Shape Optimization of Quad Mesh Elements

Transcription

1 Shape Optiization of Quad Mesh Eleents Yufei Li 1, Wenping Wang 1, Ruotian Ling 1, and Changhe Tu 2 1 The University of Hong Kong 2 Shandong University Abstract We study the proble of optiizing the face eleents of a quad esh surface, that is, resapling a given quad esh to ake it possess, as uch as possible, face eleents of soe desired aspect ratio and size. Unlike previous quad esh optiization/iproveent ethods based on local operations on a sall group of eleents, we adopt a global approach that does not introduce extra singularities and therefore preserves the original quad structure of the input esh. Starting fro a collection of quad patches extracted fro an input quad esh, two global operations, i.e. resapling and re-distribution, are perfored to optiize the nuber and spacings of grid lines in each patch. Both operations are forulated as siple optiization probles with linear constraints. Categories and Subject Descriptors (according to ACM CCS): Coputer Graphics [I.3.5]: Coputational Geoetry and Object Modeling Geoetric algoriths, languages, and systes Keywords: esh generation, quad esh optiization, esh eleent optiization, eleent aspect ratio 1. Introduction Quad eshes are used in any applications, such as B-spline surface fitting, texture synthesis [4], finite eleent analysis [17, 18] and odeling of architectural surfaces [14]. One iportant consideration about the quality of a quad esh is the shape of its face eleents. We study the proble of optiizing the shape, i.e. aspect ratio and size, of the face eleents of a quad esh according to soe pre-specified requireents, without changing the overall quad structure of the esh, that is, no extra singularities are introduced. The ain challenge of quad eleent optiization is due to the global nature of the esh connectivity. Local optiization intended to iprove the eleent shape in one region often causes ore poorly shaped eleents elsewhere. We propose a global optiization fraework for coputing esh eleents of desired aspect ratio and size for quad eshes. Refer to Fig. 1. Given an input quad esh in Fig. 1a, we first extract the patch coplex of the esh as shown in Fig. 1b. Then we forulate the aspect ratio optiization proble as an optial grid line sapling proble of the extracted quad patches. Two operations, re-sapling and re-distribution, are introduced for deterining the optial nuber and spacings of the grid lines in each patch. The final esh is constructed with the resulting grid lines, as shown in Fig. 1c. Eleent size can also be controlled by specifying desired grid line density (see Fig. 1d and 1e). When seeking an optial shape of quad eleents, we distinguish the isotropic case where all eleent aspect ratios are expected to be 1, and the anisotropic case where the eleent aspect ratios should atch soe target aspect ratios specified by users, such as conforation with surface curvatures. For the reainder of the paper, we shall explain the proble and solution ainly for the isotropic case and shall only point out at the end how to apply the sae idea to the anisotropic case Related work Quad esh optiization/iproveent Quad esh generation is a research field that has been extensively studied. Iproving the shape of esh eleents is an iportant aspect of quad esh generation. Canann et al [6] cobine Laplacian and optiization-based soothing to iniize eleent distortion. Zhang et al [23] iprove the eleent aspect ratio of quad esh by lo- Preprint subitted to SMI 2011 March 22, 2011

2 (c) (d) (e) Figure 1: An input esh. The extracted quad patch coplex. (c) The optiized esh with siilar density to the input. (d) The optiized esh at a coarse level. (e) The optiized esh at a coarser level. cally adjusting vertices in the tangent plane of the input surface. Other topology-based ethods for iproving quad finite eleent eshes include [12, 20, 5, 19]. All these ethods are based on local optiization and cannot, in general, achieve satisfactory results. 2 Isotropic quadrangulation Much research has been done for generating high-quality quad eshes directly fro triangular eshes in recent years. The reader is referred to [2, 9] for detailed surveys. There are several paraetrization-based ethods for generating sei-regular quad eshes with a sall nuber of singular vertices [8, 10, 21, 16, 11, 7, 3, 15]. The seinal work in [8] coputes a quad esh based on the Morse- Sale coplex of Laplacian eigenfunctions coputed on input surface. Their ethod was extended by [10] to provide explicit controls of the orientation and alignent of the quad eleents. The ethod in [21] eploys a user-designed singularity graph as the highlevel layout and creates quadrilateral tiling by scalar function isolines. Ray et al [16] propose a globally sooth paraetrization that is aligned with orthogonal input vector fields defined over the input surface, and deonstrated an application to quad-doinant reeshing. Kälberer et al [11] introduced an autoatic coputation algorith of global paraetrization by converting a cross field into a single vector field on a branched covering. Daniels et al [7] propose a siplificationbased technique for coputing the sei-regular quadonly reeshing fro siplified base doains. More recently, Boes et al [3] developed a quadrangulation ethod based on ixed integer optiization, which first constructs a feature-aligned sooth cross field and then coputes a sealess globally sooth paraetrization following the cross field. To further reduce the nuber of patches, Myles et al [15] propose an approach to constructing quadrilateral T-eshes while aintaining good feature alignent. Anisotropic quadrangulation It is well known that anisotropic eshes provide better approxiation quality and are preferred in soe applications. Alliez et al [1] introduced a technique of anisotropic quaddoinant reeshing by explicitly coputing integral lines of the principal direction fields. Kovacs et al [13] developed an anisotropic quadrangulation technique by cobining curvature-based surface etric into the paraetrization. Zhang et al [22] propose to reesh a surface into anisotropically sized quads by constructing a special standing wave on the surface Contributions Although any ethods entioned above could produce high-quality quad eshes with nicely shaped eleents, no one explicitly address the issue of optiizing eleent aspect ratios of a given quad esh. To fill the gap, we consider the proble that takes as input a quad esh whose esh eleent shapes need to be optiized to eet certain requireents and propose a novel and siple ethod for solving this proble. Unlike soe previous ethods which involve global paraetrization, our ethod avoids solving large scale linear systes. Both operations are forulated as siple optiization probles with linear constraints, which can be solved efficiently. We also provide analysis to the solution space for gaining insights into constraints of the proble. The fraework is flexible in that it works for both isotropic and anisotropic cases. Our ethod can be integrated with soe other quadrangulation ethods or

3 used as a post-processing step. 2. Preliinaries 2.1. Definitions Given an oriented quad esh surface M, an interior vertex of M is regular if its degree is 4; otherwise, it is a singular vertex. A boundary vertex is regular if its degree is 3; otherwise, it is singular. A poly-line is a polygon, open or closed, consisting of esh edges and satisfying the following properties: (1) any two consecutive edges of the poly-line are two opposite edges incident to a regular esh vertex; (2) if it is an open polygon, then its two endpoints are either singular vertices or boundary vertices. A singular poly-line is a poly-line such that either (1) it is a closed polygon and exactly one of its vertex is singular (e.g. the closed red curve in Fig. 2a); or (2) it is an open polygon and at least one of its endpoints is a singular vertex (e.g. the open red curves in Fig. 2a and 2b). A poly-line that is not singular is a regular poly-line (e.g. the closed blue curve in Fig. 2a and the open blue curve in Fig. 2b). Note that a regular poly-line ay be self-intersecting, as shown in Fig. 2d. the set of patch edges and F the set of patches. Clearly, these patches are quad patches, since all their interior vertices are regular. ei f e j Figure 3: The dashed lines are regular poly-lines of the sae faily. Three failies of regular poly-lines for a quad esh of a cube are shown in three different colors. (d) A faily of regular poly-lines intersect theselves at patch f. The set of regular poly-lines constructed by connecting the parallel grid lines in a sequence of adjacent patches are grouped in the sae faily (see Fig. 3a). All the failies of regular poly-lines copose a set Z. For exaple, in the quad esh of a cube in Fig. 3b there are totally three failies of regular poly-lines ( Z =3). Note that the regular poly-lines of the sae faily are locally parallel, but ay intersect theselves if considered globally (see Fig. 3c). a f (c) R=H/W b d H c (c) W Figure 2: The two red curves are singular polylines, and the blue curve is a closed regular poly-line. The red curve is a singular poly-line with one end being a boundary vertex, and the blue curve is an open regular poly-line. (c) The patch coplex of a closed esh. (d) The blue curve is a single regular poly-line that intersects itself any ties. A quad esh is segented by all its singular poly-lines into a collection of patches (see Fig. 2c), to be called a patch coplex and denoted by K = (V, E, F), where V is the set of nodes (i.e. corner vertices of the patches), E 3 Figure 4: A general quad eleent. The nodes of a patch coplex are bi-colored. (d) The definition of the aspect ratio of a quad patch, with a blue node at the origin. The aspect ratio of a rectangle is the ratio of the lengths of its two adjacent sides. For a quad eleent as shown in Fig. 4a, which is not necessarily a rectangle, we define its aspect ratio to be the ratio of a + c to b + d. There are two ways to denote this ratio either r = (a + c)/(b + d) or r = (b + d)/(a + c). We need to ake a convention as which of the two is used for our subsequent discussions. For this purpose, we first explain how to define the aspect ratio of a quad patch in a globally consistent anner. Since each patch is quad, all closed paths of the graph of the patch coplex have an even nuber of edges. It follows that the patch coplex is a bi-colored graph,

4 that is, all of its nodes can be colored with two colors, yellow and blue say, as shown in Fig. 4b, in such a way that the colors of any two adjacent nodes are different. Then for each patch with its nodes colored as such (see Fig. 4c), we define its aspect ratio R = H/W, where W, H are the average width and height of the patch (to be defined ore precisely in Section 4.2). Finally, the aspect ratio r of a quad eleent is defined in a anner consistent to its belonging patch Outline of ethod 3. Patch coplex extraction Figure 6: Trace singular poly-lines fro each singular vertex. Group regular poly-lines into different failies. (c) Figure 5: A quad patch f. Deterine the nuber of grids in the two directions. (c) Optiize the spacings of the grid lines. Consider a patch f in a patch coplex of a quad esh (see Fig. 5). The grid lines in horizontal and vertical directions define a quad esh over f, as shown in Fig. 5b. The size and aspect ratio of the eleents are dictated by the following two factors: (1) the nuber of grid lines in the two directions; and (2) the spacings of the grid lines in the two directions. Therefore, after extracting the patch coplex fro an input quad esh, our ethod for esh eleent optiization works in two stages: grid line re-sapling and grid line re-distribution. Specifically, the whole algorith has the following steps: 1. Patch coplex extraction: Construct quad patch coplex K fro the input quad esh M. 2. Grid line re-sapling: Deterine the nuber of grid lines (or equivalently the nuber of grids) for each patch by solving a linearly constrained optiization proble (see Fig. 5b). The optiization target is that the aspect ratios of the esh eleents in each patch confor with a specified aspect ratio. 3. Grid line re-distribution: Given the nuber of grid lines for all patches, as deterined fro the last step, optiize the spacings of these grid lines to achieve unifor distribution of eleent aspect ratios on each patch (see Fig. 5c). This gives the final quad esh to output. 4 The first step of patch coplex extraction is straightforward. Fro any quad esh M we can produce the coarsest esh of quad patches by extracting the collection of paths that include all singular vertices. More specifically, starting at every singular vertex we follow each of its outgoing edges until we reach another singularity or surface boundary. Consecutive edges in each path should not share a face, see Fig. 6a for exaple. The regions bounded by these paths for a patch coplex K. For grouping the regular poly-lines into different failies, we start fro all the grid lines of any patch in one direction, and keep traversing its neighboring patches along the parallel grid lines until we reach the starting patch or surface boundary. All the grid lines in the path construct a faily of regular poly-lines. This process is successively perfored until all grid lines of all patches are visited, see Fig. 6b for exaple. 4. Grid line re-sapling In this section we discuss the re-sapling operation, that is, deterining the optial nuber of grid lines for each patch. Note that we will consider only the isotropic case in this section, that is, the target aspect ratios for all esh eleents are Forulation of objective function Although the target aspect ratios are 1, in general we can only ake all eleent aspect ratios as close to 1 as possible but not exactly equal to 1, due to the esh

5 connectivity constraints. Hence, we define our objective function as F ( ) 2 J = log ri (1) r i is the aspect ratio of eleents in patch f i. Clearly, the iniu of J is non-negative, and it is 0 if and only if all the r i are equal to 1. Of course, there are other possible objective functions that could serve the sae purpose, such as J = i(r i 1) 2, for exaple. However, it will becoe clear shortly that the objective function in (1) leads to a siple quadratic optiization proble with linear constraints, which is uch easier to treat than with other objective functions Constraints Considering a quad patch f with n eleents q i j, i = 1, 2,..., and j = 1, 2,..., n. Suppose every eleent q i j in f has the sae aspect ratio r, i.e. h i j /w i j = r, w i j and h i j are the n q ij w ij h ij h ij/w ij= r Figure 7: /n = R/r width and height of q i j. Define the average width W and height H of the patch f as W = 1 w i j, H = 1 n Then the aspect ratio of the patch f is defined as R = H/W. On the one hand, n h n i j rw i j n = w n = r i j w i j On the other hand, the left hand side of the above equation is n ( h i j ) ( n w i j ) = nh W = n R Hence, n = R r h i j (2) 5 We are going to show that every interior node of the patch coplex iposes a constraint on the aspect ratios. Consider a node v V of the coplex as shown in Fig. 8, which is adjacent to three patches f i, f j, f k, for exaple. Suppose f i, f j and f k consists of n, n l and n l f i v f j f k l n Figure 8: r i r j r k = R i R j R k l eleents, whose aspect ratios are r i, r j and r k respectively. It follows fro (2) that n = R i r i, n l = R j r j, Cobining these equations yields r i r j r k = R i R j R k l = R k r k This is the constraint iposed by the node v. The case where there are ore than three patches incident to a node v can be analyzed in the sae way. In general, for every non-boundary node v V, we have r i = R i = log r i = log R i (3) where Ω(v) is the set of patches incident to node v. Note that a boundary node does not ipose any constraints. Now, denoting x i = log r i, fro (1) and (3), we ay forulate the re-sapling proble as the following optiization proble s.t. F in J = x i = where M is the boundary of M. x 2 i log R i, (v V) (v M ). (4) This is a linearly constrained quadratic optiization proble; the x i = log r i are variables and the R i are known quantities. Due to its siplicity, this proble can efficiently be solved to obtain the optial aspect ratios ri for all the patches. The desired nubers of grids for all patches are related by these optial aspect ratios ri. For exaple, supposing a patch f consists of n eleents with optial aspect ratio r, one ay first specify the value of according to soe requireent on the size of esh eleents. Then n will be

6 deterined by Eqn. (2). Then, in turn, the nubers of grids for all other patches can be deterined in the sae way. This process is illustrated with the following exaple. Exaple:. Suppose that the optial aspect ratios ri have been obtained for the six patches of a cubical quad esh in Fig. 9a. The user first specifies for the nuber of grids in one direction of patch f i. Note that this actually deterines the nuber for a faily of regular poly-lines since T-vertices are not allowed on patch boundaries. Then in Fig. 9b, the nuber of grids in another direction of f i is deterined through ri by n = ri /R i (cf. Eqn (2)). Finally, in Fig. 9c, the optial nuber l is deterined through r j by l = n r j /R j. Then the grid nubers for all the six patches are obtained. * n* f i Figure 9: The user first specifies. n is deterined through r i. (c) l is deterined through r j Analysis of solution space The constraints in (4) raise an issue about the existence of solutions. We now show that the diension of the solution space of the x i is not null. For siplicity, first suppose that the input quad esh is a closed surface. The constraints in (4) is a syste of linear equations with V equations and F unknowns. Since each patch is a quad, we have E = 2 F. Then, by Euler s forula V E + F = 2 2g, we have V F = 2 2g, where g is the genus of the input quad esh surface. Take the case of g = 0 for exaple. In this case, since V F = 2, there are ore equations than unknowns in the constraints, which eans that the linear syste could be over-deterined. Then the constraint space would be null if the constraints are independent. Fortunately, these constraints are always dependent, as can be seen as follows. Due to the special structure of the constraints, there always exists a solution x i = log R i (i.e. r i = R i ), which can be interpreted as trivially sapling the sae nuber of rows and coluns = n for every patch. For a quad esh with only one faily of f j (c) l* 6 regular poly-lines, x i = log R i is the only solution. For a quad esh with Z > 1 failies of regular poly-lines, one is free to independently choose the nuber of regular poly-lines for all the Z failies. This gives rises to Z 1 degrees of freedo for the aspect ratios r i that satisfy the constraints. Hence, the diension of solution space of the linear syste in (4) is at least Z 1. We shall use two exaples to explain the above analysis. First consider the quad esh of a cube in Fig. 9. For this exaple, it can be verified nuerically that the diension of solution space is 2. This agrees with the intuitive understanding that only two aspect ratios r 1 and r 2 of quad patches on two adjacent sides of the cube can be specified freely; those for the other patches will be deterined once r 1 and r 2 are fixed. Figure 10: A quad esh with only one faily of regular poly-lines, depicted with one of its regular and self-intersecting poly-line. The corresponding abstract patch coplex of the esh in. The second exaple is shown in Fig. 10. It is a quad esh that has only one faily of regular poly-lines, that is, Z = 1, and every patch is traversed twice by this sae faily. In this case, the solution space has diension Z 1 = 0, and there is no degree of freedo to specify the aspect ratio of any face eleents resapling regular poly-lines of this faily affects siultaneously the width and length of a quad patch. 5. Grid line re-distribution In this section, we present the re-distribution operation for optiizing the distribution of grid lines in ters of aspect ratios, supposing that the nuber of grid lines for each patch has been coputed as described in the last section. Considering the grid line connectivity constraint between the patches, the operation is defined on each faily z Z of regular poly-lines instead of being restricted in one single patch. We shall first define an

7 objective function in ters of the regular poly-line spacings to be found and then develop an iterative ethod for solving the proble Proble forulation Consider a faily of regular poly-lines z Z in Fig. 11. It divides the corresponding patches into strips s i and s i q ij d i W Figure 11: Optiize the spacings of a faily of regular poly-lines. each strip s i contains a sequence of n eleents q i j, i = 1, 2,..., and j = 1, 2,..., n. To easure the spacings of the regular poly-lines, we ay take any of its cross section (see Fig. 11). Suppose that the total width along this section is W and the widths of the strips s i are d i. Then, clearly, d i = W. d z = (d 1, d 2,..., d ) T are the unknown spacings (i.e., basic variables) to be deterined via optiization. Note that the choice of the cross section is arbitrary, which does not affect the re-distribution result. Take the eleent aspect ratio as its weight, and then the weight of each strip s i equals to t i = n r i j, where the aspect ratios r i j of eleents q i j depend on the spacings d z. To achieve unifor distribution of the eleent aspect ratios r i j, we require that all the t i be equal to each other, i = 1, 2,...,, for any z Z, which is equivalent to t i = t 1 k=1 t k. Therefore, we define the error ter for the faily z to be E(d z ) = (t i t) 2. Finally, letting D denote the collection of variables d z for all z Z, the su of these error ter E(d z ) over all the failies z Z gives the objective function, i.e. G(D) = E(d z ) (5) 5.2. Optiization algorith z Z Starting fro soe initial spacings of the regular polylines of all the failies, we iniize the objective function G(D) in Eqn. (5) by alternatively applying a gradient-descent step to all the failies. That is, in each iteration, we adopt a ultiple-pass approach, with each pass processing only one of the failies. Or, in other words, in each pass we iniize G(D) only along the direction of a subset of variables, d z D, corresponding to the regular poly-line spacings of z, while fixing the spacings of the other failies. 7 The coputation in each pass is carried out as follows. The weight t i of the strip s i can be expressed as t i = r i j = where κ i j = h i j d i h i j w i j = κ i j w i j d i = α i d i, and α i = n κ i j w i j. Here we ake use of the fact that, in the first order of approxiation, the average height h i j of q i j is linearly related to d i ; and we use κ i j to denote this linear proportion. Clearly, the error ter E(d z ) collects all the ters in G(D) that are related to the spacings d z. Hence, in each pass, we just need to iniize E(d z ), which leads to the following syste of + 1 linear equations α 1 d 1 1 α i d i = 0 α 2 d 2 1 α i d i = 0... α d 1 α i d i = 0 d 1 + d d = W where the last equation represents the constraints on d z. It is easy to see that the first equations are linearly dependent but any 1 of the are linearly independent. Hence, plus the last equation, the whole syste has a unique solution d z = (d 1, d 2,..., d n), which deterines the regular poly-line spacings of the faily z. Figure 12: Re-distribution operation: An input esh with two regular poly-line failies z 1 and z 2. Optiize the spacings of z 1. (c) Optiize the spacings of z 2. This procedure of iproving the regular poly-line spacings d z is applied successively to all the failies in ultiple passes until convergence. The terination rule is that iproveent increents t for all failies drop beneath a user-specified bound σ. Then the final quad esh is constructed using these optially deterined regular poly-line spacings di. The new regular polylines are produced by linearly interpolating the input ones. This is illustrated by the siple exaple involving two failies in Fig. 12. z 1 z 2 (c)

8 6. Anisotropic Optiization As discussed in [1], anisotropic eshes are preferred in certain applications, such as flow siulation. In addition, eshes with desired anisotropy have better approxiation quality than isotropic eshes. It has been proven in approxiation theory that a quad esh with a given nuber of eleents provides best approxiation of a sooth surface if its face eleents are oriented with the curvature tensor and with an aspect ratio equal to κ 1 / κ 2 (κ 1 and κ 2 are principal curvatures). Given an input quad esh with face eleents oriented with the curvature tensor, which can be generated by explicit ethods such as [1], we now explain how to adapt our optiization fraework to the anisotropic case, that is, resaple the input esh to ake its eleents with the desired aspect ratio ρ = κ 1 / κ 2. Patch coplex extraction This step is exactly the sae as in the isotropic case. Grid line re-sapling Consider a quad patch f with n eleents q i j, and define its desired anisotropy as A = 1 n Then the objective function for re-sapling is obtained by adapting the objective function (1) into ρ i j F ( r i ) 2 J = log (6) A i And the constraint syste reains the sae as in (4). Grid line re-distribution The proble forulation reains the sae as in (5) with the only odification of cobining desired aspect ratio into eleent weight, i.e. define the weight of q i j as r i j /ρ i j. Accordingly, the weight of strip s i is t i = n r i j ρ i j. This odification eans to achieve a unifor distribution of the eleent weights r i j /ρ i j. If J = 0 in (6) is achieved in the last re-sapling step, r i j /ρ i j is nearly 1 (i.e. r i j ρ i j ) for all eleents. A siple exaple of anisotropic optiization for a surface of revolution is shown in Fig Experiental results and discussions In this section we present experiental results. All the experients were conducted on a laptop PC with 2.4 GHz Intel(R) Core(TM)2 CPU and 4GB eory. 8 Figure 13: Anisotropic optiization of a surface of revolution. The color of each face eleent encodes its noralized weight [ r ρ ] = ax( r ρ, ρ r ). Feature preserving To preserve sharp feature edges of a quad esh, regular poly-lines containing sharp feature edges are treated as singular poly-lines, that is, they are also used in defining patch coplex. Note that this treatent does not change the four-sided property of the patches and all the algoriths are still valid. Error easureent of aspect ratios Note that the aspect ratio r of a quad eleent of the sae rectangular shape can be less than 1 or greater than 1, depending on its orientation (cf. the definition of the aspect ratio in Section 2). To eliinate this discrepancy when easuring optiization quality in ters of aspect ratios, in the following we use the noralized aspect ratio [r] = ax(r, 1 r ). Thus, a quad eleent whose average width and height are not equal always has its aspect ratio greater than 1. Likewise, we use the noralized eleent weight [ r ρ ] = ax( r ρ, ρ r ) to easure the quality of anisotropic optiization. Other exaples More exaples of isotropic optiization are shown in Fig. 14 and the corresponding statistics are listed in Table 1. Tie is the total tie for the three steps. The iperfectness ([r] 1) is ainly caused by two factors: (1) The constraint syste. If R i 1 for a node v, there is no way for all the eleent aspect ratios of the patches incident to v to reach 1. In other words, the geoetry of the patch coplex deterines how uch the eleent aspect ratios can be optiized. (2) Rounding errors. Rounding optial grid nubers to its nearest integers causes further inaccuracy. For anisotropic optiization, the results are deonstrated in Fig. 15 with statistics listed in Table 2. Note that in soe of our exaples (e.g. Fig. 15d) highquality anisotropic eleents (r i j ρ i j ) ight not be achieved everywhere due to the highly varying curvatures and the global nature of the optiization.

9 Figure 14: Exaples of isotropic optiization. In each sub-figure, left: input esh; iddle: the patch coplex of the input; right: after isotropic optiization. Table 1: Statistics for isotropic results. Liitations Our ethod didn t consider to optiize the topology of the patch coplex. Thus, little iproveent could be achieved if the extracted patch coplex consists of highly fragented, Figure 16: Fertility. sall patches. Such a difficult esh surface produced by the ixed integer quadrangulation ethod [3] is shown in Fig. 16 ( F = 3357, [r]avg. = 1.250): there are 48 singular vertices, whose singular poly-lines however segent the surface into 2271 patches. Most face eleents theselves are patches, leaving little space to define sufficiently any grids. Future work will be focused on optiizing the patch coplex, e.g. odifying the location of singularities, aligning isatched singularities, canceling adjacent singularity pairs, etc. Fig V F Z 1c 14a 14b 14c 14d 14e Tie 748s 202s 780s 172s 3510s 390s Quad In Out [r]avg. In Out Another issue is that our definition of the aspect ratio for a quad eleent does not take angles into account. Therefore, our ethod cannot iprove the orthogonality of input esh edges. For exaple, for a sheared patch, new poly-lines are generated by linearly interpolating input poly-lines in both re-sapling and redistribution steps, thus quad eleents with acute angles still exist within the sheared patch. A future research proble is optiizing the patches to generate orthogonal esh edges. 9

10 Figure 15: Exaples of anisotropic optiization. In each sub-figure, left: input esh; right: after anisotropic optiization. The color of each face eleent encodes its noralized weight [ ρr ] = ax( ρr, ρr ). Table 2: Statistics for anisotropic results. Fig V F Z 15a 15b 15c 15d 15e Tie 203s 514s 311s 2137s 640s Quad In Out Acknowledgeents [r/ρ]avg. In Out The work of Wenping Wang is partially supported by the State Key Progra of NSFC project ( ). References [1] Alliez, P., Cohen-Steiner, D., Devillers, O., Levy, B., Desbrun, M., Anisotropic polygonal reeshing. ACM Trans. Graph. 22 (3), [2] Alliez, P., Ucelli, G., Gotsan, C., Attene, M., Recent advances in reeshing of surfaces. Research report, AIM@SHAPE Network of Excellence. [3] Boes, D., Zier, H., Kobbelt, L., Mixed-integer quadrangulation. ACM Trans. Graph. 28 (3), 77:1 77:10. [4] Botsch, M., Pauly, M., Kobbelt, L., Alliez, P., Le vy, B., Geoetric odeling based on polygonal eshes. In: Eurographics Tutorial. [5] Canann, S., Muthukrishnan, S. N., Phillips, R. K., Topological iproveent procedures for quadrilateral finite eleent eshes. Engineering with Coputers 14 (2), [6] Canann, S., Tristano, J., Staten, M., An approach to cobined laplacian and optiization-based soothing for triangular, quadrilateral, and quad-doinant eshes. In: 7th International Meshing Roundtable. pp Conclusion We have introduced a novel fraework for optiizing the eleent aspect ratios of a quad esh without changing its esh structure or introducing extra singular vertices. Our fraework is siple and flexible, and can handle both isotropic and anisotropic cases. To our knowledge, this is the first attept to forulate the aspect ratio optiization proble based on global optiization. Theoretical analysis is also provided for better understanding the existence of solutions and the diension of the solution space. 10

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