Visual Curvature. 1. Introduction. y C. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), June 2007

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1 IEEE onf. on omputer Vson and Pattern Recognton (VPR June 7 Vsual urvature HaRong Lu, Longn Jan Lateck, WenYu Lu, Xang Ba HuaZhong Unversty of Scence and Technology, P.R. hna Temple Unversty, US lhrbss@gmal.com, lateck@temple.edu, luwy@hust.edu.cn, xang.ba@gmal.com bstract In ths paper, we propose a new defnton of curvature, called vsual curvature. It s based on statstcs of the extreme ponts of the heght functons computed over all drectons. By gradually gnorng relatvely small heghts, a sngle parameter mult-scale curvature s obtaned. It does not modfy the orgnal contour and the scale parameter has an obvous geometrc meanng. The theoretcal propertes and the experments presented demonstrate that mult-scale vsual curvature s stable, even n the presence of sgnfcant nose. In partcular, t can deal wth contours wth sgnfcant gaps. We also show a relaton between mult-scale vsual curvature and convexty of smple closed curves. To our best knowledge, the proposed defnton of vsual curvature s the frst ever that apples to regular curves as defned n dfferental geometry as well as to turn angles of polygonal curves. Moreover, t yelds stable curvature estmates of curves n dgtal mages even under sever dstortons.. Introducton urvatures of curves are the key to detect the salent ponts and to compute the shape descrptors. Mathematcally, curvature of a pont v s defned as followng: Δθ () () v K v = lm () where θ( s the tangental angle of the pont v and S s the arc length. When appled n dgtal mages, three problems arse: () The dgtal mages are usually dstorted by nose. Fg. (a) can be regarded as a pentagram heavly dstorted by nose; Fg. s the pentagram wthout nose. For the vsual percepton, pont s not mportant, because t should be flat there. However, the curvature computed by formula () can be very hgh. (a) Fgure. Pentagram () The mages may have dfferent level of detals. If Fg. (a) s regarded as an mage that looks lke a pentagram n global, the curvature of pont should be low n the large scale; at the same tme, because there s a very sharp turn n small scale, the curvature should be hgh. Obvously, formula () s hard to compute the curvature n dfferent scales. (3) Due to dgtalzaton, the contours of the mages are all star-lke, such as Fg. 3. In such case, formula () cannot be drectly appled. y o S θ M ' M Δθ θ + Δθ Fgure. urvature of the curve x Fgure 3. Star-lke contour From the pont of vsual percepton, the curvature estmated n certan level must get rd of the nfluence of the convex and concave parts n smaller levels. The contour can be parameterzed by arc length: () s = ( x() s, y() s ) ()

2 IEEE onf. on omputer Vson and Pattern Recognton (VPR June 7 We call x(s) the heght functon n o drecton and y(s) the heght functon n 9 o drecton. The ntuton s that x(s) measures the dstance to y-axs n o drecton. Rotate the coordnate system by angle antclockwse, the new x(s) becomes the heght functon of the contour n drecton, whch we denote H. Snce the heght functon s defned as dstances to rotated y-axs, and the drecton of y-axs does not matter, we restrct [, π). By rotatng the coordnate system by angle = π, =, K,, we obtan a seres of heght functons. H B (a) B Fgure 4. Heght functons n o, 45 o, 9 o, 45 o drectons Fg. 4(b (c (d (e) shows the heght functons of the contour n (a) n o, 45 o, 9 o, 45 o drectons, respectvely. Every heght functon reflects partal nformaton of the contour. The curvature s related to the local extreme ponts of the heght functons: In more drectons the pont s the extreme ponts, the sharper the contour s at the pont,.e. pont, the hgher s the curvature of the pont. The man dea of ths paper s to defne the curvature at a contour pont v by countng the number of drectons n whch v s an extremum of the heght functon. Obvously, all of the extreme ponts are not of the same mportance. ose may perturb the curve and cause small extreme ponts n the heght functons. However, a pont on a small concave or convex part can not become an obvous heght n any heght functons, whle a pont on a large concave or convex part wll be an obvous heght n some heght functons. For example, n Fg. 4(d and B are the very mportant extreme ponts and s not so mportant, but s a very mportant mnmum pont n Fg. 4. When the number of heght functons s suffcently large, no mportant ponts are gnored, and mportant hgh curvature ponts are detected. In ths paper, we obtan mult-scale curvature by gnorng small heghts n the heght functons. The new defnton for curvature, called vsual curvature, s based on statstcs of the extreme ponts of the heght functons computed over all drectons. Moreover, by gradually gnorng relatvely small heghts, mult-scale curvature s constructed. The mult-scale vsual curvature has the followng propertes: () It s sutable for every planar curve. When the number of the heght functons approaches nfnte, on the regular curve, ts lmt s standard curvature and on the polygonal curve, t s dentcal to turn angle. B () It forms a sngle parameter scale space. urvature s obtaned by gnorng small heghts, not by smoothng. Hence t does not modfy the orgnal curve. The related lteratures are revewed n Secton. In Secton 3, the vsual curvature s defned and ts relaton to standard curvature and turn angle has been proved. In Secton 4, a scale measure of extreme pont s defned n the pont of absolute extreme. In Secton 5, some propertes of mult-scale vsual curvature are descrbed and ther sgnfcances are dscussed. In Secton 6, mplementaton detals are analyzed and the expermental results are demonstrated. In Secton 7, we descrbe some applcatons of the vsual curvature. In Secton 8, we draw a concluson of ths paper.. Lterature Revew urvature estmaton n dgtal mages s known to be very susceptble to nose on the contour, thus t s very hard to estmate t robustly. Snce large number of methods has already been proposed for estmatng the curvature of contour, t s beyond the scope of ths paper to lst all of them. Therefore, we menton only a few begnnng wth a very nfluental method form the early days of computer vson [] through methods n [,, 3, 4, 5, 6]. Those approaches can be classfed nto three groups, accordng to defnton of curvature they are usng: tangent drecton, osculatng crcle, dervaton. Most methods use a sldng wndow, thus they n essence estmate the curvature locally. The sze of the sldng wndow s usually hard to choose and when nose s large, smply ncreasng the sze of sldng wndow usually does not work. t the same tme, these methods are all under the assumpton that there s a unque curvature at each pont whch s obvously true n pure mathematcal vew. However, n the context of computer vson, dependng on partcular goals, the curvature of a pont may take on dfferng values, e.g., dependng on whether a gven pont s regarded as nose or sgnal pont. Before descrbng the exstng mult-scale curvature technques, we characterze the desrable propertes of the curvature that s useful n computer vson: () t should be mult-scale and reflect the curvature nformaton of the contour n dfferent scale; the geometrc meanng of the scale factor should be as clearly as possble whch facltates the selecton of scale n the applcaton; () t should have proper precson; (3) t should be stable under nose; (4) t should be nvarant under rotatons and translatons. (5) t should be sutable for both smooth curves and polygonal arcs. The man goal of ths paper s to propose a new curvature defnton whch wll acheve all the above propertes. To our best knowledge, no exstng mult-scale curvature has all these propertes. Our proposed mult-scale curvature s easy to mplement, and can be computed effcently.

3 IEEE onf. on omputer Vson and Pattern Recognton (VPR June 7 The followng s a bref revew of exstng mult-scale curvature technques. In computer vson, mult-scale curvature s usually related to mult-scale shape representaton technques. Mokhtaran and Mackworth [7] proposed a mult-scale, curvature based shape representaton technque by convolvng the contour wth a Gaussan kernel. They demonstrated many of ts appealng propertes. However, ths method modfes the orgnal curve. t the same tme, the geometrc meanng of ts scale factor whch s n fact a parameter of Gaussan kernel s not obvous. smlar method [8], whch s proposed by Yu-Png Wang, convolves the contour wth a dlated-splne kernel. Snce just kernel functon s altered, they share the same problems. Lateck and Rosenfeld [9] proposed a class of planar arcs and curves whch s general enough to descrbe (parts of) the boundares of planar real objects. They analyzed the propertes of these arcs and ruled out pathologcal arcs, thus smplfy the shape representaton problem. popular way of shape representaton n dgtal grd s curvature based polygonal approxmaton [,, 3, 4, 5]. In these methods, the orgnal contour s approxmated by smplfed polygon. Obvously, n polygonal arcs, a natural measure of curvature nformaton s turn angle. The problem wth standard curvature s that t s defned on smooth curve and can not be appled to polygonal arcs drectly. Thus, they need complcated estmaton procedure to calculate curvature. To summarze, although there are many methods to obtan mult-scale curvature, they n essence estmate the curvature from the defnton of standard curvature or ts propertes. Thus, they usually need to smooth the polygonal arcs, ether by curve fttng or by convoluton. Ths results n parameters that are hard to control, such as the sze of sldng wndow, and dsplacement of contour ponts whch s usually not desrable. 3. Vsual urvature s descrbed n Secton, by rotatng the coordnate system, we can obtan a seres of heght functons H, =, =,, π K. Defnton. For a pont v on the curve, suppose S( s ts neghborhood of sze on the curve, the vsual curvature of v s defned as: #[ H ( S( = K = π (3) where #[ H ( S( represents the number of local extreme ponts of the heght functon H n the neghborhood S(. Ths defnton also ponts out how to compute the vsual curvature. For a pont v on the contour, we estmate ts curvature n ts small neghborhood S(. In every heght functon, we fnd ts extreme ponts and count the number of the extreme ponts that are n the neghborhood S(. We sum up all the numbers and calculate the curvature usng formula (3). s we wll report later n the secton on expermental results, keepng S=,.e., S(={v}, yelds the most robust curvature estmate for dgtal contours n real mages. We dd not restrct the sze S of the neghborhood of pont v n formula (3) for theoretcal reasons, n partcular, to formulate Theorem below n ts general form. Ths theorem reveals the relaton between vsual curvature and the standard curvature on the regular curve. It states that when the number of the heght functons s suffcently large, the vsual curvature approaches the standard curvature. Regular curve s a curve whch s dfferentable and the dervatve never vanshes. Theorem. For a pont v on the regular curve, we have K v lm lm K v (4) ( ) ( ) = Proof: Let θ be the tangent angle at pont v. ssume θ. If θ=, we rotate the coordnate system so that t satsfes ths assumpton. Snce the curve s regular, by properly rotatng coordnate system, there exsts a neghborhood S( such that the range of the tangent angle n ths neghborhood s a subset of the half-open nterval [,π devoted by (θ, θ ). See Fg. 5 below. y o θ S() v θ v v θ Fgure 5. Relaton between tangental angle and extreme pont If a pont v S( s the extreme pont of the heght functon H, then (θ, θ ) and vce versa. Hence the number of the extreme ponts of all the heght functons n the neghborhood S( s dentcal to the number of drecton angles that belong to the open nterval (θ, θ ). The drecton angle seres π = =, K, of the heght functons s a unform samplng of the half-open nterval [, π). Suppose n =πn/ and m =πm/ are the smallest and largest samplng drecton angles n the open nterval (θ, θ respectvely. Then π(m-n+)/ s an estmaton of θ -θ. We now prove that the lmt of π(m-n+)/ s θ -θ when approaches nfnty. We just need to show that x

4 IEEE onf. on omputer Vson and Pattern Recognton (VPR June 7 (a) m π lm = θ n π lm = θ Both (a) and can be proved n the same way, thus, we just prove (a). Because m =πm/ s the largest angle n the set π = =, K, whch s n nterval (θ, θ then m =πm/ θ and m+ =π(m+)/>θ, lm m θ lm m m+ = π lm = m π lm = lm m = θ Therefore, # [ H ( S( = θ θ lm lm K () v = lm lm π = lm = K() v The theorem below reveals the relaton between vsual curvature and turn angle of polygonal curves. We frst motve ths theorem wth an example. In Fg.6, MO s part of the polygonal curve, the turn angle at O s. t s a lne whose drectonal angle, denoted by β t, s n the nterval (,) and l s a lne whose drectonal angle, denoted by β l, s n the nterval (,π). Obvously, O s an extreme pont of heght functon n the drecton π/+β t whch s perpendcular to t, but t s not an extreme pont of heght functon n the drecton π/+β l whch s perpendcular to l. Thus, n Fg. 6, n the drecton perpendcular to β (, O s an extreme pont and total range of β s. l Fgure 6. Relaton between vsual curvature and turn angle Theorem. For a pont O wth turn angle (O) on a polygonal curve, we have O = lm K O (5) ( ) ( ), Proof: Snce =, we just need to count the number of heght functons n whch O s an extreme pont. Let us assume that there are heght functons and O s an extreme pont of M heght functons. Then πm/ s an estmaton of the range of the angle n whch drecton O s an extreme pont. s llustrated n Fg. 6, such range s. Followng the proof of theorem, we can show: M ( O) = π lm = lm K,( O) y M O x t 4. Scale Measure of Extreme Pont In Def., all extreme ponts are counted, not consderng whether they are mportant or not. Therefore, Theorem also explans why standard curvature s not robust. In fact, n a certan scale, small concave or convex parts should be gnored. By mposng a scale measure for extreme pont, the mult-scale vsual curvature can be defned as follows: Defnton. For a pont v on a curve, suppose S( s ts neghborhood of sze on the curve, the mult-scale vsual curvature of the pont v s defned to be: #[ H ( S( = K = π (6) Where s a scale factor and #[ H ( S( represents the number of the extreme ponts of the heght functon n the neghborhood S( whose scale H measure s not smaller than. In short, the mult-scale vsual curvature s computed by countng the number of relatve mportant extreme ponts. The scale measure can be defned n dfferent ways. Defnton 5 below presents our choce. The ntuton s that n every heght functon, the hgher the peak represented by the extreme pont s, the more mportant the extreme pont s. We begn wth a defnton of a measure that quantfes the heghts of peaks. Defnton 3. The nfluence regon of a local maxmum (mnmum) pont v n a heght functon H, denoted by R (v s ts maxmal neghborhood such that the heght of every pont n ths neghborhood s not hgher (lower) than the heght of the pont v. If curve s open or pont v s not an absolute extremum, R ( s dvded nto two segments R v and the rght by v, we denote the left segment by ( ) segment by R +, see Fg. 7. If curve s closed, R ( may be the whole curve, n whch case R and both represent the whole curve except pont v, n partcular, + R = R = R. Defnton 4. The heght of the peak represented by an extreme pont v n the heght functon H, denoted by r (v s defned as: + r = mn [ r, r ] (7) + + r v max H p H v p R v R + { } ( ) = ( ) ( ) ( ) r = max { H ( p) H p R } r + and r between v and the ponts belongng to R + and respectvely. are the maxmal heght dfferences R,

5 IEEE onf. on omputer Vson and Pattern Recognton (VPR June 7 h H In Fg. 7, v s a local maxmum pont, the curve segment P P whch s n red s ts nfluence regon, the curve segment vp s the left segment of the nfluence regon and curve segment vp s the rght segment of the nfluence regon. Obvously, whether a peak s mportant or not, depends not only on the heght of ths peak, but also on the scale of the contour or the mage. In the defnton below, t s compared to the heght of H, denoted by h, whch s the heght dfference between the absolute maxmum pont and the absolute mnmum pont of H. However, we could defne t n dfferent ways accordng to applcatons. Defnton 5. The scale measure of an extreme pont v n the heght functon H, denoted by (v s the heght of the peak represented by v dvded over the heght of H : r () () v v = (8) h The scale measure of a pont v represents n whch scale n drecton, v can be consdered to be mportant. ccordng to the defnton, (>. Defnton 6. The representatve scale measure of a pont v, denoted by (v s the maxmum of ts scale measures n all the heght functons. For a contour pont, n the scale larger than ts representatve scale measure, ts vsual curvature vanshes and the convex or concave part represented by ths pont s gnored. 5. Propertes of Mult-scale Vsual urvature Ths secton presents a number of mportant results on the mult-scale vsual curvature. It also dscusses the practcal sgnfcance of each of those results. The propertes below are all under the assumpton that the number of the heght functons s suffcently large, thus none of mportant extreme pont s gnored. Theorem 3. s the scale factor ncreases, the mult-scale vsual curvature of a pont s non-ncreasng. Theorem 3 s a natural result of the defnton of mult-scale vsual curvature. It shows that an unmportant contour pont n a certan scale s also unmportant n a hgher scale. v P P r R () v r + () v R + Fgure 7. Influence regon and the heght of peak Theorem 4. Mult-scale vsual curvature s nvarant under rotaton and translaton. If the sze n formula (6) s proportonal to the whole length of the contour, t s also nvarant under unform scalng. Ths nvarance property s very essental snce t make t possble to compute the shape descrptors from mult-scale vsual curvature. Theorem 5. Let be a closed planar curve and let G be the boundary curve of ts convex hull, v s a pont on the curve. Then () v G f and only f (= () v G f and only f (< In the dgtal mages, the contour s n fact a polygon wth fnte vertces {V =,,}, where s the number of the vertces. Let G be the boundary curve of the convex hull of. For a contour segment defned by vertces {V =m,,n} wth V m and V n beng ts two end ponts, we call t a concave segment f all the ponts except the two end ponts on the segment do not belong to G. V 3 V 4 V V 5 V V 8 V 7 V 6 V V V V 9 Fgure 8. The concave segments In Fg. 8, there are two concave segments, V V 8 V 7 V 6 V 5 V 4 and V 8 V 9 V V V. Obvously, by substtutng all concave segments wth the lne segments connectng ther two end ponts, we obtan the convex hull of the polygon. Defnton 7. The scale measure of a concave segment Γ, denoted by (Γ s the maxmum of the representatve scale measure of the ponts whch belong to Γ except the two end ponts. In Fg. 8, the scale measures of the two concave segments are: ( V8 V9V VV ) = max{ ( V9 ( V ( V) } ( V V8V7V6V5V4 ) = max{ ( V8 ( V7 ( V6 ( V5 )} Snce except the two end ponts, the ponts whch belong to Γ do not belong to the convex hull, accordng to Theorem 5, (Γ)<. Defnton 8. Gven a scale threshold, for a closed polygon, deletes all the vertces V where the vsual curvature vanshes and connects the remanng vertces n sequence. The new polygon s called a -scale approxmaton of, denoted by. Theorem 6. On the, all the concave segments whch scale measures are smaller than are substtuted by the lne V 6 V 5 V 4 V 7 V 3 V 8

6 IEEE onf. on omputer Vson and Pattern Recognton (VPR June 7 segments connectng ther two end ponts. Specally, = and =G. Theorem 6 s a natural result of Def. 7 and Def. 8. It shows that as ncreases, more concave segments are gnored and becomes smpler untl t converges to the boundary curve of ts convex hull. It also ponts out how to select the scale threshold n the applcatons: the scale threshold depends on the scale of the concave segments we want to gnore. 6. Implementaton Detals and Expermental Results Because of dgtalzaton, some hgh curvature ponts may dsappear. Let us consder two example cases n Fg. 9. O M P M O (a) Fgure 9. Hgh curvature pont dsappears In Fg. 9(b the turn angle at pont O s about 7 o ; however, pont O s not represented by a pxel at the same locaton. The dgtalzaton process mapped t to one of dgtal ponts M and or possbly to both of them. ether the vsual curvature at M nor at s equal to the curvature of O, but ther sum s. Ths observaton motvates the followng approach to compute vsual curvature n dgtal mages. For a gven scale and a gven threshold T, for every pont v we consder ts neghborhood U( of radus T. If the representatve scale measure ( s the largest among all ponts n U(v then the new dgtal vsual curvature at v s sum of all curvature values n U(v.e., DK K ( u) (9) ( =, Δ S u U t the same tme we set the dgtal vsual curvature value of all other ponts n U( to zero. ctually, we compute n formula (9) the total curvature over the arc determned by the neghborhood U(v and assgn t to a sngle pont. ow we llustrate on our example n Fg. 9 that there exsts a dgtal pont whose vsual curvature best represents the orgnal pont O. fter dgtalzaton, O dsappears and ts nformaton s lost. However, the turn angle of M and s hgh n a relatve hgh scale, such as =.. Pont M s selected as best representng O based on the fact that (M)>() and the dstance between M and s less then T. We show that the proposed approach modfes the curvature of M to be the curvature of O n the orgnal contnuous contour. For smplcty, let us assume that neghborhood U(M) just contan P, M,. We obtan ( DK M ) = K ( M ) + K ( ) + K ( P), Δ S and DK ) DK ( P) ( = = The computed dgtal vsual curvature of M s 7/8π. ccordng to Theorem, ths yelds correctly the value of about 7 o for the turn angle of M. The justfcaton for ths s as follows. In a heght functon H, f one of the ponts among P, M and s an extreme pont, we ncrease the count number of M. From the fgure, we can observe that f O s an extreme pont of the heght functon of the orgnal contnuous object n drecton, then ether M or wll be an extreme pont of the heght functon of the dgtal object n drecton. Thus, the computed turn angle of M s about 7 o now. We need to assgn to other ponts n ths neghborhood, such as pont and P, snce ther contrbuton s added to M. ccordng to our experment, T= s a good choce for most of the case. In our method, the man computaton load s to compute the scale measures for all the extreme ponts n all the drectons. In the worst case, the tme complexty s O(n where s the number of the heght functons and n s the number of vertces on the curve. When they have been computed, gven a scale threshold, the vsual curvature for all the ponts can be computed n the complexty of O(n). From the defnton of vsual curvature, we can see that the arc length parameterzaton s not needed, through t makes our mplementaton easer. What we need s just the order of the contour ponts, whether there are gaps or not makes no dfference. Ths makes our method can deal wth very complcated mages. In all our experments, =8 and =. B D Vsual urvature Scale (a) Fgure. Vsual curvatures n dfferent scales Fg. shows the vsual curvatures of the four ponts, B,, D n the Fg. (a) calculated n dfferent scales. Obvously, the turn angle of these four ponts on a pentagram wthout nose should be 7 o, o, o and 44 o, respectvely. Because of nose, B and have large turn angles n small scales. For example, when =., the turn angles of these ponts are 77 o, o, 4 o, 4 o, respectvely. s scale ncreases, vsual curvature decreases. Snce () < (B) < () < (D the vsual curvature of vanshes frst, then B and, the vsual curvature of D never vanshes snce t s a pont on the convex hull of the B D Turn ngle

7 IEEE onf. on omputer Vson and Pattern Recognton (VPR June 7 pentagram. s the value of ncreases, the obtaned curvature estmaton s not accurate. However, ncreasng s very useful for domnant pont detecton, e.g., as can be seen n Fg.. The most domnant pont s D and then. Vsual urvature rc Length (a) =.9 Vsual urvature rc Length = urve Evoluton s ncreases, the -scale approxmaton seres { } can be consdered as an evoluton procedure. Snce we just delete the vertces where the vsual curvature vanshes, t leads to smplfcaton of shape complexty wth no blurrng effects and no dslocaton of relevant features. Fnally, when all the concave segments dsappear, the contour converges to the boundary of ts convex hull. Fg. 3 shows the evoluton procedure of a horse by gradually deletng the ponts where vsual curvature vanshes. When s suffcently large, the horse s evolved to the boundary curve of ts convex hull Vsual urvature.8.6 Vsual urvature.8.6 (a). (c) rc Length rc Length (c) =.8 (d) =.8 Fgure. Vsual curvature for the two pentagrams n Fg. Fg. demonstrates the vsual curvature as arc length functons for two pentagrams n Fg. n two scales. Fg. (a) and Fg. (c) s the functon of Fg. (a); Fg. and Fg. (d) s the functon of Fg.. s the start pont and we follow the contour clockwse. Obvously, there are ten peaks n all graphs; the nose n Fg. (a) s suppressed, especally when the scale s large, see Fg. (a) and Fg. (c). Fg. demonstrates the mult-scale approxmaton of Fg. 4(a). s ncreases, the vsual curvature of more ponts vanshes and becomes smpler untl t converges to the boundary curve of ts convex hull. (d).7 (e).5 (f).8 Fgure 3. The evoluton procedure of a horse 7.. orner Detecton The mult-scale vsual curvature can estmate the curvatures at a contnuum of scales. t the same tme, t does not modfy the orgnal curve. So t can be utlzed to detect the corners precsely and robustly. In our method, a contour pont s descrbed both by ts vsual curvatures and correspondng scales. In a certan scale, we consder the ponts whch dgtal vsual curvature s above a threshold DK as corner ponts. Fg. 4 demonstrates the corners of a butterfly detected n dfferent scales. In ths experment, curvature threshold DK =7π/64(48 o ). =. =.3 =.8 Fgure 4. orners of a butterfly n dfferent scales Fgure. Mult-scale approxmaton of Fg. 4(a) 7. pplcatons In ths secton, we wll demonstrate some applcatons of the mult-scale vsual curvature. (a) Fgure 5. The corners of the raster scanned ponts Fg. 5(a) s a raster scanned pont set wth known order; however, there are many gaps. In Fg. 5(b we connect these ponts by lne segments and demonstrate the detected corner ponts at scale.. In ths experment,

8 IEEE onf. on omputer Vson and Pattern Recognton (VPR June 7 DK =33π/8(46 o ). Fgure 6 demonstrates the results of corner detecton on knds of objects, each knd has two mages, one s the orgnal mage and one s wth sgnfcant nose. Fgure 6. orner detecton of twenty knds of objects 8. oncluson Ths paper proposes a new curvature defnton whch can be consdered to be a geometrc explanaton of standard curvature. Based on ths defnton, a natural mult-scale curvature s ntroduced. Because the scale measure can be defned n dfferent ways, n fact we obtan a seres of mult-scale vsual curvature. The propertes of the mult-scale vsual curvature are nvestgated and ther practcal sgnfcances are analyzed. Based on these propertes, we dscussed two knds of applcatons of mult-scale vsual curvature, corner detecton and curve evoluton. The experments show that the mult-scale vsual curvature s very robust and ntutve, and thus s very sutable to vsual processng. cknowledgement Ths work was supported by the ultvatng Fond for Momentous Scentfc Innovaton Project of Hgher Educaton n hna (Grant no. 7538) and n part by the SF Grant IIS and by DOE Grant DE-FG References [] M. Worrng and.w.m. Smeulders, Dgtal urvature Estmaton, VGIP: Image Understandng, vol. 58, pp , 993 [] D. oeurjolly, M. Serge, and T. Laure. Dscrete urvature based on Osculatng rcles Estmaton. Lecture otes n omputer Scence, vol. 59, pp. 33-3, [3] Thomas Lewner, rc-length Based urvature Estmator, SIBGRPI 4, pp.5-57, October 4 [4] D. G. Lowe, Organzaton of smooth mage curves at multple scales, n Proc., IEEE IV, Tarpon Sprngs, FL, pp , 988 [5]. L. Yulle, Zero crossngs on lnes of curvature, omput. Vson Graphcs Image Process, vol. 45, pp , 989 [6] T. Lewner, J. Gomes Jr., H. Lopes, and M. razer. urvature estmaton: Theory and practce. Pr epublcac oes do Departamento de Matem atca da PU-Ro, 4 [7] Mokhtaran, F., Mackworth,.K.: theory of mult-scale, curvature-based shape representaton for planar curves. IEEE Trans. on PMI, vol. 4, pp ,99 [8] Yu-Png Wang, S. L. Lee, and Kazuo Torach, Multscale urvature-based Shape Representaton Usng B-Splne Wavelets. IEEE Transactons on Image Processng, vol. 8, no., 999 [9] L. J. Lateck and. Rosenfeld, Supportedness and Tameness Dfferentalless Geometry of Plane urves, Pattern Recognton, vol. 3, no. 5, pp. 67-6, 998 [] Rosenfeld,. and Johnston, E. ngle Detecton on Dgtal urves. IEEE Trans. omputers -, pp , 973 []. Katzr, M. Lndenbaum, and M. Porat, urve segmentaton under partal occluson, IEEE Trans. PMI, vol. 6, pp , May 994 []. nsar and E. J. Delp, On detectng domnant ponts, Pattern Recognton, vol. 4, no. 5, pp ,99. [3] Pnhero,.M.G.; Izquerdo, E.; Ghanhar, M. Shape matchng usng a curvature based polygonal approxmaton n scale-space. Internatonal onference on Image Processng, vol., pp , [4]. Bengtsson and J.-O. Eklundh, Shape Representaton by Multscale ontour pproxmaton, IEEE Trans. PMI, vol. 3, no., 99 [5] Gregory Dudekm and John K. Tsotsos, Shape Representaton and Recognton from Mult-scale urvature, VIU, vol. 68, pp. 7-89, 997

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