Laplacan Egenmap for Image Retreval Xaofe He Partha Nyog Department of Computer Scence The Unversty of Chcago, 1100 E 58 th Street, Chcago, IL 60637 ABSTRACT Dmensonalty reducton has been receved much attenton n mage retreval. Even though the dmenson of mage feature vectors s normally very hgh, the embedded dmenson s much lower. Before we utlze any learnng technque, t s benefcal to frst perform dmensonalty reducton. Prncpal component analyss s one of the popular methods used, and can be shown to be optmal when the underlyng structure of data ponts s lnear. However, t fals to dscover the nonlnear structure f the data le on a lowdmensonal manfold embedded n hgh-dmensonal feature space. In ths paper, we apply a geometrcally motvated algorthm for representng the hgh dmensonal data, whch has localty preservng propertes. Consequently, a novel relevance feedback scheme on manfold s proposed. Experment results on real-world mage collectons have shown the effectveness and robustness of our proposed approaches. Keywords: mage retreval, laplacan egenmap, dmensonalty reducton, relevance feedback
1. Introducton Due to the rapd growth n the volume of dgt mages, there s an ncreasng demand for effectve mage management tools. Consequently, content-based mage retreval (CBIR) s recevng wdespread research nterest (Cox et al., 1996; Smth and Chang, 1996; Ma and Manjunath, 1997; Ru et al., 1997; Tong and Chang, 2001; Su et al, 2001). Content based mage retreval uses features automatcally extracted from the mages themselves, rather than manually provded annotatons, to facltate the retreval of mages relevant to a user s query. However, there are stll many open research ssues to be solved before such retreval system can be put nto practce. In recent years, much research has been done to deal wth the problem caused by the hgh dmensonalty of mage feature space (Tenenbaum et al., 2000; Rowes and Saul, 2000). Typcally, the dmensons of feature vector range from few tens to several hundreds. For example, a color hstogram may contan 256 bns. Hgh dmensonalty creates several problems for mage retreval. Frst, learnng from examples s computatonally nfeasble f t has to rely on hgh-dmensonal representatons. The reason for ths s known as curse of dmenson: the number of examples necessary for relable generalzaton grows exponentally wth the number of dmensons (Stone, 1982). Learnablty thus necesstates dmensonalty reducton. Second, n large multmeda databases, hgh-dmensonal representaton s computatonal ntensve and most users do not wat around to provde a great deal of feedbacks. Hence for storage and speed concern, dmensonalty reducton s needed. Two classcal technques for dmensonalty reducton are Prncple Component Analyss (PCA) and Multdmensonal Scalng (MDS). PCA performs dmensonalty reducton by projectng the orgnal n- dmensonal data onto the m(<n) dmensonal lnear subspace spanned by the leadng egenvectors of the data s covarance matrx. Thus PCA bulds a global lnear model of the data (an m dmensonal hyperplane).
For lnearly embedded manfolds, PCA s guaranteed to dscover the dmensonalty of the manfold and produce a compact representaton n the form of an orthonormal bass. However, PCA fals to dscover the underlyng structure, f the data le on a nonlnear submanfold of the feature space. For example, the 3 covarance matrx of data sampled from a helx n R wll have full-rank and thus three prncple components. However, the helx s a one-dmensonal manfold and can be parameterzed wth a sngle number. Classcal MDS fnds an embeddng that preserves parwse dstances between data ponts. It s equvalent to PCA when those dstances are Eucldean. A key queston n mage retreval s that, how do we judge smlarty? The choce of the smlarty measure for the nputs s a deep queston that les at the core of the feld of machne learnng. If the mages le on or close to a low-dmensonal nonlnear manfold embedded n hgh-dmensonal feature space, then the Eucldean dstance n hgh-dmensonal feature space may not accurately reflect ther ntrnsc smlarty, as measured by geodesc dstance along the low-dmensonal manfold. In addton, due to the gap between semantc concepts and low-level mage features, the global topology of mages can not match human percepton. Thus, local topology of mages s much more mportant and relable. When two mages are rrelevant, the absolute value of ther dstance n feature space gves us lttle nformaton about ther smlarty. For example, t s meanngless to say that, a tger s more lke a dog than a horse. In ths paper, we apply a smple nonlnear dmensonalty reducton algorthm, whch has localty preservng propertes. A lnear dmensonalty reducton algorthm s proposed as ts lnear extenson. Consequently, a geometrcally motvated relevance feedback scheme s proposed for mage rankng, whch s conducted on the mage manfold, rather than Eucldean space. To model the geodesc paths of the mage manfold, we fnd the shortest-path dstances between mages n the database, whch s n turn used as smlarty measure. Our goal s to dscover, gven only the unordered hgh-dmensonal nputs, low-dmensonal representatons that capture the ntrnsc degrees of freedom of an mage set. The rest of ths paper s organzed as follows. Secton 2 relates a lst of prevous works to our work. Secton 3 descrbes laplacan egenmap for nonlnear dmensonalty reducton. Secton 4 descrbes ts
connectons to prncple component analyss. In secton 5, we propose a lnear dmensonalty reducton algorthm as lnear extensons of laplacan egenmap. Secton 6 descrbes the proposed method for relevance feedback on manfold. The expermental results are shown n secton 7. Fnally, we gve conclusons and future work n secton 8. 2. Prevous Work One of the most popular models used n nformaton retreval s the vector space model (Salton and McGll, 1983). Varous retreval technques have been developed for ths model, ncludng the method of relevance feedback. Most prevous researches on relevance feedback have fallen nto the followng three categores: retreval based on query pont movement (Ru et al., 1997), retreval based on re-weghtng of dfferent feature dmenson (Ishkawa et al., 1998) and retreval based on updatng the probablty dstrbuton of mages n the database (Cox et al., 2000). In recent years, some learnng-based approaches are proposed. Wu et al. (2000) proposed a Dscrmnant-EM algorthm wthn the transductve learnng framework n whch both labeled and unlabeled mages are used. Teu et. al (2000) presented a framework for mage retreval based on representng mages wth a very large set of hghly selectve features. Queres are nteractvely learned onlne wth a smple boostng algorthm. Tong et. al (2001) proposed the use of a support vector machne actve learnng algorthm for conductng effectve relevance feedback for mage retreval. Whle most machne learnng algorthms are passve n the sense that they are generally appled usng a randomly selected tranng set, the SVM actve learnng algorthm chooses the most nformatve mages wthn the database to ask the user to label. All these approaches have acheved good emprcal results. However, a common lmtaton of them s that they do not consder the underlyng structure of mage set n hgh-dmensonal feature space. There have been several studes of dmensonalty reducton for mage retreval. In (Su et al., 2001), prncple component analyss s used to reduce both nose contaned n the orgnal mage features and dmenson of feature space. The PCA process s ncorporated nto the relevance feedback framework to
extract feature subspaces n order to represent the subjectve class mpled n the postve examples. Dfferent types of features (color, texture, etc.) are allowed to have dfferent dmensons accordng ther sgnfcances and dstrbutons as mpled n user s feedbacks. Cohen et al. (2002) proposes a novel method, called prncple feature analyss, for dmenson reducton of a feature set by choosng a subset of the orgnal features that contans most of the essental nformaton, usng the same crtera as the PCA. In (Wu, 2000), weghted mult-dmensonal scalng s used for dmensonalty reducton. All of these methods do not consder the structure of the mage manfold on whch the mages are lkely to resde. 3. Laplacan Egenmap for Image Representaton Recently, there has been some renewed nterest n the problem of developng low dmensonal representatons when data les on a manfold (Tenenbaum et al., 2000; Rowes and Saul, 2000). In mage retreval, t s desrable to map the mage feature vectors nto a reduced space, due to the consderaton of learnablty, memory storage, and computatonal speed. In addton, due to the gap between semantc concepts and low-level mage features, the global topology of mages can not always match human percepton. Thus, local topology of mages s much more mportant and relable. If the dstance of two mages s large enough, we regard them as rrelevant. When two mages are rrelevant, the absolute value of ther dstance gves us lttle nformaton about ther smlarty. For example, t s meanngless to say that, a tger s more lke a dog than a horse. Thus, for mage retreval, a crteron should be satsfed that, the mappng must have localty preservng propertes. That s, the nearest neghbors of an mage n the orgnal feature space should be mapped to nearest neghbors of that mage n the reduced space. Recall that prncpal component analyss (PCA) s a lnear mappng algorthm whch maxmzes the T varance. Let y = y, y,, } be a one-dmensonal map. The objectve functon of PCA s as follows { 1 2 L y n max y ( y y) 2 where y s the sample mean.
In ths paper, we apply laplacan egenmap (Belkn and Nyog, 2001) to map the mages nto a lowdmensonal space. Consder the problem of mappng the weghted graph G to a lne so that connected ponts stay as close as possble. Let y = y, y,, } be such a map. A reasonable crteron for choosng a { 1 2 L good map s to mnmze the followng objectve functon y n T j ( y y ) j 2 W j The mnmzaton problem reduces to fndng arg mn y T y Dy= 1 y T Ly where L = D W s the Laplacan matrx. D s dagonal weght matrx; ts entres are column (or row, snce W s symmetrc) sums of W, D = W j j. Laplacan s symmetrc, postve semdefnte matrx whch can be thought of as an operator on functons defned on vertces of G. W s the weght matrx such that W j 1 = 0 f ponts x, x j are connected otherwse The objectve functon wth our choce of weghts W j ncurs a heavy penalty f neghborng ponts x and x j are mapped far apart. Therefore, mnmzng t s an attempt to ensure that f x and x j are close then y and y j are close as well. The algorthmc procedure s formally stated below: Step 1 [Constructng the adjacency graph] Nodes and j are connected by an edge f s among k nearest neghbors of j or j s among k nearest neghbors of. Step 2 [Choosng the weghts] W f and only f vertces and j are connected by an edge. = 1 j
Step 3 [Egenmaps] Assume the graph G, constructed above, s connected, otherwse proceed wth Step 3 for each connected component. Compute egenvalues and egenvectors for the generalzed egenvector problem: Ly = λdy (1) Let y 0,..., y k 1 be the solutons of equaton 1, ordered accordng to ther egenvalues, Ly Ly Ly 0 1 LL = λ Dy 1 0 = λ Dy = λ k 1 k 1 k 1 0 = λ0 λ1 L λk 1 1 0 Dy We leave out the egenvector y correspondng to egenvalue 0 and use the next m egenvectors for embeddng n m-dmensonal Eucldean space. 0 x ( y 1 ( ), L, y ( )) m 4. Connectons to Prncpal component analyss (PCA) 4.1 Prncpal component analyss Prncpal component analyss s a wdely used statstcal tool for data analyss. Gven a set of feature vectors n hgh-dmensonal space, the purpose s to fnd a low-dmensonal mappng n reduced space wth less redundancy, that would gve as good a representaton as possble. The redundancy s measured by correlatons between data elements. In the PCA transform, the feature vector x s frst centered by substractng ts mean: x x E(x)
The mean s estmated from the avalable samples x( 1), x(2), L, x( m) R n. Let us assume n the followng that the centerng has been done and thus E(x)=0. Let X denote the feature vector matrx, that s, the th row of X s the feature vector x(). So, X s a m n matrx. Thus, we can obtan the covarance matrx as follows: C = X T X By egenvector decomposton, C can be decomposed nto the product of three matrces: C = VΣV T where Σ = dag λ, λ, L, λ } are the egenvalues wth descendng order and V s a orthogonal matrx, whose { 1 2 n column vector s the correspondng egenvector of C. Thus, TPCA = V s the transformaton matrx of PCA. If we want to map these m-dmensonal feature vectors to a k-dmensonal (k<m) space, t could be done by smply set the m-k least egenvalues to be zero,.e., λ 0, L λ 0. k + 1, m By applyng Sngular Value Decomposton (SVD), we can reformulate ths problem as follows: T X = UΣV T where U and V are two orthogonal matrces, whose column vectors are the egenvector of XX and X T X (the covarance matrx C), respectvely, and Σ = dag λ, λ, L, λ } are the sngular values of X (also the { 1 2 n T egenvalues of XX and X T X ). Now, we obtan the transformed feature vector n new low-dmensonal space: X = = UΣ XT PCA Ths ndcates that the low-dmensonal representaton obtaned by PCA s just the product of two matrces: T (1) the matrx U whose column vectors are the leadng egenvectors of weght matrx W = XX : PCA W PCA y = λ y (2) PCA U = { y, y 2, L, y 1 k }
and (2) the dagonal matrx Σ whose dagonal elements are the leadng egenvalues of W, Σ = dag λ, λ, L, λ }. Note that the weght matrx W s measured by nner product of two mage PCA vectors. { 1 2 k PCA 4.2 Connectons to PCA In secton 4, we have show that the optmal embeddng for non-lnear case s obtaned by solvng a general egenvector problem below: Ly = λdy ( D W ) y = λdy Wy = (1 λ) Dy 1 D Wy = (1 λ) y 1 Let W D W, W s essentally a normalzed weght matrx preservng localty. (Note that the Laplacan = Laplacan matrx D provdes a natural measure on the vertces of the graph. The bgger the value D s, the more mportant s that vertex). We rewrte the above formula as follows: W Laplacan y = 1 λ ) y (3) ( Laplacan As we can see, the equatons (2) and (3) have the same form. Ths observaton shows that, wth the same weght matrx, Laplacan Egenmap and PCA wll yeld the same result, and λ PCA + λ Laplacan = 1. Therefore, the egenvector of W wth the largest egenvalue s just the egenvector of W wth the smallest egenvalue. PCA Laplacan The connectons also gve us a way to determne the dmensonalty of the low-dmensonal manfold, analogous to PCA. J ( k) = k (1 λ ) = 1 rank( W ) 1 = 1 (1 λ ) 100%
where J ( k) 90%. 0 = λ0 < λ1 λ2 L λ rank ( W ) 1. If we want to keep 90% nformaton, we smply choose such k that Based on above analyss, we conclude that, the essental dfference between laplacan egenmap and PCA s that they choose dfferent weght matrces. PCA uses nner product as a lnear smlarty measure, whle laplacan egenmap uses a non-lnear smlarty measure whch preserves localty. For PCA, ts advantage over laplacan egenmap s that t can produce a transform matrx T. Thus, for a new pont, t can be easly map to the new space. The dsadvantage s that, t fals to dscover the underlyng nonlnear structure of data set. PCA 5. Lnear Laplacan Egenmap As we descrbed prevously, one dsadvantage of laplacan egenmap s that t cannot produce a transformaton functon. That s, f the query mage s not n database, there s no way to map t nto the reduced space. To overcome ths problem, we propose lnear laplacan egenmap, whch s a lnear approxmaton of laplacan egenmap. Suppose A=(a 1,a 2,,a k ) s a transformaton matrx of lnear laplacan egenmap, that s, Y = XA, where the row vectors of matrx Y are the mage representatons n reduced space. We rewrte the mnmzaton problem as follows: arg mn ( XA) T L( XA) A T ( XA) D( XA) = I The vectors a that mnmze the above functon s gven by the mnmum egenvalue solutons to the generalzed egenvalue problem: L' a = λd'a where T L' = X LX, D' = X DX. T
Note that L and D are two n n matrces, where n s the dmensonalty of the orgnal space. Whle n the nonlnear case, L and D are two m m matrces, where m s the number of mages n database. In the real world applcatons, m >> n. Ths property makes lnear laplacan egenmap much more promsng than nonlnear laplacan egenmap. 6. Relevance Feedback on Image Manfold An mage retreval system should be able to rank the mages accordng to the query mage and relevance feedbacks. Tradtonally, the most wdely used dstance functon s Eucldean dstance. Ths s based on an assumpton that the mages le on a lnear hgh-dmensonal space. However, Eucldean dstance s not always a good choce, f the mages le on or close to a nonlnear manfold wth low-dmensonalty. Followng laplacan egenmap dscussed n secton 3 and lnear laplacan egenmap descrbed n secton 5, we propose a geometrcally motvated relevance feedback scheme for mage rankng, whch s conducted on manfold, rather than Eucldean space. Let R denote the set of query mage and relevant mages provded by the user. Our algorthm can be descrbed as follows: 1. Canddate generaton. For each mage x R, we fnd ts k-nearest neghbors C = { y1, y 2, L, y k } (those mages n R are excluded from selecton). Let C = C 1 U C 2 ULU C R. We call C canddate mage set. Note that R I C = φ. 2. Construct subgraph. We buld a subgraph H(V) wth weght matrx W, where V H = R U C.The dstance of any two vertces x, x j s measured as follows: dst( x, x j x ) = x j 2 f x s among x or x j s among x 's k' neareset neghbor otherwse j 's k' neareset neghbor,
Snce the mages n R are supposed to have some common semantcs, we set ther dstances to be zero. That s, dst(, x ) = 0, x, x R. Note that, k <k. x j j 3. Manfold dstance measure. To model the geodesc paths of the manfold, we fnd the shortestpath dstances between vertces n H. The length of a path n H s defned to be the sum of lnk weghts along that path. We then compute the geodesc dstance dst H ( x, x j ) (.e. shortest path length) between all pars of vertces of and j n H, usng Floyd s O ( r 3 ) algorthm. 4. Retreval based on geodesc dstance. To retreve the mages most smlar to the query, we smply sort them accordng to ther geodesc dstance to the query. The selected mages are presented to the user. 5. Update query example set. Add the relevant mages provded by the user nto R. Go back to step 1 untl the user s satsfed. From the above algorthm, we can see that the smlarty between the mage n database and the query mage s determned by ther geodesc dstance on mage manfold. The query examples (ncludng the orgnal query mage and relevant mages provded by the user) are merged nto one pont on the manfold. Thus, the geodesc dstance between query mage q and mage x can also be computed as follows: dst H ( q, x ) = mn{ dst ( r, x)} r R C where dst C ( r, x) s the length of the shortest path n C between mage r and mage x. That s, the vertces n the shortest path can only contan vertces n C. (Note that, dst H ( q, r) = 0, r R ) Thus, our algorthm s essentally dfferent from Roccho s relevance feedback scheme (1971). Roccho s formula for query refnement can be llustrated as follows: Q' = α Q 0 + β n R γ n 1 2 n S n = 1 1 = 1 2
where Q s the ntal query vector, Q ' s the refned query vector, R s the vector of relevant examples, 0 S s the vector of rrelevant examples, n and 1 n 2 are the number of relevant examples and rrelevant examples, respectvely; β and γ are sutable constants. Roccho s algorthm s essentally a perceptron-lke learnng algorthm when the nner product smlarty s used. Ths technque s mplemented n the MARS system (Ru et al., 1997). Roccho s smlarty-based relevance feedback algorthm has been wdely used n nformaton retreval, and has archved good emprcal results n text retreval doman. However, t may not be sutable for mage retreval, snce t fals to preserve localty. Due to the gap between semantc concepts and lowlevel mage features, the global topology of mages can not match human percepton. Thus, local dstrbuton of mages s much more mportant and relable. Hence the local nformaton should be preserved whle retreval. Fgure 1 llustrate a smple example: A stands for the ntal query mage; B s a relevant example provded by the user. C s the refned query by Roccho s algorthm ( α = β = 0. 5 ). Thus, mage D wll be retreved. However, by ntuton, mages E and F are preferred to mage D. In our proposed algorthm, A and B are merged nto a sngle pont. Thus, the geodesc dstance between A(B) and D s larger than the geodesc dstance between A(B) and E(F). So mage E and F wll be retreved, rather than mage D. 7. Expermental Results We performed several experments to evaluate the effectveness of the proposed approaches over a large mage database. The database we use conssts of 10,000 mages of 100 semantc categores from the Corel dataset. It s a large and heterogeneous mage set. A retreved mage s consdered correct f t belongs to the same category of the query mage. Three types of color features and three types of texture features are used n our system, whch are lsted n Table 1. The dmenson of the feature space s 435. We desgned an automatc feedback scheme to model the retreval process. At each teraton, the system makes the frst three ncorrect mages from the top 100 matches as rrelevant examples, and also selects at most 3 correct mages as relevant examples (relevant examples n the prevous teratons are
excluded from the selecton). These automatc generated feedbacks are added nto the query example set to refne the retreval. To evaluate the performance of our algorthms, we defne the retreval accuracy as follows: relevant mages retreved n top N returns Accuray = N 7.1 Three examples for 2-D data vsualzaton To compare the performance of laplacan egenmap, lnear laplacan egenmap and PCA, we show two examples. In the frst example, the dataset contans two classes. One class contans mage data of human eyes, and the other class contans mage data of non-human eyes. Each class contans 1500 mages n 400- dmensonal space. Each mage s an 8 bts (256 grey) mage of sze 20 20. In the second example, a multclass dataset of handwrtten dgts ( 0-9 ) (Blake and Merz 1998). s used. The expermental results are shown n fgure 4-5. 7.2 Image Retreval n 2-dmensonal Reduced Space The mage database we use for ths experment conssts of 10,000 mages of 100 semantc categores from the Corel dataset. Each semantc category conssts of 100 mages. We compare the retreval result n PCA space, kernel PCA space, laplacan space and lnear laplacan space wth 2 dmensons. In PCA space, the tradtonal Roccho s relevance feedback scheme s used. In laplacan space and lnear laplacan space, our relevance feedback scheme on manfold s used. Fg 2 shows the experment results. As can be seen, our proposed relevance feedback scheme n laplacan space performs much better than Roccho s relevance feedback scheme n PCA space. 7.3 Image Retreval n Reduced Space wth dfferent dmensons In ths secton, we compared the performance of mage retreval n reduced space wth dfferent dmensons. Fgure 3 shows the experment results. As the feature space s drastcally reduced by PCA, the performance decreases fast. In PCA space wth 2 dmensons, only about 10% accuracy s acheved after
three teratons. Whle n 2-dmensonal laplacan space, we can stll acheve more than 40% accuracy after one teraton. Ths observaton shows that, our proposed algorthm s especally sutable for the case when drastc dmensonalty reducton needs to be performed. 8. Conclusons and Future Work Whle nonlnear dmensonalty reducton on manfold has receved much attenton, to the best of our knowledge we are not aware of any applcaton to mage representaton and retreval yet. In ths paper, we descrbed a nonlnear dmensonalty reducton algorthm and ts lnear extenson for mage retreval. The proposed scheme nfers a reduced space that preserves localty. A correspondng relevance feedback scheme s proposed. It s conducted on the mage manfold, rather than Eucldean space. As can be seen from the experments, our proposed algorthm performs much better than prncpal component analyss, especally when drastc dmensonalty reducton s performed. A fundamental problem n mage retreval s the gap between hgh-level semantc concepts and lowlevel mage features. Most exstng relevance feedback technques focus on mprovng the retreval performance of current query sesson, and the knowledge obtaned form the past user nteractons wth the system s forgotten. A possble extenson of our work s to somehow adjust the local topology of mages from user s relevance feedback, so the system wll gradually mprove ts retreval performance through accumulated user nteractons. We are currently explorng the effect and mpact of ths extenson.
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Color-1 Color hstogram n HSV space wth quantzaton 256 Color-2 Frst and second moments n Lab space Color-3 Color coherence vector n LUV space wth quantzaton 64 Texture-1 Texture-2 Texture-3 Tamura coarseness hstogram Tamura drectonary Pyramd wavelet texture feature Table 1. Image features Fgure 1. A case where Roccho s relevance feedback gves a bad result. Image A s the ntal query example. Image B s a relevant example. Image C s the refned query. By Roccho s relevance feedback scheme, mage D wll be retreved, rather than E or F. Accuracy 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Retreval accuracy n reduced space wth 2 dmensons (top 10) number of rounds 1 2 3 4 5 6 PCA space wth 2 dmensons Laplacan space wth 2 dmensons Orgnal space wth 435 dmensons Lnear Laplacan space wth 2 dmensons
0.45 0.4 0.35 Retreval accuracy n reduced space wth 2 dmensons (top 20) Accuracy 0.3 0.25 0.2 0.15 0.1 0.05 1 2 3 4 5 PCA space wth 2 dmensons Laplacan space wth 2 dmensons Orgnal space wth 435 dmensons Lnear Laplacan space wth 2 dmensons number of rounds 6 Fgure 2. Image retreval n 2-dmensonal reduced space. Roccho s relevance feedback scheme s used n PCA space, whle our proposed algorthm s used n laplacan space, lnear laplacan space, and orgnal space. Accuracy 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Retreval accuracy n PCA space wth dfferent dmensons (top 10) 0 PCA 1 space wth 100 dmensons 2 PCA space wth 80 dmensons PCA space wth 40 dmensons PCA space wth 20 dmensons PCA space wth 10 dmensons PCA space wth 5 dmensons PCA space wth 2 dmensons number of rounds 3
Accuracy 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 Retreval accuracy n Laplacan space wth dfferent dmensons (top 10) number of rounds 0 1 2 Laplacan space wth 100 dmensons 3 Laplacan space wth 80 dmensons Laplacan space wth 40 dmensons Laplacan space wth 20 dmensons Laplacan space wth 10 dmensons Laplacan space wth 5 dmensons Laplacan space wth 2 dmensons Accuracy 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 Retreval accuracy n Lnear Laplacan space wth dfferent dmensons (top 10) number of rounds 0 Lnear 1Laplacan space wth 100 2dmensons 3 Lnear Laplacan space wth 80 dmensons Lnear Laplacan space wth 40 dmensons Lnear Laplacan space wth 20 dmensons Lnear Laplacan space wth 10 dmensons Lnear Laplacan space wth 5 dmensons Lnear Laplacan space wth 2 dmensons Fgure 3. Image retreval n reduced space wth dfferent dmensons. Roccho s relevance feedback scheme s used n PCA space, whle our proposed algorthm s used n laplacan space and lnear laplacan space.
Fgure 4. Laplacan egenmap vs. PCA. (Two classes)
Fgure 5. Laplacan egenmap vs. PCA (10 classes, dgts 0-9 )