Anisotropic filtering on normal field and curvature tensor field using optimal estimation theory

Transcription

1 Aniotropic filtering on normal field and curvature tenor field uing optimal etimation theory Min Liu Yuhen Liu and Karthik Ramani Purdue Univerity, Wet Lafayette, Indiana, USA {liu66 liu28 Abtract In thi paper, we tudy the problem of meh denoiing for improving the ingle pa urface etimation on normal and curvature tenor. We focu mainly on the engineering object repreented a dene triangle mehe. In particular, a two run non-linear diffuion algorithm baed on optimal etimation theory i propoed to adaptively filter out the undeired dicontinuitie introduced by noie while preerving the underlying feature. We how that the propoed filter can uccefully improve the local urface etimate while preerving the deired feature in term of tangential and curvature dicontinuitie. 1. Introduction Smoothne refer to the mathematical notion of continuou differentiability, or continuity. When a mooth urface i approximated a a triangle meh, it maintain only the poitional continuity (no gap) of the underlying urface, while the tangential continuity (no harp angle) and the curvature continuity (no harp radiu change) are lot due to the dicrete nature of the meh. Thi bring about the problem of etimating the dicrete urface differential propertie for the mehed object. Dicrete etimation of urface differential parameter i eential for many meh baed application, but the problem i difficult if the underlying geometry contain natural urface dicontinuitie. The problem get even harder when a meh i contructed from canned data ince they carry meaurement and quantization error. The differential propertie uch a principal curvature and direction are very enitive [13] to thoe error. A ingle pa etimation, baed on either local urface fitting uing the polynomial [15, 4, 7, 20] or the etimation of a curvature tenor in the local neighborhood [17, 11, 1, 14] are error prone, due to the preence of the natural dicontinuitie a well a the noie in the data. In practice, thi mean that the etimate computed on a local bai mut be improved at a later tage. The technique related to iotropic denoiing aume that there are no underlying urface dicontinuitie, therefore the error in the etimate reult from only noie which i randomly and uniformly ditributed. Repreentative iotropic denoiing method include Laplacian moothing [18], mean curvature flow [3], and curvature conitency framework [15]. The firt two method [18, 3] achieve moother urface by changing the vertex poition to relax all the curvature peak, with improvement in the urface normal and curvature etimation a a byproduct; while the lat method [15] improve the curvature etimation by minimizing a functional form related to a minimum variation of curvature etimate. In real cae, a typical meh contain random noie a well a urface dicontinuitie. Aniotropic denoiing method, firt introduced in image proceing and later extended to geometric problem, were developed to preerve or enhance feature like harp edge or corner. Some method moothen height field by controlling the weight in mean curvature flow [2, 5, 8]. Other method [19, 16, 12] ue diffuion filter to moothen the normal field and then integrate thi to height field to get the moother urface. In thi work, we tudy the problem of meh denoiing for improving the ingle pa urface etimate. We target mainly the canned engineering object which are repreented a dene triangle mehe, though it can be applied for general hape too. Our propoed method, derived from a ignal filter, originate from the optimal etimation theory [6]. In tead of modifying the vertex poition, we improve the normal and curvature etimate by arguing that thoe differential propertie are more important than removing noie in the geometry becaue, the typical two tage in revere engineering: egmentation and urface fitting relie more on the good etimate of urface differential propertie. The focu of our work target three problem: An etimation of dicrete hape operator. We propoe a imple per-face dicrete curvature etimation method in term of curvature tenor. It benefit

2 the edge-baed meh proceing ince the harp variation of differential propertie between the two neighboring triangle potentially define a feature edge. An aniotropic filter on the normal and curvature field. We adapt the optimal etimation theory into a two run non-linear diffuion proce; one for normal conitency in which the filter i applied on a vector field (normal). And the other i for curvature conitency in which the filter i applied on a matrix field (curvature tenor). Each tage trie to preerve the underlying dicontinuitie (feature) while moothing the ingle pa normal and curvature etimate. A general framework for extending the diffuion proce into the differential parameter field with higher order. We apply the non-linear diffuion filter both on the normal field and on the curvature field to handle engineering object. We alo how that it can be generalized to handle higher order differential propertie. 2. The local geometry etimation for per-face For each triangle f in a meh, it normal direction n f i well defined. The normal direction at each vertex p in the meh i etimated by the weighted average of the normal of face adjacent to p (1-ring neighborhood face). We ue the Nelon Max weighting method [10] for weight (area of f divided by the quare of the length of the two edge that touch vertex p) a thi produce more accurate normal etimate than other weighting approache. We etimate the per-face curvature by oberving the normal variation along it three edge. For each triangle f, there i an aociated hape operator in matrix format called curvature tenor Λ: Λ f = ( D u n D v n ) ( ) nf = u u n f v u n f u v n f v v, where (u,v) i a pair of orthogonal unit vector decribing an orthonormal coordinate ytem in the tangent frame of f. Multiplying a curvature tenor by any vector in the tangent plane give the directional derivative of the normal in that direction: Λ t = D t n. The curvature tenor for each facet i contructed uing method introduced in [14]. For a facet f, there are three well-defined direction (the edge) together with the difference in normal along thoe direction (computed from the per-vertex normal), refer Figure 1. In thi figure, n i (i = 0,1,2) are the etimated normal on three vertice of the triangle f. The equation in Figure 1 provide a et of linear contraint on the element of the econd fundamental tenor, which may then be determined uing leat quare fitting. Notice that the principal curvature (k 1, k 2 ), and the principle direction (e 1, e 2 ) of the triangle f correpond to the eigenvalue and eigenvector of Λ f, while the determinant of the tenor Det(Λ f ) and the half trace of the tenor Tr(Λ f )/2 can be conidered a a local etimation of Gauian curvature K f and Mean curvature H f. n0 e2 v u e0 n2 e1 n1 Figure 1. Etimate per-face curvature tenor 3 Aniotropic filtering baed on the optimal etimation theory The local differential propertie etimated o far (refer Section 2) are error-prone due to the exitence of urface dicontinuitie which violate the moothne aumption for the underlying geometrie. The general objective of an aniotropic filtering i to mooth out the urface dicontinuitie introduced by the random noie and preerve the true urface continuitie. The optimal etimation theory, originally introduced in [6] for ignal proceing and adapted by [9] in a curvature conitency framework for the edge detection on a range image, offer a good method for integrating meaurement from multiple ource with different noie to predict a contant quantity. The main idea i that, given a noiy ource, it contribution to predict the behavior of a target in the next iteration i dependent on the variance of it prediction error in the pat. A ource with a lower variance of etimation error i conidered to have better fitne and will be given higher weight in the next prediction. Thi mechanim work elegantly for our purpoe of aniotropic filtering on the initially etimated urface parameter. We ue the propertie of neighboring facet to redefine the propertie of a target facet. At the beginning of optimal etimation flow, the full variational relaxation take place. Thi relaxation moothe out random noie and tart to ditort the true urface dicontinuitie meanwhile. However, a the iteration progree, the real nature of the local urface i learned by recording the etimation error variance in each tep. In ubequent iteration, neighbor believed to be in the ame

3 continuou region (i.e. the neighbor with low etimation error variance) will dominate the relaxation proce. The relaxation acro dicontinuou region will be penalized ince they introduce large variance of etimation error. Now we reformulate the above idea to our per-face geometry etimator. For each triangle f, conider it 1-ring neighborhood N( f ) (which are triangle haring common edge or vertice with f ) a a noiy prediction ource. During the optimal etimation flow, differential propertie of f, at a new iteration l +1 ( f l+1 ) i etimated by the weighted average of prediction from it neighbor. Each of it neighboring facet, cat a vote on f. f l+1 i given by the following equation; f l+1 = N( f ) w l N( f ) w l, (1) where w l i the weight aigned to a neighbor for meauring how much the prediction of will contribute to the behavior of the target. can be equal to but it i not neceary. If equal weight are aigned to each of f neighbor, the above iterative proce (Equation (1)) provide an iotropic filtering which i not deired for preerving underlying feature in our cae. For aigning weight to adjut the contribution of prediction from each neighbor, we ue the following function that are baed on the optimal etimation theory [9]; w l = exp( σ γ l ), (2) σ 1 = k γ l = 2 N( f ) l l=0 N( f ) ε, (3) σ, (4) where σ i the variance of prediction error in lth iteration and γ l i the moothing control parameter which i et by taking the mean of the error variance value among all the triangle in N( f ). ε meaure the prediction error, i.e. difference between and f. In equation (2), the weight i defined a a function of prediction error variance (σ ), which i normalized by the mean variance (γ l ) among all neighbor who participate in current prediction. The exponential function i choen to avoid a ingle neighbor from taking over the relaxation proce by bonding the weight a the error variance approach zero. A deired by the optimal etimation theory, thi weighting function give higher weight to the neighbor with low error variance. In addition, it provide imilar weight to the neighbor which are correlated with the target facet in the ame ubregion. A key problem i, how to etimate the error o that the weight can be correctly adjuted. In the curvature conitency framework [15, 9], thi iterative proce i conducted in a ingle pa on the Augmented Darboux frame [15] (contain the normal and the curvature element) of a vertex p, the diffuion i applied on both the normal and the curvature parameter and the error metric ε imply combine the etimation error of all element: ε = n l f n l l + e 1 f e 1 + k 1 f k l 1 + k 2 f k l 2 2. (5) Thi error metric caue a major problem ince it imultaneouly fue on two calar propertie (k 1 and k 2 ), and two vector propertie (n, e 1 ). Cae exit where the calar element dominate the error function o that the vector element never contribute in thi optimal etimation framework. We apply the diffuion proce to different run correponding to different order of differential parameter etimation o that in each run, the error variance can be meaured in the ame field. We firt apply the optimal etimation proce on urface normal. Equation (1) in thi run become; n l+1 f = N( f ) w l n l N( f ) w l. (6) and the error metric i given by the angular ditance between etimated normal of f and the normal of neighboring facet : ε l = 1 n f l n l n l f n l. (7) In the econd run, the curvature tenor Λ etimated on f, i conidered for the optimal etimation. Since the tenor Λ f i defined on a local parameter pace aociated with three orthogonal direction (n f,u f,v f ), the tenor Λ of it neighboring facet, which i defined on (n,u,v ), cannot directly cat the vote on f. Therefore, a neighboring tenor ha to be aligned in the pace of the target facet (refer to Figure 2). For each Λ, thi alignment form a rotated tenor Λ, which will be ued for prediction. f = N( f ) w l Λ l N( f ) w l. (8) Λ l+1 Two error metric can be ued for calculating variance and weight. One i the variation between principal curvature of f and thoe of it neighbor and the other i the determinant of difference matrix between the etimated tenor Λ f and Λ. Variation between principal curvature contain two element; ε l 1 = (k1 l f kl 1 )2 + (k f kl 2 )2 and ε l 2 = inθe. l (9) The firt element in Equation (9) i the Euclidean ditance between (k 1 f, k 2 f ) and (k 1,k 2 ), and the other i the

4 n f II f n ' n II v f u f II ' v ' v u u ' Figure 2. Rotate local pace of tenor Λ to the pace of neighbor triangle tenor Λ f,. Thi form a rotated tenor Λ. 4 Reult We have implemented the aniotropic meh denoiing algorithm a decribed in the previou ection, and compared our reult to that of two popular aniotropic denoiing method, bilateral meh denoiing algorithm [5] and adaptive moothing algorithm [12]. We alo compared our reult of curvature denoiing to the method baed on curvature conitency framework; iotropic denoiing method propoed in [15], and the aniotropic cheme propoed in [9]. Thee two method denoie the meh on both the normal and the curvature field in a ingle pa. We ue the following color map (ee Figure 3) to how the two etimated principal curvature through all the reult. angular ditance between one of the principal direction e 1 f and e 1. θe l i the angle between e 1 f and e 1 in the lth iteration. We ue inθe l to treat the acute angle the ame a the obtue angle becaue the principal direction i actually a line field rather than a vector field. The above two error metric are not on the ame cale and therefore hould not be ummed up directly into one error function. The variance of each element will be calculated eparately to get two correponding σ 1, γl 1 and σ 2, γl 2 which correpond to the variance of principal curvature variation and the variance of principle direction variation. The final weight function (Equation (2)) i: w l = exp( σ 1 γ1 l σ 2 γ l 2 ). (10) Another option for the error metric for curvature tenor diffuion i a impler uniformed metric which i defined by the determinant of the tenor difference, ε l = Λ f l Λ l (11) Generally peaking, the firt error metric (Equation (10)) capture the ditance of principal curvature and principal direction among neighboring triangle. The econd one (Equation (11)) can be conidered a a meaurement of total variation of curvature along all the direction in the parameter pace of triangle f. Both the error metric have their own diadvantage. The firt metric will have problem in the vicinity of umbilical point, ince the etimation of the principal curvature direction get untable on umbilical point. The econd error metric vanihe for ingular difference matrice. Conequently, different curvature tenor may have a zero ditance. We leave the improvement of the error metric for curvature tenor a a future work. k 1 >0 k 1 <0 k 1 =0 k 2 >0 k 2 =0 k 2 <0 Figure 3. The color map of the curvature value. Figure 4 how the reult of applying our aniotropic filter for denoiing the Fandik model (with triangle, vertice, edge). Thi model contain noie near the feature region a well a global gauian noie (Figure 4(a)). After applying five iteration of our denoiing filter for normal, the etimated normal are moothed while the underlying harp variation on the normal are kept (ee Figure 4(b) for the rendering effect of thi moothing). Figure 4(c) and 4(d) how the normal ditribution (normal are rendered a red line) of a mall region (rectangle region in Figure 4(a)) before and after applying our normal filter. The harp edge (colored a light blue) behave more regularly and directionally a the reult of aniotropic denoiing on normal field. Figure 4(e) and 4(f) how the curvature ditribution of the model before and after applying our filter on the curvature tenor field, the mall line troke denote the minimum curvature direction. Figure 6 give the reult of two other aniotropic denoiing algorithm, Figure 6(a) and 6(b) are the reult of applying five iteration of bilateral meh denoiing propoed in [5], Figure 6(c) and 6(d) are the reult of applying ame number of iteration uing the adaptive meh denoiing propoed in [12]. For thi model, the bilateral denoiing doe not produce very good denoiing reult. Denoiing effect on the curvature

5 etimate, a the byproduct of denoiing the height field are hown in Figure 6(b) and 6(d). Notice the difference in the curvature ditribution compared to the reult of our method in Figure 4(f). Figure 5 how how our algorithm work on a noiy rocker arm model (Figure 5(a)). After applying our filter on the normal field, the moothing reult on the urface normal i hown in Figure 5(b). Figure 5(c) i the initial etimation of curvature, denoiing on the normal field will improve the curvature etimation too, ee the reult in Figure 5(d). Figure 5(e) and 5(f) how the reult of applying five iteration of our filter on the curvature tenor field, uing two error metric propoed in Section 3, (Equation (11) and (9) repectively). They both work well and reult in imilar denoiing effect (refer to Figure 5(e) and 5(f)). Since Equation (11) give a impler computation for error metric, we recommend the reader to ue it. We compared our reult uing aniotropic denoiing on curvature tenor field to the following method that improve the curvature etimate; one i the aniotropic cheme propoed in [9] (Figure 7(a)) and the other i the iotropic cheme propoed in [15] (Figure 7(b)). Our method achieve better reult of moothing curvature etimation while preerving the underlying dicontinuitie (refer to Figure 5(f)). Oberve the difference in the curvature ditribution along the flat region on the rocker arm model. We alo teted our method on ome general hape (ee Figure 8 for example). The tet how that our aniotropic filter work well for the general hape too. 5 A general aniotropic filtering framework In theory, the framework of our aniotropic diffuion algorithm can be generalized for treating higher order feature. The firt tep i to extend the per-face local geometry etimation to higher order differential propertie. We how here how to extend it to third order differential propertie. The third order differential propertie can be defined a a derivative of curvature tenor that give the change in curvature along the urface. It i a rank-3 tenor with four unique entrie: C = (D u Λ,D v Λ) = ( ( a b b c )( b c c d ) ). To compute the above tenor, we need to etimate the curvature tenor Λ p at each vertex p on the meh. Thi can be done by the weighted average of the curvature tenor of it triangle neighbor: Λ p = m i=1 w iλ i m i=1 w in i, where the Λ i are the rotated tenor (ee Figure 2) of the triangle in the 1-ring neighborhood of p, The tenor i rotated to align with the parameter pace of vertex p. Weight w i denote how much of the curvature tenor hould be accumulated at each vertex p. The Voronoi area weighting cheme that take the weight be proportion of the area of f that lie cloe to vertex p [11] i a good choice. The derivative of curvature tenor of a triangle can then be etimated with a leat-quare fit to the difference in the curvature tenor along three edge of each triangle, in the ame manner a we did for curvature tenor etimation. For the aniotropic diffuion filter for the third order tenor, the iteration equation can be defined a, C f l+1 = N( f ) w l C l N( f ) w l, (12) where C l denote the rotated derivative of curvature tenor, and the error metric can then be defined a: ε l = C f l C l. (13) 6 Concluion and future direction We have preented a feature-preerving filter for the normal and the curvature etimated on the dene meh model. For ome revere engineering application, the propoed filter i generally a ufficient preproceing tep for further geometric operation uch a egmentation and urface fitting. We alo howed that thi aniotropic filter can be extended to improve higher order differential parameter. In future, teting the method for denoiing on higher order field will be carried out. Our aniotropic filter of normal and curvature can be ued for table extraction of feature edge correponding to C 1 dicontinuitie and C 2 dicontinuitie that can help in egmenting dene mehe. The current implementation of the iterative diffuion proce imply pecifie the number of iteration a a uer defined parameter. In future, the iteration will be controlled by tracking the convergence of the differential propertie. Reference [1] D. Cohen-Steiner and J.-M. Morvan. Retricted delaunay triangulation and normal cycle. In SCG 03: Proceeding of the nineteenth annual ympoium on Computational geometry, page , New York, NY, USA, ACM Pre. [2] M. Debrun, M. Meyer, P. Schröder, and A. H. Barr. Aniotropic Feature-Preerving denoiing of height field and image. In Graphic Interface, page [3] M. Debrun, M. Meyer, P. Schröder, and A. H. Barr. Implicit fairing of irregular mehe uing diffuion and curvature flow. In SIGGRAPH 99: Proceeding of the 26th annual conference on Computer graphic and interactive technique, page , New York, NY, USA, ACM Pre/Addion-Weley Publihing Co.

6 [4] F. P. Ferrie, J. Lagarde, and P. Whaite. Darboux frame, nake, and uper-quadric: Geometry from the bottom up. IEEE Tran. Pattern Anal. Mach. Intell., 15(8): , [5] S. Fleihman, I. Drori, and D. Cohen-Or. Bilateral meh denoiing. ACM Tran. Graph., 22(3): , [6] A. Gelb. Applied Optimal Etimation. MIT Pre, Cambridge, Ma., [7] B. Hamann. Curvature approximation for triangulated urface. page , [8] K. Hildebrant and K. Polthier. Aniotropic filtering of non-linear urface feature. Computer Graphic Forum, 23(3): , [9] S. Mathur and F. P. Ferrie. Edge localization in urface recontruction uing optimal etimation theory. In CVPR 97: Proceeding of the 1997 Conference on Computer Viion and Pattern Recognition (CVPR 97), page 833, Wahington, DC, USA, IEEE Computer Society. [10] N. Max. Weight for computing vertex normal from facet normal. J. Graph. Tool, 4(2):1 6, [11] M. Meyer, M. Debrun, P. Schr, and A. Barr. Dicrete differentialgeometry operator for triangulated 2-manifold. In Viualization and Mathematic III, edited by H.C. Hege and K. Polthier, page 35 37, Heidelberg, Springer. [12] Y. Ohtake, A. G. Belyaev, and I. A. Bogaevki. Meh regularization and adaptive moothing. Computer-Aided Deign, 33(4): , [13] S. Petitjean. A urvey of method for recovering quadric in triangle mehe. ACM Comput. Surv., 34(2): , [14] S. Ruinkiewicz. Etimating curvature and their derivative on triangle mehe. In 3DPVT 04: Proceeding of the 3D Data Proceing, Viualization, and Tranmiion, 2nd International Sympoium on (3DPVT 04), page , Wahington, DC, USA, IEEE Computer Society. [15] P. T. Sander and S. W. Zucker. Inferring urface trace and differential tructure from 3-d image. IEEE Tran. Pattern Anal. Mach. Intell., 12(9): , [16] T. Tadizen, R. Whitaker, P. Burchard, and S. Oher. Geometric urface moothing via aniotropic diffuion of normal. In VIS 02: Proceeding of the conference on Viualization 02, page , Wahington, DC, USA, IEEE Computer Society. [17] G. Taubin. Etimating the tenor of curvature of a urface from a polyhedral approximation. In ICCV 95: Proceeding of the Fifth International Conference on Computer Viion, page 902, Wahington, DC, USA, IEEE Computer Society. [18] G. Taubin. A ignal proceing approach to fair urface deign. In SIGGRAPH 95: Proceeding of the 22nd annual conference on Computer graphic and interactive technique, page , New York, NY, USA, ACM Pre. [19] G. Taubin. Linear aniotropic meh filtering. IBM Reearch Report RC2213, [20] S. Yohizawa, A. Belyaev, and H.-P. Seidel. Fat and robut detection of cret line on mehe. In SPM 05: Proceeding of the 2005 ACM ympoium on Solid and phyical modeling, page , New York, NY, USA, ACM Pre.

7 (a) Input noiy model (23964 triangle, vertice, edge) (b) Five iteration of our filter applied on the normal field (c) Initially etimated of urface normal (d) Normal etimate after aniotropic filtering on normal field (e) Initial etimation of curvature value and direction, the mall line troke how the minimum curvature direction (f) Curvature etimate after applying 5 iteration of our aniotropic filter on curvature tenor field Figure 4. Our aniotropic filter applied on the normal field and the curvature tenor field for the fan dik model.

8 (a) The original model(80354 triangle, vertice and edge) (b) Five iteration of our filter applied on the normal field (c) Initial etimated curvature value and minimum curvature direction (d) The curvature ditribution after applying five iteration of normal diffuion (e) The curvature ditribution after applying five iteration of curvature tenor diffuion uing error metric in equation 11 (f) The curvature ditribution after applying five iteration of curvature tenor diffuion uing error metric in equation 9 Figure 5. Five iteration of our filter applied on the normal field and then the curvature tenor field of a noiy rocker arm model (model courtey of Leif Kobbelt)

9 (a) Denoiing reult uing the bilateral filter [5] (5 iteration, σ f = 4, σ g = 4 ) (b) Curvature ditribution after applying the bilateral filter (c) Denoiing reult uing the adaptive moothing [12] (5 iteration or normal diffuion + 5 iteration on the hight field, c = 2) (d) Curvature ditribution after applying the adaptive moothing filter Figure 6. Comparion to the reult two other method for the hight field denoiing applied on the Fandik model. (a) Aniotropic denoiing on the Augmented Darboux frame uing the error metric in Equation 5 [9] (b) Iotropic denoiing on the Augmented Darboux frame under the curvature conitency framework [15] Figure 7. Comparion to the reult of two other method for the curvature field denoiing applied on the rocker arm model.

10 (a) The input noiy model (b) The reult of two iteration of our filter applied on normal field (c) The curvature value etimated uing initial perface curvature etimation (d) The byproduct of curvature value ditribution after applying 2 iteration of our filter on the facet normal (e) The curvature value ditribution after applying 2 iteration of our filter on curvature tenor field Figure 8. Illutration of our aniotropic filter applied on a general hape (data courtey of Alexander Belyaev).