Adaptive Fairing of Surface Meshes by Geometric Diffusion

Transcription

1 Adaptve Farng of Surface Meshes by Geometrc Dffuson Chandrajt L. Bajaj Department of Computer Scences, Unversty of Texas, Austn, TX Emal: Guolang Xu State Key Lab. of Scentfc and Engneerng Computng, ICMSEC, Chnese Academy of Scences, Bejng Emal: Abstract In trangulated surface meshes, there are often very notceable sze varances (the vertces are dstrbuted unevenly). The presented nose of such surface meshes s therefore composte of vast frequences. In ths paper, we solve a dffuson partal dfferental equaton numercally for nose removal of arbtrary trangular manfolds usng an adaptve tme dscretzaton. The proposed approach s smple and s easy to ncorporate nto any unform tmestep dffuson mplementaton wth sgnfcant mprovements over evoluton results wth the unform tmesteps. As an addtonal alternatve to the adaptve dscretzaton n the tme drecton, we also provde an approach for the choce of an adaptve dffuson tensor n the dffuson equaton. Key words: Adaptve dffuson, Loop s subdvson, Heat equaton. 1 Introducton The soluton for trangular surface mesh denosng (farng) s acheved by solvng a partal dfferental equaton (PDE), whch s a generalzaton of the heat equaton customzed to surfaces. The heat equaton has been successfully used n the mage processng for about two decades. The lterature on ths PDE based approach to mage processng s large (see [1, 10, 11, 17]). It s well known that the soluton of heat equaton t ρ ρ = 0, based on the Laplacan, at tme T for a gven ntal mage ρ 0 s the same as takng a convoluton of the Gauss flter G σ (x) = 1 2πσ ) exp ( x 2 2 2σ 2 wth standard devaton σ = 2T and mage ρ 0. Takng the convoluton of G σ and mage ρ 0 s performng a weghted averagng process to ρ 0. When the standard devaton σ become larger, the averagng s taken Supported n part by NSF grants ACI and KDI- DMS Currently vstng Center for Computatonal Vsualzaton, Unversty of Texas, Austn, TX Fg 1.1: The left fgure n the top row shows the ntal geometry mesh. The rght top fgure llustrates the result of the adaptve tmestep smoothng after 4 farng steps wth τ = The two fgures n the bottom row are the results of the unform tmestep t = smoothng after 1, and 4 farng steps, respectvely. over a larger area. Ths explans the flterng effect of the heat equaton to nosy mages. The generalzaton of the heat equaton for a surface formulaton has recently been proposed n [4, 5] and shown to be very effectve even for hgher-order methods [3]. The counterpart of the Laplacan s the Laplace-Beltram operator M (see [7]) for a surface M. However, un- 1

2 lke the 2D mages, where the grds are often structured, the dscretzed trangular surfaces are often unstructured. Certan regons of the surface meshes are often very dense, wth a wde spectrum of nose dstrbuton. Applyng a sngle Gauss-lke flter to such surface meshes would have the followng sde effects: (1) the lower frequency nose s not fltered (under-farng) f the evoluton perod of tme s sutable for removng hgh frequency nose, (2) detaled features are removed unfortunately, as hgher frequency nose (over-farng) f the evoluton perod of tme s sutable for removng low frequency nosy components. The bottom row of Fg 1.1 llustrates ths under-farng and over-farng effects. For the nput mesh on the top-left, two farng results are presented at two tme scales. The frst fgure exhbts the under-farng for the head. The second fgure exhbts the over-farng for the ears, eyes, lps and nose. Hence, a phenomena that often appears for the trangular surface mesh denosng s that whenever the desrable smoothng results are acheved for larger features, the smaller features are lost. Pror work has attempted to solve the over-farng problem by usng an ansotropc dffuson tensor n the dffuson equaton [3, 4]. However, ths s far from satsfactory. The am of ths paper s to overcome the under-farng and over-farng dlemma n solvng the dffuson equaton. There are several stuatons where the produced trangular surface meshes have varyng trangle densty. One typcal case s geometrc modelng, where the detaled structures are captured by several small trangles whle the smpler shapes are represented by fewer large ones. We may call such trangular meshes as featureadaptve. Another case s the results of physcal smulaton, n whch the researcher s nterested n certan regons of the mesh. In these regons, accurate solutons are desred, and qute often fner meshes are used. For example, n the acoustc pressure smulaton [15], the nterestng regon s the ear canal for human hearng. Hence, more accurate and fner meshes are used there. We call such meshes as error-adaptve. One addtonal case arses from the multresoluton representaton of surfaces, for example usng wavelet transforms or drect mesh smplfcatons. Each resoluton of the representaton s a surface mesh that approxmates the hghest resoluton surface. The approxmaton error s usually adaptve and can vary over the entre surface. For nstance, the mesh smplfcaton scheme n [2], whch s drven by the surface normal varaton, results n meshes that are both feature-adaptve and error-adaptve. Prevous Work. For PDE based surface farng or smoothng, there are several methods that have been proposed (see [3, 4, 5, 6]) recently. Desbrun et al n [5, 6] also use Laplacan, whch s dscretzed as the umbrella operator n the spatal drecton. In the tme drecton dscretzaton, they propose to use the semmplct Euler method to obtan a stable numercal scheme. Clarenz et al n [4] generalze the Laplacan to the Laplace-Beltram operator M, and use lnear fnte elements to dscretze the equaton. In [3], the problem s reformulated for 2-dmensonal Remannan manfold embedded n IR k amng at smooth geometrc surfaces and functons on surfaces smultaneously. The C 1 hgher-order fnte element space used s defned by the Loop s subdvson (box splne). One of the shortcomngs of all these proposed methods that we address here s ther non-adaptvty. All of them use unform tmesteps. Hence they qute often suffer from under-farng or over-farng problems. Our Approach. For a feature-adaptve or erroradaptve mesh, the deal evoluton strategy would be to correlate the evoluton speed relatve to the mesh densty. In short, we desre the lower frequency errors use a faster evoluton rate and the hgher frequency errors succumbs to a slower evoluton rate. To acheve ths goal, we present a dscretzaton n the tme drecton whch s mesh adaptve. We use a tmestep T (x) whch depends on the poston x of the surface. The part of the surface that s coarse uses larger T (x). The dea s smple and t s easy to ncorporate t nto any unform tmestep dffuson mplementaton. The mprovements acheved over the evoluton results wth unform tmestep are sgnfcant. The top row of Fg 1.1 shows ths adaptve tme evoluton mprovement over the unform tmestep evoluton results, shown n the bottom row. The rght top fgure s the smoothng result of the mesh on the left top after 4 farng steps. As an alternatve to the adaptve dscretzaton n tme drecton, we also provde an approach for the adaptve choce of the dffuson tensor n the dffuson PDE equaton. The remanng of the paper s organzed as follows: Secton 2 summarzes the dffuson PDE model used, followed by the dscretzaton secton 3. In the spatal drecton, the dscretzaton s realzed usng the C 1 smooth fnte element space defned by the lmt functon of Loop s subdvson (box splne), whle the dscretzaton n the tme drecton s adaptve. The concluson secton 4 provdes examples showng the superorty of the adaptve scheme. 2 Geometrc Dffuson Equaton We shall solve the followng nonlnear system of parabolc dfferental equatons (see [3, 4]): t x(t) M(t) x(t)) = 0, (2.1) 2

3 where M(t) = dv M(t) M(t) s the Laplace-Beltram operator on M(t), M(t) s the soluton surface at tme t and x(t) s a pont on the surface. M(t) s the gradent operator on the surface. Ths equaton s a generalzaton of heat equaton: t ρ ρ = 0 to surfaces, where s the Laplacan. To enhance sharp features, a dffuson tensor D, actng on the gradent, has been ntroduced (see [3, 4]). Then (2.1) becomes t x(t) dv M(t) (D M(t) x(t)) = 0. (2.2) The dffuson tensor D := D(x) s a symmetrc and postve defnte operator from T x M to T x M. Here T x M s the tangent space of M at x. The detaled dscusson for choosng the dffuson tensor can be found n [3, 4] for enhancng sharp features. In ths paper, we do not address the problem of enhancng sharp features. However, we shall use a scalar dffuson tensor for achevng an adaptve dffuson effect. The dvergence dv M ψ for a vector feld ψ T M s defned as the dual operator of the gradent (see [12]): (dv M v, φ) M = (v, M φ) T M (2.3) for any φ C 0 (M), where C 0 (M) s a subspace of C (M), whose elements have compact support. T M s the tangent bundle, whch s a collecton of all the tangent spaces. The nner product (φ, ψ) M and (u, v) T M are defned by the ntegraton of φψ and u T v over M, respectvely. The gradent of a smooth functon f on M s gven by M f = [ t 1, t 2 ]G 1 [ (f x) ξ 1, (f x) ξ 2 ] T, (2.4) where G 1 = 1 [ ] [ ] g 22 g 12 g11 g, G = 12, det G g 21 g 11 g 21 g 22 ( ) T and g j = ξ ξ j. x(ξ 1, ξ 2 ) s a local parameterzaton of the surface. G s known as the frst fundamental form. For a vector feld X = 2 =1 X ξ T M, an explct expresson for the dvergence s gven by (see [8], page 84) dv M X = 1 det G 2 =1 Then t s easy to derve that ξ ( det GX ). dv M (h M f) = ( M f) T M h + h M f, (2.5) where f, h are smooth functon on M. From (2.4), (2.5) and the fact that M x = 2H(x)n(x), we could rewrte (2.2) as t x(t) = D(x) + 2D(x)H(x)n(x), (2.6) where H(x) and n(x) are the mean curvature and the unt normal of M, respectvely. Equaton (2.6ples that the moton of the surface M(t) can be decomposed nto two parts: One s the tangental dsplacement caused by D(x), and the other s the normal dsplacement (mean curvature moton) caused by 2D(x)H(x)n(x). Usng (2.3), the dffuson problem (2.2) could be reformulated nto the followng varatonal form Fnd a smooth x(t) such that ( t x(t), θ) M(t) + (D M(t) x(t), M(t) θ) T M(t) = 0, M(0) = M, (2.7) for any θ C0 (M(t)). Ths varatonal form s the startng pont for the dscretzaton. We already know that the equaton (2.1) descrbes the mean curvature moton. Its regularzaton effect could be seen from the followng equaton (see [4], [13]) d dt Area(M(t)) = M(t) H2 dx, d dt Volume(M(t)) = M(t) Hdx, (2.8) where Area(M(t)) and Volume(M(t)) are the area of M(t) and volume enclosed by M(t), respectvely, H s the mean curvature. From these equatons, we see that the evoluton speed depends on the mean curvature of the surface but not on the densty of the mesh. Hence f the mesh s spatally adaptve, the dense parts that have detaled structures, have larger curvatures, whch very possbly be over-fared. 3 Dscretzaton We dscretze equaton (2.7) n the tme drecton frst and then n the spatal drecton. Gven an ntal value x(0), we wsh to have a soluton x(t) of (2.7) at t = T (x(0)). Usng a sem-mplct Euler scheme, we have the followng tme drecton dscretzaton: ( Fnd a smooth ) x(t ) such that x(t ) x(0) T, θ + M(0) (D M(0) x(t ), M(0) θ) T M(0) = 0, (3.1) for any θ C 0 (M(0)). If we want to go further along the tme drecton, we could treat the soluton at t = T (x) as the ntal value and repeat the same process. Hence, we consder only one tme step n our analyss. 3.1 Spatal Dscretzaton The functon n our fnte element space s locally parameterzed as the mage of the unt trangle T = {(ξ 1, ξ 2 ) IR 2 : ξ 1 0, ξ 2 0, ξ 1 + ξ 2 1}. 3

4 That s, (1 ξ 1 ξ 2, ξ 1, ξ 2 ) are the barycentrc coordnate of the trangle. Usng ths parameterzaton, our dscretzed representaton of M s M = k α=1 T α, Tα T β = for α β, where T α s the nteror of T α. Each trangular patch s assumed to be parameterzed locally as x α : T T α ; (ξ 1, ξ 2 ) x α (ξ 1, ξ 2 ). Under ths parameterzaton, tangents and gradents can be computed drectly. The ntegraton on surface M s gven by M fdx := α T f(x α (ξ 1, ξ 2 )) det(g j )dξ 1 dξ 2. The ntegraton on trangle T s computed adaptvely by numercal methods Fnte Element functon Space Let M d be the gven ntal trangular mesh, x, = 1,, m be ts vertces. We shall use C 1 smooth quartc Box splne bass functons to span our fnte element space. The pecewse quartc bass functon at vertex x, denoted by φ, s defned by the lmt of Loop s subdvson for the zero control values everywhere except at x where t s one (see [3] for detaled descrpton of ths). For smplcty, we call t the Loop s bass. Let e j, j = 1,, m be the 2-rng neghborhood elements. Then f e j s regular (meanng ts three vertces have valence 6), explct Box-splne expressons exst (see [14, 16]) for φ on e j. Usng these explct Boxsplne expresson, we derve the BB-form expresson for the bass functons φ. These expressons could be used to evaluate φ n formng the lnear system (3.3). If e s rregular, local subdvson s needed around e untl the parameter values of nterest are nteror to a regular patch. An effcent evaluaton method, that we have mplemented, s the one proposed by Stam [14]. Compared wth the lnear fnte element space, usng the hgher-order C 1 smooth fnte element space spanned by Loop s bass does have advantages. The bass functons of ths space have compact support (wthn 2-rngs of the vertces). Ths support s bgger than the support (wthn 1-rng of the vertces) of hat bass functons that are used for the lnear dscrete surface model. Such a dfference n the sze of support of bass functons makes our evoluton more effcent than those prevously reported, due to the ncreased bandwdth of the affected frequences. The reducton speed of hgh frequency nose n our approach s not that drastc, but stll fast, whle the reducton speed of lower frequency nose s not slow. Hence, the bandwdth of affected frequences s wder. A comparatve result showng the superorty of the Loop s bass functon s gven n [3]. Let V M(0) be the fnte dmensonal space spanned by the Loop s bass functons {φ } m =1. Then V M(0) C 1 (M(0)). Let x(0) = m =1 x (0)φ M(0), x(t ) = m =1 x (T )φ, and θ = φ j. Then equaton (3.1) s dscretzed n VM(0) 3 as m =1 (x (T ) x (0)) ( ) T 1 φ, φ j M(0) + m =1 x (T ) ( D M(0) φ, M(0) φ j )T = 0 (3.2) M(0) for j = 1,, m, where x (0) := x s the -th vertex of the nput mesh M d, T = T ( x (0)) and x (0) s a surface pont correspondng to vertex x (0). Equaton (3.2) s a lnear system for unknowns x (T ). 3.2 Adaptve Tmestep We frst use adaptve tmesteps to acheve the adaptve evoluton effect. In ths case the dffuson tensor D s chosen to be dentty, but T (x) s not a constant functon. Now (3.2) can be wrtten n the followng matrx form: (M + L)X(T ) = MX(0), (3.3) where X(T ) = [x 1 (T 1 ),, x m (T m )] T, X(0) = [x 1 (0),, x m (0)] T and M = ( (T 1 φ, φ j ) M(0),j=1, L = ( ( M(0) φ, M(0) φ j ) T M(0),j=1. (3.4) Note that both M and L are symmetrc. Snce φ 1, φ 2,, φ m are lnearly ndependent and have compact support, M s sparse and postve defnte. Smlarly, L s symmetrc and nonnegatve defnte. Hence, M + L s symmetrc and postve defnte. The coeffcent matrx of system (3.3) s hghly sparse. An teratve method for solvng such a system s desrable. We solve t by the conjugate gradent method wth a dagonal precondtonng. Defnng adaptve tmesteps. Now we llustrate how T (x) s defned. At each vertex x of the mesh M d, we frst compute a value d > 0, whch measures the densty of the mesh around x. We propose two approaches for computng t: 1. d s defned as the average of the dstance from x to ts neghbor vertces. 2. d s defned as the sum of the areas of the trangles surroundng x. To make the d s relatve to the densty of the mesh but not the geometrc sze, we always resze the mesh nto the box [ 3, 3] 3. The experments show that both approaches work well, and the evoluton results have no sgnfcant dfference. Ths value d s used as control 4

5 value for defnng tmestep that s the same as defnng the surface pont: T (x) = τd(x), D(x) = m d φ, (3.5) =1 where τ > 0 s a user specfed constant. Hence, T s a functon n the fnte element space V M(0). Note that snce T s not a constant any more, t s nvolved n the ntegraton n computng the stffness matrx M. Snce T (x) V M(0), t s C 2, except at the extraordnary vertces, where t s C 1. However, T (xay also be nosy, snce t s computed from the nosy data. To obtan a smoother T (x), we smooth repeatedly the control value d at the vertex x by the followng rule: d (k+1) = (1 n l )d (k) n + l j=1 d (k) j, (3.6) where d (0) = d for = 1,, m, d (k) j n the sum are the control values at the one-rng neghbor vertces of x, n s the valence of x, l and a(n ) are gven as follows: [ ( ) ] 2 1 l = n, a(n +3/8a(n ) ) = 1 5 n π 4cos n. The smoothng rule (3.6) s n fact for computng the lmt value of Loop s subdvson (see [9], pp 41-42) applyng to the control values d (k) at the vertces. In our examples, we apply ths rule three tmes. Experments show that even more tmes of smoothng of d are not harmful, but the nfluence to the evoluton results are mnor. The smoothng effect of (3.6) could be seen by rewrtng t n the followng form d (k+1) n l d (k) = 1 n n (d (k) j j=1 d (k) ). The left-handed sde could be regarded as the result of applyng the forward Euler method to the functon d (t), the rght-handed sde s the umbrella operator (see [5]). Hence, (3.6) s a dscretzaton of the equaton D t = D. Snce n l < 1, the stablty crteron for (3.6) s satsfed. A dfferent vew of adaptve tmestep approach Consder the followng dffuson PDE t x(t) D(x) M(t) x(t)) = 0, (3.7) where D(x) s a functon defned by (3.5). Agan, equaton (3.7) descrbes the mean curvature moton wth a compresson factor D(x). If we use a sem-mplct Euler scheme to dscretze the equaton wth constant tmestep τ, we could arrve at the same lnear system as (3.3). Hence, solvng the equaton (2.1) wth an adaptve tmestep τ D(x) s equvalent to solvng equaton (3.7) wth a unform tmestep τ. But (3.7ay be easer to handle n the theoretcal analyss. 3.3 Unform Tmestep and Adaptve Dffuson Tensor Now we use unform tmestep τ but a non-dentty dffuson tensor D(x). Ths D(x) s the same as the one defned n (3.5), but we should regard t as D(x)I, where I s the dentty dffuson tensor. The dscretzed equaton (3.2) then becomes m =1 (x (τ) x (0)) (φ, φ j ) M(0) + m =1 x (τ) ( τd M(0) φ, M(0) φ j )T M(0) = 0. (3.8) From ths, a smlar lnear system as (3.3) s obtaned wth M = ( (φ, φ j ) M(0),j=1, L = ( (τd M(0) φ, M(0) φ j ) T M(0),j=1. (3.9) We know that D(x) s a smooth postve functon that characterzes the densty of the surface mesh. The effect of ths dffuson tensor s suppressng the gradent where the mesh s dense, and hence slows down the evoluton speed. Comparng equaton (3.3) wth equaton (3.8), we fnd that they are smlar (snce τd = T ), though not equvalent. Indeed, f D(x) s a constant on each trangle of M, then they are equvalent. In general, D(x) s not a constant, but approxmately a constant on each trangle, hence the observed behavor of (3.3) and (3.8) are often smlar. The bottom row fgures n Fg 4.1 exhbt ths smlarty, where the left and rght fgures are the evoluton results usng an adaptve tmestep and an adaptve dffuson tensor, respectvely. Snce the results of the two adaptve approaches are very close, n the other examples provded n ths paper, we use only the adaptve tmestep approach. Homogenzaton Effect of D It follows from (2.6) that the non-constant dffuson tensor D(x) causes tangental dsplacement of the vertces. For the dffuson tensor D(x) defned n the last sub-secton, we know that t s adaptve to the densty of the mesh n the sense that t takes smaller values at denser regons of the mesh. Consder a case where a small trangle s surrounded by large trangles. In such a case, functon D(x) s small on the trangle and larger elsewhere. Ths mples that the gradent of D(x) on 5

6 the small trangle ponts to the outsde drecton, and the tangental dsplacement makes the small trangle become enlarged. If the densty of the mesh s even, then D(x) s near a constant. Then the tangental dsplacement s mnor. Hence, the adaptve dffuson tensor we use has homogenzng effect. Such an effect s nce and mportant, as t avods producng collapsed or tny trangles n the fared meshes. 3.4 Algorthm Summary For a gven ntal mesh, stoppng control threshold values ɛ > 0, = 1, 2 and τ > 0, the adaptve tmestep evoluton algorthm could be summarzed va the followng pseudo-code: Compute functon and dervatve values of φ on the ntegraton ponts; do { Compute d ; Smooth d by (3.6); Compute matrces M and L by (3.4); Solve lnear system (3.3); Compute H(t); } whle (none of (3.10) (3.11) s satsfed); Note that the evoluton process does not change the topology of the mesh. Hence the bass functons could be computed before the multple teratons. We use two of the three stoppng crtera proposed n [3] for termnatng the evoluton process: Let / H(t) = H(t, x) 2 dx H(0, x) 2 dx, M(0) M(t) where H(t, x) s the mean curvature vector at the pont x and tme td(x). The stoppng crtera are H (t) ɛ 1, or (3.10) H(t) ɛ 2, (3.11) where ɛ are user specfed control constants, H (t) s computed by dvded dfferences. 4 Conclusons and Examples We have proposed two smple adaptve approaches n solvng the dffuson PDE by the fnte element dscretzaton n the spatal drecton and the sem-mplct dscretzaton n the tme drecton, amng to solvng the under-farng/over-smooth problems that beset the unform dffuson schemes. The mplementaton shows that the proposed adaptve schemes work very well. Fg 4.1: The left top fgure s the ntal geometry mesh. The rght top fgure s the fared mesh after 3 farng teratons wth unform tmestep t = The left and rght fgures n the bottom row are the fared meshes after 3 farng teratons wth adaptve tmestep and alternatvely adaptve dffuson tensor wth unform tmestep τ = 0.025, respectvely. Fg 4.1 and Fg 4.2 are used to llustrate the dfference between the unform tmestep evoluton and the adaptve tmestep evoluton. Snce the adaptve tmestep s not unform, we cannot compare the evoluton results for the same tme. The comparng crteron we adopted here s we evolve the surface, startng from the same nput, to arrve at smlar smoothness for the rough/detaled features and compare the detaled/rough features. In Fg 4.1, the left fgure n the top row s the nput mesh, the rght top fgure uses unform tmestep, the left and rght fgures n the bottom row use adaptve tmestep and adaptve dffuson tensor wth a unform tmestep, respectvely. Comparng the three smoothng results, we can see that the 6

7 References Fg 4.2: The top fgure s the ntal geometry mesh. The second and the thrd fgures are the fared meshes after 2 and 4 farng teratons wth unform tmestep t = The last two are the fared meshes after 2 and 4 farng teratons wth adaptve tmestep and τ = large features look smlar, but the toes of the foot are very dfferent. The evoluton results of the adaptve tmestep and the adaptve dffuson tensor are much more desrable. Fg 4.2 exhbts the same effect. The top fgure s the nput, the next two are the results of the unform tmestep evoluton. Comparng these to the bottom two fgures, whch are the results of the adaptve tmestep evoluton, many detaled features on the back and the snout of the crocodle are preserved by the adaptve approach. Furthermore, the large features of the unform tmestep evoluton (compare the tals of the crocodles) are less farer than that of the adaptve tmestep evoluton, even though the detaled features are already over-fared. [1] Ed B. Harr Romeny. Geometry Drven Dffuson n Computer Vson. Boston, MA: Kluwer, [2] C. Bajaj and G. Xu. Smooth Adaptve Reconstructon and Deformaton of Free-Form Fat Surfaces. TICAM Report 99-08, March, 1999, Texas Insttute for Computatonal and Appled Mathematcs, The Unversty of Texas at Austn, 1999, [3] C. Bajaj and G. Xu. Ansotropc Dffuson of Nosy Surfaces and Nosy Functons on Surfaces. TICAM Report 01-07, February 2001, Texas Insttute for Computatonal and Appled Mathematcs, The Unversty of Texas at Austn, 2001, [4] U. Clarenz, U. Dewald, and M. Rumpf. Ansotropc Geometrc Dffuson n Surface Processng. In Proceedngs of Vz2000, IEEE Vsualzaton, pages , Salt Lake Cty, Utah, [5] M. Desbrun, M. Meyer, P. Schröder, and A. H. Barr. Implct Farng of Irregular Meshes usng Dffuson and Curvature Flow. SIGGRAPH99, pages , [6] M. Desbrun, M. Meyer, P. Schröder, and A. H. Barr. Dscrete Dfferental-Geometry Operators n nd, [7] M. do Carmo. Remannan Geometry. Boston, [8] J. Jost. Remannan Geometry and Geometrc Analyss, Second Edton. Sprnger, [9] C. T. Loop. Smooth subdvson surfaces based on trangles. Master s thess. Techncal report, Department of Mathematces, Unversty of Utah, [10] P. Perona and J. Malk. Scale space and edge detecton usng ansotropc dffuson. In IEEE Computer Socety Workshop on Computer Vson, [11] T. Preußer and M. Rumpf. An adaptve fnte element method for large scale mage processng. In Scale-Space Theores n Computer Vson, pages , [12] S. Rosenberg. The Laplacan on a Remannan Manfold. Cambrdge, Uvversty Press, [13] G. Sapro. Geometrc Partal Dfferental Equatons and Image Analyss. Cambrdge, Unversty Press, [14] J. Stam. Fast Evaluaton of Loop Trangular Subdvson Surfaces at Arbtrary Parameter Values. In SIG- GRAPH 98 Proceedngs, CD-ROM supplement, [15] T. F. Walsh. HP Boundary Element Modelng of the Acoustcal Transfer Propertes of the Human Head/Ear. PhD thess, Tcam, The Unversty of Texas at Austn, [16] J. Warren. Subdvson method for geometrc desgn, [17] J. Weckert. Ansotropc Dffuson n Image Processng. B. G. Teubner Stuttgart,