Polyhedral Surface Smoothing with Simultaneous Mesh Regularization

Transcription

1 olyhedral Surface Smoothng wth Smultaneous Mesh Regularzaton Yutaka Ohtake The Unversty of Azu Azu-Wakamatsu Cty Fukushma Japan Alexander G. Belyaev The Unversty of Azu Azu-Wakamatsu Cty Fukushma Japan Ila A. Bogaevsk Independent Unversty of Moscow Bolsho Vlasevsky ereulok Moscow, Russa Abstract A computer graphcs object reconstructed from real-world data often contans undesrable nose and small-scale oscllatons. An mportant problem s how to remove the nose and oscllatons whle preservng desrable geometrc features of the object. Ths paper develops methods for polyhedral surface smoothng and denosng wth smultaneous ncreasng mesh regularty. We also propose an adaptve smoothng method allowng to reduce possble oversmoothng. Roughly speakng, our smoothng schemes consst of movng every vertex n the drecton defned by the Laplacan flow wth speed equal to a properly chosen functon of the mean curvature at the vertex. Keywords: mesh smoothng and denosng, mesh regularty, Laplacan smoothng, mean curvature flow Introducton Recent advances n 3D data-acquston hardware call for development of methods to rapdly remove undesrable nose and small scale oscllatons from an rregularly trangulated data. Two the most popular approaches for smoothng and denosng of polyhedral surfaces are mnmzng energy functonals assocated wth dfferental-geometrc surface characterstcs and Laplacan smoothng. Mnmzng an energy functonal s usually a computatonally expensve task. Moreover, t lacks a local shape control. Laplacan smoothng s smple, fast, and so far the most common technque for mesh smoothng. The Laplacan smoothng flow, n ts smplest form, moves recursvely each vertex of the mesh by a dsplacement equal to a postve scale factor tmes the average of the neghborng vertces. Actually, the Laplacan smoothng flow can be consdered as the gradent descent flow for a smple quadratc energy functonal. However, due to ts smplcty, the Laplacan flow opens many ways for modfcatons and mprovements. Taubn n [8] proposed to alternate two scale factors of opposte sgns wth the negatve factor of larger magntude n a weghted Laplacan smoothng flow. Such smoothng does not produces shrnkage and suppresses hgh frequences of a dscrete Laplacan operator defned on the mesh, whle enhancng low frequences. When the scale factors are equal n magntude, the Taubn smoothng scheme turns to the blaplacan smoothng flow [4] whch can be consdered as a dscrete approxmaton of the gradent descent flow for the thn-plate energy functonal. Another non-shrnkng modfcaton of the Laplacan smoothng flow was very recently proposed n [9]. A sgnfcant development of the Laplacan smoothng method, mesh smoothng by the mean curvature flow [3, 6] (see also references theren), came from mathematcs and materal scence. The dscrete mean curvature flow moves every vertex n the normal drecton wth speed equal to a dscrete approxmaton of the mean curvature at the vertex. Smoothng by the mean curvature flow and ts varous modfcatons have became extremely popular n geometrc mage processng (see, for nstance, [6] for references). Laplacan smoothng, Taubn smoothng, and the dscrete mean curvature flow contan a number of drawbacks. The Laplacan smoothng flow ncreases the mesh regularty but develops unnatural deformatons whle beng appled to a hghly rregular mesh. Smoothng by the dscrete mean curvature flow s relatvely ndependent of the mesh samplng rate but ncreases the mesh rregularty. Both the Laplacan and mean curvature flows do not decelerate the smoothng process and may lead to oversmoothng and loosng desrable geometrc features. The Taubn smoothng scheme lacks a local shape control and enhances low frequency surface features. In ths paper we propose smple and effectve polyhedral

2 (a) (b) (c) (d) (e) (f) Fgure. The top row: (a) a polyhedral sphere; (b) the sphere wth a unform nose added; (c) Laplacan smoothng develops unnatural deformatons; (d) smoothng by the Taubn method converts hgh-frequency surface oscllatons nto low-frequency waves; (e) smoothng by the mean curvature ow ncreases the mesh rregularty; (f) smoothng accordng to a method proposed n ths paper, see Secton 3. The bottom row: (a) the Stanford bunny; (b) the bunny wth a unform nose added; (c) Laplacan smoothng wth a number of teratons chosen to acheve `a good lookng' result: extra teratons wll lead to oversmoothng; (d) Taubn smoothng reduces hgh-frequency surface oscllatons but enhances low-frequency oscllatons: extra teratons wll lead to enhancng of surface wrnkles; (e) smoothng by the mean curvature ow wth a number of teratons chosen to acheve `a good lookng' result: extra teratons wll lead to oversmoothng; (f) smoothng accordng to a method proposed n ths paper: the smoothng process slows down automatcally and extra teratons wll produce almost the same appearance, see Secton 4. surface smoothng schemes whch combne the best propertes of the Laplacan smoothng flow and dscrete mean curvature flow, often outperform best exstng smoothng methods, and, n addton, reduce possble oversmoothng. Fg. demonstrates some of our results. The paper s organzed as follows. In Secton 2, we ntroduce the Laplacan flow, the Taubn smoothng method, and the dscrete mean curvature flow. In Secton 3, our new technque whch combnes the best propertes of the Laplacan and dscrete mean curvature flow s explaned. In Secton 4, we descrbe our technque to reduce possble oversmoothng. We dscuss applcaton of smoothng methods for stable detecton of rdges and ravnes on a polyhedral surfaces n Secton 5. We conclude and sketch drectons for future research n Secton 6. mean curvature flow [3]. Laplacan Smoothng. Let us consder a trangulated surface and, for any vertex, let us defne the so-called umbrella-operator [4] U( )= w w, () where summaton s taken over all neghbors of, w are postve weghts. See Fg. 2. old 2 Laplacan Flow, Taubn Method, Blaplacan Flow, Mean Curvature Flow In ths secton we wll ntroduce and analyze four methods for polyhedral surface smoothng: Laplacan smoothng, Taubn smoothng [8], the blaplacan flow [4], and the n Fgure 2. new 2

3 The local update rule, old + U( old ) (2) appled typcally to every nner pont of the trangulated surface s called Laplacan smoothng of the surface. Here s a small postve number and the process (2) s executed repeatedly. The Laplacan smoothng algorthm reduces the hgh frequency surface nformaton and tends to flatten the surface. The weghts can be chosen n many dfferent ways. The smplest choce s to set the weghts equal to each other: w =, U 0 ( )= n, ; (3) where n s the number of neghbors. Another choce that produces good results [8] s set the weghts as the nverse dstances between and ts neghbors: U ( )= w w, ; w = k, k, : (4) Note that smoothng wth U 0 mproves the mesh samplng rate, whereas smoothng wth U worsens the rate. To demonstrate ths, let us consder a plane curve r(s) parameterzed by arclength parameter s. Consder three ponts on the curve A = r(s, ); O = r(s); B = r(s + ) wth dstances a = joaj and b = jobj between them. Let dr=ds = t and n = t? compose the Frenet frame at O, see Fg. 3. Smple manpulatons wth Taylor seres expansons n r(s) t O r (s ) r(s+ ) a b A B Fgure 3. and Frenet formulas show that ",,!,,! # 2 OA OB a + b a + = n b + t k + b, a 3 k0 + O(a; b) 2 + a, b 4 k2 + O(a; b) 2 ; (5) h 2,,!,,! OA + OB a 2 + b 2 = k + b3, a 3 = n + t 3(a 2 + b 2 ) k0 + O(a; b) 2 2(b, a) a 2 + b 2 + a3, b 3 4(a 2 + b 2 ) k2 + O(a; b) 2 : If, for example, pont O s located closer to A than to B, a < b, then, due to the tangent components n the above expansons, one step of Laplacan smoothng wth U 0 shfts O closer to B and one step of the Laplacan smoothng wth U shfts O closer to A. However Laplacan smoothng wth U 0 develops unnatural deformatons, see Fg. c (the top row). Taubn Smoothng. Taubn n [8] proposed to alternate two scale factors of opposte sgns wth the negatve factor of larger magntude n the Laplacan smoothng flow. Such smoothng suppresses hgh frequences of the umbrella operator (), whle preservng and enhancng ts low frequences [8]. Combnng the two successve steps of the Taubn method n one local update rule we arrve at, (, U)( + U) old = (6) = old, (, ) U( old ), U 2 ( old ); where >>0, U 2 s the squared umbrella operator U 2 ( )= w + w U( ),U( ): Accordng to [8], the best smoothng wth (6) s obtaned when U = U 0 or U = U. In our experments, Taubn smoothng wth U = U 0 works much better than wth U = U. However, even the frst Taubn smoothng scheme often produces poor results, see for example Fg. d (the top row) where we used =0:3 and =, = =0:. In Fg. d (the bottom row) the Taubn flterng scheme demonstrates a good performance. Nevertheless, one can note enhancng of low-frequency surface wrnkles. Blaplacan flow. The blaplacan flow, old + U 2 ( old ) s a dscrete analog of the steepest descent flow for the the thn-plate energy functonal [4]. It can be obtaned from the Taubn smoothng scheme f the postve and negatve scale factors are equal n magntude. The blaplacan flow does not enhance low-frequency surface features. In our mplementaton of the blaplacan flow we use the plan umbrella operator (3). 3

4 (a) (b) (c) (d) (e) (f) (g) (h) Fgure 4. (a) A torus wth consstng of two parts wth derent samplng rates; (b) a magned vew of a part of the torus; (c) the torus wth a unform nose added; (d) Laplacan smoothng deforms the ntal shape; (e) smoothng by the Taubn method reduces hgh-frequency surface oscllatons but develops low-frequency surface waves; (f) the blaplacan ow smoothes well but slghtly deforms the ntal shape; (g) the mean curvature ow smoothes well but produces rregular mesh; (h) smoothng accordng to (). Mean curvature flow. Recently t was reported [3] that the smoothng procedure, old + H( old ) n( old ); (7) where H s a dscrete verson of the mean curvature and n s the unt normal vector, produces better results than Laplacan smoothng (2). Accordng to [3], a good estmaton of the mean curvature vector at a vertex s gven by Hn =, ra 2A ; where A = A s the sum of the areas of the trangles surroundng. Calculatons [3] show that Hn( )= 4A (cot + cot )(, ); (8) where and are the two angles opposte to the edge, see Fg. 5. The two dmensonal analog of ths approxmaton s gven by the left hand-sde of (5), snce ",,!,,! # OA OB r(a + b) =ra + rb =, a + b + Fgure 5. Thus, smlar to U, smoothng by dscrete curvature flow (7), (8) worsens the mesh samplng rate. See Fg. e (the top row) where the mean curvature flow demonstrates a good performance n smoothng a nosy sphere but produces uneven dstrbuton of vertces. 3 Modfed Mean Curvature Flow Let us consder a famly of smooth surfaces S(u; v; t), where (u; v) parameterze the surface and t parameterzes the famly. We suppose (u; v) to be ndependent of t. Let us assume that ths famly evolves accordng to the evoluton 4

5 (a) (b) (c) (d) (e) (f) Fgure 6. (a) A polyhedral two-holed torus gven as a mesh havng rregular connectvty and consstng of parts wth derent samplng rates, a small nose s also added; (b) Laplacan smoothng mproves the mesh samplng rate but deforms the ntal shape; (c) smoothng by the Taubn method substantally deforms the ntal shape n unnatural way; (d) the blaplacan ow also deforms the ntal shape n an rregular manner. (e) the mean curvature ow produces rregular mesh; (f) smoothng accordng to (2), (3) produces a regular meshng surface whch shape s close to the shape produced by the mean curvature ow. v; t) = F n; S(u; v; 0) = S (0) (u; v); where n(u; v; t) s the unt normal vector for S(u; v; t), F s a speed functon, S (0) (u; v) s an ntal surface. The famly parameter t can be consdered as the tme duraton of the evoluton. Equaton (9) means that the surface S(u; v; t) moves along ts normals wth speed equal to F. Consder now the v; t) = F n + G t; S(u; v; 0) = S (0) (u; v); where t s a vector tangent to the surface and G s a gven functon. Note that the tangent speed component does not affect the geometry of the evolvng surface and changes only surface parameterzaton. Solvng (9) for a polyhedral surface by an explct Euler scheme s more stable for polyhedral surfaces wth unform mesh samplng rates and, therefore, allows to use larger tme steps to acheve faster smoothng. Note that a backward scheme (t was proposed and used for smoothng n [3]) s no better than Euler s scheme. For larger tme steps where Euler s scheme s unstable the backward scheme s naccurate [5]. Roughly speakng, our man dea of smultaneous mesh smoothng and regularzaton conssts of usng the normal speed component F n for polyhedral surface smoothng and the tangent speed component to mprove the mesh samplng rate. For polygonal curve evolutons ths dea was proposed and used n []. Let us use the dscrete mean curvature flow (7), (8) for smoothng and the Laplacan flow (2), (3) for mprovng the mesh samplng rate. One possble mplementaton s to take F n = H n and G t = C [U 0, (U 0 n) U 0 ] ; where U 0 s the umbrella vector (3), C s a postve constant, 5

6 (b) (c) (d) (a) (e) (f) (g) (h) () (j) Fgure 7. (a) A polyhedral surface consstng of two parts wth derent samplng rates. (b){(d) Deformaton of (a) nto a at patch by the Laplacan ow. (e){(g) Deformaton of (a) nto a at patch by the mean curvature ow. (h){(j) Deformaton of (a) nto a at patch by our method (2), (3). and the dot stands for the scalar product. One can see that U 0,(U 0 n) U 0 s the projecton of the umbrella vector onto the tangent plane. It leads to the local update rule n H( old )n( old )+ () + C U 0 ( old ), (U 0 ( old ) n( old )) U 0 ( old ) o, old + Smoothng by () produces shapes of the same qualty as the mean curvature flow but wth unform dstrbuton of vertces. See Fg. 4. One can also defne the parameter C as a functon of surface curvatures to acheve a hgher mesh samplng rate n the curved surface regons. However, accordng to our experments, a smlar smoothng scheme produces better results. Let m = U 0 =ku 0 k and s the angle between the mean curvature vector Hn and m : cos = m Hn=jHj. Vector m defnes a 3D analog of 2D medan drecton. Our basc dea s to move the vertces n the medan drecton such that the normal speed component s equal to the mean curvature. However, snce for saddle vertces the medan drecton vector m and the mean curvature vector H n may have opposte normal components (.e., > =2), see Fg. 8b, we use the followng flow where F = 8 >< >:, old + F( old ); (2) jhjm cos 2 H n, jhj m f cos >" f cos <," cos 0 f j cos j " (3) Here " s a small postve parameter. Geometrc deas behnd (2) are explaned n Fg. 8. If the normal and medan vectors are almost orthogonal to each other at a vertex (j cos j "), we do not move the vertex at all. 6

7 Hn θ medan drecton (a) m H cosθ m H m -cos θ -Hn tangent plane (a) (b) θ Hn H 2H n - m cos θ (b) Hn (c) Fgure 8. (a) The case when the medan and normal vectors le on the same sde from the tangent plane: movng n the medan drecton wth normal speed component equal to the mean curvature. (b) The case the medan and normal vectors le on the opposte sdes from the tangent plane may happen for saddle vertces. (c) Computaton of the speed vector n (b). (c) (d) Fgure 9. (a) The orgnal Stanford bunny; (b) the bunny wth a unform nose added; (c) smoothng by (2) (Laplacan and mean curvature ows produce almost the same results); (d) smoothng by (4) wth the same number of teratons as n (c). Accordng to our experments, choosng " = 0: produces good results ndependently of the mesh samplng rate. Our smoothng scheme (2) demonstrates better results than the Laplacan flow, the mean curvature flow, and the Taubn smoothng scheme, see Fg.. See also Fg. 7 where a polyhedral surface consstng of two parts wth dfferent samplng rates s smoothed by the Laplacan flow, the mean curvature flow, and our scheme (2) and (3). 4 How to Avod Oversmoothng Laplacan smoothng, the mean curvature flow, and our new technque presented n the prevous secton smooth polyhedral surfaces by suppressng hgh frequency surface oscllatons. Lackng a local surface control, they may lead to oversmoothng and, therefore, destroyng desrable surface features. To ncorporate a local control for smoothng and to reduce possble oversmoothng due to too large number of teratons, let us consder a smple but very useful modfcaton allowng to slow down the smoothng process adaptvely: wth ef( )= 8 < :, old + e F(old ) (4), kf( )k,t F( ) f kf( )k >T jjf( )jj 0 f kf( )k T where T s a postve threshold selected by a user. Accordng to ths modfcaton, the smoothng s performed only on those mesh vertces where kf( )k >T. See Fg. 9 to compare performance of smoothng schemes (2), (3) and (4) wth F( ) defned n (3). Of course the above smple modfcaton can be appled to the any smoothng flow consdered before. In our experments exposed n Fg. 9 the parameter T n (4) s the same for all vertces. It s natural to allow T be dependent on shape characterstcs at the vertces: T = T ( ). In our experments we defne the threshold T ( ) at the vertex as the arthmetc mean of the mean curvatures computed at the frst rng of neghbors of or at the frst and second rngs of neghbors of. Fg. 0 demonstrates 7

8 (a) (b) (c) (d) (e) (f) Fgure. (a) A Godzlla model. (b) - (f) The rdges (black) and ravnes (whte) detected on the Godzlla model smoothed by varous smoothng schemes. (b) Smoothng by the Laplacan ow. (c) Smoothng by the Taubn method. (d) Smoothng by the blaplacan ow. (e) Smoothng by the mean curvature ow. (f) Smoothng by our method (2), (3). our experments wth smoothng a Noh mask trangulated model reconstructed from a cloud of ponts. Accordng to our experments, the best smoothng strategy conssts of recomputng perodcally the threshold T ( ) for each vertex after a fxed number of smoothng teratons. Fg. f (the bottom row) shows advantages of such threshold recomputng scheme. 5 Applcatons (a) (b) (c) Fgure 0. (a) A trangulated Noh mask model reconstructed from a scattered data generated by a laser scanner system; (b) smoothng accordng to (4) wth constant T chosen manually; (c) smoothng accordng to (4) wth T ( ) equal to the arthmetc mean of the mean curvatures of at the rst and second rngs of neghbors of. We apply our smoothng technque (2), (3) to detect rdges and ravnes on a smooth surface approxmated by a trangular mesh. Let us defne the rdges as the locus of ponts where the maxmal prncpal curvature attans a postve maxmum along ts curvature lne and the ravnes as the locus of ponts where the mnmal prncpal curvature attans a negatve mnmum along ts curvature lne [2]. ractcal detecton of the rdges and ravnes on a surface nvolves estmaton of hgh-order surface dervatves and, therefore, requres a careful surface smoothng before detecton. It seems natural to choose a smoothng procedure mnmzng vertces drft over the surface. On the other hand, a smoothng scheme ncreasng the mesh regularty mproves curvature estmaton. For practcal detecton of the rdge and ravne vertces on the mesh, we frst smooth the mesh and then estmate the normal, the prncpal curvatures, and the prncpal drectons at the mesh vertces. We use the method proposed n [7]. To check whether a gven vertex s a rdge vertex, we fnd the ntersecton between the polygon composed the rng of the frst neghbors of and the normal plane generated by the normal vector and maxmal prncpal drecton 8

9 at. We estmate the maxmal prncpal curvature at the ntersecton ponts by lnear nterpolaton. Fnally we compare the the values of the maxmal prncpal curvature at and the ntersecton ponts. Some results of our experments wth varous smoothng schemes are shown n Fg. where the rdges are colored n black and ravnes are colored n whte. We thnk that our smoothng scheme produces one of the best results. 6 Concluson In ths paper we propose smple and effectve polyhedral surface smoothng schemes whch combne the best propertes of the Laplacan smoothng flow and dscrete mean curvature flow, are good for hghly rregular meshes, outperform exstng methods, and reduce possble oversmoothng. In future we are plannng to extend deas presented n the paper to varous dscrete mplementatons of the Laplacan and mean curvature flows. Also we want to examne how the developed methods can be enhanced by blaplacan flows and surface dffuson processes. We also hope to make our smoothng methods faster and more accurate by usng the mplct Crank-Ncolson method. Another nterestng drecton for future research s shape enhancement, the opposte operaton to smoothng. Smple nvertng of a smoothng flow s usually unstable. However, stablty can be acheved by ncorporatng a curvature-based local control smlar to that we developed n Secton 4. Our prelmnary results are encouragng. [4] L. Kobbelt, S. Campagna, J. Vorsatz, and H.-. Sedel. Interactve multresoluton modelng on arbtrary meshes. In Computer Graphcs (SIGGRAH 98 roceedngs), pages 05 4, 998. [5] R. Mohtar and L. Segerlnd. Accuracy-based tme step crtera for solvng parabolc equatons. In Modelng, Mesh Generaton, and Adaptve Numercal Methods for artal Dfferental Equatons, IMA Volume 75, pages Sprnger, 995. [6] J. A. Sethan. Level Set Methods and Fast Marchng Methods. Cambrdge Unv. ress, 999. [7] G. Taubn. Estmatng the tensor of curvature of a surface from a polyhedral approxmaton. In roc. ICCV 95, pages , 985. [8] G. Taubn. A sgnal processng approach to far surface desgn. In Computer Graphcs (SIGGRAH 95 roceedngs), pages , 985. [9] J. Vollmer, R. Mencl, and H. Muller. Improved laplacan smoothng of nosy surface meshes. Computer Graphcs Forum (roc. Eurographcs 999), 8(3):3 38, 999. Acknowledgments We thank the anonymous revewers for ther careful readng and helpful suggestons. Ila Bogaevsk s grateful to the Unversty of Azu where he spent one year as a vstng researcher n 998/99 when ths research was manly conducted. Thanks to Shn Yoshzawa for the Godzlla 3D data set. References [] A. G. Belyaev, E. V. Anoshkna, S. Yoshzawa, and M. Yano. olygonal curve evolutons for planar shape modelng and analyss. Internatonal Journal of Shape Modelng, to appear. [2] A. G. Belyaev and Y. Ohtake. An mage processng approach to detecton of rdges and ravnes on polyhedral surfaces. Submtted for publcaton, [3] M. Desbrun, M. Meyer,. Schröder, and A. H. Barr. Implct farng of rregular meshes usng dffuson and curvature flow. Computer Graphcs (SIGGRAH 99 roceedngs), pages ,