行政院國家科學委員會專題研究計畫 期中進度報告
三維網格參數化及其應用之研究(1/2)
計畫類別： 個別型計畫 計畫編號： NSC92-2213-E-009-083- 執行期間： 92 年 08 月 01 日至 93 年 07 月 31 日 執行單位： 國立交通大學資訊工程學系 計畫主持人： 莊榮宏 報告類型： 精簡報告 處理方式： 本計畫可公開查詢中 華 民 國 93 年 6 月 1 日
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We propose a systematic method, called
recur-sive quinary subdivision, to efficiently find a
dis-section for an object with little user input. The quinary subdivision is a process that recursively dissects an highly stretched patch into five new patches. The process can be easily extended to multiple objects, taking into account the align-ment of extra feature points, to derive a common dissection. Based on the dissection, we imple-ment applications such as remeshing to yield a set of semi-regular meshes, and morphing between two or more objects.
Keywords: Mesh dissection, Parameterization,
Remeshing, Morphing, Multiresolution model-ing.
1
Introduction
Recently, semi-regular meshes are getting more and more popular as representations of complex objects in computer graphics and geometric mod-eling. Such meshes are multiresolution represen-tations formed by starting from a coarse irregu-lar base domain and applying recursive reguirregu-lar re-finement. Due to their regular structures, parame-ter and connectivity information can be predicted [12], and efficient tree or array based data struc-ture can be used in such a way that only geome-try information needs to be stored. Moreover, sig-nal processing algorithms such as wavelet asig-naly- analy-sis can be employed [4, 17, 16].
Remeshing is a process that for a given input
ir-regular mesh, resamples its geometry information and constructs a semi-regular mesh which approx-imates the original irregular one. The state-of-the-art algorithms of remeshing involve initially dis-secting the original irregular meshes into a set of topological disk-like patches, called base domain. These patches are later parametrized to compute a bijection between the 3D domain and the parame-ter domain. Obviously, the patch layout generated by initial dissection is one of the vital factor that dominate the quality of the remeshing.
A single mesh can be remeshed to yield a semi-regular one. Can we extend the idea to multi-ple objects? The answer is not true unless they have a common initial dissection in which each patch of one object corresponds exactly to one patch in all other objects. In consequence, they will possess a same base domain and their pa-rameterizations will be consistent. The diffculty is that the common initial dissections are not easily found, because a good dissection of one object might be bad for another object. Previ-ous works [19, 18, 11, 7, 21] leave this prob-lem to users, requiring the user manually pro-vides a common initial dissection and many cor-responding feature points. Besides, many appli-cations such as metamorphosis (or morphing) and DGP (Digital Geometry Processing) applications [19], which require the establishment of corre-spondences among multiple objects, benefit from consistent parameterizations.
We propose a systematic method, called
quinary subdivision, to find a common initial
dis-section for multiple objects of genus-zero with lit-tle user interventions. The quinary subdivision scheme is a process that recursively subdivide an undesirable patch into five new patches, that possess better parameterization than their parent patch. The quinary subdivision scheme can be ex-tended to multiple objects and guarantees to yield a common initial dissection. Extra feature corre-spondences can also be provided by users. The alignment of feature correspondences during the quinary subdivision is also taken care of by using a foldover-free warping. After the common ini-tial dissection is found, we compute parameteriza-tion for each patch, begin the remeshing process and finally obtain a set of semi-regular meshes. The remeshing process can be either uniform or
adaptive. Based on the parameterization, a set of
normal maps, which capture geometry details, can
also be easily resampled. This improves the real-time rendering quality of semi-regular meshes in coarse levels. A multiresolution 3D mesh morph-ing is demonstrated as an application of our pro-posed approach.
2
Related Work
Remeshing process begins with an input irregu-lar connectivity model, dissecting the model into a set of patches and then computing a parameter-ization for each patch. Eck et al. [4] proposed a method that produces a semi-regular mesh fully automatically by employing a Voronoi-like algo-rithm coupled with a Delaunay triangulation to yield the initial dissection, and parametrizing each patch using harmonic mapping. In contrast, the MAPS scheme proposed by Lee et al. [15] em-ployed a mesh simplification algorithm to yield an initial dissection. Their algorithm is also au-tomatic and more practical than Eck’s. The
con-formal mapping is performed during mesh
simpli-fication and a patch parameterization is obtained. Guskov et al. [8] proposed a new remeshing pro-cess to construct a compact representation called
Normal Mesh. The above works focus on single
objects. Praun et al. [19] illustrated that shortest paths probably lead to problems for finding an ini-tial dissection, and proposed a modified shortest path algorithm to trace curves between given fea-ture points and yeild a base domain. They also focus on multiple objects, but users are required to provide a patch layout for the common base domain and feature points identificaton on each objects.
The correspondence problem in the mesh mor-phing is naturally related to parameterizations. The survey of 3D morphing can be referred to [13], and for mesh morphing, an extensive review can be found in [1]. Particularly, Alexa pointed out in [1] that the remeshing approach is appeal-ing for morphappeal-ing applications, because it allows to scale the size of the representation mesh. On the contrary, the conventional merging approach generates a more complicated intermediate repre-sentation. Kanai et al. [10] used a single patch and the patch will be parametrized by harmonic mapping. In their recent works [11], the user first
defines a set of corresponding features vertices and applied their approximate shortest path algo-rithm [9] to find an initial dissection. The works of Gregory et al. [7] and Z¨ockler et al. [21] also require users to manually provide initial dissec-tions. Among the methods proposed, Lee et al. utilized their MAPS to the mesh morphing ap-plication [14]. But the base domains of the two input meshes are different—they are not consis-tently parametrized. They need a “meta-mesh” as an intermediate representation. Unfortunately this algorithm does not scale well, because the meta-mesh is more complicated than the original two models. Michikawa et al. [18] proposed a
mul-tiresolution interpolation meshes(MIMesh)
repre-sentation for mesh morphing. An interface is de-signed for users to define a common patch lay-out on both source and target meshes. Each patch is then parametrized and a surface fitting is per-formed to produce a semi-regular mesh for the in-termediate mesh. The method can be extended to multi-target morphing.
3
Consistent Mesh
Parameterization
3.1
Quinary Patch Subdivision
We use the parameterization scheme proposed by Floater [5] for the parameterization of patches. The L2 stretch metric used in TMPM [20] is adopted to evaluate the patch’s stretch ratio. If the stretch ratio of a patch P, say L2(P), exceeds a pre-defined thresholdτ, it will be further subdi-vided into five patches by our quinary subdivision. Fig. 1 illustrates the quinary subdivision. The
ini-v1 v2 v3 v4 v7 v0 v6 vpeak v5 (a) 3D domain p2 p0 p1 p3 ppeak p6 p7 p5 p4 (b) 2D parameter domain
Figure 1: Quinary subdivision illustration
tial patch is the region with corners v0, v1, v2, and
p1 = u(v1), p2 = u(v2), and p3 = u(v3), And
the corresponding parameterization in R2 is de-noted as S{0,1,2,3} where p0= u(v0), p1= u(v1),
p2= u(v2), and p3= u(v3). To apply our quinary
subdivision to the patch S{0,1,2,3}, we first find the greatest stretched face tpeakin R2with
correspond-ing Tpeak in R3, and the centriod ppeak of tpeak is
the peak point. For each corner pi, i = 0, 1, 2, 3, of
the patch, an intermediate point pi+4 on the line
segment connecting ppeak and pi is computed as
follows:
pi+4= (1 −ωi)pi+ωippeak,
i = {0, 1, 2, 3}, 0 ≤ωi≤ 1 (1)
And a constantω forωi, i = 0, 1, 2, 3 is used as
ω =ψ+ log10(L2(t
peak)), (2)
whereψ is a bias and is taken as 0.85 in our cur-rent implementation.
The four new corners are used to subdivide the patch into five patches S{0,1,5,4}, S{1,2,6,5},
S{2,3,7,6}, S{3,0,4,7}, and S{4,5,6,7}. As in [8], a straight line, which may go through faces, in the parameter domain will be a fair curve in 3D do-main. By using the inverse mapping from 2D to 3D, the boundary curves of the newly created patches can be determined. Fig. 2 shows the pro-cess of quinary subdivision on a cat head model.
3.2
Dissection on a Single Object
In order to perform quinary subdivision, we have to first divide a genus-zero model into two patches. This step requires users to specify four “seed points” on the surface, and the close loop of the four points are computed by Dijkstra’s short-est path algorithm [3] based on geodesic distance to dissect the input mesh. For each patch, if the stretch ratio exceeding a user-specified threshold, subdivide it into five patches using the quinary subdivision scheme and check the stretch ratios of the five new patches recursively until all patches satisfy the user-specified threshold. Then the dis-section of a single object is found. Fig. 3 illus-trates the dissection on a single object.
We can also perform patch boundary relaxation and global vertex relaxation (similar to [8]) to yield a better dissection. Fig. 4 shows the relax-ation of a boundary curve. To improve the bound-ary curve with endpoints v2 and v5, the two
inci-dent patches of the boundary curve are used to to
(a) Four seed points specified.
(b) Two seed patches cre-ated.
(c) Subdivision on the larger patch.
(d) Result of recursive quinary subdivision. Figure 3: Dissection on the venus head model.
(a) The original model. (b) The initial parameter-ization.
(c) The face stretch ratio. (d) The quinary subdivi-sion.
Figure 2: Quinary subdivision on cat head model.
construct the parameterization. The 3D curve on the mesh corresponding to the straight line from
u(v2) to u(v5), which will be the new boundary
curve. The position of a corner can be reposi-tioned with similar process as shown in Fig. 5.
v1
u(v4) u(v5) u(v6) v4
v5 v6 u(v1) u(v3) v3 v2 u(v2) u u−1 v2 v3 v4 v5 v6 v1
Figure 4: Relaxation of a boundary curve.
u−1 u
Figure 5: Relaxation of a corner vertex.
3.3
Common Dissection for
Multi-ple Objects
The recursive quinary subdivision for the seed patches can be recorded as a “quinary tree”. In order to yield a common dissection for multi-ple objects, the recursive subdivision processes must be the same. We simply take the union of these quinary trees, denoted as Qunion, and the
initial seed points act as feature corresponding points. For an object and its respected quinary tree Qi, we examine the difference between Qi
and Qunion, and further deliberately subdivide the
patch if there is a newly added node or subtree. The quinary subsivision should be performed on all corresponding patches of all objects if there is one patch with stretch ratio exceeds the threshold.
3.3.1 Extra Feature Correspondences
If the users require to specify extra feature corre-sponding points on the models for better perfor-mance on applications such as mesh morphing, the quinary subdivision rule can be enhanced to take into account the alignment of those additional feature points. Insufficient feature points may result in unexpected morphing sequence in such applications. Moreover, without proper align-ment, the specified feature points in correpon-dence could belong to different dissected patches. Such cases often occur when feature points are specified too close or the input models are too dis-similar,
To prevent from this situation, a foldover-free
warping proposed by [6] is used in the parameter
domain before each quinary subdivision. Given patches Si, i = 1, ..., n, in R2and the associated set of feature points { f1i, ..., fei}, where n is the num-ber of input meshes and e is the numnum-ber of feature points in Si, we first compute the averaged feature point position as fk= 1 n n
∑
i=1 fki, k = 1, ..., e.For each Si, we first construct a warp mesh for it by a 2D Delaunay triangulation which takes four corners, and f1i,..., and feias input. All other points will be marshaled into their respected enclosing triangles and their barycentric coordinates are also computed. The objective of a warping in param-eter domain of Si is to move feature point fki to
fk for all i and k and recompute all other
param-eters which will still keep it as a bijective map-ping. Therefore, the algorithm is just performed to each individual patch, not to all patches
simul-taneously. During the deformation of the warp mesh by the movement of the feature points, some triangles of the warp mesh may degenerate and start folding over. We call such situation an event. In order to prevent triangles from folding over, an algorithm called triangulation over time will be performed. Before starting the deformation, we detect if there will be an event by employ-ing a binary search between the source position and the destination position of an arbitrary fea-ture point ft. If an intermediate position of the
event is found, we alter the local triangulation and recompute all barycentric coordinates affected by this alteration. Then we move ft to the position
of the event, which was detected but will not oc-cur this time. Then all parameters are recom-puted by using the new barycentric coordinates. The influence range of the warping is not global and only parameters marshaled in this range will be affected. Furthermore, if there are more than one feature point, the triangulation over time al-gorithm will process them one by one in an arbi-trary order. Although the processing order will af-fect the final distribution of the non-feature points, what we wish is to keep the mapping bijective. After processing one feature point, we set the po-sition as the source popo-sition and repeat the event detection for other feature points. The process ter-minates when all feature points reach their desti-nation positions. If no event is found, simply de-form the warp mesh to destination and recompute all parameters. Because the barycentric coordi-nates make the parameters inside a triangle bijec-tive and the triangulation of the warp mesh is also bijective, we can assure that the warped param-eters are still bijective. An example is shown in Fig. 6, in which, besides four limbs, more feature points are specified for mouths, tail of the tricer-atops and the pig, horn of the tricertricer-atops and ear of the pig.
4
Applications
With the result of the parameterization, lots of ap-plications can be performed. Two apap-plications are discussed here, which are the remeshing and mesh morphing.
Figure 6: Common dissection and base domains of a pig model and a triceratops model.
(a) Pig: M0 (b) Triceratops: M0
(c) Pig: M3 (d) Triceratops: M3
Figure 7: The uniform remeshing derived from the dissection in Fig. 6.
4.1
Remeshing
4.1.1 Uniform Remeshing
With the parameterization ready for each patch of the base domain, we can start remeshing pro-cess. For each patch, we simply take regular grid point parameters and compute their inverse mapped 3D position. Fig. 7 shows the result of uniform remeshing derived from the dissection in Fig. 6.
4.1.2 Adaptive Remeshing
Uniform remeshing has the drawback that in or-der to resolve a small local feature on the original mesh, one may need to subdivide to a very fine level. This total number of faces will be quadru-pled. We now describe a simple method to build the adaptive remesh within a conservative error
bound.
For a given input mesh M , a base domain consisting of some quad-faces corresponding to patches are constructed. For a given quad-face
q, we find a best-fitting plane g, and measure the
minimum Euclid distance between the plane g and each vertex in the patch Pq associated with
quad-face q. Let d(v) be the minimum Eucilid distance for each v ∈ Pqand each p ∈ g.
d(v) = min
p∈g|v − p|
We define the error function E(q) for each quad-face q as the maximum of d(v) for all v ∈ Pq; i.e.,
E(q) = max v∈Pq
d(v) (3)
Fig. 8 illustrates the error function. The error
p
Patch Pq
Best-fitting plane g of the quad-face q
d(v) v
Figure 8: Error function definition.
function can be normalized by the diagonal length of the bounding box of the input mesh, denoted as
B(M ); i.e.,
EN(q) =
E(q)
B(M ) (4)
We begin the remeshing process with base domain mesh and construct a quadtree root for each quad-face of base domain mesh. Then we evaluate the error of quad-faces in each quadtree based on the error function EN(q). If the error of a quad-face q,
EN(q), is exceeding a pre-defined error boundε, the quad-face is further refined and its geometry informations are resampled. In other words, we take the quad-face to next finer level and four new children will be attached into the quadtree. The process is performed recursively and the adaptive remesh is constructed. However, there will be lots of T-vertices, which appear along the bound-aries between quad-faces of different levels. We first force the level difference between neighbor-ing quad-faces to be at most one by deliberately refining the quad-face of coarser level. Then per-form adaptive subdivision to quad-face of coarser level.
(a) pig (b) triceratops
Figure 9: Adaptive remeshing (ε = 0.0025). Now consider to perform adaptive remeshing consistently on multiple objects, we can simply take the maximum of the error of each corre-sponding quad-faces in the multiple objects. The result will still be consistent.
EC= max
1≤i≤n(EN(qi)) (5)
where n is the number of objects. This will also result in adaptive remeshes with the same connec-tivity structure. If there are some local geomet-ric features in one object, which correspond to a flat region in another object, the flat region will be forced to refined and still result in lots of faces. Fig. 9 shows the result of the consistently adaptive remeshing.
4.1.3 Normal Mapping
Normal map is an image storing quantized
nor-mal vectors of surfaces. By mapping (nx, ny, nz) in the range [−1, 1] to (r, g, b) in the range [0, 255], it’s possible to recover the detailed geometry fea-ture from this map and improve the rendering vi-sual effects. This is appealing while a remesh of coarse level is rendered in real-time applications. Normal vectors are also geometry information and can be resampled as 3D position resampling. With barycentric coordinates, the intermediate interpo-lated normal vectors within a face can also be re-sampled. No further packing algorithm such as pull-push algorithm in [20] is needed since the patches are all square in parameter domain. Fig. 10 shows the result with and without normal map-ping.
4.1.4 Results of Remeshing
We have implemented quinary subdivision and remeshing as described above on a PC with Athlon 900Mhz CPU and NVidia geforce 2 graph-ics card. The remeshing results are evaluated by
(a) Original model: 69664 faces
(b) Base domain: 34 faces
(c) M3: 2176 faces (d) M3 with normal
map-ping
Figure 10: Normal mapping example.
IRI-CNR Metro tool [2]. We show the percentage of mean square error (L2) normalized by the diag-onal length of the bounding box of the models.
Table 1 shows the result of uniform and adap-tive remeshing. Both single object adaptive remeshing and multiple object adaptive remesh-ing are shown. Table 2 shows the approximate errors. Table 3 shows calculation time.
Model
size
original uniform remeshing adaptive remeshing M0 M1 M2 M3 M4 M5 M6 (ε= 0.0025) venus 10000 14 56 224 896 3854 14336 57344 8656 isis pig 7164 66 264 1056 4224 16896 67584 270336 10730 isis 5660 body 1418 14 56 224 896 3854 14336 57344 14624 venus 5000 isis 5000
Table 1: Statistics of polygon numbers.
Model
L2error(%)
uniform remeshing adaptive remeshing
M0 M1 M2 M3 M4 M5 M6 (ε= 0.0025) venus 7.97 2.46 0.83 0.26 0.086 0.033 0.012 0.047 isis 2.64 1.32 0.66 0.19 0.073 0.026 0.009 0.037 pig 3.09 1.73 1.16 0.55 0.287 0.124 0.058 0.085 triceratops 2.80 1.68 0.85 0.73 0.369 0.114 0.072 0.105 body 4.59 2.12 0.96 0.39 0.210 0.085 0.022 0.042 venus 5.73 2.87 0.99 0.38 0.143 0.057 0.021 0.054 isis 3.97 1.47 0.65 0.33 0.136 0.046 0.015 0.040
Table 2: Statistics of errors.
4.2
Mesh Morphing
4.2.1 Morphing Among Two Models
The semi-regular meshes constructed from the remeshing process for multiple objects have the
Model size M0 time(sec)
dissection remeshing venus+isis 10000+10000 14 4.715 1.688 horse+human 5000+5000 106 5.276 3.751 horse+triceratops 2000+5660 70 4.760 2.532 bunny 69664 34 108.729 8.714 body+venus+isis 1418+5000+5000 14 1.621 1.145 pig+triceratops 7164+5660 66 5.034 2.783
Table 3: Statistics of calculation time. The remeshing is uniform and up to level 5.
same connectivity. We can simply linearly inter-polate their positions. A sequence of intermediate morphed objects will be generated, and they still have multiresolution structures. Fig. 11 shows the morphing sequence from a pig model to a tricer-atops model.
Figure 11: Morphing from a pig model to a tricer-atops model.
4.2.2 Multi-Target Morphing
The common dissection for multiple objects can be established. Based on the constructed semi-regular meshes, we can produce any morphing se-quence among these objects. Fig. 12 shows the result.
5
Conclusion
The correspondence establishment among multi-ple objects is a versatile algorithm in computer graphics and geometry computing, especially in the morphing applications. Other applications such as geometry processing also benefit from the
correspondences establishment. However, large efforts required by users suppress the establish-ment. A simpler and intuitive framework is nec-essary for alleviating the effors.
We have proposed a systematic method, called recursive quinary subdivision, to find a common dissection for multiple objects with only four fea-ture points specified by the user. Extra feafea-ture points in correspondences can also be specified for semantics and aligned during the subdivision. Based on this dissection, uniform and adaptive remeshing can be performed to yeild a set of semi-regular meshes. Moveover, geometric details can easily be resampled and stored as normal maps to improve the visual effects using the modern graphics hardware. We have demonstrated the 3D mesh morphing application between two or more objects using the correspondence established by the common dissection and remeshing. In ad-dition to morphing in spatial domain, scheduled morphing between objects in wavelet domain is also demonstrated.
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