Former weighting schemes proposed by Kho et al. [22] and Pojar et al. [5] increase mesh resolution in the interested region by directly multiplying the cost of edge collapse in the interested region with a user-specified multiplier. However, such method has two shortcomings. First, because the value of the multiplier has no direct relation to the resulting simplification, the multiplier is usually chosen in a trial and error basis. Second, for some error metrics, such as QEM, the cost of edge collapse grows rapidly later in the PM sequence, so the effect of multiplying the costs with a multiplier will diminish when simplified to low polygon counts.
The proposed weighting scheme differs from the previous methods in that the weighting values are automatically derived based on how many more vertices users expect to have in the interested regions 1. No multiplier needs to be specified, what the user needs to input is the number of vertices he or she expects in a region. As a consequence, such a weighting scheme is equally effective for a reduced model of any polygon count.
The basic concept of our weighting scheme is to shift the execution order of all edge collapses in the interested region 2, so that a user-expected number of vertices will be allocated in that region after the model is simplified to a user-specified simplification target. Let’s assume that the user-specified simplification target is N vertices, the number of vertices in the interested region is C when the model is simplified to N vertices, and, moreover, the user-expected number of vertices to be allocated in the interested region is C∗.
Figure 3.2: Notations of the weighting scheme.
For a given original mesh Mn of n vertices, let’s denote all the edge collapses from the original mesh Mn to the most simplified mesh M1 (consisting of one vertex) are ec1, ec2,...,ecn−1, and among them the edge collapses in the interested region are ecr1,
1each region is identified as a set of vertices in the original mesh
2an edge collapse is in a region if it collapses some vertices belong to that region
3.2 Weighting Simplification Cost 23 ecr2,...,ecrl, where l is the number of original vertices in that region. The above notations are illustrated in Figure 3.2, where the horizontal line presents the execution order of edge collapses along the PM sequence and triangular dots mark edge collapses in the interested region.
For a given simplified mesh MN, N < n, the weighting scheme consists of two steps.
In the first step, the new execution order of ecr1,...,ecrl are determined, and in the second step, we adjust the simplification cost of these edge collapses so that these edge collapses will be executed at the desired order when the original mesh is re-simplified. The two steps are stated as follows:
1. Shift the execution order of ecr1,ecr2,...ecrl such that there are C∗ edge collapses ecr(l−C∗+1),...,ecrl avoid to be performed when the original mesh is re-simplified to MN. Because the number of remaining vertices in the interested region is determined by the number of edge collapses not are not yet performed in the region, by doing so we can assure that there will be C∗ vertices allocated in the interested region after the original mesh is re-simplified to MN. This stage involves
(a) Find the new place where ecr(l−C∗+1) is shifted to. Since the given simplified mesh, MN of N vertices, is obtained by doing edge collapses ec1, ec2,...,ecn−N −1, the edge collapse ecr(l−C∗+1)shall be shifted to the place for ecn−N +(C∗−C). Doing so, we will assure that there will be C∗ edge collapses in the interested region that are not yet performed after the original mesh is re-simplified to MN. This stage is illustrated in Figure 3.3.
(b) Shift all edge collapses ecr1, ecr2,...,ecr(l−C∗),ecr(l−C∗+2),...,ecrl in a magnitude
This stage is illustrated in Figure 3.4.
2. The new simplification cost for each of these shifted edge collapses ecr1,...,ecrl will be the cost of the edge collapse at the shifted place. For example, if ecri is shifted to the place of ecrj, the cost of ecri will be the cost of ecrj.
M
NC*edge collapses in the interested region
C*edge collapses in the interested region not performed until the original mesh is simplified to MN
Figure 3.3: Illustration of the first step in the weighting scheme.
Figure 3.5 depicts an example for the proposed weighting scheme, in which n is 36, N is 10, l is 8, C is 3 and C∗ is 5. In the first stage, we shift ecr4(= ecr(l−C∗+1)), marked
When the weighting is applied to more than one region simultaneously, the effect of weighting in each region would be less accurate because the shifted edge collapses for one region may conflict with the shifted edge collapses for another region. We take this issue into account in the re-simplification process. Once edge collapses ecr(l−C∗+1),...,ecrl are shifted to appropriate places after the weighting scheme is applied in a region, they will not be performed in subsequent applications of weighting scheme in other regions unless the expected simplification target cannot be reached. When this happens, the prohibited edge collapses will be performed until the simplification target is reached.
3.2 Weighting Simplification Cost 25
Figure 3.4: Illustration of the second step in the weighting scheme.
ecr1 ecr2 ecr3 ecr6 ecr7
Figure 3.5: An example of weighting.