MOTION BLUR ESTIMATION AT CORNERS

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1 Gacomo Boracch and Vncenzo Caglot Dpartmento d Elettronca e Informazone, Poltecnco d Mlano, Va Ponzo, 34/ MILANO boracch@elet.polm.t, caglot@elet.polm.t Keywords: Abstract: Pont Spread Functon Parameter Estmaton, Moton Blur, Space Varyng Blur, Blurred Image Analyss, Optcal Flow. In ths paper we propose a novel algorthm to estmate moton parameters from a sngle blurred mage, explotng geometrcal relatons between mage ntenstes at pxels of a regon that contans a corner. Corners are sgnfcant both for scene and moton understandng snce they permt a unvocal nterpretaton of moton parameters. Moton parameters are estmated locally n mage regons, wthout assumng unform blur on mage so that the algorthm works also wth blur produced by camera rotaton and, more n general, wth space varant blur. 1 INTRODUCTION Moton estmaton s a key problem both n mage processng and computer vson. It s usually performed comparng frames from a vdeo sequence or a par of stll mages. However, n case of fast moton or long exposure mages, moton can be also estmated by analyzng only a sngle blurred mage. Algorthms that consder one mage have to face a more challengng problem, because lttle nformaton s avalable, snce both mage content and blur characterstcs are unknown. In ths paper we ntroduce an algorthm to estmate moton from a sngle blurred mage, explotng moton drecton and length at mage corners. Several algorthms that estmate moton blur from a sngle mage have been proposed; most of them process the mage Fourer transform, assumng unform blur (Cho et al., 1998), (Kawamura et al., 2002). Reklets (Reklets, 1996) estmates locally moton parameters from a blurred mage by defnng an mage tessellaton, and then analyzng Fourer transform of each regon separately. However frequency doman based algorthms are not able to manage blur when moton parameters are varyng through the mage. Moreover, moton estmaton from Fourer doman s partcularly dffcult at mage corners because Fourer coeffcents are mostly nfluenced by the presence of edges than from blur. Our algorthm consders mage regons contanng a blurred corner and estmates moton drecton and length by explotng geometrcal relatons between pxels ntensty values. Besde blnd deconvoluton, moton estmaton from a sngle mage has been addressed for several other purposes. Reklets estmates the optcal flow (Reklets, 1996), Ln determnes vehcle and ball speed (Ln and Chang, 2005), (Ln, 2005) and more recently Klen (Klen and Drummond, 2005) suggested a vsual gyroscope based on estmaton of rotatonal blur. The paper s organzed as follows: n Secton 2 the blur model and the corner model are ntroduced, n Secton 3 we present the algorthm core dea and n Secton 4 we descrbe a robust soluton based on a votng algorthm. Secton 5 descrbes the algorthm detals and presents expermental results. 2 PROBLEM FORMULATION Our goal s to estmate blur drecton and extent at some salent ponts, whch are pxels where t s possble to unvocally nterpret the moton. For example, pxels where the mage s smooth as well as 296

2 Fgure 1: Blurred corner synthetcally generated. pxels along a blurred edge do not allow an unvocal moton nterpretaton: gven a blurred edge or a blurred smooth area there are potentally nfnte scene dsplacements that could have caused the same blur (see regon B n Fgure 1). Corners, nstead, offer a clear nterpretaton of moton drecton and extent and that s the reason why we desgn an algorthm to estmate moton specfcally at corners. We consder mage I modelled as follows I(x) = K ( y+ξ ) (x)+η(x), x = (x 1,x 2 ) (1) where x s a mult ndex representng mage coordnates varyng on a dscrete doman X, y s the orgnal and unknown mage and K s the blur operator. We ntroduce two dfferent sources of whte nose, ξ and η. In our model η represents electronc and quantzaton nose, whle ξ has been ntroduced to attenuate dfferences between corners n real mages and the bnary corner model that we present n the next secton. Therefore ξ plays a crucal role only when a regon contanng a corner s analyzed. 2.1 The Blur Model Here we model the blur operator K on the whole mage, so that we do not need to consder ξ whch s relevant only at mage corners. Our goal s to determne the blur operator K whch can be wrtten as (Bertero and Boccacc, 1998) K ( y ) (x) = X k(x, µ)y(µ)dµ. (2) Usually, K s consdered space nvarant, so that equaton (2) becomes a convoluton wth a kernel v, called pont spread functon (PSF) K ( y ) (x) = X v(x µ)y(µ)dµ = (v y)(x). (3) Ths assumpton s too restrctve for our purpose, because often scene ponts follow dfferent trajectores wth respect to the camera vewpont and are ndeed dfferently blurred. Equaton (3) does not concern, for nstance, scenes where there are objects followng dfferent trajectores, scenes wth a movng target on a stll background and statc scenes captured by a rotatng camera. Fgure 2: Example of moton blur psf wth drecton 30 and 60 degrees respectvely and length 30 pxels. On the other hand, solvng (2) s a dffcult nverse problem: to reduce ts complexty we assume that the blur functonal K s locally approxmated as a shft nvarant blur,.e. x 0 X, U 0 X, x 0 U 0 and a PSF v 0 such that K ( y ) (x) X v 0 (x µ)y(µ)dµ x U 0. (4) Furthermore, we consder only moton blur PSF defned over an 1-D lnear support: they can be wrtten as ( ) v 0 = R (θ) sl (x) θ [0,2π],l N 1/(2l + 1), l x 1 l s l (x 1,x 2 ) = x 2 = 0 0, else where θ and l are moton drecton and length respectvely and R (θ) ( sl ) s functon sl rotated by θ degrees on X. Fgure 2 shows examples of moton blur PSF. 2.2 The Corner Model Our corner model reles on two assumptons. Frstly, we assume that y s a grayscale mage or, equvalently, an mage plane n a color representaton whch s constant at corner pxels and at background pxels. Ths means that gven D X, neghborhood of an mage corner, we have y(d) = {b,c}, where b and c are the mage values for the background and for the corner, respectvely. Moreover, the sets of pxels belongng to the background B = y 1 ({b}) and the set of pxels belongng to the corner C = y 1 ({c}), have to be separated by two straght segments (havng a common endpont). Fgure 3 shows the corner model. Then, let us defne as the corner dsplacement vec- Fgure 3: The Corner Model. 297

3 VISAPP Internatonal Conference on Computer Vson Theory and Applcatons tor: ths vector has the orgn at mage corner and drecton θ and length l equal to drecton and length of the PSF v 0 whch locally approxmates the blur operator. Let γ be the angle between a reference axs and the corner bsectng lne, let α be the corner angle, and θ be the angle between and the reference axs, then θ [γ α/2,γ+α/2] + kπ k N. Fgure 4.a shows a corner dsplacement vector satsfyng ths assumpton, whle Fgure 4.b a corner that does not. a b I I α 2 γ θ Fgure 4: Two possble cases for corner dsplacements, a agrees wth our model whle b does not. 3 PROBLEM SOLUTION In ths secton we derve the core equatons for moton estmaton at a blurred corner that satsfes assumptons of Sectons 2.1 and 2.2. We frst consder nose η only, then we explot how ξ corrupts the proposed soluton. 3.1 Bnary Corners Let us examne an mage regon contanng a bnary corner, lke the one depcted n Fgure 3, and let us assume that nose ξ s null. Let d 1 and d 2 be the frst order dervatve flters w.r.t. x 1 and x 2. The mage gradent s defned as I(x) = [ I1 (x) I 2 (x) α 2 γ θ ] = K ( y ) (x)+ η(x), where I 1 = (I d 1 ) and I 2 = (I d 2 ). If = c b s the mage ntensty dfference between the corner and the background, t follows, as llustrated n Fgure 5, that = K ( y ) (x), x D 0, (5) where D 0 = {x D K ( y ) (x) [0,0] T }. Equaton (5) s undetermnate as we do not know and K ( y ) but only I, whch s corrupted by η. Smlar stuatons can be solved takng nto account, x D 0, several nstances of (5), evaluated at neghborng pxels. Fgure 5: Intensty values of n box A of Fgure 1. We call w a wndow descrbed by ts weght w, n < < n, and we solve the followng system A(x) = [w n,...,w 0,...,w n ] T (6) where A s defned as A(x) = w n I(x n ) T... w 0 I(x) T... w n I(x n ) T. In our experment we choose w as a squared wndow havng gaussan dstrbuted weghts. A soluton of system (6) s gven by = argmn v A(x)v [w n,...,w 0,...,w n ] T 2 (7) whch yelds H = = H 1 (x)a T (x) [w n,...,w 0,...,w n ] (8) w 2 I 1(x ) 2 w 2 I 2(x )I 1 (x ) w 2 I 2(x )I 1 (x ) w 2 I 2(x ) 2. H corresponds to Harrs Matrx (Harrs and Stephens, 1988), whose determnant and trace are used as corner detectors n many feature extracton algorthms, see (Mkolajczyk et al., 2005). If w does not contan any mage corner, H s sngular and consequently the system (8) does not admt a unque soluton. Therefore, when the wndow w ntersects only one blurred edge (lke regon B n Fgure 1), system (8) admts an nfnte number of solutons and the moton parameters can not be estmated. On the contrary, H s nonsngular when w ntersects 298

4 two blurred edges (lke box A n Fgure 1) and n ths case the system (8) can be unvocally solved. The least square soluton (8) performs optmally n case of gaussan whte nose. Here we assume that η s whte nose, wthout specfyng any dstrbuton because η would not be whte anymore. However, n case of nose wth standard devaton sgnfcantly smaller than, equaton (8) represent a suboptmal soluton. 3.2 Nosy Corners The proposed algorthm works when y contans a bnary corner, that takes only two ntensty values. These cartoon world corners are far from beng smlar to corners of real mages. It s reasonable to expect corners to be dstngushable from ther background, but hardly they would be unform. More often ther ntensty values would be varyng, for example, as there are texture or detals. However, snce the observed mage I s blurred, we do not expect a bg dfference between a blurred texture and a blurred whte nose ξ, added on a blurred corner. Let then consder how equaton (5) changes f ξ 0. We have I(x) = K ( y ) (x)+ K ( ξ ) (x), and (5) holds for K ( y ) (x), whle t does not for K ( ξ ) (x). However the blur operator K ( ξ ), whch s locally a convoluton wth a PSF, produces a correlaton of ξ samples along the moton drecton (Ytzhaky and Kopeka, 1996), so that K ( ξ ) (x) 0, (9) whch means that the more blur nduces correlaton among random values of ξ, the more our algorthm wll work wth corners whch are not bnary. 4 ROBUST SOLUTION Although the equaton (9) assures that the proposed algorthm would work for most of pxels, even n presence of nose ξ, we expect that outlers would heavly nfluence the soluton (8), snce t s an l 2 norm mnmzaton (7). Besde pxels where K ( ξ ) (x) 0 there could be several other nose factors that are not consdered n our model but that we should be aware of. For example compressed mages often present artfacts at edges such as alasng and blockng, corners on y are usually smoothed and edges are not perfectly straght lnes. However, f we assume that outlers are a relatvely small percentage of pxels, we can stll obtan a relable soluton usng a robust technque. We do not look for a vector that satsfy the equaton (5) at each pxel or that mnmze the l 2 error norm (7): rather we look for a value of that satsfes a sgnfcant percentage of equatons n system (6), dsregardng how s far from the soluton of the remanng equatons. 4.1 The Votng Approach If we defne, for every pxel, the vector N(x) as N(x) = I(x), (10) I(x) 2 we have that N(x) corresponds to the component along I(x) drecton, x D 0. The endpont of any vector, soluton of (5), les on the straght lne perpendcular to N(x), gong through ts endpont. Then, the locus l x (u) of the possble endponts, compatble wth a gven datum I(x), s a lne (see Fgure 6). As n usual Hough approaches, the 2-D parameter space of endponts s subdvded nto cells of sutable sze (e.g. 1 pxel); a vote s assgned to any cell that contans (at least) a value of satsfyng an nstance of equaton (5). The most voted cells represent values of that satsfy a sgnfcant number of equatons (6). 4.2 Neghborhood Constructon In order to reduce the approxmaton errors due to the dscrete parameter space and to take nto account η, we assgn a full vote (e.g 1) to each parameter pars that solve (5), (the lne of Fgure 6), and a fracton of vote to the neghborng parameter pars. We defne the followng functon [ ( u ) 2 2 ] l(u 1,u 2 ) = exp, (11) 1+k u 1 σ η u 2 N(x) l x (u) u 1 Fgure 6: l(x) set of possble endpont for. 299

5 VISAPP Internatonal Conference on Computer Vson Theory and Applcatons where σ η s η standard devaton and k s a tunng parameter. l has the followng propertes: t s constant and equal to 1 on u 1 axs, (.e. l(u 1,0) = 1), and when evaluated on a vertcal lne, (u 1 = const), t s a gaussan functon havng standard devaton that depends on u 1,.e. l(u 1,u 2 ) = N(0,1+k u 1 σ η )(u 2 ). We select ths functon as a prototype of the vote map, gven I(x), the votes dstrbuted n the parameter space are the values of an opportunely translated and scaled verson of l(u 1,u 2 ). The straght lne of Fgure 6, l x (u), s therefore replaced by functon l rotated by ( π 2 θ) degrees and translated so that ts orgn s n N(x) endpont,.e. l x (u) = R ( π 2 θ)( l ) (u N(x)), (12) where θ s I(x) drecton and R ( π 2 θ) s the rotaton of ( π 2 θ) degrees. In such a way, we gve a full vote to parameter pars whch are exact solutons of (5) and we ncrease the spread of votes as the dstance from N(x) endpont ncreases. Fgure 7(a) shows how votes are dstrbuted n parameter space for a vector N(x). Fgure 7(b) shows parameter space after havng assgned all votes, the arrow ndcates the vector estmated. (a) Fgure 7: (a) Neghborhood l x (u) used to assgn votes n parameter space. Vector represent N(x). (b) Sum of votes n parameters space, the vector drawn s. (b) 5 EXPERIMENTAL RESULT 5.1 Algorthm Detals Gven a wndow contanng a blurred corner, we proceed as follows Defne D 0, the set of consdered pxels as D 0 = {x s.t. I(x) > T }, where T > 0 s a fxed threshold. In such a way we exclude those pxels where mage y s constant but gradent s non zero because of ξ and η. Estmate σ η usng the lnear flterng procedure proposed n (Immerkær, 1996). Estmate as = max(d 0 ) mn(d 0 ) +3 σ η. Votng: x D 0 dstrbute votes n parameter space computng l x (u) and addng them to the prevous votes. The k parameter used n (11) s chosen between [0.02, 0.04]. The soluton of (6),, s the vector havng endpont n the most voted coordnates par. Whenever several parameter pars receve the maxmum vote, ther center of mass s selected as endpont. To speed up the algorthm, we eventually consder gradent values only at even coordnate pars. 5.2 The Experments In order to evaluate our approach we made several experments, both on synthetc and real mages Synthetc Images We generate synthetc mages accordng to (1), usng a bnary corner (lke that of Secton 2.2) takng y constantly equal to 0 at background and equal to 1 at corner pxels and wth η and ξ havng gaussan dstrbuton. Moton parameters have been estmated on several mages wth values of the standard devatons σ η [0,0.02] and σ ξ [0,0.08]. Blur was gven by a convoluton wth a PSF v havng drecton 10 degrees and length 20 pxels n the frst case and 70 degrees and 30 pxels n the second case. Fgure 8 and Fgure 9 show some test mages and Table 1 and Table 2 present algorthm performances n terms of dstance, n pxel unt, between the endponts of the estmated,, and the true dsplacement vector v, expressed as a percentage w.r.t psf length. Comparng the frst rows of Table 1 and Table 2, we notce the correlaton produced by the blur on ξ samples, as expressed n equaton (9). In fact, as the blur extent ncreases, the mpact of ξ s reduced. Table 1: Result on synthetc mages: v has drecton 10 degrees and length 20 pxels, σ η [0,0.02] and σ ξ [0,0.08]. σ η σ ξ % 2.37% 1.67% 3.26% 5.40% % 2.98% 1.67% 4.21% 1.68% % 7.57% 5.40% 3.97% 3.35% 300

6 Table 2: Result on synthetc mages: v has drecton 70 degrees and length 30 pxels, σ η [0,0.02] and σ ξ [0,0.08]. σ η σ ξ % 1.08% 1.95% 2.23% 0.98% % 0.31% 3.99% 1.43% 2.54% % 10.11% 6.55% 7.65% 7.50% Fgure 9: Example of synthetc test mages used, psf was drected 70 degrees and length 30 pxels, n a σ η = 0 and σ ξ = 0.08, whle n b σ η = 0.02 and σ ξ = 0. Fgure 8: Synthetc test mages used psf drected 10 degrees and length 20 pxels, n a σ η = 0 and σ ξ = 0.08, whle n b σ η = 0.02 and σ ξ = Real Images We perform two tests on real mages 1 ; n the frst test we replace y+ξ wth a stll camera pcture, we blur t usng a convoluton wth a PSF and we fnally add gaussan whte nose η. We take house as the orgnal mage and we manually select fve squared wndows of sde 30 pxels at some corners. Fgure 10 shows the orgnal and the blurred house mage (usng psf wth drecton 30 degrees and length 25 pxels) and the analyzed regons. Fgure 11 shows two vectors n pxel coordnates, the estmated (dashed lne) and the vector havng true moton parameters (sold lne), for each selected regon. Table 3 shows dstance between the endponts of the two vectors. Fgure 10: Orgnal and blurred house mage. Blur have drecton 30 degrees and 25 pxels length, regons analyzed are numbered. Table 3: Estmaton error: dstance between endpont and dsplacement vector, expressed n pxels, on each mage regon r. σ η r 1 r 2 r 3 r 4 r , We perform a second experment usng a sequence of camera mages, captured accordng to the followng scheme a stll mage, at the ntal camera poston. a blurred mage, captured whle the camera was movng. a stll mage, at the fnal camera poston. We estmated moton blur at some manually selected corners n the blurred mage and we compare results 1 Further mages and expermental result are avalable at Fgure 11: Dsplacement vectors estmated n selected regons of camera mages. The sold lne s the true dsplacement vector, whle the dotted lne represents the estmated vector. wth the ground truth, gven by matchng corner found by Harrs detector n the mages taken at the ntal and at the fnal camera poston. Clearly, the accuracy obtaned n moton estmaton from a sngle blurred mage s lower than that obtaned wth methods based on two well focused vews. However prelmnary results show good accuracy. For example, moton parameters estmated n regon r 2 are close to the ground truth, even f the corner s consderably smooth, as t s taken from a common swvel char. As Fgure 13.2 shows, the votes n parameter space are more spread around the soluton than n Fgure 13.1, where the corner s close to the model of Secton 2.2. Table 4 shows result usng the same crtera of Table 3. Results are less accurate than n prevous experments 301

7 VISAPP Internatonal Conference on Computer Vson Theory and Applcatons because accordng to expermental settngs, moton PSF could be not perfectly straght or not perfectly unform, because of camera movement. Ths affects algorthm performances snce t approxmates moton blur to a vectoral PSF. Table 4: Estmaton error expressed n pxel unt on each mage regon r. r 1 r 2 r 3 r 4 r corners n real mages. Ths s mostly due to background and corner non unformty because of shadows, occlusons or because the orgnal mage tself shows sgnfcant ntensty varatons. We are actually nvestgatng a procedure to automatcally detect blurred corners n a gven mage and to adaptvely select mage regons around them. In ths paper we use squared regons but there are no restrctons on ther shape, whch could be adaptvely selected to exclude background elements whch would be consdered n D 0. We beleve that estmatng blur on adaptvely selected regons could sgnfcantly mprove the algorthm performance on real mages. We are also nvestgatng an extenson of our algorthm to deal wth corners whch are movng lke Fgure 4 b or at least to dscern whch corners satsfy our mage model. Fnally, we are lookng for a crtera to estmate the goodness of an estmate, as up to now, we consder the value of the maxmum voted parameter pars. REFERENCES Fgure 12: Dsplacement vectors estmated n camera mages. In each plot, the sold lne ndcates the true dsplacement vector obtaned by matchng corners of pctures at ntal and fnal camera poston. Dotted lne represents the estmated dsplacement vector. Fgure 13: Fgure a Orgnal corner n mage b blurred corner, c set D 0 of consdered pxels and d votes n the space parameter. 6 ONGOING WORK AND CONCLUDING REMARKS Bertero, M. and Boccacc, P. (1998). Introducton to Inverse Problems n Imagng. Insttute of Physcs Publshng. Cho, J. W., Kang, M. G., and Park, K. T. (1998). An algorthm to extract camera-shakng degree and nose varance n the peak-trace doman. Harrs, C. and Stephens, M. (1988). A combned corner and edge detector. Immerkær, J. (1996). Fast nose varance estmaton. Kawamura, S., Kondo, K., Konsh, Y., and Ishgak, H. (2002). Estmaton of moton usng moton blur for trackng vson system. Klen, G. and Drummond, T. (2005). A sngle-frame vsual gyroscope. Ln, H.-Y. (2005). Vehcle speed detecton and dentfcaton from a sngle moton blurred mage. Ln, H.-Y. and Chang, C.-H. (2005). Automatc speed measurements of sphercal objects usng an off-the-shelf dgtal camera. Mkolajczyk, K., Tuytelaars, T., Schmd, C., Zsserman, A., Matas, J., Schaffaltzky, F., Kadr, T., and Gool, L. V. (2005). A comparson of affne regon detectors. Reklets, I. (1996). Steerable flters and cepstral analyss for optcal flow calculaton from a sngle blurred mage. Ytzhaky, Y. and Kopeka, N. S. (1996). Identfcaton of blur parameters from moton-blurred mages. Results from the experments, performed both on synthetc and natural mages, show that the mage at blurred corners has been sutably modelled and that the soluton proposed s robust enough to cope wth artfcal nose and to deal wth real mages. However, we notced that there are only a few useful 302

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