- Email: [email protected]
A concept-based query evaluation with indefinite fuzzy triples
Information Processing Letters 74 (2000) 209–214
A concept-based query evaluation with indefinite fuzzy triples Jae Dong Yang 1 Department of Computer Science, Chonbuk National University, Chonju, 561-756, South Korea Received 28 March 1999; received in revised form 10 February 2000 Communicated by F. Dehne
Abstract Triple indexing technique provides us with more understandable specification of the spatial structure of images without compromising retrieval time by a well-designed hash function. However, this technique has a serious drawback; it accommodates neither a concept-based image retrieval facility nor does it allow indefinite object labeling. 2000 Elsevier Science B.V. All rights reserved. Keywords: Information retrieval; Fuzzy logic; Query evaluation
1. Introduction Triple representation [1] is a variation of the 2D (dimensional) string [2–4], which is one of the most promising techniques used for indexing images. This triple representation was proposed to enhance the semantic expressiveness of the 2D strings by translating them into conceptually equivalent triples. The novelty of this technique is to provide us with more understandable specification of the spatial structure of images without compromising retrieval time by a welldesigned hash function. However, the technique has a serious drawback; it accommodates neither a conceptbased image retrieval facility nor does it allow indefinite object labeling. Retrieving images based on concepts is crucial to get answers relevant to user queries, whereas allowing the indefinite labeling is indispensable to compensate for the poor recognition power of extant image analyzers. The purpose of this paper is to propose a query evaluation supporting a conceptbased image retrieval as well as the indefinite label1 Email: [email protected].
ing. The evaluation is based on the I-triple (Indefinitetriple) framework adopting a fuzzy matching [5].
2. Image retrieval by I-triples According to [1], the iconic image p in Fig. 1 may be indexed by ordinary triples: {hw, c, northwesti, hr, w, easti, hr, c, northi, hw, s, northi, hr, s, northeasti} where w is ‘working table’, r is ‘radio’, s is ‘speaker’ and c is ‘clock’. However, if it is possible that the radio can also be labeled as ‘recorder’, then indefinite triples (I-triples) should be introduced and hr, c, northi may be represented as hr ∨ re, c, northi where re is ‘recorder’. Additionally, we need to specify objects by their names in order to deal with an I-triple. It helps us deal with occurrences of an object, which share the same name. The objects should be named by the following terms. Definition 2.1. A fuzzy linguistic term (or simply a term) T is a fuzzy set characterized by the membership
0020-0190/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 0 - 0 1 9 0 ( 0 0 ) 0 0 0 6 6 - 1
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J.D. Yang / Information Processing Letters 74 (2000) 209–214
Fig. 1. Iconic image p.
function µT : N → [0, 1] where N is a set of terms. c in N is a crisp term if there exists c0 such that µC (c0 ) = 1, for c0 = c and µC (c0 ) = 0, otherwise. A term that is not crisp is referred to as fuzzy term. Definition 2.2. Let N be a set of terms, IDB be a set of all images in an image database and Op be a set of all objects which can occur in an image p ∈ IDB. Then a name function, fNAME is defined as follows. For all o ∈ Op , there exists T ∈ N such that fNAME (o) = T . We can now define an indefinite triple with this function. Definition 2.3. Let Op be the set of all objects in p ∈ IDB and D = {east, northeast, north, . . .} be the set of eight directions. Then a set of all indefinite triples (I-triples) for p, I _Tp is given by (* m + m2 1 _ _ fNAME (oi ), fNAME (oj ), rij I _Tp = i=1
j =1
rij ∈ D is a relative direction of oj with respect to oi for oi , oj ∈ Op ,
*m _1
fNAME (oi ),
m2 _
+ fNAME (oj ), rij ,
j =1
i=1
if it is given as + *m m1 2 _ _ fNAME (oj ), fNAME (oi ), rij , j =1
i=1
where rij is the reverse direction of rij . We next need term predicates to make our I-triple framework complete. The following is a prerequisite. Definition 2.4. A fuzzy linguistic term predicate (or simply a term predicate) T corresponding to the term T in N is defined by T : N → [0, 1], where T (c) =
1 if c = T , µT (c) for each c 6= T ∈ N.
For example, furniture(X) is a term predicate, whereas furniture is its corresponding term. furniture(c) for some c ∈ N is therefore interpreted as the degree of satisfaction of c to furniture.
)
1 6 i 6 m1 , 1 6 j 6 m2 . Note that the number of I-triples for n primitive objects of an image is n∗ (n − 1)/2, since we treat an I-triple identical to
The following disjunction of term predicates enables us to introduce the final I-triple version where terms are replaced by their corresponding term predicates. The term predicates serve as templates which can make it possible for terms to match each other based on concepts.
J.D. Yang / Information Processing Letters 74 (2000) 209–214
W Definition 2.5. Let d_T (A) = m i=1 Ti (A) be a disjunction of term predicates. Then an I-triple t ∈ I _Tp is defined as follows. (1) Let each of d_T1 , and d_T2 be a disjunction of term predicates and d ∈ D. Then hd_T1 , d_T2 , di is an I-triple. (2) Nothing else is an I-triple. If an I-triple t is of the form, t = hT1 , T2 , di where each of T1 and T2 is a term predicate, then we call it, simply, triple.
3. Exploiting thesauri for a concept-based match The term thesaurus in Fig. 2 is used to evaluate term predicates. It contains crisp terms in leaf nodes and in the other ones, fuzzy terms, each taking lower level fuzzy terms as its members with degrees specified on the corresponding edges. Any degree between them is assumed 0 if unspecified. Additionally, a composed membership function is provided for obtaining membership values between two terms indirectly related. Definition 3.1 [5]. Let T be a fuzzy term. Then µT (c) = max min(µai (c), α) for all c ∈ N, where α = µT (ai ) and ai is a fuzzy term. Note that since ai is a fuzzy term, µai (c), ∀c ∈ N can also be calculated by the recursive application of Definition 3.1 even if c is not directly connected with ai . Note also that the degree of conceptual closeness between two terms not related with any edge cannot be obtained by this definition. For example, it does
211
not define an edge of certain degree between electronic appliance and audio. In this paper, we do not consider a similarity relation which may be used to quantify such degree. Example 3.1. The degree of conceptual closeness between furniture and radio is calculated by µfurniture(radio) = max min µelectronic_appliance(radio),
µfurniture(electronic_appliance) , min µaudio(radio), µfurniture(audio) = max min(0.96, 0.87), min(0.7, 0.91) = 0.87. Hence, furniture(radio) = 0.87. In the above example, the max operator enables µfurniture(radio) to take the most informative value 0.87 by passing through ‘electronic appliance’ between ‘radio’ and ‘furniture’ instead of passing through ‘audio’. Moreover, it states that either of the two paths is available to quantify µfurniture(radio). We are now in a position to formally define our image retrieval system. Definition 3.2. An image retrieval system I_IR is defined as follows. I_IR = hIDB, I _T , Tr, Ini, where IDB is a set of all images, I _T is a set of all (I _Tp )s for p ∈ IDB, Tr is a term thesaurus, and In is an inverted file. We define In as a set of entries, each formed by attaching to t ∈ I _Tp , every image, p satisfying t ∈ I _Tp , i.e., In = {ht, {p}i}. In is used to search for images indexed by t.
Fig. 2. Term thesaurus.
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J.D. Yang / Information Processing Letters 74 (2000) 209–214
4. Query evaluation Evaluation of a user query to retrieve images involves translating it into equivalent triples and then matching the I-triple in each entry of In. As a first step toward developing such an evaluation, we provide a definition to specify our query structure. Definition 4.1. Let each of QTj , j = 1, . . . , s, be a triple used as a basic unit of a query. Then the following disjunction of the s triples is called disjunctive query QD . QD =
s _
QTj .
j =1
Definition 4.2. Let QDi , i = 1, . . . , n, be n disjunctive queries. Then a conjunctive normal query Q (or simply query) is defined as follows. Q=
n ^
QDi .
i=1
We now formalize a fuzzy match involving more than one disjunction of term predicates. Wm
Definition 4.3. Let d_T = i=1 Ti be a disjunction of term predicates. Then for all c ∈ N , d_T (c) = max(Ti (c), i = 1, . . . , m). Definition 4.4. Let d_T be a disjunction of term predicates. Then |d_T | = {c ∈ N | d_T (c) = 1}. Definition 4.5. Let each of d_T1 and d_T2 be a disjunction of term predicates. Then d_T1 is more general with a degree α ∈ [0, 1] than d_T2 iff min d_T1 (c) for all c ∈ |d_T2 | = α > 0. It is denoted by d_T2 ⊆α d_T1 . Example 4.1. Let d_T1 = electronic_appliance and d_T2 = radio ∨ recorder. Then d_T2 ⊆0.92 d_T1 since min d_T1 (c) for all c ∈ |d_T2 | = {radio, recorder} = min electronic_appliance(radio),
electronic_appliance(recorder)
= 0.92 > 0.
0 Definition 4.6. Let WsQ be a sub-query of the disjunctive query QD = j =1 QTj given as follows; 0
Q =
ik _
QTj ,
QTj = hT1,j , T2,j , di
j =i1
for j = i1 , i2 , . . . , ik ∈ {1, 2, . . . , s}, sharing d with an I-triple t = hd_T1 , d_T2 , di. Then we call it common direction disjunctive query, or briefly c-d query for t. W Definition 4.7. Let Q0 = sj =1 hT1,j , T2,j , di be a c-d query for t = hd_T1 , d_T2 , di. Then t ⊆α Q0 with α = min(α1 , α2 ) > 0 ⇔ s s _ _ T1,j and d_T2 ⊆α2 T2,j . d_T1 ⊆α1 j =1
j =1
In the sequel, the following W proposition is provided for testing d_T1 ⊆α1 sj =1 T1,j or d_T2 ⊆α2 Ws j =1 T2,j . be a disProposition 4.1. Let each of d_T1 and d_T2 W junction of term predicates. Suppose d_T1 = m i=1 Ti . Then d_T2 ⊆α d_T1 iff min(max(Ti (c/A), i = 1, 2, . . . , m) for all c ∈ |d_T2 |) = α > 0. Proof. It can be directly proved by Definitions 4.3 and 4.5. We omit its proof. 2 Example 4.2. Let d_T1 = radio∨receiver and d_T2 = electronic_appliance ∨ audio. Then d_T1 ⊆0.9 d_T2 , since min max(electronic_appliance(radio), audio(radio)), max(electronic_appliance(receiver), audio(receiver) = min(0.96, 0.9) = 0.9 > 0 by Proposition 4.1. Consider next the condition that an I-triple t ∈ I _Tp exactly matches a query Q guaranteeing a minimum possibility regardless of its indefiniteness. Detecting
J.D. Yang / Information Processing Letters 74 (2000) 209–214
the condition may be crucial to obtain the following exact answer set of Q. Definition 4.8. Let QD be a c-d query for t ∈ I _Tp . Then I _IR `α ∗ QD (p)
iff t ⊆α QD ,
α > 0.
Definition 4.9. kQkα ∗ = {p | I _IR `α ∗ Q(p), α > 0} is a set of exact answers satisfying a query Q with a threshold value α > 0. Example 4.3. Let a query be given as “search for images where an electronic appliance locates at the northeast of furniture”. Then it is converted into QT1 = helectronic_appliance, furniture, northeasti. Now, for t = hd_T1 , d_T2 , northeasti ∈ I _Tp1 where d_T1 = radio ∨ recorder, d_T2 = working_table, t ⊆0.91 QT1 since d_T1 ⊆0.92 electronic_appliance (see Example 4.1) and working_table ⊆0.91 furniture. Hence, I _IR `0.91∗ QT1 (p1 ). Example 4.4. Let the query QD1 be “search for images where an electronic_appliance or audio is at the northeast of furniture”. Then QD1 = QT1 ∨ QT2 where QT1 is given in Example 4.3 and QT2 = haudio, furniture, northeasti. Now, for t = hd_T1 , d_T2 , northeasti ∈ I _Tp2 where d_T1 = radio ∨ receiver, d_T2 = table, the result is I _IR `0.9∗ QD1 (p2 ) since d_T1 ⊆0.9 electronic_appliance ∨ audio (see Example 4.2) and table ⊆0.91 furniture. W Proposition 4.2. Let QD = sj =1 QTj be a disjunctive query. Then for all p ∈ IDB, p ∈ kQD kα ∗ ,
α = max(α1 , α2 , . . . , αs ) > 0,
iff there exists at least one c-d query, Q0i satisfying p ∈ kQ0i kαi∗ ,
αi > 0, 1 6 i 6 s. kQ0i kαi∗ ,
there should exist at least Proof. To be p ∈ one ti ∈ I _Tp that satisfies ti ⊆αi Q0i , αi > 0, 1 6 i 6 s. Without loss of generality, we can assume such c-d queries are Q01 and Q02 for t1 , t2 ∈ I _Tp , respectively. If t1 ⊆α1 Q01 or t2 ⊆α2 Q02 , then p ∈ kQ01 kα1∗ , or p ∈ kQ02 kα2∗ , α1 , α2 > 0 from Definitions 4.8 and 4.9. We can now prove this proposition since either p ∈ kQ01 kα1∗ or p ∈ kQ02 kα2∗ implies p ∈ kQD kα ∗ , α = max(α1 , α2 ) > 0. The same is true even if t1 ⊆α1 Q01
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and t2 ⊆α2 Q02 . It is possible since p can be indexed simultaneously by t1 and t2 which are not necessarily the same with each other. 2 Theorem 4.1. Let QD = query. Then s [
Ws
i=1 QTi
be a disjunctive
kQTi kαi∗ ⊆ kQD kα ∗ ,
i=1
α = max(α1 , α2 , . . . , αs ) > 0. Proof. By using Proposition 4.2, it is straightforward to show if p ∈ kQTi kαi∗ , i = 1, 2, . . . , s, then p ∈ kQD kα ∗ . Hence, we only prove s [
kQTi kαi∗ 6= kQD kα ∗
i=1
by showing that there exists some p ∈ kQD kα ∗ but p∈ / kQTi kαi∗ , i = 1, 2, . . . , s. Let Q0 =
ik _
hT1,j , T2,j , di
j =i1
be a c-d query of QD for an I-triple, t = hd_T1 , d_T2 , di ∈ I _Tp . If such a c-d query does not exist for all t, any p cannot be considered as an answer. Hence φ ⊆ φ, which trivially holds. By Definition 4.8 and Proposition 4.2, we can find the p if it is indexed by t that satisfies t ⊆α Q0 with α > 0, but, none of t ⊆αi QTi , i = 1, 2, . . . , s, holds. Without loss of generality, let Q0 = QT1 ∨ QT2 = hT1 , T2 , di ∨ hT10 , T20 , di and t = hT1 ∨ T10 , T2 ∨ T20 , di. Then, obviously, t ⊆1 Q0 but t ⊆0 QTi , i = 1, 2. Therefore, we can always find some p satisfying p ∈ / kQTi kαi∗ , i = 1, 2, . . . , s. 2 kQD kα ∗ but p ∈ Theorem 4.2. Let Q = for all p ∈ IDB,
Vn
i=1 QDi
be a query. Then
p ∈ kQkα ∗ with α = min(α1 , α2 , . . . , αn ) ⇔ p ∈ kQDi kαi∗ , αi > 0, i = 1, 2, . . . , n. Proof. (⇒) Suppose p ∈ kQkα ∗ . Then by Definition 4.8, I _IR `α ∗ QD1 (p) ∧ QD2 (p) ∧ · · · ∧ QDn (p),
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which means I _IR `αi∗ QDi (p) for all i = 1, 2, . . . , n, with α = min(α1 , α2 , . . . , αn ). Accordingly, p ∈ kQDi kαi∗ . (⇐) Similarly, we can prove the (⇐) part. We omit it. 2 V Theorem 4.3. Let Q = ni=1 QDi , and QDi = W si ∗ j =1 QTij be a query. Then p ∈ kQkα with α = min max(α11 , . . . , α1s1 ), . . . , max(αi1, . . . , αisi ), . . . , max(αn1 , . . . , αnsn )
5. Conclusion In this paper, we proposed a concept-based query evaluation against images indexed with indefinite triples. As further research, the similarity between terms needs to be incorporated into the query evaluation for more sophisticated concept-based matching. References
such that for i, 1 6 i 6 n, p ∈ kQTij kαij , j = 1, . . . , si . Proof. This theorem can be easily proved by Proposition 4.2 and Theorem 4.2. We omit its proof. 2 Example 4.5. Let the query Q be QD1 ∧ QD2 where QD2 = hfurniture, table, northeasti. Then, since p2 ∈ kQD1 k0.9 (see Example 4.4) and p2 ∈ kQD2 k0.84 from min(0.84, 1) = 0.84, we get p2 ∈ kQk0.84 .
[1] C.C. Chang, S.Y. Lee, Retrieval of similar pictures on pictorial databases, Pattern Recognition (1991) 675–680. [2] S.K. Chang, E. Jungert, C. Tortora, Intelligent Image Database Systems, Series on Software Engineering and Knowledge Engineering, Vol. 5, World Scientific, Singapore, 1996. [3] E. Jungert, S.-K. Chang, An image algebra for pictorial data manipulation, CVGIP: Image Understanding 58 (2) (1993) 147– 160. [4] D. Papadias, T. Sellis, The semantics of relations in 2D space using representative points: Spatial indexes, in: Proc. European Conference on Spatial Information Theory, 1993. [5] J.D. Yang, D.G. Lee, Incorporating concept-based match into fuzzy production rules, Inform. Sci. 104 (1998) 213–239.
A concept-based query evaluation with indefinite fuzzy triples Jae Dong Yang 1 Department of Computer Science, Chonbuk National University, Chonju, 561-756, South Korea Received 28 March 1999; received in revised form 10 February 2000 Communicated by F. Dehne
Abstract Triple indexing technique provides us with more understandable specification of the spatial structure of images without compromising retrieval time by a well-designed hash function. However, this technique has a serious drawback; it accommodates neither a concept-based image retrieval facility nor does it allow indefinite object labeling. 2000 Elsevier Science B.V. All rights reserved. Keywords: Information retrieval; Fuzzy logic; Query evaluation
1. Introduction Triple representation [1] is a variation of the 2D (dimensional) string [2–4], which is one of the most promising techniques used for indexing images. This triple representation was proposed to enhance the semantic expressiveness of the 2D strings by translating them into conceptually equivalent triples. The novelty of this technique is to provide us with more understandable specification of the spatial structure of images without compromising retrieval time by a welldesigned hash function. However, the technique has a serious drawback; it accommodates neither a conceptbased image retrieval facility nor does it allow indefinite object labeling. Retrieving images based on concepts is crucial to get answers relevant to user queries, whereas allowing the indefinite labeling is indispensable to compensate for the poor recognition power of extant image analyzers. The purpose of this paper is to propose a query evaluation supporting a conceptbased image retrieval as well as the indefinite label1 Email: [email protected].
ing. The evaluation is based on the I-triple (Indefinitetriple) framework adopting a fuzzy matching [5].
2. Image retrieval by I-triples According to [1], the iconic image p in Fig. 1 may be indexed by ordinary triples: {hw, c, northwesti, hr, w, easti, hr, c, northi, hw, s, northi, hr, s, northeasti} where w is ‘working table’, r is ‘radio’, s is ‘speaker’ and c is ‘clock’. However, if it is possible that the radio can also be labeled as ‘recorder’, then indefinite triples (I-triples) should be introduced and hr, c, northi may be represented as hr ∨ re, c, northi where re is ‘recorder’. Additionally, we need to specify objects by their names in order to deal with an I-triple. It helps us deal with occurrences of an object, which share the same name. The objects should be named by the following terms. Definition 2.1. A fuzzy linguistic term (or simply a term) T is a fuzzy set characterized by the membership
0020-0190/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 0 - 0 1 9 0 ( 0 0 ) 0 0 0 6 6 - 1
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J.D. Yang / Information Processing Letters 74 (2000) 209–214
Fig. 1. Iconic image p.
function µT : N → [0, 1] where N is a set of terms. c in N is a crisp term if there exists c0 such that µC (c0 ) = 1, for c0 = c and µC (c0 ) = 0, otherwise. A term that is not crisp is referred to as fuzzy term. Definition 2.2. Let N be a set of terms, IDB be a set of all images in an image database and Op be a set of all objects which can occur in an image p ∈ IDB. Then a name function, fNAME is defined as follows. For all o ∈ Op , there exists T ∈ N such that fNAME (o) = T . We can now define an indefinite triple with this function. Definition 2.3. Let Op be the set of all objects in p ∈ IDB and D = {east, northeast, north, . . .} be the set of eight directions. Then a set of all indefinite triples (I-triples) for p, I _Tp is given by (* m + m2 1 _ _ fNAME (oi ), fNAME (oj ), rij I _Tp = i=1
j =1
rij ∈ D is a relative direction of oj with respect to oi for oi , oj ∈ Op ,
*m _1
fNAME (oi ),
m2 _
+ fNAME (oj ), rij ,
j =1
i=1
if it is given as + *m m1 2 _ _ fNAME (oj ), fNAME (oi ), rij , j =1
i=1
where rij is the reverse direction of rij . We next need term predicates to make our I-triple framework complete. The following is a prerequisite. Definition 2.4. A fuzzy linguistic term predicate (or simply a term predicate) T corresponding to the term T in N is defined by T : N → [0, 1], where T (c) =
1 if c = T , µT (c) for each c 6= T ∈ N.
For example, furniture(X) is a term predicate, whereas furniture is its corresponding term. furniture(c) for some c ∈ N is therefore interpreted as the degree of satisfaction of c to furniture.
)
1 6 i 6 m1 , 1 6 j 6 m2 . Note that the number of I-triples for n primitive objects of an image is n∗ (n − 1)/2, since we treat an I-triple identical to
The following disjunction of term predicates enables us to introduce the final I-triple version where terms are replaced by their corresponding term predicates. The term predicates serve as templates which can make it possible for terms to match each other based on concepts.
J.D. Yang / Information Processing Letters 74 (2000) 209–214
W Definition 2.5. Let d_T (A) = m i=1 Ti (A) be a disjunction of term predicates. Then an I-triple t ∈ I _Tp is defined as follows. (1) Let each of d_T1 , and d_T2 be a disjunction of term predicates and d ∈ D. Then hd_T1 , d_T2 , di is an I-triple. (2) Nothing else is an I-triple. If an I-triple t is of the form, t = hT1 , T2 , di where each of T1 and T2 is a term predicate, then we call it, simply, triple.
3. Exploiting thesauri for a concept-based match The term thesaurus in Fig. 2 is used to evaluate term predicates. It contains crisp terms in leaf nodes and in the other ones, fuzzy terms, each taking lower level fuzzy terms as its members with degrees specified on the corresponding edges. Any degree between them is assumed 0 if unspecified. Additionally, a composed membership function is provided for obtaining membership values between two terms indirectly related. Definition 3.1 [5]. Let T be a fuzzy term. Then µT (c) = max min(µai (c), α) for all c ∈ N, where α = µT (ai ) and ai is a fuzzy term. Note that since ai is a fuzzy term, µai (c), ∀c ∈ N can also be calculated by the recursive application of Definition 3.1 even if c is not directly connected with ai . Note also that the degree of conceptual closeness between two terms not related with any edge cannot be obtained by this definition. For example, it does
211
not define an edge of certain degree between electronic appliance and audio. In this paper, we do not consider a similarity relation which may be used to quantify such degree. Example 3.1. The degree of conceptual closeness between furniture and radio is calculated by µfurniture(radio) = max min µelectronic_appliance(radio),
µfurniture(electronic_appliance) , min µaudio(radio), µfurniture(audio) = max min(0.96, 0.87), min(0.7, 0.91) = 0.87. Hence, furniture(radio) = 0.87. In the above example, the max operator enables µfurniture(radio) to take the most informative value 0.87 by passing through ‘electronic appliance’ between ‘radio’ and ‘furniture’ instead of passing through ‘audio’. Moreover, it states that either of the two paths is available to quantify µfurniture(radio). We are now in a position to formally define our image retrieval system. Definition 3.2. An image retrieval system I_IR is defined as follows. I_IR = hIDB, I _T , Tr, Ini, where IDB is a set of all images, I _T is a set of all (I _Tp )s for p ∈ IDB, Tr is a term thesaurus, and In is an inverted file. We define In as a set of entries, each formed by attaching to t ∈ I _Tp , every image, p satisfying t ∈ I _Tp , i.e., In = {ht, {p}i}. In is used to search for images indexed by t.
Fig. 2. Term thesaurus.
212
J.D. Yang / Information Processing Letters 74 (2000) 209–214
4. Query evaluation Evaluation of a user query to retrieve images involves translating it into equivalent triples and then matching the I-triple in each entry of In. As a first step toward developing such an evaluation, we provide a definition to specify our query structure. Definition 4.1. Let each of QTj , j = 1, . . . , s, be a triple used as a basic unit of a query. Then the following disjunction of the s triples is called disjunctive query QD . QD =
s _
QTj .
j =1
Definition 4.2. Let QDi , i = 1, . . . , n, be n disjunctive queries. Then a conjunctive normal query Q (or simply query) is defined as follows. Q=
n ^
QDi .
i=1
We now formalize a fuzzy match involving more than one disjunction of term predicates. Wm
Definition 4.3. Let d_T = i=1 Ti be a disjunction of term predicates. Then for all c ∈ N , d_T (c) = max(Ti (c), i = 1, . . . , m). Definition 4.4. Let d_T be a disjunction of term predicates. Then |d_T | = {c ∈ N | d_T (c) = 1}. Definition 4.5. Let each of d_T1 and d_T2 be a disjunction of term predicates. Then d_T1 is more general with a degree α ∈ [0, 1] than d_T2 iff min d_T1 (c) for all c ∈ |d_T2 | = α > 0. It is denoted by d_T2 ⊆α d_T1 . Example 4.1. Let d_T1 = electronic_appliance and d_T2 = radio ∨ recorder. Then d_T2 ⊆0.92 d_T1 since min d_T1 (c) for all c ∈ |d_T2 | = {radio, recorder} = min electronic_appliance(radio),
electronic_appliance(recorder)
= 0.92 > 0.
0 Definition 4.6. Let WsQ be a sub-query of the disjunctive query QD = j =1 QTj given as follows; 0
Q =
ik _
QTj ,
QTj = hT1,j , T2,j , di
j =i1
for j = i1 , i2 , . . . , ik ∈ {1, 2, . . . , s}, sharing d with an I-triple t = hd_T1 , d_T2 , di. Then we call it common direction disjunctive query, or briefly c-d query for t. W Definition 4.7. Let Q0 = sj =1 hT1,j , T2,j , di be a c-d query for t = hd_T1 , d_T2 , di. Then t ⊆α Q0 with α = min(α1 , α2 ) > 0 ⇔ s s _ _ T1,j and d_T2 ⊆α2 T2,j . d_T1 ⊆α1 j =1
j =1
In the sequel, the following W proposition is provided for testing d_T1 ⊆α1 sj =1 T1,j or d_T2 ⊆α2 Ws j =1 T2,j . be a disProposition 4.1. Let each of d_T1 and d_T2 W junction of term predicates. Suppose d_T1 = m i=1 Ti . Then d_T2 ⊆α d_T1 iff min(max(Ti (c/A), i = 1, 2, . . . , m) for all c ∈ |d_T2 |) = α > 0. Proof. It can be directly proved by Definitions 4.3 and 4.5. We omit its proof. 2 Example 4.2. Let d_T1 = radio∨receiver and d_T2 = electronic_appliance ∨ audio. Then d_T1 ⊆0.9 d_T2 , since min max(electronic_appliance(radio), audio(radio)), max(electronic_appliance(receiver), audio(receiver) = min(0.96, 0.9) = 0.9 > 0 by Proposition 4.1. Consider next the condition that an I-triple t ∈ I _Tp exactly matches a query Q guaranteeing a minimum possibility regardless of its indefiniteness. Detecting
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the condition may be crucial to obtain the following exact answer set of Q. Definition 4.8. Let QD be a c-d query for t ∈ I _Tp . Then I _IR `α ∗ QD (p)
iff t ⊆α QD ,
α > 0.
Definition 4.9. kQkα ∗ = {p | I _IR `α ∗ Q(p), α > 0} is a set of exact answers satisfying a query Q with a threshold value α > 0. Example 4.3. Let a query be given as “search for images where an electronic appliance locates at the northeast of furniture”. Then it is converted into QT1 = helectronic_appliance, furniture, northeasti. Now, for t = hd_T1 , d_T2 , northeasti ∈ I _Tp1 where d_T1 = radio ∨ recorder, d_T2 = working_table, t ⊆0.91 QT1 since d_T1 ⊆0.92 electronic_appliance (see Example 4.1) and working_table ⊆0.91 furniture. Hence, I _IR `0.91∗ QT1 (p1 ). Example 4.4. Let the query QD1 be “search for images where an electronic_appliance or audio is at the northeast of furniture”. Then QD1 = QT1 ∨ QT2 where QT1 is given in Example 4.3 and QT2 = haudio, furniture, northeasti. Now, for t = hd_T1 , d_T2 , northeasti ∈ I _Tp2 where d_T1 = radio ∨ receiver, d_T2 = table, the result is I _IR `0.9∗ QD1 (p2 ) since d_T1 ⊆0.9 electronic_appliance ∨ audio (see Example 4.2) and table ⊆0.91 furniture. W Proposition 4.2. Let QD = sj =1 QTj be a disjunctive query. Then for all p ∈ IDB, p ∈ kQD kα ∗ ,
α = max(α1 , α2 , . . . , αs ) > 0,
iff there exists at least one c-d query, Q0i satisfying p ∈ kQ0i kαi∗ ,
αi > 0, 1 6 i 6 s. kQ0i kαi∗ ,
there should exist at least Proof. To be p ∈ one ti ∈ I _Tp that satisfies ti ⊆αi Q0i , αi > 0, 1 6 i 6 s. Without loss of generality, we can assume such c-d queries are Q01 and Q02 for t1 , t2 ∈ I _Tp , respectively. If t1 ⊆α1 Q01 or t2 ⊆α2 Q02 , then p ∈ kQ01 kα1∗ , or p ∈ kQ02 kα2∗ , α1 , α2 > 0 from Definitions 4.8 and 4.9. We can now prove this proposition since either p ∈ kQ01 kα1∗ or p ∈ kQ02 kα2∗ implies p ∈ kQD kα ∗ , α = max(α1 , α2 ) > 0. The same is true even if t1 ⊆α1 Q01
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and t2 ⊆α2 Q02 . It is possible since p can be indexed simultaneously by t1 and t2 which are not necessarily the same with each other. 2 Theorem 4.1. Let QD = query. Then s [
Ws
i=1 QTi
be a disjunctive
kQTi kαi∗ ⊆ kQD kα ∗ ,
i=1
α = max(α1 , α2 , . . . , αs ) > 0. Proof. By using Proposition 4.2, it is straightforward to show if p ∈ kQTi kαi∗ , i = 1, 2, . . . , s, then p ∈ kQD kα ∗ . Hence, we only prove s [
kQTi kαi∗ 6= kQD kα ∗
i=1
by showing that there exists some p ∈ kQD kα ∗ but p∈ / kQTi kαi∗ , i = 1, 2, . . . , s. Let Q0 =
ik _
hT1,j , T2,j , di
j =i1
be a c-d query of QD for an I-triple, t = hd_T1 , d_T2 , di ∈ I _Tp . If such a c-d query does not exist for all t, any p cannot be considered as an answer. Hence φ ⊆ φ, which trivially holds. By Definition 4.8 and Proposition 4.2, we can find the p if it is indexed by t that satisfies t ⊆α Q0 with α > 0, but, none of t ⊆αi QTi , i = 1, 2, . . . , s, holds. Without loss of generality, let Q0 = QT1 ∨ QT2 = hT1 , T2 , di ∨ hT10 , T20 , di and t = hT1 ∨ T10 , T2 ∨ T20 , di. Then, obviously, t ⊆1 Q0 but t ⊆0 QTi , i = 1, 2. Therefore, we can always find some p satisfying p ∈ / kQTi kαi∗ , i = 1, 2, . . . , s. 2 kQD kα ∗ but p ∈ Theorem 4.2. Let Q = for all p ∈ IDB,
Vn
i=1 QDi
be a query. Then
p ∈ kQkα ∗ with α = min(α1 , α2 , . . . , αn ) ⇔ p ∈ kQDi kαi∗ , αi > 0, i = 1, 2, . . . , n. Proof. (⇒) Suppose p ∈ kQkα ∗ . Then by Definition 4.8, I _IR `α ∗ QD1 (p) ∧ QD2 (p) ∧ · · · ∧ QDn (p),
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which means I _IR `αi∗ QDi (p) for all i = 1, 2, . . . , n, with α = min(α1 , α2 , . . . , αn ). Accordingly, p ∈ kQDi kαi∗ . (⇐) Similarly, we can prove the (⇐) part. We omit it. 2 V Theorem 4.3. Let Q = ni=1 QDi , and QDi = W si ∗ j =1 QTij be a query. Then p ∈ kQkα with α = min max(α11 , . . . , α1s1 ), . . . , max(αi1, . . . , αisi ), . . . , max(αn1 , . . . , αnsn )
5. Conclusion In this paper, we proposed a concept-based query evaluation against images indexed with indefinite triples. As further research, the similarity between terms needs to be incorporated into the query evaluation for more sophisticated concept-based matching. References
such that for i, 1 6 i 6 n, p ∈ kQTij kαij , j = 1, . . . , si . Proof. This theorem can be easily proved by Proposition 4.2 and Theorem 4.2. We omit its proof. 2 Example 4.5. Let the query Q be QD1 ∧ QD2 where QD2 = hfurniture, table, northeasti. Then, since p2 ∈ kQD1 k0.9 (see Example 4.4) and p2 ∈ kQD2 k0.84 from min(0.84, 1) = 0.84, we get p2 ∈ kQk0.84 .
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