Basic Image Features (BIFs) arising from approximate Symmetry Type

Transcription

1 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: Bsi Imge Fetures (BIFs) rising from pproximte Symmetry Type Lewis D. Griffin 1, M Lillholm 1, M Crosier 1, vn Snde (1) Computer Siene, University College London, London WC1E 6BT, UK. () Biomedil Engineering, Eindhoven University of Tehnology, the Netherlnds. l.griffin@s.ul..uk Abstrt. We onsider detetion of lol imge symmetry using liner filters. We prove simple riterion for determining if filter is sensitive to group of symmetries. We show tht derivtive-of-gussin (DtG) filters re exellent t deteting lol imge symmetry. Building on this, we propose very simple lgorithm tht, bsed on the responses of bnk of six DtG filters, lssifies eh lotion of n imge into one of seven Bsi Imge Fetures (BIFs). This effetively nd effiiently relizes Mrr s proposl for n imge priml sketh. We summrize results on the use of BIFs for texture lssifition, objet tegory detetion, nd pixel lssifition. Keywords: Gussin Derivtives; Hermite Trnsform; Group Theory. 1 Introdution Previous shemes for detetion of imge symmetry re firly omplex [1-6]; requiring, for exmple, omprison of the outputs of filters t multiple positions. Herein we show tht symmetries my be deteted by single liner filters. Building on this we present simple lgorithm tht omputes Mrr-type priml sketh [7] by tegorizing lol imge struture ording to its pproximte symmetry. The pper is orgnized s follows. In setion we present results on imge symmetries. In 3 we show how to test whether liner filter is sensitive to symmetry. In 4 we review imge mesurement with derivtive-of-gussin (DtG) filters. In 5 we onsider the symmetry-sensitivity of these DtG filters. In 6 we show how this sensitivity gives rise to system of Bsi Imge Fetures (BIFs). In 7 we summrize results on using BIFs for texture tegoriztion, objet tegory detetion nd pixel lssifition. In 8 we onlude. Setions -5 re distilltion of work published, in press nd under review in fuller form elsewhere [8-14]; 6 is new; prts of 7 hve been presented or re under review in fuller form elsewhere [9, 11].

2 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: Imge Symmetries Symmetry of struture (X) is lwys reltive to some lss of dmissible trnsformtions. A struture is sid to hve symmetry when non-trivil group of dmissible trnsformtions, known s the utomorphism group, eh leve it Aut X : t t X X. indistinguishble from the originl. This is denoted by Considering imges, n obvious lss of trnsformtions re the sptil isometries; nd the possible symmetries, reltive to this lss, hve long been tlogued [15-17]. A broder lss of trnsformtions, where eh sptil isometry is ombined with permuttion of finite set of imge olour vlues, hs lso been onsidered. These llow the symmetries of, for exmple, Esher s Reptiles to be expressed [18]. The gmut of possible olour symmetries hs been fully determined [19, 0]. We propose tht the lss of imge isometries, defined s sptil isometry ombined with n intensity isometry, is pproprite for imges. We write n imge g i, s, where i : is n intensity isometry, nd s : is isometry s sptil isometry. Suh n imge isometry is pplied to n imge ording to g I i Is _ i Is_ I :. Choosing lss of trnsformtions is tntmount to hoosing geometry [1], nd the geometry tht orresponds to the lss of imge isometries hs previously been onsidered for imges [] nd muh erlier, bstrtly, s one of lrger lss of possible geometries [3]. We hve employed method for determining the possible utomorphism groups of imges, reltive to the lss of imge isometries. The method relies on two results. First, tht the projetion of group of imge isometries onto their sptil or intensity omponents in both ses mkes group. Seond, tht (exept for speil se) the intensity projetion group must be isomorphi to ftor group of the sptil projetion group [8]. Using the method, we hve determined the possible utomorphism groups of -D imges, exept for ses tht ontin disrete periodi trnsltions. A summry of these possible symmetries, together with our nottionl system is shown in fig. 1. The symmetries inlude: fmilir ones, suh s refletionl (,1 ), reflet-nd-negte ( 6,1 ), nd Yin-Yng type ( 7, ); simple but often ignored ones, suh s vrition in one diretion only ( 3 ); simple but novel, suh s ontinuous trnslte-nd-inrement in one diretion, plus line of refletion prllel to tht diretion ( 11 ); nd some wholly novel, suh s ontinuous trnslte-nd-inrement in one diretion, plus ontinuous line of entres of Yin-Yng type symmetry ( 1 ). 3 Sensitivity of Liner Filters to Symmetries Detetion of symmetry seems to require multiple mesurements, but this is inorret. Consider +1/-1 filter, suh s used in finite-differene shemes. When positioned so tht it strddles puttive line of refletion, neessry riterion for the symmetry is tht the filter gives 0 response. We generlizes this: filter F is sensitive to symmetry K if it gives the sme response to ll imges tht hve the

3 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: symmetry (i.e. f AutI K F I f ). This definition is imprtil beuse it requires ssessment ross ll imges. However, we hve found neessry nd suffiient test tht requires only single integrl to be omputed. We present this below in Theorem 1, fter introduing some nottion.

4 6,3, 6,5, Out[81]= 1,3, 1,5, Out[90]= 6,4, 6,6, Out[8]= 1,4, 1,6, Out[85]= Out[80]=,4,,6,,3,,5, Out[9]= 8,4, 8,6, 7,4, 7,6, Fig. 1. The group/subgroup lttie of the possible imge symmetries, exluding those with disrete periodi trnsltion. Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: Out[93]= Out[94]= Out[96]= 6, 6,1 Out[89]= Out[86]= 5 Out[84]= 1, Out[97]= Out[83]= Out[79]= Out[99]= onst Out[9]= 4 9, Out[95]= 8, Out[101]= Out[87]= Out[91]=,1 3 7, 0 Out[103]= 11 Out[100]= slope Out[10]= 10 1

5 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: We use n inner produt nottion ( F I : F x I x ) to denote the mesurement of n imge opertor F K i, sk x I : by filter F : ; nd we define n : i s F whih, roughly speking, smers filter by group. Theorem - Symmetry-Sensitivity Test for Filters F is sensitive to K if nd only if K FF 0 Proof A forml proof will be published elsewhere [14]. Intuitively the truth of the theorem n be understood s follows. The signl tht filter sees best is opy of itself. Of ll the symmetri signls, symmetrised version of the filter should be the most esily seen. If the filter nnot see symmetrised version of itself, then it is insensitive to the symmetry. 4 Gussin Derivtive Filters Gussin Derivtive (DtG) filters re defined in 1-D by x n n d 1 x n n 1 n :, : G x e G x G x H G x dx where H n is the nth Hermite polynomil; nd in -D by m n m n, :, G x y G x G y. They re used s generl-purpose method to probe n imge lotion (whih for simpliity we ssume is t the origin 0 ) by omputtion of inner produts mn mn, j G I. Typilly, one mesures with fmily of DtG filters up to some order e.g. the nd order fmily mn, G 0 m n. Sle-normlized filter responses pq : p q jpq mke lter equtions simpler. The suitbility of DtGs s the front-end of n unommitted omputtionl vision system rises from the symmetries tht individul filters nd fmilies possess [4]. First mongst these is sle symmetry, whih mnifests s hnge of size, but not of shpe, when DtG is resled by blurring with Gussin kernel. Seond is tht the liner spn of fmily of DtGs is rottionlly symmetri. The responses of bnk of DtG filters entngle intrinsi nd extrinsi spets of imge struture. For exmple, n in-plne rottion of the imge bout the mesurement point uses the DtG responses to hnge. A representtion tht disentngles these spets for mesurement up to nd order hs been developed [13]. The representtion works by ftoring out of the 6-D nd order DtG response spe the hnges due to the group of imge isometries tht fix the mesurement point nd do

6 Out[81]= Out[80]= Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: not invert the intensity xis whih we denote D A = imge pth nd order DtG fmily point in 6-D jet spe 6-D jet spe D A The group of entred rottions nd refletions, nd positive ffine intensity re-slings = The nd order lol-imge-struture orbifold Fig.. The top prt illustrtes shemtilly the probing of n imge pth by bnk of DtG filters resulting in point in jet spe; the bottom, the ftoring of the jet spe by group of trnsformtions resulting in the lol-imge-struture orbifold. The result is n orbifold type of mnifold with boundries, reses nd orners llowed onsisting of 3-D nd 0-D omponent (figure ). The intrinsi spet of 6-tuple of filter responses orresponds to prtiulr lotion in the orbifold, nd is invrint to rotting the imge bout the mesurement point, refleting it in line through the mesurement point, or ffinely sling the intensity. When the responses of the 1 st nd nd order DtG filters re ll zero, the intrinsi spet is the 0-D prt of the orbifold; ll other responses mp to the 3-D omponent. A oordinte system l, b,, 0, 0, for the 3-D omponent is given by [13]: l rtn 4 4, b rtn, rtn 4, The orbifold hs been equipped with metri, indued by one on the filter response spe, whih expressed s line element in the lb-system is 1 ds dl os l db d 5 3osb sin b. The orbifold is intrinsilly urved, but it n be embedded into Euliden 3-spe with only mild distortion.

7 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: Symmetry-sensitivity of DtG Filters Using the elements of setions -4, we n determine whih DtG filters re sensitive to whih symmetries. We onsider not just nonil filter forms (e.g. n x-derivtive) but ny liner ombintion of filters in the nd order filter fmily. This llows us to determine the symmetry-sensitivity of the entire filter fmily, independent of the prtiulr bsis filters used. For exmple, while the x-derivtive filter is sensitive to refletionl symmetry with vertil mirror line through the mesurement point, the x- nd y-derivtives together re sensitive to ny refletionl symmetry in line through the mesurement point. 6, 3, 7, 6, 8,4 6, 6, 7, 1, 8, 6, onst 1, 4, 1, 3,,3 5 6, 8,4 4 7, 7,4 8, 9 1 slope, 1, 11 3 g 4 g 9,1, 8, 5 g g 1 6,1 6, 8, SS is the exterior only 3 0 SS is the entire volume g g 1,,1 g g 6,1 7, Fig 3. The sensitivity-submnifolds (SS) of different symmetry types re shown in red. The different possible SS re rrnged in lttie indued by inlusion reltions. The symmetry type lbels orrespond to those used in figure 1. Supersripts indite the sptil reltionship between the symmetry nd the origin: indites origin-entred rottion; n + tht the origin is ontined in line of refletion, but is not entre of rottion; similrly for - nd nti-refletions; g indites generl position, neither entred nor ligned. All symmetries lbelled in box hve the indited SS; those on the left re miniml. g 8, The filter fmily sensitivities n be projeted into the orbifold to determine where the intrinsi omponent of the jet responses must lie whenever the imge hs ny of

8 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: lss of symmetries equivlent by onjugtion with n element of D A 1 0. We ll the restrited set of possible responses the sensitivity-submnifold (SS). For exmple the SS is the orbifold exterior ( 0 ) for refletionl symmetry in mirror through the mesurement point. The results re summrized in fig 3. 6 Symmetry-bsed Bsi Imge Fetures (BIFs) We hve used the symmetry sensitivities of the DtG filters s strting point in defining set of Bsi Imge Fetures (BIFs) tht relize Mrr s ide of priml sketh of imge struture, in omputtionlly simple sheme. We do not lim tht the sheme is derived s rigorously s the results on symmetry sensitivity. Our sheme works by onsidering the orbifold projetion of jets, nd lssifying them ording to the SS tht they re losest to i.e. we define Voronoi ell prtitioning of the orbifold with the SS s ell entres. We find tht this works best when only seven 0-D SS (the first nd seond rows of figure 3) re used, though we nnot justify this beyond tht it produes nie results. The resulting orbifold prtitioning is shown in the top-left of figure 4. Fig. 4. Top left: the prtitioning of the orbifold into BIF tegories. Bottom left: BIFs lulted ross rnge of sles for simple imge of figure 8 ; in eh ube sle inreses right-to-left. Lower ubes setioned for visulistion. Right: n exmple omplex greysle imge, with BIFs lulted t one prtiulr sle.

9 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: The orbifold distne to the six of these SS tht lie in the 3-D omponent of the orbifold re simple to ompute; for exmple, the distne to the 7, SS is tn. To find whih distne is shortest it is omputtionlly equivlent but simpler to find whih of six quntities is mximum. The distne to the seventh SS, whih orresponds to the origin of jet spe where ll the 1 st nd nd derivtive filters hve zero response, is not well-defined. We inorporte it into our sheme by using multiple of the 0 th order jet response. The full resulting sheme for omputing BIFs is s follows. i) ompute sle-normlized DtG filter responses s desribed in setion ii) ompute : nd : iii) lssify ording to whih is the lrgest of M.,,,,,,. In our sheme the only free prmeters, tht hve to be tuned to the pplition re the filter sle nd whih ontrols the mount of imge lssified s flt; setting of 0.05 is n effetive defult. For disply purposes we find the following olour sheme effetive: if.00 is the lrgest of M then olour the pixel pink; if is lrgest olour it grey; then blk, white, blue, yellow nd green. 7 Exmple pplitions using BIFs We summrize results on using BIFs for texture, objet nd pixel lssifition. 7.1 Texture lssifition Textures re often lssified bsed on representtion of them by histogrm over texton vobulry [5-9]. Textons re tegoril pth lssifitions [5, 30]. To define the texton vobulry, spe of pth desriptions is typilly Voronoi prtitioned into on-the-order-of 1000 texton tegories, usully round entres found by k-mens lustering of the responses from mny imges. Textures re then lssified by nerest-neighbour mthing of histogrms. We hve investigted the lssifition performne of n pproh in whih imges re lbelled using sptil omplexes of BIFs insted of Voronoi ells in lol desription vetor spe. Our pproh is (i) simpler beuse we hve eliminted the lustering step needed to produe ditionry of fetures, nd (ii) fster beuse we ssign imge pthes to histogrm bins without hving to use high-dimensionl nerest-neighbour omputtion. We ll the sptil omplexes of BIFs tht we use nlogously to textons, Bsi Imge Ptterns (BIPs). The type of BIP tht we hve found effetive for texture

10 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: desription is sle-templte of the BIFs t the sme lotion but t four, otve-seprted sles. Unlike sptil-templte BIPs, these sle-templtes retin the rottion invrine of BIFs, whih hs been shown [30] to be dvntgeous in texture lssifition tsks. For textures, we do not use the pink/flt BIF tegory, so four sles produes 6 4 =196 bin histogrm representtion, whih seems to pture the right trde-off between speifiity nd generlity (see figure 5) Fig. 5. Left: An imge from the CUReT 'polyester' texture lss. Centre: BIFs omputed t four otve-seprted sles, stked to form n rry of 'olumn-bips'. Right: Ourrene histogrm of olumn-bips from every position in the imge" Our method hs been tested on the CUReT texture dtset [31]. As reported in [9], the simple olumn-bip representtion nd nerest-neighbour mthing using the Bhtthryy distne orretly lssifies 98.±0.1% of the remining 49 imges per lss, whih is t lest s good s other methods using nerest-neighbour lssifiers. Extending this method by using multi-sle histogrm omprison [9] results in n improvement to 98.6±0.1% on CUReT, whih is omprble to methods [7] using SVMs for lssifition; nd produes wht re, to the best of our knowledge, the best reported results [9] on the more hllenging UIUCTex [3] nd KTH-TIPS [7] dtsets, whih inlude vritions in sle. 7. Objet Ctegoriztion Texton pprohes hve lso been shown to be useful for objet tegoriztion [8, 33]. Similr to texture, the stndrd pproh is to prtition pth desriptor spe, suh s tht used by SIFT [34, 35], into on-the-order-of 1000 tegories (visul words) nd then to desribe eh imge to be nlyzed by wht visul words it ontins, nd to use mhine lerning tehniques to determine lssifier tht n predit the tegory of objet bsed on suh desriptions. We hve onduted preliminry experiments to ssess whether visul words built from BIFs ould be used rther thn SIFT-spe tegories. As with texture this would be simpler nd fster. For our initil experiments, we hve lbelled pixels ording to their BIF type nd, inspired by SIFT, with n orienttion, quntized t the 4 level. The orienttion depends on the BIF type: grey BIFs hve one of eight possible orienttions bsed on 1 st order struture; yellow, green nd blue BIFs hve

11 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: one of four possible orienttions bsed on nd order struture; blk, white nd pink BIFs re unoriented. Thus we hve twenty-three possible orienttion-ugmented BIF (obif) lbels. obifs re nturl nd order generliztion of the grdient orienttion lphbet typilly used in SIFT [34, 35]. See figure 6 for n exmple imge nd lulted obifs. Out[70]= Out[17]= Fig. 6. The top row (left) shows n imge from the PASCAL hllenge, lbeled with diretion-ugmented BIFs t right. On the bottom re shown the 44 templte BIPs whose ourrene in n imge most informtively signls the presene of r. We hve tested three different types of visul word, whih when built from BIFs or obifs we ll Bsi Imge Ptterns (BIPs); two bsed on geometril prtitioning of pth spe nd one bsed on more stndrd dt-driven quntiztion. Eh BIP system hs been used with simple un-optimised of-the-shelf lssifiers nd pplied to the 0-lss PASCAL VOC 008 [33] objet reognition hllenge dtset. Our sore in figure 7 is bsed on lte fusion of the three shemes nd is mid-field: bove other first-time entrnts nd below well-optimised vetern entries. Using the PASCAL VOC 008 [33] dtset, exmples of 4x4 templte BIPs whose presene in imges is pproximtely independent, nd whih re mximlly informtive for the r tegory re shown in figure 6.

12 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: SurreyUv A SRKDA Uv A TreeSFS LEAR shotgun Uv A FullSFS Uv A Soft5ColorSift LEAR flt XRCE TKK ALL SFBS TKK MAXVAL BerlinFIRSTNikon UCL ECPLIAMA CASIA LinSVM INRIASly CMA CASIA NonLinSVM INRIASly MEVO FIRST SCST FIRST SC1C CASIA NeurlNet Fig. 7. Results for the PASCAL VOC 008 hllenge. Eh br in the hrt is hllenge entry - our result is highlighted. 7.3 Pixel Clssifition Mny imge problems involve inferring one of smll lss of lbels for eh pixel of n imge. For omplex imges with unpreditble globl struture, most pprohes blne the likelihood of the lbels, given the lol imge struture, nd the likelihood of the lol rrngement of inferred lbels. In both ses the likelihoods re omputed on the bsis of sttistis lernt from groundtruth-lbelled trining dt. We hve experimented with the use of BIFs in the omputtion of lbel likelihoods given the imge i.e. ignoring the likelihood of rrngements of inferred lbels. For our experiments we hve used -D Eletron Mirosopy imges of neuronl grey mtter tissue, stined to enhne neuronl membrnes. We trined on four imges with hnd-drwn groundtruth dt, inditing the position of membrnes, nd evluted on further four imges. We use k-nerest Neighbour (k-nn) pproh to lssifition. NN lssifition strts by ompiling list of desriptors of ll the pthes in the trining dt, together with the groundtruth lbel of the pixel in the pth entre. The lssifier is used by extrting pth round eh pixel to be lssified, forming desription of it, ompring the desription to eh the ompiled desriptions, finding the k whih re most similr, nd ssigning the pixel being nlyzed with the lbel ssoited with the mjority of the k. We evluted bseline solution bsed on pixel vlues. The distne between two pth desriptions is simply the Euliden distne between their blurred pixel vlues, minimized over llowing one pth to be rotted or refleted into eight onfigurtions. We jointly optimize blur, pth size nd k. The best settings tht we find re: no blur, 77 pthes, nd k=14. At these settings membrne-lbelled pixels overlp (intersetion divided by union) with the groundtruth by 48%. Our solution uses pth of BIF lbels s pth desriptor. The distne between two desriptors is simply the number of pixels where the lbel does not gree. As in

13 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: the bseline, we minimize the distne over one of the pthes being rotted or refleted. We jointly optimize the sle ( ) t whih the BIFs re omputed, the prmeter whih ontrols the mount of the flt BIF lss, pth size nd k. The best settings tht we find re 1., 0.15, 99 pthes, nd k=10. At these settings we hieve n overlp of 55%. See figure 8. imge groundtruth greylevel-bsed lssifition BIF-bsed lssifition BIFs Fig. 8. Typil results of our pixel-lssifition system. So, using BIF- rther thn greylevel-desription rises the sore from 48% to 55%. Computtion is lso fster beuse the knn lookup domintes the ost of omputing pth desriptions, nd with BIFs the distnes tht need to be omputed re of Hmming rther thn Euliden type. 8 Conlusions We hve derived sheme for lssifying imge struture into one of seven BIF types bsed on the outputs of bnk of six DtG filters. Applied to n entire imge, the output relizes Mrr s notion of n imge priml sketh. Presented results show tht BIF desription is simple, fst nd effetive for texture, objet nd pixel lssifition. The BIF system ws derived by onsidering the sensitivity of DtG filters to imge symmetry. Although the finl lgorithm is plesingly simple, there re some wek points in the derivtion of the BIFs from symmetry sensitivities. Speifilly, why re only the 0-D SS onsidered, how extly does orbifold distne orrespond to degree of filure of symmetry, why should lest-pproximte lol symmetry be n effetive feture lbel? We hope tht the foundtion of symmetry-sensitivity of DtGs n eventully nswer ll of these questions in sheme where rbitrry hoie hs been eliminted. Suh sheme will be extendble to higher-order filter fmilies (where ppel to visul evidene nd pst prtie re less effetive), for whih riher lphbet of feture lbels is to be expeted. We predit tht suh riher lphbet will give more effetive solutions in the pplition res tht we hve reviewed. Referenes 1. Liu, Y.X., R.T. Collins, nd Y.H. Tsin, A omputtionl model for periodi pttern pereption bsed on frieze nd wllpper groups. IEEE Trnstions on

14 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: Pttern Anlysis nd Mhine Intelligene, (3): p Sognmillo, R., et l., A feture-bsed model of symmetry detetion. Proeedings of the Royl Soiety of London Series B-Biologil Sienes, (155): p Mellor, M. nd M. Brdy, A new tehnique for lol symmetry estimtion, in Pro. Sle Spe & PDE Methods in Comp Vis. LNCS vol p Bonneh, Y., D. Reisfeld, nd Y. Yeshurun, Quntifition of lol symmetry - pplition to texture-disrimintion. Sptil Vision, (4): p Mnini, S., S.L. Slly, nd R. Gurnsey, Detetion of symmetry nd ntisymmetry. Vision Reserh, (16): p Bylis, G.C. nd. Driver, Pereption of symmetry nd repetition within nd ross visul shpes: Prt-desriptions nd objet-bsed ttention. Visul Cognition, (): p Mrr, D., Vision. 198, New York: W H Freemn & o. 8. Griffin, L.D., Symmetries of 1-D Imges. ournl of Mthemtil Imging nd Vision, (-3): p Crosier, M. nd L.D. Griffin, Texture lssifition with ditionry of bsi imge fetures, in CVPR ' , IEEE. 10. Lillholm, M. nd L.D. Griffin, Sttistis nd tegory systems for the shpe index desriptor of lol imge. Imge nd Vision Computing, 008. in press. 11. Lillholm, M. nd L.D. Griffin, Novel imge feture lphbets for objet reognition, in ICPR ' Griffin, L.D., Symmetries of D imges: ses without periodi trnsltion. ournl of Mthemtil Imging nd Vision, in press. 13. Griffin, L.D., The nd order lol-imge-struture solid. IEEE Trnstions on Pttern Anlysis nd Mhine Intelligene, (8): p Griffin, L.D. nd M. Lillholm, Symmetry-sensitivity of derivtive of gussin filters. IEEE Trnstions on Pttern Anlysis nd Mhine Intelligene, in press. 15. Bieberbh, L., Über die bewegungsgruppen der euklidishen rume I. Mthemtishe Annlen, : p Conwy,.H., et l., On three-dimensionl spe groups. Contributions to Algebr nd Geometry, (): p Grünbum, B. nd G.C. Shephrd, Tilings nd Ptterns. 1987, New York: WH Freemn & o. 18. Shttshneider, D., MC Esher. Visions of Symmetry. 1990: Plenum Press. 19. Holser, W.T., Clssifition of symmetry groups. At Crystllogrphi, : p Loeb, A.A., Color nd Symmetry. 1978: Robert E. Krieger. 1. Klein, F., A omprtive review of reent reserhes in geometry (trns. by MW Hskell). Bulletin of the New York Mthemtil Soiety, 189. : p Koenderink,.. nd A.. vn Doorn. Imge proessing done right. in ECCV Copenhgen: Springer. 3. Cyley, A., Sixth memoir upon the quntis. Philosophil Trnstions of the Royl Soiety, : p Koenderink,.. nd A.. vn Doorn, Generi Neighborhood Opertors. Ieee Trnstions on Pttern Anlysis nd Mhine Intelligene, (6): p. 597-

15 Author s Version: Griffin LD, Lillholm M, Crosier M & vn Snde (009) Bsi Imge Fetures (BIFs) rising from lol symmetry type. In: Pro. SSVM 09. LNCS vol. 5567: Vrm, M. nd A. Zissermn, Texture lssifition: re filter bnks neessry?, in CVPR ' , IEEE. 6. Vrm, M. nd A. Zissermn, A sttistil pproh to texture lssifition from single imges. Interntionl ournl of Computer Vision, (1): p Hymn, E., et l., On the signifigne of rel-world onditions for mteril lssifition, in ECCV ' , Springer. p Zhng,., et l., Lol fetures nd kernels for lssifition of texture nd objet tegories: omprehensive study, in CVPR ' Perronnin, F., et l., Adpted vobulries for generi visul tegoriztion, in ECCV ' p Vrm, M. nd A. Zissermn, Unifying Sttistil Texture Clssifition Frmeworks. Imge nd Vision Computing, 005. In Press. 31. Cul, O.G. nd K.. Dn, Compt representtion of bidiretionl texture funtions, in CVPR ' , IEEE. 3. Lzebnik, S.C., C. Shmid, nd. Pone, A spre texture representtion using lol ffine regions. IEEE Trnstions on Pttern Anlysis nd Mhine Intelligene, (8): p Csurk, G., et l., Visul tegoriztion with bg of keypoints, in ECCV ' p Lowe, D.G., Towrds omputtionl model for objet reognition in IT ortex, in Biologilly Motivted Computer Vision, Proeeding p Lowe, D.G., Distintive imge fetures from sle-invrint keypoints. Interntionl ournl of Computer Vision, (): p