Feature-Preserving Mesh Denoising via Bilateral Normal Filtering

Transcription

1 Feature-Preservng Mesh Denosng va Blateral Normal Flterng Ka-Wah Lee, Wen-Png Wang Computer Graphcs Group Department of Computer Scence, The Unversty of Hong Kong Abstract In ths paper, we propose a feature-preservng mesh denosng algorthm whch s effectve, smple and easy to mplement. The proposed method s a two-stage procedure wth a blateral surface normal flterng followed by ntegraton of the normals for least squares error (LSE vertex poston updates. It s well-known that normal varatons offer more ntutve geometrc meanng than vertex poston varatons. A smooth surface can be descrbed as one havng smoothly varyng normals whereas features such as edges and corners appear as dscontnutes n the normals. Thus we cast featurepreservng mesh denosng as a robust surface normal estmaton usng blateral flterng. Our defnton of ntensty dfference used n the nfluence weghtng functon of the blateral flter robustly prevents features such as sharp edges and corners from beng washed out. We wll demonstrate ths capablty by comparng the results from smoothng CAD-lke models wth other smoothng algorthms. 1. Introducton The ease-of-use and affordable 3D scannng devces [12], [13] have been wdely used n varous doman of applcatons rangng from reverse engneerng to character modelng n anmaton producton. These models are often represented as trangular meshes as hardware graphcs cards are optmzed for trangle renderng. Despte of usng hgh-fdelty scannng, undesrable nose s nevtably ntroduced from varous sources such as local measurements, lmted samplng resoluton and algorthmc errors. Models extracted from CT or MRI [14] volumetrc data also result n detaled meshes wth sgnfcant amount of nose. Thus, detaled nosy models need to be smoothed or denosed before any subsequent mesh processng such as smplfcaton [15] and compresson [16] could be successfully appled. Mesh denosng has been an actve research area snce the poneer work done by Taubn [1]. Research efforts were ntally focused n surface farng n whch surface Fgure 1. Denosng a model: The leftmost column s the orgnal nose-free model. Gaussan nose s added n the mddle column. The mesh s smoothed wth our proposed algorthm. The top row shows the mesh detals whle the bottom row dsplays the mean curvature vsualzaton of the mesh. smoothness s enhanced at the expense of sharp features beng blurred [1], [3], [4]. Attentons are then turned to smoothng wth features preserved. Feature-preservng flterng technques used n 2D mage denosng are extended to address the denosng problems wth 3D mesh. Mostly notable technques are ansotropc dffuson [7], robust estmaton [18] and blateral flterng [9]. Researches [10], [18] have shown that these three technques do have strong connectons wth each other. In ths paper, we choose to frst apply blateral flterng on the surface normals and then evolve the mesh wth least squares error wth respect to the fltered normal feld. Though recent researches of blateral flterng on 3D mesh [5], [6] have delvered promsng results n general, preservatons of sharp edges and corners on CAD-lke models are stll far from satsfactory. Snce surface varatons are best descrbed wth the frst-order normal varaton, we propose to apply the blateral flterng on the surface normals frst and evolve the mesh wth least squares error update wth respect to the fltered normal feld. Our major contrbuton s the formulaton of the ntensty dfference on the normal feld so that blateral flterng on surface normals can be properly appled. The rest of the paper s organzed as follows: Secton 2 wll present a bref overvew of related work n the

2 lterature. The mesh notaton used n ths paper together wth the workng prncple of blateral flterng wll be explaned n Secton 3. In Secton 4, we wll explan the applcaton of blateral flterng on surface normals together wth our novel formulaton of ntensty dfference. We wll also dscuss how to update the mesh vertex postons based on the smoothed surface normals. We wll outlne our algorthm n pseudo-code. Results and comparson wth other smoothng algorthms are done n Secton 5 and fnally conclusons are drawn n Secton Related Work Mesh denosng has been an on-gong research problem and a wde varety of algorthms have been proposed. Taubn [1] poneered a sgnal processng approach to mesh smoothng based on the defnton of Laplacan on mesh. He proposed an λ µ algorthm whch uses alternatve sgned smoothng to prevent shrnkage problem assocated wth Laplacan smoothng. Desbrun [3] then proposed a geometrc dffuson algorthm whch performs smoothng n the normal drecton and nhbts vertex shft n flat regons. The rate of smoothng s determned by the mean curvature. Guskov et al. [4] ntroduced a smoothng applcaton from the desgn of a general sgnal processng framework based on subdvson. These algorthms mentoned so far are sotropc n nature. Thus, nose and salent features such as edges and corners are ndscrmnately smoothed. To address ths shortcomng, ansotropc dffuson schemes have been recently proposed. Taubn [2] proposed a two-phase lnear sotropc mesh flterng method. In hs approach, surface normals are frst fltered by applyng a rotaton determned by the weghted sum of neghborng surface normals. Then the vertex postons are updated by solvng a system of lnear equatons usng the least squares error method. He refers ths as ansotropc Laplacan smoothng as the weghts appled to the neghborng vertces are matrces rather than scalars. Smlarly, other researchers lke Yagou, Ohtake and Belyaev [11] perform medan smoothng on the surface normals frst and compute a mesh evoluton to match the new surface normal feld. Imagng denosng s a major research area n mage processng and computer vson. Recently, Fleshman et al. [6] and Jones et a. [5] have ndependently extended the blateral flterng [9] from mage denosng to mesh denosng and acheve satsfactory results. Fleshman approached the smoothng problem by teratvely movng the vertces n the normal drecton wth an offset determned from blateral flterng of the heghts of neghborng vertces over the tangent plane. On the other hand, Jones et al. compute the projectons of current vertces on neghborng tangent planes and apply blateral flterng to vertex predctons to obtan a robust estmate of vertex postons. The latest work from Hldebrandt et al. [19] not only consdered sharp edges preservaton but also protected and recovered non-lnear surface features through ther proposed prescrbed mean curvature flow. Tasdzen [8] ponted out that surface normals play an mportant role n surface denosng as surface features are best descrbed wth the frst-order surface normals. 3. Basc Concepts In ths secton, we wll frst defne the notaton used n ths paper for mesh representaton. Then the workng prncple of blateral flterng on 2D mage denosng s ntroduced Mesh Representaton Geometrcally, a trangle mesh s a pecewse lnear surface consstng of trangular faces pasted together along ther edges. The mesh geometry can be denoted by a tuple ( K, V, where K s a smplcal complex specfyng the connectvty of the mesh smplces and V = { v1,, v m } s the set of vertex postons defnng the 3 shape of the mesh n R. The three dfferent types of smplex are: 0-smplex, 1-smplex and 2-smplex. A 0- smplex, represented by v = { v }, s a vertex, a 1-smplex, e = { v, v }, s an edge and a 2-smplex, j f = { v, vj, vk}, s a face. In ths paper, we represent the vertces, edges and faces by ther correspondng ndces. Besdes, the terms face and trangle, surface and mesh are used nterchangeably Blateral Flterng The blateral flter s a nonlnear, feature preservng mage flter, proposed by Smth and Brady [17], and separately by Tomas and Manduch [9]. Although, the flter s ntally desgned to be an alternatve to ansotropc dffuson [7], recent researches demonstrate that t has close connectons wth robust estmaton and ansotropc dffuson [10], [18]. Followng the formulaton of Tomas and Manduch [9], the blateral flterng for an mage I ( u, at u = x, y, s defned as: coordnate ( I ( u = p N ( s( Wc p u W I( u I( p I( p ( u, Wc( p u Ws( I( u I( p N( u p (1

3 nˆ a nˆ nˆ a nˆ a nˆ a c a c nˆ b c b nˆ b nˆ b a nˆ b b (a (b (c Fgure 2. (a Nosy mesh. (b Normals wthn a unt crcle. (c Projectons of the normal dfferences ˆ na and ˆ nb along the nˆ drecton. where N( u s the neghborhood of u and defned to be the set of ponts { q : u q < ρ = 2 σ c }. The spatal smoothng functon s a standard Gaussan flter 2 2 x /2σc Wc ( x = e wth the standard devaton σ c and the nfluence functon s also chosen to be a standard 2 2 x /2σs Gaussan flter Ws ( x = e wth the standard devaton σ s. The output of the flter s the weghted average of the nput where the weght of each pxel s computed usng a standard Gaussan functon W c n the spatal doman multpled by an nfluence functon W s n the ntensty doman that decreases the weght of pxels wth large ntensty dfferences. Therefore, the value at a pxel u s nfluenced manly by pxels that are spatally close and have a smlar ntensty. Snce large ntensty dfferences are regarded as mage features and penalzed by the nfluence functonw s, so smoothng across features are nhbted. 4. The Algorthm In ths secton, we wll frst show how to apply blateral flterng on surface normals and our formulaton of the ntensty dfference. Then, we wll ntroduce the least squares error update of the vertex postons wth respect to the fltered normal feld Surface Normal Flterng We extend the blateral flterng appled on 2D mages to flter the surface normals of 3D trangular meshes. For a mesh face wth unt surface normal nˆ and centrod at pont c, the blateral fltered normal n at the face s defned as: Fgure 3. Left: Nosy pyramnd model. Mddle: Model smoothed wth Fleshman's blteral flterng on vertex postons (note the corrupted edges. Rght: Model smoothed wth our proposed algorthm. Sharp edges are preserved. n = where ( W ( c c W ( d ˆ n c j s j j j N( Wc( cj c Ws( dj j N( N = { j : j < ρ = 2σc }, (2 c c s the set of neghborhood faces j of face wth unt surface normal n ˆ j and d j s the ntensty dfference between the two face normals nˆ and n ˆ j. The ntensty dfference s defned to be the projecton of the normal dfference vector ˆ n j on the surface normal n ˆ,.e. d = nˆ nˆ (3 j j In Fgure 2a, face neghborhoods are consdered f the Eucldean dstance between the face centers are wthn a defned radus. In Fgure 2b and 2c, normal dfferences and ther projectons along the face normal nˆ are computed. Though the formulaton s smple, t provdes an effectve measure of the degree of dsperson of the

4 neghborhood face normals n ˆ j at the face normal n ˆ. Fgure 3 llustrates the advantage of applyng blateral flterng on surface normals over Fleshman s approach n whch blateral flter s used to determne the vertex postons from the heghts of the neghborng vertces defned over the tangent plane. Sharp edges along the pyramd model are properly preserved when usng our proposed algorthm. For each face, we use the blateral flterng n equaton (2 to compute the fltered normal n, then normalze to n and use t as the smoothed normal. The blateral flterng and normalzaton operatons can be terated to acheve a desred level of smoothng. Fgure 4 above shows the results from dfferent stages of the smoothng process. Fgure 4a s the orgnal cube wth surface normals dsplayed. Each vertex s then dsplaced along the normal by zero-mean Gaussan nose wth σ nose = 0.1 of the mean edge length as shown n Fgure 4b. In Fgure 4c, the nosy surface normals are smoothed wth the blateral flter. Notce that the fltered normals are very close to the orgnal normals. Ths llustrates that our formulaton of ntensty dfference together wth the blateral flterng s effectve at removng nose from surface normals whle preservng features such as edges and corners. The fnal smoothed model as shown n Fgure 4d s obtaned va least squares error (LSE update of the vertex postons. The LSE vertex postons update s dscussed n the next secton LSE Vertex Poston Update Snce a face normal should be perpendcular to the three edges of a trangular face, so once the fltered face normal n of the face s obtaned, the correspondng trangle vertces ( v, vj, v k are then updated under the followng famly of smultaneous lnear equatons: n ( vj v = 0 n ( vk vj = 0 n ( v vk = 0, (4 The system of equatons (4 n reconstructng the vertex postons wth respect to a gven feld of face normals has no soluton for general meshes accordng to the analyss from Taubn n hs paper [2]. He proposed to fnd the least squares soluton of equaton (4 whch s equvalent to mnmze the followng cost functon defned on the mesh (a (b (c (d Fgure 4. Blateral Flterng of Surface Normals ( 2 ψ ( v1, v2,, v = n ( v v, (5 m f j f {, j} f where f denotes the set of edges that consttute face f. The gradent of ψ ( v1, v2,, v m wth respect to v s 1 2 T m f f j j f F v ψ ( v, v,, v = 2 n n ( v v, where Fj denotes the set of faces that are adjacent to the edge {, j }. Based on ths metrc and dervatves, the vertex postons can be updated as j (6 v v + λ n n T ( v v, = 1,2,, m, (7 f f j * j f F j where λ s the teraton step sze defned n Taubn s paper [2]. Fgure 1d shows the smoothed mesh from the vertex postons update accordng to equaton (6 for the gven fltered surface normal feld n Fgure 1c. We can now state the overall mesh denosng algorthm n the followng pseudo-code: For each face normal flterng teraton N f : For each mesh face : Collect neghbor faces j s.t. cj c < ρ = 2σ c

5 Compute normalzed blateral fltered normal n at face For each vertex poston update teraton N v : For each mesh vertex : Compute new vertex poston wth LSE update The parameters of the algorthm are: σ c, σ s, λ, and the number of teratons N f and N v. Approprate values for the standard devatons σ s and σ s are crtcal n features preservaton. We follow the nteractve approach proposed by Fleshman n settng the value of σ c. User selects a face center on the mesh where the surface s expected to be smooth and defne a radus of neghborhood from that pont. Ths radus s then assgned to σ c. For the value of σ s, we set t to the standard devaton of the normal dfference projected on the surface normal at the selected pont wthn the defned radus. In our experment, we set the values for N f, Nv and λ to be 3, 10 and 0.01 n order to obtan qualty smoothng results. 5. Results We have mplemented the mesh denosng algorthm as descrbed n the prevous secton. All meshes are rendered wth flat shadng to show facetng. To demonstrate our feature preservng capablty on CAD models, we compare our results to the results of the mesh medan flterng from Yagou [11], blateral mesh flterng from Fleshman [6] and the mean curvature flow (MCF from Desbrun [3]. We summarze the mesh statstcs and the denosng tme collected on a 1.5GHz Pentum (M n Table 1. In Fgure 5 top row, we can see that our algorthm can delver qualty smoothng n mostly flat regon whle preservng the sharp edges of the tube model. Unlke the case wth Fleshman s algorthm, n whch large noses along the edges are mstakenly treated as features and result n corrupted edges. Though medan flterng by Yagou [11] can preserve the edges, crspy appearance s unavodable owng to the nature of order statstc-based flter. Sharp edges are completely smoothed out wth MCF as feature-preservng s not consdered n the orgnal algorthm desgn. Smlarly n Fgure 5 bottom row, we can see that our algorthm performs equally well n preservng corner features of the Fandsk model. Our ntensty dfference formulaton can not only offer a strong edge and corner preservng capablty, but also be able to smooth models wthout losng the fne detals. We test our algorthm on the Stanford Bunny model to llustrate ths pont. In the Fgure 6c, t can be seen that the detals around the eye regon are properly preserved and so are the detals near the nose, mouth and the leg. The mean curvature vsualzatons from Fgure 6d to 6c provde a better comparson between the orgnal and the smoothed model. Table 1. Mesh statstcs and denosng tmes. (a, (b, (c and (d are the denosng tmes collected from Fleshman's, Desbrun's, Yagou's and our algorthm accordngly. Faces/ Denosng Tme (seconds Model Vertces (a (b (c (d Tube 16128/ FanDsk 12946/ Concluson Blateral flterng s proven to be a robust and effcent denosng technque n 2D mages and 3D meshes. As surface normals are better n descrbng the surface varaton, our applcaton of blateral flter on surface normals can best preserve the sharp edges and corners whle delver promsng smoothng results. Ths applcaton s made possble wth our formulaton of the ntensty dfference whch helps n penalzng averagng of normals across surface features. We have shown that our algorthm s smple, easy to understand and relatvely effcent as compared wth other recently developed smoothng algorthms. References [1] G. Taubn, A Sgnal Processng Approach for Far Surface Desgn, SIGGRAPH 95 Conf. Proc., pp , Aug [2] G. Taubn, Lnear Ansotropc Mesh Flterng, Tech. Rep. IBM Research Report RC2213, Oct [3] M. Desbrun, M. Meyer, P. Schroder, and A.H. Barr, Implct Farng of Irregular Meshes Usng Dffuson and Curvature Flow, SIGGRAPH 99 Conf. Proc., pp , May [4] I. Guskov, W. Sweldens, and P. Schroder, Multresoluton Sgnal Processng for Meshes, SIGGRAPH 99 Conf. Proc., pp , Aug [5] T.R. Jones, F. Durand, and M. Desbrun, Non-teratve, Feature-Preservng Mesh Smoothng, ACM Trans. Graphcs, vol. 22, pp , July [6] S. Fleshman, I. Dror, and D. Cohen-Or, Blateral Mesh Denosng, ACM Trans Graphcs, vol. 22, pp , July [7] P. Persona and J. Malk, Scale-Space and Edge Detecton Usng Ansotropc Dffuson, IEEE Trans. Pattern Analyss and Machne Intellgence, pp , July [8] T. Tasdzen, R. Whtaker, P. Burchard, and S. Osher, Geometrc Surface Processng va Normal Maps,

6 Nosy tube model Fleshman algorthm Desbrun s algorthm Yagou s algorthm Our proposed algorthm Nosy fandsk model Fleshman s algorthm Desbrun s algorthm Yagou s algorthm Our proposed algorthm Fgure 5. Comparson of our method wth other smoothng algorthms. Both nosy tube and fandsk models are prepared wth Gaussan nose wth σ nose =0.1 of the mean edge length s added. Parameters n each algorthm are chosen to smooth the mostly flat regons effectvely. It can be seen that our algorthm can delver promsng smoothng qualty wth strong edge and corner preservng capablty. (a (b (c (d (e (f Fgure 6. (a Nose-free Stanford Bunny. (b Gaussan nose added. (c Smoothed model wth our proposed algorthm. (d,(e,(f Mean curvature vsualzaton of the correspondng mesh ACM Trans. Graphcs, vol. 22, no. 4, pp , [9] C. Tomas and R. Manduch, Blateral flterng for gray and color mages, Proc. IEEE Int. Conf. on Computer Vson, pp , [10] D. Barash, A fundamental relatonshp between blateral flterng, adaptve smoothng and the nonlnear dffuson equaton, IEEE Trans. Pattern Analyss and Machne Intellgence, vol. 24, no. 6, [11] H. Yagou, Y. Ohtake, and A. Belyaev, Mesh Smoothng va Mean and Medan Flterng Appled to Face Normals, Geometrc Modelng and Processng, pp , [12] M. Levoy, K. Pull, B. Curless, S. Rusnkewcz, D. Koller, L. Perera, M. Gnzton, S. Anderson, J. Davs, J. Gnsberg, J. Shade and D. Fulk, The Dgtal Mchelangelo Project: 3D Scannng of Large Statues, SIGGRAPH 00 Conf. Proc., pp , July [13] S. Rusnkewcz, O. Hall-Holt, and M. Levoy, Real- Tme 3D Model Acquston, ACM Trans. Graphcs, vol. 21, pp , [14] W.E. Lorensen and H.E. Clne, Marchng Cubes: A Hgh Resoluton 3D Surface Constructon Algorthm, SIGGRAPH 87 Conf. Proc., pp , [15] H. Hoppe, Progressve Mesh, SIGGRAPH 96 Conf. Proc., pp , [16] M. Deerng, Geometry Compresson, SIGGRAPH 95 Conf. Proc., pp , Aug [17] S.M. Smth and J.M. Brady, SUSAN-A New Approach to Low Level Image Processng, IJCV, vol. 23, pp , [18] M.J. Black, G. Sapro, D.H. Marmont, and D. Heeger, Robust ansotropc dffuson, IEEE Trans. On Image Processng, vol. 7, no. 3, pp , [19] K. Hldebrandt, and K. Polther, Ansotropc Flterng of Non-Lnear Surface Features, Eurographcs 04, vol. 23, no. 3, 2004.