A PrimeNumb erBased Matrix Strategy for Ecient Iconic Indexing of Symb olic Pictures YeIn Chang and BiYen Yang Dept of Applied Mathematics National Sun YatSen University Kaohsiung Taiwan ROC fEmail changyimathnsysuedutwg fTel ext g fFax g Abstract In the previous approaches to represent pictorial data as the complexity of the represen tation strategy is increased the more spatial relationships can b e represented which also results in a more complex strategy for query pro cessing and a limited typ es of queries that can b e answered In this pap er we prop ose an ecient iconic indexing scheme called PrimeNumberbased Matrix PN Matrix for symb olic pictures whichcombines the ad vantages of the D Cstring and the DLT matrix Basically the prop osed strategy can represent those complex relationships which are represented in D Cstrings in a matrix and an ecient mo dulebased op eration can b e used to supp ort pictorial query spatial reasoning and similarity retrieval In the prop osed scheme we classify spatial relation ships b etween two ob jects in D space into ve spatial categories and dene a category rule for each of those ve spatial categories Those category rules are mo duleop erationbased therefore they are ecient enough as compared to the previous approaches Following those category rules we prop ose algorithms to eciently supp ort spatial reasoning picture queries and similarity retrieval based on a data structure of a PrimeNumb erbased Matrix PN Matrix Keywords D string D Cstring image databases pictorial query similarity retrieval spatial reasoning symb olic databases This researchwas supp orted in part by the National Science Council of Republic of China under Grant No NSCE Intro duction The design of image databases has attracted much attention over the past few years Ap plications which use image databases include oce automation computer aided design rob otics and medical pictorial archiving A common requirement of these systems is to mo del and access pictorial data with ease Thus one of the most imp ortant problems in the design of image database systems is how images are stored in the image database In traditional database systems the use of indexing to allow database accessing has b een well established Analogously picture indexing techniques are needed to make ease pictorial information retrieval from a pictorial database Over the last deco de many approaches to represent symbol data have b een prop osed as shown in Figure Chang et al prop ose a pictorial data structure D stringusing symb olic pro jections to representsymb olic pictures preserving spatial relationships among ob jects The basic idea is to pro ject the ob jects of a picture along the xaxis and y axis to form two strings representing the relative p ositions of ob jects in the xaxis and y axis resp ectively A picture query can also b e sp ecied as a D string Based on D strings several algorithms in pictorial querying spatial reasoning and similarity retrieval are prop osed where picture querying allows the users to query images with a sp ecied spatial relationship spatial reasoning means the inference of a consistent set of spatial relationships among the ob jects in an image and the target of similar retrieval is to retrieve the images that are similar to the query image Based on the D string longest common subsequence strategy Lee et al prop ose an algorithm for similarity retrieval Based on an objects and spatial relationship OSR table strategyLiaw also prop oses an algorithm for similarity retrieval which distinguishes whether two images are similar by comparing two sets of real numb ers that are asso ciated to these images resp ectively Besides Costagliola et al intro duce a variation of D string representation for symb olic pictures the non redundant D stringwhich is a more compact representation than the D string However the representation of D strings is not sucient enough to describ e pictures and Chang et al in of arbitrary complexity completelyFor this reason Jungert tro duce more spatial op erators to handle more typ es of spatial relationships among ob jects in image databases Using these extended spatial op erators D Gstring representation Symbolic Picture Representation 2D string 9DLT-matrix [Chang 87] [Chang 91] * * non-redundant 2D string longest 2D G-string fast spatial match 2D string common subsequence [Chang 88] accessing scheme [Costagliola 95] [Lee 89] [Chang 90] 2D C-string * * [Lee 90] object’s and spatial module-oriented relationship (OSR) table signature extraction [Liaw 90] + [Chang 90] 2D C- string [Huang 94] *: Query Processing Strategies Figure Approaches to symb olic picture representation facilitates spatial reasoning ab out shap es and relative p ositions of ob jects But a D G string representation scheme is not ideally economic for complex images in terms of storage space eciency and navigation complexity in spatial reasoning Therefore Lee and Hsu prop ose a D Cstring representation scheme Since the numb er of subparts generated by this new cutting mechanism is reduced signicantly the lengths of the strings representing pictures are much shorter while still preserving the spatial relationships among ob jects However the previous abstractions ignore the relative sizes and lo cations of ob jects Thus similarity retrieval of images maybeambiguous and manytyp es of queries concerning sizes lo cations and distances cannot b e answered due to this inadequacy Therefore Huang and Jean prop ose a D C string representation scheme which extends the work of Lee and Hsu D Cstring by including relative metric information ab out the picture into the strings As describ ed b efore based on D string representation the problem of picture query turns out to b e the matching of D subsequences which takes nonp olynomial time com plexity This makes the picture retrieval metho d inappropriate for implementation esp e cially when the numb er of ob jects in an image is large Therefore Chang et al prop ose an ecient approach of iconic indexing byanine direction lowertriangular DLT ma trix Moreover to handle large amounts of image databases a simple algorithm for spatial match retrieval of symb olic pictures based up on DLT matrices is prop osed by Chang et al by using the concept of superimposedcoding which reduces the ratio of false match when a query picture o ccurs Next Chang et al prop ose a moduleoriented signature extraction strategy in which prime numb ers are used to comp ose the signatures and the mo dule op eration will b e applied when the query happ ens In the previous approaches to represent pictorial data as the complexity of the rep resentation strategy is increased the more spatial relationships can b e represented which also results in a more complex strategy for query pro cessing and a limited typ es of queries that can b e answered In this pap er we prop ose an ecient iconic indexing scheme called PrimeNumberbased Matrix PN Matrix for symb olic pictures whichcombines the ad vantages of the D Cstring and the DLT matrix Basically the prop osed strategy can represent those complex spatial relationships that are represented in D Cstrings in a matrix while it do es not need any cutting strategy and complex pro cedures to do spa tial reasoning Moreover the prop osed strategy can b e considered as an extended DLT matrix strategy in which more than spatial relationships can b e represented and an ef cient mo dulebased op eration can b e used to supp ort pictorial query spatial reasoning and similarity retrieval In the prop osed scheme we classify those spatial relationships between two ob jects in D space as observed by Lee and Hsu into ve spatial categories and dene a category rule for each of those ve spatial categories Those category rules are mo duleop erationbased therefore they are ecient enough as compared to the previous approaches Following those category rules we prop ose algorithms to eciently supp ort spatial reasoning picture queries and similarity retrieval based on a PrimeNumb erbased Matrix PN Matrix strategy In a PN Matrix the relationships b etween two ob jects along the xaxis or y axis is recorded in a numb er which is a pro duct of some prime numbers Therefore spatial reasoning can b e done very straightforwardlyFor answering a pictorial query some mo dule op erations or memb ership checking of a set of numb ers are applied In similarity retrieval some new matrix op erations are applied The rest of the pap er is organized as follows In Section wegive a brief description d b c e a f Figure A picture f ab out the previous symb olic picture representations In Section we will present the prop osed ecient iconic indexing scheme for symb olic pictures Finally Section gives a conclusion Background In this Section we briey describ e several data structures for symb olic picture representa tion including D string D Gstring D Cstring and DLT matrix Moreover we will briey describ e the strategy for spatial reasoning based on the D Cstring representation esp ecially D string For pictorial information retrieval Chang et al presentanewway of representing a picture bya D string and a picture query can also b e sp ecied as a D string The problem of pictorial information retrieval then b ecomes a problem of D subsequence matching Let V b e a set of symb ols where eachsymb ol could represent a pictorial ob ject or a pixel Let A b e the set fgwhere and are three sp ecial symb ols not in V For example consider the picture shown in Figure V fa b c d e f g The D string representing