1. Introduction. 2. Related Work

Transcription

1 ASIAGRAPH 008 PROCEEDINGS Curvature Estmaton of 3D Pont Cloud Surfaces Through the Fttng of Normal Secton Curvatures Xaopeng Zhang* /LIAMA-NLPR, Insttute of Automaton, CAS; Hongjun L /LIAMA-NLPR, Insttute of Automaton, CAS; Zhangln Cheng 3 /LIAMA-NLPR, Insttute of Automaton, CAS * xpzhang@nlpr.a.ac.cn, hjl@nlpr.a.ac.cn, 3 zhangln.cheng@gmal.com Abstract: Keywords: As the techncal development of laser scannng and mage based modelng, more and more pont cloud data are obtaned to represent 3D geometrc shapes of natural objects. Calculaton of dfferental propertes of 3D dscrete geometry becomes one fundamental work. Through the relaton of dscrete normal curvatures and prncpal curvatures, a new algorthm s presented on estmatng the prncpal curvatures and prncpal drectons 3D pont cloud surface. Based on the local fttng of each normal secton crcle propertes wth the poston and the normal at a neghbor pont, prncpal curvatures and prncpal drectons are estmated from the contrbuton of these neghbor ponts. Optmzaton of ths estmaton s converted as a lnear system by least squares fttng to all dscrete normal curvatures correspondng to ts neghbor ponts. A local feature curve, called as normal curvature ndex lnes, s constructed to show the effcency of ths work. Ths curve s ntutve and equvalent to Dupn ndex lne. Experments are desgned on Gaussan curvature, mean curvature and prncpal drectons for an analytcal surface and dscrete surfaces of pont cloud data. Expermental results show that ths work s more advantageous than smlar approaches, ad have applcatons to shape analyss and measurements. Normal secton curvature, prncpal curvatures, prncpal drectons, least squares fttng.. Introducton In recent years, the ncreasng avalablty and power of range scanners has enabled us to scan larger and more complex objects, to obtan rch detal features of objects and to get larger quantty of pont cloud data, whch s helpful to the reconstructon of geometrc objects, shape modelng and shape feature analyss. Some basc dfferental geometrc propertes should stll be better estmated for object reconstructon, shape modelng or analyss. Two of these most mportant propertes are the man curvatures together wth prncpal drectons on an estmated surface. If ponts are from a known analytc surface, the curvature at every pont can be precsely calculated by classc dfferental geometrc methods. However, f ponts were sampled from an unknown surface, wth a laser-canner for example, estmatng man curvatures and prncpal drectons of every pont would be an nterestng and challengeable topc. Estmatng curvatures of 3D B-Rep (Boundary Representaton) models has been cared snce 980s. Conventonal methods often nclude pont cloud denosng, mesh generaton and curvature estmaton. Some preprocessng may be performed at frst, such as denosng and algnment. Accordng to the mathematc tools adopted for geometrc models, these methods can be dvded nto three categores. The frst category s surface fttng wth a polynomal surface at a local area. The surface can be a quadrc surface [Sander990; Stokely99; Hamann993; Krsek998; Krsek997], a cubc surface [Goldfeather004], or a general polynomal surface. The second category s to compute the curvatures at each vertex of a mesh model by measurng the angle of each polygon passng through ths vertex [Dyn00; Km00]. The thrd category s to calculate the normal curvature of a drecton from neghbor ponts, and all the normal curvatures are weghtly averaged wth central angle or wth of tangent vectors [Taubn995]. These methods work well for mesh models, and they can be appled to pont cloud model after an extenson. We wll present a new fast method for pont cloud models, where both mesh reconstructon and surface fttng are all avoded. Our method just uses neghbor ponts and normal vectors to estmate the normal curvature, where a normal secton curve s thought of as spanned by one neghbor pont and correspondng normal vector there. Prncpal curvatures and prncpal drectons are estmated through the least square fttng of all normal curvatures related to all neghbor ponts. It can be shown that ths new approach s better to some degree than other smlar methods, and t can be sued for shape analyss and shape measurements. To show the effect of curvature estmaton and to compare dfferent approaches, we wll construct a new knd of graph, called as normal curvature ndex lnes, abbrevated as NCIL. Ths graph s smple and ntutve. Wth NCIL, t wll be llustrated that our method s more precse and faster than the method by Taubn [Taubn995], and t s better than the approach of Goldfeather [Goldfeather004] also. The organzaton of ths paper s as flows. State-of-art work of ths topc s descrbed n Sesson. The man algorthm s reasoned and descrbed n Sesson 3. Experments and analyss on analytc data are presented n Sesson 4, and on dscrete data are n Sesson 5. Conclusons and further work are descrbed at last n Sesson 6.. Related Work Varous algorthms to estmate curvature have been proposed n lteratures n recent years. They can be smply classfed as two categores: drect curvature fttng and ndrect curvature fttng though surface fttng. Two approaches are selected as representatves of two types

2 approaches, Taubn s approach [Taobn995] and Jack Goldfeather s approach [Goldfeather004]. The curvature tensor of a surface at each pont of a polyhedral approxmaton s estmated n Taubn s approach. Prncpal curvatures are obtaned by computng the egenvalues and egenvectors of certan 3 3 symmetrc matrces defned by ntegral formulas. Ths algorthm s a functon of the number of ponts and t s lnear n tme and n space. Ths approach does not employ normal vector, so large errors may be produced. It wll be mproved n ths paper. Goldfeather presented three methods n hs work [Goldfeather004]: normal curvature approxmaton, quadratc surface approxmaton and adjacent-normal cubc approxmaton. Snce the thrd one s the best among these three, we wll choose t for an analyss and result comparson. The objectve of the adjacent-normal cubc approxmaton method s to ft the neghbor pont wth a cubc polynomal surface () by neghbor ponts and correspondng normal vectors. a c 3 3 z ( x, y) = x + bxy + y + dx + ex y + fxy + gy () For each gven pont p wth normal N, ts m neghbor ponts are q wth normal vector M, =,,,m. In the coordnate system wth p as the orgn and N as z-axs, the coordnate of q s (x, y, z ), and M s (A, B, C ). The estmaton need subject to ax + by + 3dx + ex y + fy = A C () bx + cy + ex + fx y + 3gy = B C Ths s an overdetermned equaton system, and t can be wrtten n the followng matrx form M μ = R (3) where μ = (a b c d e f g) T M 3m 7 x = x 0 x y y x y 0 y 3 x 3x 0 x y x y x x y y x y ( z A C B C ) T 3 y 0 3y R3 m = Least square fttng wll be appled to fnd the best soluton of (3),, where only a, b and c are useful to the Wengarten matrx (4) a b W = (4) b c Wth the egenvalues and the egenvectors of (4), the prncpal curvatures and prncpal drecton vectors all can be calculated drectly. From Goldfeather s the adjacentnormal cubc approxmaton method, t can be safely deduce that the normal vectors of neghbor ponts are mportant factors, but the complexty of the system (3) s as three tmes more complex that of the system of the nput pont coordnates, whch wll ncrease computer tme and space for computaton. Our method wll mprove n ths complexty whle mantanng the optmzaton. Accordng to a comparson of all curvature estmaton methods n [Magd007] wthout consderng normal vectors, Taubn s approach s nether the best and nor the worst for ponts from an ellpsod. But f ponts are from a plane or a sphere, the Taubn s approach s the best among all other methods ndexed n [Magd007]. Therefore, we just choose the Taubn approach as representatve. Jack Goldfeather s method [Goldfeather004] employs normal vectors and yelds much better results, and t s probably the best n recent years. The movng least square used for curvature estmaton [Yang007] s too complex n determnng Gaussan parameter. The robust statstcal estmaton [Kalogeraks007] needs more tme for computaton. The ansotropc flterng approach on normal feld and curvature tensor feld [Lu007] focused more on pre-processng. Curvature-doman shape processng [Egensatz008] emphaszed on post processng, especally geometry processng operatons n the curvature doman. Most papers provde comparson between dfferent methods. In these comparsons, error analyss s very mportant. Tradtonally, data-error or nose-error fgure and table are employed, even n specal paper of comparson methods [Magd007; Mee000; Mokhtaran00]. We wll adopt these fgures or tables too. Meanwhle, we wll gve a new comparson fgure, whch s very ntutve for the effect of normal curvature computaton. 3. Curvature Estmaton To estmate the curvatures at a pont on a dscrete surface, the dstrbuton of neghbor ponts are used. Man steps of ths method ncluded usng the estmaton of normal secton lnes for normal curvature and the optmzaton of all these normal curvatures. Normal drecton at each sample pont s regarded as the nput data of the method presented here. If no normal vector nformaton s gven, we can select one method, for example the Max method [Max999] to ft the normal vectors at frst, whch s not dscussed n ths paper. 3.. Prncple All neghbor ponts of a specfc pont on a pont cloud surface determne the local shape. Bg errors may be generated f curvatures are estmated through surface fttng. The contrbuton of normal vectors should be consdered. To estmate the curvature on a pont, we wll consder the contrbuton of one neghbor pont only. Ths contrbuton s converted as the constructon of a normal secton curve. We wll construct a normal secton crcle and estmate the normal curvature from the postons and normal vectors of two ponts, the object pont and one of ts neghbors. 3.. Local Fttng for Normal Curvatures For each pont p n the pont cloud, let N be the unt normal vector obtaned elsewhere. We wll use pont coordnates and normal vectors to estmate normal curvatures at pont p. Suppose there are m ponts n the neghbor of the pont p and let q be the -th ( =,,..., m ) neghbor pont. The

3 normal vector correspondng to q s M. Let {p, X, Y, N} be an orthogonal coordnates system, called local coordnates L at pont p. N denotes the normal vector at p. X and Y are orthogonal unt vectors and they are not needed to be specfed gven here. In L, the coordnates of p, q, and M can be p s (0,0,0), q s (x, y, z ), M s (n x,,, n y,,, n z,, ). Then we can estmate normal curvature kn of pont p wth an osculatng crcle passng through pont p and q wth normal N and M. Fgure shows the geometrc relaton of these varables. N p α q M β S X O q β O Fgure Trangle defned by Osculatng crcle, neghbor pont and normal vectors. Local coordnates system L The normal curvature can be estmated wth the radus at pont q. sn β kn = (5) pq snα Where, α s the ncluded angle between vectors -N and pq, and β between vectors N and M. An approxmaton of equaton (5) s gven by: nxy kn = (6) nxy + nz x + y where x nx, + y ny, nxy =, n z = n z,. x + y Ths method employs chord, neghbor normal vector and osculatng crcle, so we call ths method as Chord And Normal vectors (CAN) method. The advantage of ths approach s that the normal vectors of neghborng ponts are used to estmate the man curvatures of a pont Least Square Fttng wth Euler Equaton The relaton of all normal curvatures wth man curvatures s analyzed here. In order to estmate prncple curvatures at a pont wth equaton (6), all coordnates of neghbor ponts are transformed nto the coordnates of local coordnates system. Gven the normal vector N at p N = ( nx, p, ny, p, nz, p ) and X = ( snϕ, cosϕ, 0) (7) Y = (cosψ cosϕ, cosψ snϕ, snψ ) (8) where, ψ = arccos( n z, p ), ϕ = arctan( n y, p nx, p ). The local coordnates system L at pont p becomes {p, X, Y, N} (Fgure ). e p N e Y Q M Let S be the plane through pont p wth normal vector N. Let e and e are prncpal drectons at pont p, correspondng prncpal curvatures k and k, and both are unknown. Let unknown parameter θ be the ncluded angle between vectors e and X, θ be the ncluded angle between vectors X and the vector pq obtaned by projectng the vector pq onto the plane S. θ can be calculated wth local coordnate of pont q. We employ Euler formula k n = k cos ( θ + θ ) + k sn ( θ + θ ), =,..., m (9) The task can be wrtten as a optmze queston: mn k, k, θ Snce m = [ k cos ( θ θ ) k sn ( θ θ ) kn ] k cos ( θ + θ ) + k sn ( θ + θ ) = cos θ ( k cos θ + k + cosθ snθ ( k + sn θ ( k sn θ + k cos θ ) Expresson (0) can be wrtten as the matrx form mn Mμ R μ sn θ ) cosθ snθ k cosθ snθ ) (0) () where cos θ cosθ snθ sn θ kn M = = m 3 cos θ cosθ snθ sn θ, Rm kn m cos θm cosθm snθm sn θm kn T μ = ( A, B, C) and A = k cos θ + k sn θ B = ( k k)cosθ snθ C = k sn θ + k cos θ After the least square fttng of (), the estmaton value of μ can be obtaned accordngly. It can be deduced that A B = B C k cosθ + k snθ ( k k)cosθ snθ ( k k)cosθ snθ k snθ + k cosθ cosθ snθ k = 0 cosθ snθ snθ cosθ 0 k snθ cosθ So k and k are Egen values of the matrx A B W = B C If we transform the unt egenvectors of W nto global coordnate system wth local coordnate system L, we obtan the prncpal drecton vectors. Experments wll be desgned n the next two sectons on analytc surfaces and on dscrete models to show the effcency of the new method. Dfferental nvarant propertes such as Gaussan and mean curvatures are one of the most essental features n practce. The results about Gaussan and mean curvatures wll be compared wth dfferent methods.

4 4. Experments on Analytc Data The precson of ths approach s shown on analytc surface n ths secton, where the value of prncpal curvatures and prncpal drectons are known before the estmaton. Fgure : Pont cloud torus model wth unform nose.. Nose level h=0.0.. Nose level h=.0. The procedure of the experment wth CAN method s performed n two steps: calculaton of normal curvature related to each neghbor pont n (6) and least square fttng (n secton 3.3) for prncpal curvatures and prncpal drectons. Gaussan curvature and mean curvature are obtaned from prncpal curvatures. Experments are performed wth C language programmng n a PC wth the confguraton of Intel(R) Core(TM) CPU, G,.99GHz, and G memory. absolute error Taubn Goldfeath CAN H nose level absolute error K Taobn Goldfeath CAN nose level Fgure 3: Average of the absolute error produced by dfferent methods (5 neghbor ponts).. Gaussan curvature.. Mean curvature. 4. Analytc Surface We use a torus as the representatve of analytc surfaces, snce rch geometrc curvature phenomena are ncluded, lke convexty and concavty. The equaton of the torus s n (). r ( u, v) = ( ( R + r cosu) cos v, ( R + r cosu)sn v, r sn u) ( 0 u π, 0 v π ) () where parameters are specfed as R=, r= ponts are sampled from () accordng to a unform dstrbuton. Wth analytc method, normal vector and prncpal curvature of each pont are calculated analytcally. Along normal drecton, nose s added to each pont whch subjects to unform dstrbuton U (-mh, mh), where parameter m s the medan of dstances of all par of ponts. Nose level was under the control of postve real number h. The values of h are 0., 0.,...,.0 respectvely. To dsplay clearly the shape varaton of the pont data surface, a mesh model s constructed and rendered flled. Topologcal relatons of vertexes n the mesh model s never used here for curvature estmaton, the. Fgure shows a mesh model of 000 sample ponts selected from the 5000 ones wth two nose levels, level h=0.0 and level h= Comparson on Absolute Errors In order to show a comparson of errors statstcally of each approach, experments are repeated 30 tmes at every nose level for each pont cloud data set n secton 4.. The average error wll be adopted for comparson n each case. The experment results of Taubn method (legend as the Taubn), the Adjacent-normal cubc approxmaton method (legend as the Goldfeath) and our method (legend as CAN) are explaned n Fgure 3 and Fgure 4. absolute error H absolute error Taubn Goldfeath CAN nose level Taubn Goldfeath CAN K nose level.0. Fgure 4: Average of the absolute error produced by dfferent methods (30 neghbor ponts).. Gaussan curvature.. Mean curvature. Fgure 3 and Fgure 4 show that the absolute error of estmatng Gaussan and mean curvature produced by the Taubn method s far bgger than that by the Goldfeath method and our method CAN. Both the Goldfeath and the CAN are robust to nose. Table shows that the CAN tends

5 to produce better results than the Goldfeath, especally when the nose becomes larger. Table : A part of average absolute average error produced by the Goldfeath and CAN method Nose Gaussan curvature error mean curvature error level Goldfeath CAN Goldfeath CAN Absolute error levels of each pont data are shown n Fgure 5 and Fgure 6. All ponts are samples from the torus (). The redder, the bgger the error s. The blacker, the more probable that the error s zero. It s clearly that Fgure 5 has lots of hgh error ponts. We just compare Fgure 5 wth Fgure 5(c). Two blue crcles n Fgure 5 and 6 mean that larger absolute errors are produced by the Goldfeath at some ponts, but smaller error produced by the CAN. 0 = mn error max = Fgure 5: Absolute error of Gaussan curvature estmaton n dfferent color.. The Taubn.. The Goldfeath. (c). The CAN. have error. In order to have an ntutonstc vew about the result, an ndex lne s defned and constructed. Inspred by Dupn ndex lne, normal curvature ndex lnes (NCIL) are desgned here. In Dupn ndex lne, ts polar coordnates equaton s ρ = / kn. A polar coordnate equaton of NCIL s desgned as ρ = kn n Fgure 7. Along any drecton n tangent plane of pont p on a surface, the normal curvature can be approxmately calculated accordngly. The way to draw an NCIL s to take the relaton of the ncluded angle and the normal curvature correspondng to a neghbor pont. The curve of NCIL comes from sample ponts obtaned and connected. p e ρ = k n θ q e Fgure 7: Normal curvature ndex lnes. Accordng to the above expermental data, a pont p accompaned by ts 30 neghbor ponts s randomly selected, and 30 pars of (θ, k n ) at pont p are obtaned then. Prncpal curvatures and prncpal drectons are estmated respectvely wth three methods descrbed above. So we wll get three curves when correspondng pars (θ, k n ) are connected sequentally as NCILs for the three methods. A NCIL wll be drawn wth real values of (θ, k n ) for each of three approaches compared n polar system n Fgure 8 and rectangular system n Fgure 9. 0 = mn error max = Fgure 6: Absolute error of mean curvature estmaton n dfferent color.. The Taubn.. The Goldfeath. (c). The CAN. In addton, compare wth the Goldfeath method, the CAN algorthm s good n both tme and memory complexty. The memory complexty can be explaned from the system (43) n see secton wth ts coeffcent matrx of 3m rows and 7 columns, but the system () (see secton 3.3) wth m rows and 3 columns. In an experment to calculate curvatures of 0000 ponts whle adoptng 30 neghbor ponts as neghbors, the Goldfeath spent 538 ms, but the CAN 34 ms. 4.3 Normal curvature ndex lnes In fact, dfferent method uses dfferent normal curvature estmaton, whch means that at each pont, prncpal curvatures (k, k ) obtaned by dfferent approaches must Fgure 8: Normal curvature ndex lnes of three approaches n a polar system Fgure 9: Normal curvature ndex lnes of three approaches n an orthogonal system Fgure 8 shows that the NCIL (blue) obtaned by the CAN s more close to the real curve (black). But the stuaton of the green one for the Taubn s very serous,

6 whch may cause large errors. The red curve for the Goldfeath s the moderate. It can be seen n Fgure 9 that the Taubn method generates large errors also. One reason of these large errors s that the mean of k n s large and the ampltude s small. Normal curvature ndex lnes can be used to at least two thngs. One s an analyss of the effect of curvature fttng, whch helps to evaluate curvature estmaton method. The other s to compare error of several normal curvature approxmatons, whch helps to select the best one among exstng algorthms. drectons on the surface of the mddle part of thumb back are n order n Fgure, whch s close to the drectons n Fgure 0 and Fgure 0(c). Chaos of prncpal drectons can be seen n Fgure (c), whch s qute dfferent from Fgure 0 and Fgure 0(c). Ths means that the errors of prncpal drectons estmated by the CAN s far smaller than that estmated by the Goldfeath. Therefore, the CAN s more robust than the Goldfeath. 5. Experments on dscrete models In order to show the accuracy and the robust of the CAN algorthm, we apply t to a dscrete model hand, where the heght of the model hand s 0 unts. Ths hand model n Fgure 0 has 658 ponts wth normal vectors. Ths model s a mesh model and we only use the vertexes to perform experments. The polygon nformaton here s only adopted to vsualze the prncpal drectons estmated. We do the experment wth dfferent nose levels, free nose n Fgure 0, nose level 0.5 unts (h=0.5) n Fgure, and nose level.0 unts (h=.0) n Fgure. The effect of nose level can be seen n Fgure. Noses are added accordng to unform dstrbuton U(-h, h) along the normal drecton of each pont. The prncple curvatures and drectons are estmated by the CAN approach n Fgure 0, Fgure and Fgure, and by the Goldfeath algorthm Fgure 0(c), Fgure (c) and Fgure (c) respectvely. Snce the Taubn algorthm produces large curvature error, t s not used here. Fgure 0 shows that the estmated prncpal drectons are good wth the Goldfeath and the CAN thanks to the pont cloud hand model free of nose. Fgure : Pont cloud hand model wth nose level 0.5 and prncpal drectons estmated by two approaches.. Pont cloud (dsplay as mesh only for watch).. CAN. (c). Fgure : Pont cloud hand model wth nose level.0 and prncpal drectons estmated by two approaches.. Pont cloud (dsplay as mesh only for watch).. CAN. (c). Goldfeath. Fgure 0: Pont cloud hand model free of nose and prncpal drectons estmated by two approaches.. Pont cloud (dsplay as mesh only for a vew). (s). CAN. (c). Goldfeath Fgure shows that wth low nose level, both CAN and Goldfeath produce good results. The prncpal drectons are organzed n order n both cases. It can be seen that the prncpal drectons on the surface of the mddle part of thumb back n Fgure (c) are dfferent from the ones n Fgure and Fgure 0 (c). Therefore, the Goldfeath does not work well for ths level of nose. Fgure shows that wth large nose level, the CAN s better than the Goldfeath. It can be seen that the prncpal Another dscrete model elephant was used also, where the heght of ths model s 75 unts. Ths model conssted of 6859 ponts wth normal vectors. The expermental results of dfferent nose levels are presented n Fgure 3, Fgure 3 4 and Fgure 5. Mesh models n Fgure 3, Fgure 4, and Fgure 5 are dsplayed only for data vsualzaton. From Fgure 5, t can be seen that the Fgure 3: Pont cloud model elephant free of nose and prncpal drectons estmated by two approaches.. Pont cloud surface (dsplay as mesh only for data vsualzaton).. CAN. (c). Goldfeath.

7 Fgure 4: Pont model elephant wth nose level 0.50 and prncpal drectons estmated by two approaches.. Pont cloud surface.. CAN. (c). Goldfeath. Fgure 7 shows the estmated prncpal drectons wth the CAN on the tree branch surface. Fgure 7 s a global vew. Fgure 7 shows that the gradual changes of estmated prncpal drectons at branch ramfcaton node are reasonable. If the surface of tree branch s concave, the max prncpal drectons wll snk together, whch can be seen n two areas n Fgure 7(c), where the two local ellptc ponts are on the bark surface. From Fgure 7(d), t can be seen that the blue lnes as the lattude are nearly the cross-sectonal drectons, and the green ones as the longtude, are nearly the growth drectons of branches. Ths example demonstrates that the CAN can be used for the shape analyss of tree branch surfaces and tree skeleton extracton [Cheng007], whch should be a brdge to tree reconstructon and measurements. Fgure 5: Pont model elephant wth nose level.0 and prncpal drectons estmated by two approaches.. Pont cloud surface.. CAN. (c). Goldfeath. prncpal drectons on the surface of the rght sde of the elephant back are n order n Fgure 5, whch s close to the drectons n Fgure 3 and Fgure 3(c). Chaos of prncpal drectons can be seen n Fgure 5 (c), whch s qute dfferent from Fgure 3 and Fgure 3(c). Ths means that n ths comparson, the performance of the CAN approach are sgnfcantly better than that of the Goldfeath. Above experments llustrate that f the nose level s low, both CAN and Goldfeath methods produce accurately prncpal drectons; f the nose level s hgh, the CAN has sgnfcant advantages n estmatng prncpal curvature and drecton. It s nterestng and challengeable to use the CAN algorthm to scanned data of complex objects. The pont data of a tree s obtaned wth a laser scanner of a trucked tree, and the result s range mage shown n Fgure 6. Ths range mage ponts are clustered accordng the dfference of the dstance of neghborng ponts, so four dfferent branch blocks are obtaned then. Ths heght of ths model s meter ponts are ncluded. (c) (d) Fgure 7: Curvature drectons estmated by the CAN.. A global vew; Zoom n to the mddle rectangle area of ; (c) Zoom n to the lower rectangle area of ; (d) Zoom n to the upper rectangle area of. Brefly speakng, the CAN method can be used to estmate prncpal curvatures and drectons wth a hgh precson from pont cloud coordnates accompaned by normal vectors. Ths approach s robust for pont cloud data wth nose. 6. Conclusons and Further Work Fgure 6: Pont cloud tree branch model.. Ponts.. Rendered model when ponts are connected to a mesh model accordng pont neghborhood n branch postons. A new algorthm s presented for estmatng the prncpal curvatures and prncpal drectons on a dscrete pont cloud data. Ths algorthm, called as CAN, can be used to scattered sampled pont data, and mesh vertexes data as well.

8 Experments show that ths new algorthm can be used to extract local dfferental propertes wth hgh relablty, and the effects are better than other smlar approaches. The complexty of ths new algorthm s lower than that of the neghbor-normal cubc approxmaton method [Goldfeather004], both n tme cost and n computer memory cost. The CAN algorthm s robust to data wth strong nose also. We present an ndex tool for error analyss,.e. Normal curvature ndex lnes (NCIL). It s nterestng and ntutve. The errors generated by curvature estmaton can be shown drectly wth NCIL. In the future, we would lke to apply CAN extract hgher features from pont cloud data, lke hgh dervatves, and further be used to construct curvature lnes, create lnes and the parameterzaton of pont cloud data. In the applcaton to specfc problems, the CAN method could be used to reconstruct real tree geometry of the data scanned wth noses, whch s valuable to the measurement of structural and functonal nformaton of trees. Acknowledgement Ths work s supported n part by Natonal Natural Scence Foundaton of Chna wth projects No ; n part by the Natonal Hgh Technology Development 863 Program of Chna under Grant No. 006AA0Z30, 008AA0Z30, and by Bejng Muncpal Natural Scence Foundaton under Grant No References Cheng Z., Zhang X., Chen B Smple reconstructon of tree branches from a sngle range mage. Journal of Computer Scence and Technology, Vol., No. 6, pp Dyn, N., Hormann, K., Km, S., and Levn, D. 00. Optmzng 3D trangulatons usng dscrete curvature analyss. In Mathematcal Methods For Curves and Surfaces: Oslo 000, T. Lyche and L. L. Schumaker, Eds. ed. Vanderblt Unv. Press Innovatons In Appled Mathematcs Seres. Vanderblt Unversty, Nashvlle, TN, Egensatz, M. Sumner, R. and Pauly, M Curvature- Doman Shape Processng, EUROGRAPHICS 008, Volume 7 (008), Number Goldfeather, J. and Interrante, V A novel cubc-order algorthm for approxmatng prncpal drecton vectors. ACM Trans. Graph. 3, (Jan. 004), Hamann, B Curvature approxmaton for trangulated surfaces. In Geometrc Modellng, G. Farn, H. Hagen, H. Noltemeer, and W. Knödel, Eds. Sprnger Computng Supplementum, vol. 8. Sprnger-Verlag, London, Kalogeraks, E., Smar, P., Nowrouzezahra, D., and Sngh, K Robust statstcal estmaton of curvature on dscretzed surfaces. In Proceedngs of the Ffth Eurographcs Symposum on Geometry Processng (Barcelona, Span, July 04-06, 007). Km, S.J. Km, C.-H. and Levn, D. 00. Surface smplfcaton usng dscrete curvature norm, n: The Thrd Israel-Korea Bnatonal Conference on Geometrc Modelng and Computer Graphcs, Seoul, Korea, October 00. Krsek, P, Luka cs, C. and Martn, R. R Algorthms for computng curvatures from range data. n: The Mathematcs of Surfaces VIII, Informaton Geometers, n: A. Ball et al. (Eds.), 998, 6 Krsek, P. Pajdla, T. and Hlava c V Estmaton of dfferental parameters on trangulated surface, n: The st Workshop of the Austran Assocaton for Pattern Recognton, May 997 Lu, M., Lu, Y., and Raman, K Ansotropc flterng on normal feld and curvature tensor feld usng optmal estmaton theory. In Proceedngs of the IEEE nternatonal Conference on Shape Modelng and Applcatons 007 (June 3-5, 007). Magd, E., Soldea, O., and Rvln, E A comparson of Gaussan and mean curvature estmaton methods on trangular meshes of range mage data. Comput. Vs. Image Underst. 07, 3 (Sep. 007), Max N Weghts for computng vertex normals from facet normals. Journal of Graphcs Tools 4,,-6 Meek, D. S. and Walton, D. J On surface normal and Gaussan curvature approxmatons gven data sampled from a smooth surface. Comput. Aded Geom. Des. 7, 6 (Jul. 000), Mokhtaran, F., Khall, N., and Yuen, P. 00. Estmaton of Error n Curvature Computaton on Mult-Scale Free- Form Surfaces. Int. J. Comput. Vson 48, (Jul. 00), Sander990, P. T. and Zucker, S. W Inferrng Surface Trace and Dfferental Structure from 3-D Images. IEEE Trans. Pattern Anal. Mach. Intell., 9 (Sep. 990), Stokely, E. M. and Wu, S. Y. 99. Surface Parametrzaton and Curvature Measurement of Arbtrary 3-D Objects: Fve Practcal Methods. IEEE Trans. Pattern Anal. Mach. Intell. Vol. 4, No. 8. Taubn, G Estmatng the tensor of curvature of a surface from a polyhedral approxmaton. In Proceedngs of the Ffth nternatonal Conference on Computer Vson (June 0-3, 995). ICCV. Watanabe, K. and Belyaev A.G.00. Detecton of salent curvature features on polygonal surfaces, n: Eurographcs 00, vol. 0, No. 3, 00. Yang, P. and Qan, X Drect Computng of Surface Curvatures for Pont-Set Surfaces, Eurographcs Symposum on Pont-Based Graphcs (007) submsson.