Rotation Invariant Shape Contexts based on Feature-space Fourier Transformation

Transcription

1 Fourth Internatonal Conference on Image and Graphcs Rotaton Invarant Shape Contexts based on Feature-space Fourer Transformaton Su Yang 1, Yuanyuan Wang Dept of Computer Scence and Engneerng, Fudan Unversty, Shangha 00433, Chna Dept of Electronc Engneerng, Fudan Unversty, Shangha 00433, Chna E-mal: Abstract We propose a new pxel-level shape descrptor Frst, shape contexts are computed Then, D FFT s performed on each D hstogram from shape contexts Such a scheme solves the rotaton-nvarance problem of shape contexts based on the shft theorem of Fourer Transformaton whle does not ncrease the computatonal complexty Theoretcal proof and expermental valdaton are provded 1 Introducton Shape matchng plays an mportant role n a varety of applcatons n computer vson and pattern recognton The key problem for shape analyss s how to capture and descrbe the characterstcs of a shape Shape context s a recently proposed descrptor [1], whch has receved much attenton to date An deal shape descrptor should possess the followng propertes: (1 good dscrmnatve power, ( rotaton and scale-nvarant, (3 robust aganst deformatons and occlusons The advantages of shape contexts are as follows (1 The computaton of shape contexts s performed drectly on pxels, avodng the preprocessng to detect structures n pxels, whch s usually regarded an error-prone process Ths promses the robustness of shape contexts The dscrmnatve power of shape contexts are very good because t fgures out how the other pont confgure n reference to every pont The weakness of shape contexts les n the rotaton nvarance The rotaton nvarance of shape contexts s dependent on the tangent at every boundary pont To compute the tangent at every pxel, the boundary ponts must be computed and organzed n order Ths s aganst the fundamental sprt of shape contexts One man advantage of shape contexts s that t can be drectly appled to pxels wthout any errorprone preprocessng such as the perceptual organzaton of the boundary ponts nto a pont sequence The tangent based rotaton nvarance s unstable n that perturbatons may arse from both the outsde outlers and the error-prone perceptual organzaton of the boundary ponts Moreover, the perceptual organzaton of boundary ponts prevents the shape contexts from beng appled to more general cases other than boundary based shape representatons It s known that shape can be represented n many ways, not ust the boundary of obects For example, skeleton ponts are the more generally appled prmtves n bnary mage classfcaton [] Besdes, pont set matchng, whch s a classcal problem n computer vson and pattern recognton, tackles more general cases, not ust boundary ponts In vew of such lmt of shape contexts, we propose a new descrptor n [], namely, statstcal ntegraton of pxel-level constrant hstograms It preserves the man advantages of shape contexts, dscrmnatve power and robustness, whle solves the rotaton-nvarance problem However, ths descrptor s lmted n that ts computatonal complexty s hgh, roughly O(N 3 To mprove the rotaton-nvarance of shape contexts whle not ntroduce hgher computatonal complexty, we propose n study a new scheme for pxel-level shape descrpton We refer to ths new descrptor as Rotaton-nvarant Shape Contexts based on FFT (FFT- RISC The key s to perform -dmensonal FFT on the orgnal Shape Contexts Then, let the modulus of the FFT transformaton of Shape Contexts be the sgnature to characterze how the other ponts dstrbute around every pont Based on the shft theorem of Fourer Transformaton, n ths paper, we have theoretcally proved that the proposed FFT-RISC feature s nvarant under any affne transformaton Some examples are also provded to expermentally valdate the nvarance of the FFT-RISC feature A related work can be found n [3] The dfference between ths study and [3] les n two aspects: (1 The descrptor proposed n [3] reles on the computaton of a center for a pont set, whch degrades ts /07 $ IEEE DOI /ICIG

2 dscrmnatve power ( Ths study proposes to use FFT to solve the shft problem encountered n [3] Rotaton-nvarant shape contexts mplemented by FFT In the followng, we frst present how to compute the rotaton-nvarant shape contexts va FFT (FFT- RISC Then, we descrbe the method to match two pont sets usng the FFT-RISC feature Fnally, we prove the nvarance of the FFT-RISC feature under affne transformatons 1 Computaton of the FFT-RISC feature Defnton 1: nt(r functons to preserve only the nteger porton of a real number r Defnton : c means the modulus of a complex value c Notes: The comments are enclosed between /* and */ Subroutne 1: Compute the feature matrx for every pont n a gven pont set P{P 1,,,L} Input: P{P (x,y 1,,,L} /*A pont set of nterest*/ Output: {F(P 1,,,L} /*The feature matrx n assocaton wth every pont P, 1,,,L*/ Step 1: For,1,, L and, compute l ( x x + ( y y (1 Step : Compute r0 mn{log( l, 1,,, L; } ( Step 3: For,1,, L and, compute r logl r 0 (3 Step 4: For,1,, L and, compute y y α arctan (4 y + y Step 5: Call subroutne to compute the feature matrx F(P of every pont P, 1,, L Step 6: Return {F(P 1,,,L} Note that: In subroutne, M wll affect the resoluton of FFT Subroutne : Compute the feature matrx for a gven pont P n a pont set P{P 1,,,L} Input: {(r,α 1,,,L; } Parameters: M, r and α /*M must be greater than nt(r m / r +1, where r m max{r 1,,,L; }*/ Output: F(P /*The feature matrx of P */ Step 1: Let h 0 for p1,,,m; q1,,,n /*Construct a D hstogram [h ]*/ Step : For 1,,,L and, compute pnt(r / r and qnt(α / α, and then let h h +1 /* r and α are two parameters determnng the sze of every block of the D hstogram Correspondngly, p and q are the ndces to the D hstogram*/ Step 3: For p1,,,m and q1,,,n, let h h /(L-1 /*Normalze the D hstogram*/ Step 4: Perform D FFT on the D hstogram [h ], whch results n a new matrx [f ] wth the same dmenson M N Note that f s a complex number, p1,,,m; q1,,,n Step 5: Let F(P [a ], where a s the modulus of f, p1,,,m; q1,,,n Step 6: Return F(P For a gven pont P, the step 1 to step 3 n subroutne results n a D hstogram as follows h L 1 δ ( r, α,, α, p, q, (5 p1,,,m and q1,,,n, where 1 δ ( r, α, r, α, p, q 0 r r [( p 1 r, p r ] α [( q 1 α, q α ] else (6 Followng subroutne 1 and, a shape descrptor F(P can be obtaned wth regard to each pont n a pont set Snce the computaton of FFT s very fast, t can be regarded that the above shape descrptor mplemented by FFT does not lead to obvously hgher computng cost n contrast to the orgnal Shape Contexts Because F(P fgures our how the other ponts n the same pont set dstrbute around P, the pont correspondences between two pont sets can then be computed based on such a pxel-level shape descrptor Matchng of pont sets Subroutne 3: Compute the pont correspondence between two gven pont sets P{P 1,,,L} and Q{Q 1,,,L }, where we assume L L wthout losng the generalty 576

3 Input: {F(P 1,,,L} and {F(Q 1,,, L }, where ( P [ a F ] and F ( Q [ b ], p1,,,m; q1,,,n /*Input the feature matrces of the two gven pont sets*/ Output: ψ /*A set conssts of the ndces of matched pars Step 1: Intalze ψ as an empty set Step : Compute D{ d M N p 1 q 1 ( a b 1,,,L; 1,,, L } /*The dstance between the two feature matrces F(P and F(Q Step 3: Search ( s, t arg mn{ d } ; Add a new, element (s,t to ψ; Update D by removng the elements {d k k1,,,l} and {d k k1,,, L } Step 4: If ψ <L, go to step Else, return ψ (7 In subroutne 3, the dstance between two ponts s defned as the Eucldean dstance between the correspondng two feature matrces In each teraton n subroutne 3, the two ponts wth the closest dstance are selected to match Of course, there must exst L-L unmatched ponts n Q 3 Proof of nvarance In the followng, we wll show that F(P s a rotaton-nvarant pxel-level shape descrptor If we apply an affne transformaton to every pxel P (x,y n pont set P, then, we can obtan a new pont set Q{Q (X,Y 1,,,L}, where (X,Y s transformed from (x,y va X snθ x x0 s + Y snθ y y0 (8 s x + ssnθy + x0 s snθx + s y + y0 In the above transformaton, (x 0,y 0, s, and β are the parameters to determne the translaton, scale, and rotaton, respectvely Accordng the above equaton, t s easy to see that L ( X X + ( Y Y s ( x x + ( y y l It follows that (9 R0 mn{log( L } mn{log( l + log( s},, mn{log( l } + log( s r + log( s, 0 (10 Due to Eq (9 and Eq (10, R log( L R0 [log( l + log( s] (11 [ r0 + log( s] log( l r0 r Eq (11 shows that scalng does not affect the computaton of the feature matrx Accordng to Eq (4, Y Y tan β X X snθ ( x ( x x + ( y x + snθ ( y snθ + tanα + snθ tanα snθ cosα + snα cosα + snθ snα sn( α θ tan( α θ cos( α θ β α θ ± π y y (1 It s know that α [0,π], β [0,π], and θ (0,π So, α -θ (-π,π Eq (1 shows that the dfference between β and α s a constant f α -θ 0 If α -θ<0, then, β α -θ+π Because of Eq (11 and Eq (1, we know that (R,β (r,α -θ+c, (13 where Cπ or C0 for 1,,,L and Due to Eq (11~(13, we have (R / r,β / α (r / r,(α -θ+c/ α, (14 Denote the Fourer transformaton (FT of (r / r,α / α as f(r / r,α / α F(u,v (15 In accordance wth the shft theorem of Fourer transformaton, we hold f(r / r,β / α f(r / r,(α -θ+c/ α exp{-[π(θ-c/ α ]v}f(u,v (16 Ths leads to f(r,β F(u,v f(r,α (17 where * means the modulus of a complex value On account of subroutne and Eq (17, we can conclude that the proposed feature s nvarant under affne transformaton Because we apply (nt(r / r and nt(α / α n constructng the D hstogram and we use D FFT nstead of D FT n computng the feature matrx, the affne nvarance property of the proposed feature only holds approxmately 577

4 3 Experments We demonstrate the nvarance of the FFT-RISC shape descrptor va the followng experments We conducted 3 tests n total The mage to be tested s shown n Fg 1, whch s a Chnese character consstng of a pont set In the frst test, we generate a new pont set by applyng an affne transformaton to the pont set shown n Fg 1 The new pont set and the orgnal pont set are llustrated n Fg In the second test, we frstly add 50 nosy ponts to the pont set shown n Fg 1 The pont set contanng nosy ponts s shown n Fg 3 Then, we apply the same affne transformaton to the pont set shown n Fg 3 Fg 4 llustrates both the pont set shown n Fg 3 and the new pont set transformed from ths pont set Fg 5 and Fg 6 are smlar to Fg 3 and Fg 4 whle the number of nosy ponts s ncreased to be 100 In each of the above 3 tests, we compute the FFT-RISC feature wth two groups of parameters The parameters of the frst group are: M64, r 01, and α 10 The parameters of the second group are: M18, r 01, and α 10 We only alert M n the two groups snce we want to observe how the FFT computaton affects the feature After we obtaned the feature matrx for each pont followng the procedure descrbed n subroutne and 3, we matched the two pont sets n each test va the method descrbed n subroutne 3 The expermental results are shown n Table 1, whch shows the rato of the number of ncorrectly matched pars to the total pars n every test It can be seen that nvarance can be guaranteed by the proposed shape descrptor f there are no nosy ponts When the number of nosy ponts ncrease, the matchng becomes worse Also, bgger M results n better matchng of ponts because the resoluton of FFT ncreases wth M These tests confrm the nvarance of the FFT-RISC descrptor ntally Yet, there are a lot of works to be done to refne the proposed scheme Nevertheless, t s an nterestng step n mprovng the nvarance of Shape Contexts The advantage of the propose scheme s: It promses rotaton-nvarance wth lttle addtonal computatonal cost because the computaton of FFT s very fast In the experments, the computaton of the FFT-RISC feature s very fast So, the proposed scheme merts further nvestgatons for refnement Fg 1: Pont set A Fg : Pont set A and ts affne transformaton Fg 3: Pont set B (Pont set A plus 50 addtonal nosy ponts 578

5 transformaton Table 1 Rato of erroneous matchng Nosy ponts M64 0 7/105 31/105 M18 0 1/105 15/105 4 Summary Fg 4: Pont set B and ts affne transformaton Ths paper proposes a FFT based scheme to mprove the rotaton-nvarance of shape contexts The man contrbuton s: The rotaton nvarance s acheved at very lttle cost because of the fast computaton of FFT The theoretcal proof of the nvarance of the proposed descrptor s provded The experments ntally evaluated the proposed FFT- RISC shape descrptor Yet, there are a lot of works to be done to refne the proposed scheme, whch wll be our future endeavor Acknowledgements: Ths work s supported by Natural Scence Foundaton of Chna under grant , Chna/Ireland Scence and Technology Research Collaboraton Fund under grant CI , and Natonal Basc Research Program of Chna under grant 006CB References Fg 5: Pont set C (Pont set A plus 100 addtonal nosy ponts [1] S Beloge, J Malk, J Puzcha, Shape Matchng and Obect Recognton usng Shape Contexts IEEE Trans PAMI, 00, vol 4, pp [] S Yang: Symbol Recognton va Statstcal Integraton of Pxel-Level Constrant Hstograms: A New Descrptor, IEEE Trans PAMI, 005, Vol 7, No, pp [3] W-P Cho, K-M Lam, and W-C Su: A Robust Lnefeature-based Hausdorff Dstance for Shape Matchng, PCM 001, LNCS 195, 001, pp Fg 6: Pont set C and ts affne 579