User-Controllable Mesh Simplification
6.5 Local Refinement
6.5 Local Refinement
The weighting scheme, including previous methods, generally cannot recover sharp fea-tures, such as sharp edges and corners. The second stage of our user-controllable simpli-fication framework is a local refinement scheme aiming to provide an effective tool for recovering local sharp features. The proposed refinement operation is similar to the se-lective refinement and simplification in view-dependent level-of-detail modeling [31]. The selective refinement (simplification) refines (simplifies) a mesh by moving down (up) the active cut of the vertex hierarchy.
Given a simplified mesh with its progressive mesh sequence, normally the result of the first stage, the system constructs the corresponding vertex hierarchy with collapsing cost recorded on each vertex and the active cut associated with the given simplified mesh. To do the local refinement, user selects a set of vertices and the system will perform vertex split on these vertices, and in the meantime do the vertex collapsing on some vertices to maintain the polygon count. Those vertices that have the lowest collapsing cost are the candidate vertices for edge collapsing. Note that the vertex split or collapsing are just the moving down or up of the active cut.
One thing worth mentioning is that the vertex split dependency problem may limit the ability of local refinement since a vertex can be split only if all its neighboring vertices after split are reachable. In our implementation, such problems are overcome by applying the approach proposed in [39]. Another issue needs to be addressed is that, after local re-finement, vertices resulting from a vertex split normally have costs lower than their parent.
After a sequence of vertex splits applied to a vertex v, the subtree originates from v may have leaf vertices whose costs are relatively lower than that of vertices in the active cut.
This implies that the split vertices may soon be collapsed when vertices in other region are split. To prevent this problem, we need to adjust the costs of split vertices such that they have about the same magnitude as the cost of v. Further, the cost difference for vertices in the subtree should be maintained to preserve the local features.
The cost adjustment is done along with the vertex split operations in the local refinement process. Let cvbe the cost of a vertex v to be split, and c1and c2be the costs of the children
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of v. Suppose c1≥ c2, c1and c2are adjusted to c∗1and c∗2as follows:
c∗1 = cv+c1−c2 2 c∗2 = cv−c1−c2 2.
(6.4)
Note that Eq. 6.4 ensures that the average cost of the split vertices is the same as their parent and the cost difference between the split vertices is maintained.
v1
Figure 6.5: Cost adjustment in local refinement process.
As shown in Fig. 6.5, the costs of v31 and v32 are adjusted after v13 is split, and the average cost of v31 and v32 are the same as their parent v13. Moreover, the difference between v31and v32remains the same after local refinement.
6.6 Results
In the implementation, the proposed user-controllable mesh simplification framework sup-ports QEM [20] and APS [14] as the cost measure for the edge collapsing. To preserve the simplification styles of the employed error metric, the simplification after applying weighting is executed in the same way as the automatic simplification process with the error metric, except that the edge collapses associated with the weighted vertices are not performed until the delayed orders are encountered.
Several experimental tests are performed to demonstrate the effectiveness of the pro-posed weighting schemes. First example is a cow model of 5, 804 polygons (Fig. 6.6(a)), which is simplified to a mesh of 1, 160 polygons (20% of the original mesh) by using QEM [20] (Fig. 6.6(b)). Different weighting values are applied to the region of left eye using uniform weighting scheme as shown in Fig. 6.6(a) by red color. Fig. 6.6(c) and Fig. 6.6(d) depict simplified result and the refined meshes after applying weighting values
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(a) Original mesh (b) Simplified mesh (8 vertices in the selected
Figure 6.6: Apply uniform weighting on the cow model using different weighting values.
(a) Original mesh (b) Simplified mesh (c) After applying weighting scheme
(d) After local refine-ment
Figure 6.7: Two-stage user-controllable simplification (with uniform weighting) on the dragon model.
2 and 3, respectively. Fig. 6.7 and Fig. 6.8 illustrate the effectiveness of two-stage user-controllable simplification on the dragon model of 50, 000 polygons and male model of 151k polygons. Both models are first simplified to meshes of 1, 500 polygons using QEM.
For the dragon model, the uniform weighting scheme with weight value of 3 is applied to the regions of eyes, with the resultant mesh shown in Fig. 6.7(c). Local refinement is then applied to areas of teeth and nose, producing refined mesh shown in Fig. 6.7(d). For the male model, the uniform weighting scheme with weight value of 3 is applied to regions of eyes, lips, and nose, and local refinement is applied to recover sharp features such as eye-balls, eyebrows, and nose. Table 6.1 and Table 6.2 list the geometry and normal deviations, respectively, before and after the user-controllable simplification for the male model. The errors are measured using MeshDev, which a mesh comparison tool using attribute devia-tion metric [69]. Although the mean errors after applying user-controllable simplificadevia-tion are slightly increased, the errors are diffused over the regions that are considered perceptu-ally less important. Fig. 6.9 visualizes the distributions of geometry and normal deviations for the male model. Noticeable improvements can be found in the selected regions, and the compensative error introduced by the proposed scheme is almost invisible and diffused over the less-important regions.
Fig. 6.10 illustrates the result of nonuniform weighting scheme and local refinement
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(a) Original mesh (b) Simplified mesh (c) After applying weighting scheme
(d) After local refine-ment
Figure 6.8: Two-stage user-controllable simplification (with uniform weighting) on the male model.
Figure 6.9: Visualization of the error distributions for simplified male model before (top) and after apply user-controllable simplification (bottom). The left column are the shaded models, the middle column shows the distributions of geometry deviation, and the right col-umn visualizes the normal deviation. Both deviations are measured by using MeshDev [69].
applied to the buste model of originally 511k polygons. It is first simplified to 1, 500 poly-gons using QEM. Three different levels of nonuniform weights are applied to the model according to the significance in perception; as shown in Fig. 6.10(c) on which the green, yellow, and red colors represent the weighting value 2, 3, and 4, respectively. Then, the local refinement is applied to the eyes, nose, and lips to recover crease features.
Fig. 6.11 compares the effectiveness of the proposed uniform weighting and nonuni-form weighting schemes against the one proposed in [38]. The cow model is simplified to 1, 160 polygons (20% of the original mesh) using QEM; as shown in Fig. 6.11(a). Weight-ing value 3 is assigned to the left eye as the red region shown in Fig. 6.6(a). The proposed uniform and nonuniform weighting schemes yield similar resolution increment for that re-gion, namely increasing from 8 vertices to 27 and 25 vertices, respectively; see Fig. 6.11(b) and Fig. 6.11(c), respectively . The weighting scheme of [38] reorders the edge collapse
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Table 6.1: Geometry deviation of the simplified male model before and after applying the user-controllable simplification.
Table 6.2: Normal deviation of the simplified male model before and after applying the user-controllable simplification.
sequence by directly multiplying the weighting values to the corresponding quadric er-rors. Since modification of quadric error has no direct link to the resolution improvement, the resolution improvement is not predictable. In this test case, the number of vertices remain unchanged; as shown in Fig. 6.11(d). Next, we compare the effectiveness of the proposed uniform weighting, nonuniform weighting schemes, and the weighting scheme in [38] using the buste model. The buste model is first simplified to the meshes of 1, 500 polygons. Uniform weighting with values 2 and 3 is applied to the selected regions as shown in Fig. 6.10. For nonuniform weighting, values similar to that in Fig. 6.10 are ap-plied to the selected regions. As shown in Fig. 6.12, the uniform weighting with value 2 may not preserve the eyes well (Fig. 6.12(b)) while the uniform weighting with value 3 seems over-preserve the eyes (Fig. 6.12(c)). The nonuniform weighting scheme is more capable of adapting to the expectation of users. Again, the weighting scheme of [38] with weighting value of 3 performs badly in this case.
Both of our proposed weighting schemes are independent on the resolution of the given meshes, meaning that similar resolution increment in the selected regions is achieved for the given simplified meshes in different resolutions. Fig. 6.13 depicts the results of ap-plying the uniform weighting with value 3 to the cow models with polygon counts 500, 1, 160, 1, 739, and 2, 321. The selected region is the same as the one in Fig.6.6. The in-creased number of vertices in the selected region is shown in Table 6.3, which also includes
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Figure 6.10: Two-stage user-controllable simplification (with nonuniform weighting) on the Buste model.
Figure 6.11: Comparison of the proposed uniform weighting and nonuniform weighting schemes against the weighting scheme in [38].
the performance of the proposed nonuniform weighting scheme and the scheme proposed in [38]. The three numbers in each item of the ”vertex count in the selected region” indi-cate the vertex counts resulting from the uniform weighting scheme (top), the nonuniform weighting scheme (middle), and the weighting scheme of [38] (bottom). We observe that the performance of the uniforming and nonuniform weighting scheme is quite close to what we expect. Note that the small inaccuracy in hitting the expected target is due to the de-pendency problem in the dede-pendency hierarchy that occurs on the boundary of the selected region. On the other hand, the resolution improvement of the weighting scheme of [38]
is unpredictable. It is usually hard for users to specify the weight value for an expected resolution improvement.
The proposed weighting schemes are also independent on the error metric used in the mesh simplification. Since APS is a texture-deviation error metric, we consider the Parasaur model with texture mapped. The Parasaur model of 7685 polygons is first simpli-fied to 750 polygons using QEM and APS. Then we apply the uniform weighting scheme with values 2 and 3 to the region of left eye. Table 6.4 shows the resolution increments in the region of left eye after we apply the uniform weighting scheme with values 2 and 3 to the region. As we can see that the obtained resolution increments are close to what we expect for both metrics. The proposed weighting schemes can be applied to models
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Figure 6.12: Comparison of the proposed uniform weighting, nonuniform weighting schemes, and the weighting scheme in [38].
(a) 500 (8.6%)
Figure 6.13: Results of applying the uniform weighting with value 3 to the cow model of resolutions in different resolutions.
with texture mapped to reduce the texture distortion. Fig. 6.14 shows the result of applying the two-stage user controllable simplification scheme to the Parasaur model with texture mapped. Again, the Parasaur model is simplified to a mesh of 750 polygons using APS, on which noticeable texture distortion can be found; as shown in Fig. 6.14(b). Fig. 6.14(c) and Fig. 6.14(d) depict a great reduction in texture distortion after applying uniform weight-ing scheme with weightweight-ing value of 3 around the left eye and then local refinement on the texture boundaries.
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Table 6.3: Comparison on the resolution improvement obtained by the proposed weighting schemes and the weighting scheme of [38].The three numbers in each item of the ”vertex count in the selected region” indicate the vertex counts resulting from the uniform weight-ing scheme (top), the nonuniform weightweight-ing scheme (middle), and the weightweight-ing scheme of [38] (bottom).
Polygon count Vertex count in the selected region of Without Weighting Weighting simplified mesh weighting value = 2 value = 3
500 (8.6%)
Table 6.4: Resolution improvement after applying constant weighting scheme on different error metrics.
Error metric
Vertex count in the selected region W/O Weighting Weighting weighting value = 2 value = 3
QEM 15 33 49
APS 6 15 22
(a) Original mesh (b) Simplified mesh (c) After applying uniform weighting scheme
(d) After local refine-ment
Figure 6.14: Applying two-stage user-controllable simplification to the Parasaur model.