus, it is preferable to set the reference point at P or Q, which are points at both ends of the line PQ, or at a point on an extension of that line. Thus, the most suitable reference point varies according to the distribution of object points in a multidimensional space. Knowing how the object points are distributed as in the case of FIG. 11 or 12 will also make it possible to determine the most suitable reference point. In general, however, in many cases it is expected to be difficult to determine the most suitable reference point regardless of whether the distribution is known or not. With actual data, it is expected that the object points may increase or decrease in number, or changes in the features of the object points may require changes in their position. Changing the reference point on all such occasions could complicate the processing. If, when the system is commercially available, such setting were imposed on users, they might well think it troublesome. Consider which way to set the reference point even through it is not the most suitable is generally desirable. The above examples suggest the following two requests: 1. It is desirable to take the reference point outside the range in which object points are distributed. 2. It is desirable to allow the distance values to extend over a wide range. Taking the request 1 too seriously, from a sense that the reference point is to be taken outside the range in which the object points are distributed, it follows that the reference point is to be set at the outermost point in a multidimensional space no matter how object points are distributed. For example, with a multidimensional space in the form of a super cube, a vertex of the cube is desirable as the reference point. Considering the above points into consideration, vertexes and points on an extension of a straight line are listed as candidates for the reference point. However, since the requests 1 and 2 are not necessarily the best, the ways to take the reference point are merely exemplary. First, a description is given of the case where a vertex is selected as the reference point. In a space consisting of a super cube each side of which is a in length, for example, the following vertexes are selected as the reference points: (0, 0, 0, . . . , 0) (a, 0, 0, . . . , 0) (0, a, 0, . . . , 0) (0, 0, 0, . . . , a) When the reference points are set at the vertexes of an n-dimensional super cube, the width of the distribution range of distance values becomes maximum and equals the diameter of that cube, a√ . Next, a description is given of the case where the reference point is set on an extension of a straight line. Suppose a certain straight line and project object points onto it. At this point, select a straight line such that the distribution range of the projected object points is maximum. Take the reference point outside the distribution range. This example includes the example of FIG. 12. When a plurality of reference points is taken, a plurality of such straight lines is selected. In this case, the second and subsequent straight lines do generally not necessarily show a maximum distribution range; however, they should be selected so that the distribution range becomes as wide as possible independently of the previously selected straight line or lines. The method of taking the reference points at vertexes can also be regarded as taking the reference points at ends of diagonal lines of a super cube. For example, the origin (0, 0, . . . , 0) corresponds to a point at the end of the diagonal line connecting the origin and point (a, a, . . . , a). The point (a, 0, 0, . . . , 0) corresponds to the point at one end of the diagonal line connecting that point and point (0, a, a, . . . , a). In addition to taking the reference points at the vertexes, they may be taken on extensions of these diagonal lines. Here, where to take the reference point on a straight line becomes a problem. One candidate is a point which, like a vertex, is present at one end of the distribution range. Another candidate is a point on an extension of the straight line. Still another candidate is the point at infinity on the extension. Bring the reference point near to an infinitely far point on the extension of a straight line results in a spherical surface centered at the reference point approximating a plane. With the reference point set at the point at infinity, the spherical surface centered at that point will become a plane. That is, a space becomes divided into two or more bands by a plane perpendicular to a straight line. In this case, since the original distance becomes infinite, handling needs care. To define discrete distances, object points are projected onto a straight line and the range of projections is determined. Suppose the projections are distributed between points P and Q. Then, it is required to divide the line PQ with n points: P0 (=P), P1, P2, . . . , Pn-1, Pn (=Q) and, with projected points of object points t on the straight line as t', to redefine the discretized distances as follows:
Position of t'
distance Discretized
P0-P1 (including P1) 0
P1-P2 (including P2 but not P1) 1
P2-P3 (including P3 but not P2) 2
Pn-2-Pn-1 (including Pn-1 but not Pn-2) n-2
Pn-1-Pn (including Pn but not Pn-1) n-1
As a special case, a coordinate axis may be chosen as a straight line and divided in the above manner. This method will be described later in detail as a projection-based discrete concatenated distance index. As described previously, the principles of the invention are applicable to any distance which satisfies the property of distance. In general, the so-called Euclidean distance is frequently assumed. It is thus difficult to image other distance than the Euclidean distance. As such distance, the Manhattan distance which is used frequently in addition to the Euclidean distance will be described below. FIG. 13 illustrates the relationship between d-distance neighborhood and a set of candidate points when the Manhattan distance is used and corresponds to FIG. 3 which is based on the Euclidean distance. In the Manhattan distance in a two-dimensional space, the distance between two points (x1, y1) and (x2, y2) is represented by .vertline.x1-x2.vertline.+.vertline.y1-y2.vertline. This distance corresponds to actual distance to go from a point to a point in the case where streets are constructed in a check pattern as in Manhattan. The Manhattan distance, also called the L1 distance, is used in integrated circuits as well in which interconnect lines are parallel to the x-axis and y-axis. In contrast, the Euclidean distance, also called the L2 distance, is represented by (.vertline.x1-x2.vertline.^2+.vertline.y1-y2 .vertline.^2)^(1/2) In general, the Lm distance is defined by (.vertline.x1-x2.vertline.^m+.vertline.y1-y2.vertline.^m)^(1/m) and satisfies the property of distance. Accordingly, the present invention is also applicable to the Lm distance. In practice, however, the L1 distance and the L2 distance, i.e., the Manhattan distance and the Euclidean distance, are frequently used. In FIG. 13, a set of object points contained in the intervening region between two squares 161 and 162 forms a set T' of candidate points. A set of object points contained in the square 163 forms the d-distance neighborhood. Using the Manhattan distance, a region represented as a circle in Euclidean distance becomes a square. Points at the same distance from the reference point swill be put on the sides of the square. Next, similarity retrieval using projection-based discrete concatenated distance index will be described. With this method, the way to determine the original distance differs from with the aforementioned discrete concatenated distance index. The other processes remain basically unchanged. In the aforementioned method, the discretized distances were determined on the basis of distances from the reference points to the object points; in this method, a reference line is used to determine distances. First, a specific line in a multidimensional space is selected as a projection reference line. It is not required that object points lie on the projection reference line. Next, a specific point s on the selected projection reference line is selected, which is referred to as the projection reference point in distinction from the previously defined reference point. A point resulting from projection of an object point onto the projection reference line is referred to as the object projected point. The distance from the projection reference point s to the object projected point is referred to as the projection reference point distance. The projection reference point distance is represented by original distance. It is also possible to select two or more projection reference lines. Also, it is possible to use two types of discretized distances in combination; that is, some discretized distances are determined based on the normal reference point and the other discretized distances are determined based on the projection reference point. Although there is a problem of how to select the projection reference line and the projection reference point, they can be determined in the same way as the reference point is determined. When distances from the reference point s are used, the necessary condition for allowing object points t to be contained in a sphere is, using discretized distance function .delta.(t)=.gamma.(d(s, t)) and with D=.delta.(p), .gamma.(D-d)<=.delta.(t)<=.gamma.(D+d) (F1) Here, let L be the projection reference line, s be the projection reference point, and the projected point t' of point t onto L be represented by t'=proj(t, L) Then, the discretized distance function can be written as .delta.(t)=.gamma.(d(s, proj(t, L)) Using this function, the condition under which, when the projection reference point is used, object points t are contained in a sphere becomes exactly the same as the condition (F1). Therefore, the use of either the reference point, the projection reference point, or both in determining the discretized distance allows the d-distance neighborhood to be determined in the aforementioned way. FIG. 14 shows the discretization of the two-dimensional space of FIG. 6 using the projection reference line. The diagonal extending from s1 to the upper right and the diagonal extending from s2 to the upper left are selected as the projection reference lines. The points s1 and s2 form the projection reference points. Using the coordinate axes as the projection reference lines in place of the diagonals, the space becomes descretized as depicted in FIG. 15. Here, the x-axis and the y-axis are used as the projection reference lines. For either of the coordinate axes, the origin s is used as the projection reference point. FIG. 16 shows another discretization method using three projection reference lines. Here, a straight line L1 that corresponds to the x-axis and two broken straight lines L2 and L3 are used as projection reference lines. Each of L1 and L2 intersects with L3 at an angle of 60.degree.. The projection reference points on L1, L2 and L3 are s1, s1, and s2, respectively. Thus, even with two-dimensional space it is possible to define three or more discretized distances, thereby making the filtering of candidate points more easy. However, since increasing the number of discretized distances involves additional overhead, it is required to select suitably the number of discretized distances according to data circumstances at that point. In FIGS. 14, 15 and 16, only distances from the projection reference points are used; however, as described previously, they may be used together with the distance from the normal reference point. Next, how to produce a distance index using bit patterns (bit pattern index) and how to determine d-distance neighborhood using that index will be described. On the basis of information concerning discretized distances obtained according to the method described so far, bit patterns indicating distances are produced and then an index is produced using them. In this case, an existing index such as a B-tree is not necessarily used and bit patterns are appended to individual records. The procedure of producing a bit pattern for each object point is as follows: Assume that the distance from one reference point to an object point is d and the bit pattern is composed of m bits. When a1<a2< . . . <am-1, the discretized distance is defined as follows:
Band Discretized distance d'
d < a1 0
a1 <= d < a2 1
a2 <= d < a3 2
am-2 <= d < am-1 m-2
am-1 <= d m-1
Each distance bit pattern is produced by setting the d'-th bit of m bits. In this case, the d'-th bit is set to one and the other bits are set to zero. Distance bit patterns corresponding to distances from two or more reference points are concatenated, thereby producing the overall bit pattern (concatenated bit pattern) for one object point. FIG. 17 shows bit patterns for the three object points t1, t2 and t3 shown in FIG. 6. In this case, the bit pattern of discretized distance d1 from the reference point s1 and the bit pattern of discretized distance d2 from the reference point s2 are concatenated to produce the bit pattern for one object point. The bits in the bit pattern are counted from left to right as bit 0, bit 1, . . . In this example, m is set to 8, that is, basically the bit pattern consists of eight bits. For the object point t1, since d1=4 and d2=3, bit 4 in the d1 bit pattern and bit 3 in the d2 bit pattern are both set to one. The same holds for the object points t2 and t3. FIG. 18 is a flowchart for the distance neighborhood calculation using such bit patterns. The procedure is as follows: 1. Production of Bit Pattern for Sphere When the discretized distances of bands which intersect a sphere representing the d-distance neighborhood are in the range of d'lower to d'upper, the d'lower-th to d'upper-th bits are set, producing each distance bit pattern. Further, distance patterns based on distances from two or more reference points are concatenated, whereby the overall bit pattern for the sphere is produced (step S21). FIG. 19 shows the bit pattern for the sphere q shown in FIG. 6. Since the sphere spans the bands in which d1=3 and 4 with respect to the discretized distance d1 from the reference point s1, bit 3 and bit 4 in the d1 bit pattern are set. On the other hand, with respect to the discretized distance d2 from the reference point s2, the sphere spans the d2=4 and 5 bands; thus, bit 4 and bit 5 in the d2 bit patterns are set. 2. Matching Based on Logical Products and Filtering For corresponding distance bit patterns in the bit pattern for the sphere and the bit pattern for each object point, two corresponding bits are ANDed (step S22). With respect to all the distances (d1 and d2) of an object point, when any of the logical products of corresponding bits is 1, there is the possibility that the object point is contained in the sphere; thus, that object point is added to the set T' of candidate points. Conversely, when the logical products are all 0s, the object point will not be contained in the sphere; thus, it is not added to T'. In this manner, filtering of object points is performed, i.e., the object points are narrowed down to candidate points. The logical products of corresponding bits in the bit pattern for each object point shown in FIG. 17 and the bit pattern for the sphere shown in FIG. 19 are shown as follows:
d1 d2
t1 AND q 00001000 00000000
t2 AND q 00010000 00000100
t3 AND q 00010000 00000100
In this case, the logical products of corresponding bits for t1 and q contain 1 for d1 but not for d2; thus, we can see that the object point t1 is not contained in the sphere q. From FIG. 6 as well, it is sure that the object point t1 is not contained in the sphere q. In contrast, for t1 and t3, the logical products for both d1 and d2 contain 1; thus, it is possible that they are contained in the sphere. The object points t2 and t3 are therefore added to the set of candidate points. In calculation efficiency, the logical products are taken from left to right in the bit patterns and, when even one distance for which all the logical products are 0s appears, the processing is stopped. 3. Calculation of Distance For each object point t which has been left in the candidate point set T' through filtering, the distance d(p, t) from the specified point p is calculated and a set of object points for which d(p, t)<=d is determined (step S23). This set is the d-distance neighborhood to be determined. In the example of FIG. 17, as can be seen from FIG. 6 as well, only the object point t3 will be left in the d-distance neighborhood. The bit pattern index has advantages of requiring less storage capacity and providing faster retrieval in comparison with the discrete concatenated distance index. The previously described discrete concatenated distance index stores only the discretized distances and pointers. In order to acquire the feature parameters of object points, therefore, it is required to make access to the feature parameters through the use of the pointers. In general, there is the possibility that the secondary storage unit may store information of object points in locations apart from that for the index. There is also the possibility that pieces of information of the object points may be stored in different locations. Such storage methods would cause the disk head to move in all directions, which might adversely affect the performance of the system. One way to prevent this will be to append not only the discretized distances and pointers but also all or part of the feature parameters to the index. According to such a method, the use of the index alone allows feature parameters of each object point to be acquired. Further, the distance between each object point and the specified point can also be calculated. Clustering of index information could provide fast processing. Next, an example of constructing a discrete concatenated distance index by a relational database will be described. Here, it is supposed that each record has 64 feature parameters each of which is represented by a 4-byte floating-point number. Each feature parameter is in the range of 0 to 1. In this case, the number of dimensions of the multidimensional space is 64. Those feature parameters can be stored in the secondary storage unit in the form of such a table as shown in FIG. 20. Here, feature parameter 1, feature parameter 2, . . . , feature parameter 64 are stored in fields c1, c2, . . . , and c64, respectively. Also, 64 reference points s1, s2, . . . , and s64 are set as follows: s1=(1, 0, 0, . . . , 0) s2=(0, 1, 0, . . . , 0) s3=(0, 0, 1, . . . , 0) s64=(0, 0, 0, . . . , 1) The distance di from the i-th reference point is calculated for each object point t and the discretized distance d'i is calculated as follows: d'i=floor(256*di/dmax) if di<dmax=255 if di=dmax where floor (x) stands for a maximum integer not greater than x and dmax is sqrt(64), the square root of 64. In this case, dmax corresponding to the length of the diagonal of a 64-dimensional super cube each side of which is 1 in length and is the maximum value the distance di can take. d'i thus obtained is an integer in the range of 0 to 255 and hence can be represented by one byte. By concatenating 64 discretized distances a 64-byte key is produced, which is stored in the field d in FIG. 20. Normal relational databases would allow an index of B-tree to be readily set up on the field d. When a bit pattern index is constructed by a relational database, 64 feature parameters and a bit pattern are stored in the form of such a table as shown in FIG. 21. In this case, the field b corresponds to the bit pattern index. The reference point is taken in the same way as with the discrete concatenated distance index. The discretized distance d'i is calculated as follows: d'i=floor(8*di/dmax) if di<dmax=7 if di=dmax Next, the d'i-th bit in one byte is set to produce a bit pattern for each distance. The resulting 64 bit patterns are concatenated to produce a 64-byte bit pattern. The bit patterns thus produced for individual object points are stored in the field b. In determining the d-distance neighborhood in similarity retrieval, a bit pattern for a given sphere is produced and then corresponding bits in that pattern and the bit pattern for each object point stored in the field b are ANDed. If all the bytes of logical products contain a numeral value other than 0, then the feature parameters in the fields c1, c2, . . . , c64 are taken out and the distance between the object point and the specified point is calculated. If the distance is less than d, then the record is extracted as an element of the d-distance neighborhood. With this method, ANDing of corresponding bits in the bit pattern for each object pattern and the bit pattern for the sphere requires data input/output operations between the main storage and the secondary storage. One way to avoid this is to, when the number of object points is small, expand the bit patterns for all the object points as an array on the main storage and make access to the feature patterns in the secondary storage unit using pointers in the array. This approach requires no data input/output operation for filtering; data input/output operations are performed only when a decision is made as to whether the distance is less than d. With the storage method shown in FIG. 20, the feature parameters and the discretized distances are stored in relation. The relation is normally stored on pages linked in list form. In addition to this, an index having a B-tree structure is setup on the field storing the discretized distances. In contrast, with the aforementioned method to append the feature parameters to the index, the feature parameters will also be stored in that B-tree. The retrieval system of FIG. 2 is implemented by an information processing unit (computer) as shown in FIG. 22. The information processing unit of FIG. 22 includes a CPU (central processing unit) 171, a memory 172, an input device 173, an output device 174, an external storage unit 175, a medium driver 176, and a network adapter 177, which are interconnected by a bus 178. The memory 172 includes a ROM (read only memory), a RAM (random access memory), etc. and stores processing programs and data. The CPU 171 carries out the programs through the use of the memory 172 to perform necessary processing. The multidimensional space management section 111 and the database system 112 in FIG. 1 are implemented by the programs stored in the memory. The input device 173 is a keyboard, pointing device, touch panel, or the like and is used by a user to enter commands and information. Input information includes specified points, the distance, d, in d-distance neighborhood, the number, k, of elements in k-order neighborhood, and so on. The output device 174 is a display, printer, loudspeaker, or the like and used to output inquiries and the results of retrieval to the user. The external storage device 175 is a magnetic disk device, optical disk device, optomagnetic disk device, or tape device. The information processing unit loads the programs and data stored in the external storage unit 175 into the memory 172 when necessary for subsequent use. The external storage unit is also used as the secondary storage unit 113 in FIG. 1. The medium driver 176 drives a portable recording medium 179 to access its recorded contents. As the portable recording medium use may be made of a memory card, flexible disk, CD-ROM (compact disk read only memory), optical disk, optomagnetic disk, or any other computer-readable recording medium. The user is allowed to store the programs and data on the portable recording medium and load them into the memory 172 when necessary. The network adapter 177 is attached to a communications network, such as a LAN (local area network) or the Internet, and provides data conversion involved in communications. The information processing unit receives the programs and data from another unit via the network adapter 177 and loads them into the memory 172 when necessary. FIG. 23 shows a computer-readable recording medium which can feed programs and data into the information processing unit of FIG. 22. The programs and data retained on the portable recording medium 179 or in a database 181 of a server 180 are loaded into the memory 172. At this point, the server 180 produces carrier signals to carry the programs and data and sends them to the information processing unit over a transmission medium on the network. The CPU 171 then carries out the programs using the data to implement necessary processing. According to the embodiment described above, the following advantages can be expected: .cndot. System Simplicity From a User's Viewpoint A function that common databases have, a B-tree, can be used. Therefore, when a commercial database already used by a user has no multidimensional index, an index for a multidimensional space can be built on the database with relative ease. From a Database System Developer'Viewpoint Since a function that common databases have, a B-tree, can be used, the inventive techniques can be incorporated into a database to construct a retrieval system of the invention with ease. Also, the development cost and the maintenance cost can be reduced. In contrast, the addition of a new access method, such as an R-tree, is expected to make the system considerably complex in view of influence on optimization. .cndot. Space Efficiency When object points are allowed to be small in number, there is the possibility that the B-tree index requires less disk space than an R-tree which is a typical multidimensional index. This tendency becomes more pronounced as the dimensions become higher. .cndot. High-speed Performance Since the space efficiency is good as described above, there is the possibility that the number of input/output operations is smaller than with a multidimensional index such as a R-tree, depending on the distribution of object points in a multidimensional space. In addition, the B-tree itself is originally a high-speed access method. It is possible that the overall performance is also more excellent than with a multidimensional index, such as an R-tree, depending on the distribution of object points. The performance of the inventive processing is subject to the process of determining a set of candidate points. The closer the set of candidate points approaches to the distance neighborhood, the higher the performance becomes and vice versa. An improvement is made by taking two or more reference points; however, too many reference points may degrade the performance. The processing performance depends also on the distribution of object points. The advantageous distributions to the invention include: (1) A distribution such that object points are distributed uniformly with respect to the reference point distance. (2) A distribution such that similar object points are collected locally (as opposed to the spatially uniform distribution); i.e., a spatially biased distribution. The spatially uniform distribution reduces the effectiveness of the reference points; however, with the biased distribution, the possibility of making the reference points effective increases. Adaptability to High Dimensions .cndot. The inventive processing requires no much overhead even if the number of dimensions is increased. For example, as the number of dimensions increases, the reference points may increase in number, which requires only the amount of disk space and computation to be increased. In this sense, the adaptability to high dimensions is expected to be high. According to the present invention, in similarity retrieval for determining a set of points close to a specified point in a multidimensional space of feature parameters, an index can be constructed relatively easily, allowing efficient retrieval processing to be implemented.
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