We present the results of our method for computing shape correspondence and then discuss its limitations.
Results. In the preprocessing stage, we computed the skeletons of the models and AGD of the vertices. We then applied our method for a variety of models. The dog skele-ton is the source. The skeleskele-ton correspondence results are shown in Fig.12. The matched node pairs are drawn in the same colors but the unmatched nodes are not shown. All of the semantic parts (e.g. legs, heads, and ears) are matched correctly. Since our method performs junction node merging and branch merging, there may be one-to-many or many-to-many correspondences between junction nodes and also
between branches. This is different from the previous ap-proaches [HSKK01,BMSF06,ATCO∗10] which establish only one-to-one mapping. For example, a junction node at the chest of the dog is matched with two junction nodes at the chest of the goat. Our method can handle the skeletons of chairs with loops, as shown in Fig.13.
Comparisons.We compared our method with the method by Au et al. [ATCO∗10]. Fig.14shows two sets of corre-spondence results. For the top example, their method mis-matches an ear of the horse to the jaw of the dog. For the bottom example, their method mismatches the ears of the pig to the upper jaw of the dragon. Our method produces correct results due to branch clustering. The ears form clusters and they cannot be matched with the non-clustered jaws.
Compared to [BL08], our method can match the internal nodes of the skeletons. Our method matches the core tion nodes, and then match the remaining branches and junc-tion nodes in an interleaving manner. So that our method can match the internal nodes.
Model #Tri. AGD MSP SE MAS
Function Function
Dog(source) 18976 157.71 25.86 33.23
-Deer 7402 22.23 5.04 3.57 0.20
Human 11258 43.05 5.10 9.74 0.22
Armadillo 20000 193.32 30.10 23.63 0.24
Dragon 16000 199.02 27.79 28.16 0.19
Triceratops 15764 104.33 18.17 27.44 0.21
Elephant 30000 500.30 52.73 146.08 0.30
Asian dragon 28198 490.70 44.54 37.28 0.41
Table 2: Model complexities, the timings (sec) of the pre-processing tasks and the timings of matching the augmented skeletons. SE: Skeleton extraction; MAS: Matching the aug-mented skeletons.
Timing information.All the results were performed on Intel Core 2 Duo CPU E8400 3.0GHz with 4GB memory, us-ing a sus-ingle thread implementation. We precomputed AGD function, MSP function, and skeleton extraction. Table 2 shows the timing information of the preprocessing computa-tion and the timings of performing the shape correspondence (i.e. matching the augmented skeletons of the objects). The computation time for shape correspondence is under 0.5 sec-ond in all our experiments. Table3shows the information of the correspondence trees which were constructed for match-ing the augmented skeletons of the objects.
The searching space of our method is smaller than the combinatorial searching method [ZSCO∗08,ATCO∗10] as we perform branch clustering. The total number of the pos-sible matchings between the two skeletons reduce dramati-cally. Fig.15shows three leaf nodes of the correspondence tree for matching the skeletons of the dog and giraffe mod-els. The best result is shown in Fig. 15(a). The cost of the best node is smaller if more feature nodes and skeletal branches are matched. Similar to other existing techniques, our method requires the manual operations to change the pa-rameters if the results are not good. It usually takes a few
c 2011 The Author(s) c
2011 The Eurographics Association and Blackwell Publishing Ltd.
Online submission ID:1077 / A Skeleton-Based Approach for Shape Correspondence
Figure 12: Shape correspondence results between a dog and a variety of models: goat, camel, deer, giraffe, cat, pig, human, armadillo, dinosaur, triceratops and Asian dragon. The matched node pairs are drawn in the same colors.
Figure 13: Shape correspondence results for ant, teddy and chair models.
minutes for making several attempts to obtain the desired results.
Model #Tree Tree #Core #Branch #Branch #Leaf
Nodes Height Junction and Node Clusters Nodes Nodes Merging
Dog(source) - - - -
-Goat 18 4 3 12 20 4
Deer 18 4 3 15 26 5
Giraffe 23 4 3 8 36 7
Cat 18 4 3 8 22 5
Pig 11 4 3 0 16 3
Human 12 3 2 4 16 4
Armadillo 18 4 3 0 24 5
Dragon 14 4 3 0 18 4
Ant 42 5 4 50 24 10
Teddy 13 4 3 0 24 4
Chair 6 6 5 0 2 1
Table 3: Information of the correspondence trees.
Discussions and limitations.Our method may mismatch the branch clusters (e.g. mapping the ears of the horse to the horns of the cows) because of a lack of the semantic knowl-edge. In addition, some objects having flat shape around the junction nodes may not be inferred the front or back side by skeletons. A flipping correspondence may be obtained.
Figure 15: Some leaf nodes of the correspondence tree of matching the skeletons of the dog and giraffe models. (a):
Best result; (b): the chest of the dog is matched with the belly of the giraffe; (c): the head of the dog is matched with the mouth of the giraffe.
Our method requires manual operations to set the parame-ters and it may take several attempts to obtain acceptable re-sults. Finally, we cannot construct good skeletons for some man-made models (e.g., cups, sphere ball, or engineering models) and the quality of shape correspondence results is affected. If the skeletons do not capture the shape of the meshes faithfully, our method may not work properly.
c
2011 The Author(s) c
2011 The Eurographics Association and Blackwell Publishing Ltd.
Online submission ID:1077 / A Skeleton-Based Approach for Shape Correspondence
Figure 14: Comparison with the method [ATCO∗10]. Left: the results of [ATCO∗10]; Right: our results. Their method mis-matches the nodes at the heads (highlighted by the red squares).
7. Conclusions
We have presented a novel skeleton-based approach for com-puting shape correspondence. Our method may merge skele-tal nodes and branches so that it may compute 1-1 corre-spondences and many-many correcorre-spondences. Our method has been tested for a variety of similar objects and they may have different poses. The experimental results show that our method can compute acceptable shape correspondence.
Our method relies on the association between the skeletal nodes and object vertices, but we do not consider the actual shape of the parts for matching. Our method may fail to com-pute a reasonable shape correspondence if the objects are not similar. One possible extension of our work is to consider the actual shape of the parts so that partial correspondence can be computed. Another possible extension is that instead of selecting the core junction nodes, the parts of the objects are first extracted by employing skeletons. After that we can match the parts and build the entire shape correspondence.
Furthermore, we would like to apply prior knowledge with our method for computing shape correspondence.
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c 2011 The Author(s) c
2011 The Eurographics Association and Blackwell Publishing Ltd.
SAI-KEUNG WONG, JAU-AN YANG, TAN-CHI HO AND JUNG-HONG CHUANG