Structure-oriented Networks of Shape Collections

Transcription

1 Structure-oriented Networks of Shape Collections Noa Fish 1 Oliver van Kaick 2 Amit Bermano 3 Daniel Cohen-Or 1 1 Tel Aviv University 2 Carleton University 3 Princeton University 1 pplementary material In this supplementary material, we provide additional evaluation and comparisons to our method. Dataset Decomposition Matching Lifting Chair (large) 9min 1.5h Table 5min 0.8h Car 8min 1h Plane 8min 1.4h Lamp 3min 0.4h Guitar 50s 4min Faucet 1min 5min Chair (small) 28s 3min Vase 25s 2min Individual 0.2s 0.05s 0.4s Table 1: Timing. mmary of running times on batch tasks as well as average individual execution durations per task. The lifting task is not executed in batch as it is invoked on-demand during propagation. 1.1 Timing and implementation We summarize the run times for each step of our approach in Table 1. We provide overall batch run times measured for the decomposition and matching tasks on each dataset, as well as average individual run times for each task. All steps are implemented in Matlab, and batch tasks are executed on 4 parallel workers where possible. Our method performs simple and local operations and is more efficient than our implementation of Wang et al. [2013], which took 3.8 and 7.3 minutes to decompose and match the COSEG chair set, respectively (under similar implementation conditions). The parameters we employ throughout the computation mainly control the characteristics of the formed regions, balancing intraregion complexity and overall decomposition complexity. We require our regions to be simple, such that noticeable structural changes define borders between them. Simultaneously, we aim for a compact decomposition that is meaningful and easy to match to others. Our experiments were carried out with the same configuration of parameters and we noted that small changes do not affect the results significantly. For a full implementation of our method please see our project page underhttp:// noafish/corrnet. 1.2 Decomposition and matching Figure 1 features several pairs of shapes undergoing decomposition and matching. 1.3 Correspondence networks Comparison on BHCP. Figure 2 contains a quantitative comparison on three out of the four sets of the BHCP correspondence benchmark [Kim et al. 2013] (chair, plane, helicopter). A subset of the shapes in each of these sets is accompanied by ground-truth feature points for correspondence prediction evaluation. For each of these subsets, we apply our decomposition and matching method in an all-pairs manner. That is, each shape is matched against all other shapes in the subset. Next, we run two experiments where we compute correspondence accuracy according to the ground-truth feature points. In the first experiment, face-level correspondences are inferred directly from the matching between any pair of shapes. In the second experiment, we first compute correspondence propagation routes based on our distance matrix, and then compute a face-level correspondence for each pair of shapes, by propagating correspondences along the minimax path (as described in the paper). Since our matching may be partial by design, we emphasize that we are not necessarily able to match all points in the set of groundtruth correspondences. Indeed, in all three sets, we computed only a portion of the given correspondences. In the chair subset, we were able to compute 73.4% of the correspondences in the first experiment (direct comparison), and 89.4% in the second experiment (propagation). In the plane set, we computed 90.2% and 96.4%, and in the helicopter set 86.7% and 93.8%, respectively. For each unmatched correspondence, we set the predicted distance to as a penalty. Examining the curves in Figure 2, we note the substantial gap between our two experiments, in favor of the second. This result supports our central claim that correspondence inference via similar shapes can provide a more reliable estimation. Additionally, we observe the steep rise of the curve of our method compared to those of previous methods ( [Kim et al. 2013; Huang et al. 2014]), and attribute it to two characteristics: 1) Our method assumes a prealigned set; 2) Our method allows many-to-many face correspondences. Finally, we note that all three subsets are of a relatively small size ( 100 shapes), but still contain considerable shape variations. For the construction of a meaningful and well-connected correspondence network, our method requires a reasonably dense set when high variation is observed. Thus, a greater contribution can be achieved within sets that are more densely populated in terms of their similarities. Figures 3 4 feature an example extracted from the subset of chairs, where a directly-computed correspondence between two chairs is unsatisfactory. In comparison, we show that the by-proxy correspondence inference, computed along a propagation path through two 3rd-party shapes, provides a correspondence of higher quality, with more well-matched feature points. Furthermore, we note that despite some partial matches along the path, resulting from differences in part sizes and structure, the multi-view nature of the matching is able to make up for some of the missing information. Correspondence subtrees. Figure 5 contains several examples of correspondence subtrees, each being a portion of a minimum spanning tree of a network of correspondences constructed for a set of shapes.

2 1.4 Segmentation transfer Figure 6 features examples for segmentation transfer through propagation on various shapes. 1.5 Isometry-based correspondence experiment In this experiment, we evaluate whether correspondence methods designed for nearly isometric shapes are suitable for structurevarying man-made shapes. We combine the recent shape correspondence method of Solomon et al. [2016] with a segmentation propagation scheme to obtain a labeling for the COSEG chairs set. Since this method does not provide an estimation of similarity between shapes, we utilize distances to select 15 seeds (as described in the paper) and determine propagation routes. In this experiment, we observe that although this method is robust to multimodal data and approximates semantic correspondences under nonrigid articulation-like deformations, it is not suitable for man-made shapes. Our shape class incorporates large variation in size and structure within a part, which adds ambiguity to correspondence estimation. This type of behavior is not well aligned with the assumptions made by nearly-isometric correspondence matching methods, therefore they are unlikely to be suitable in this context. The tested approach achieves 60% accuracy on the examined set, and Figure 7 features an example where the method fails to infer a reliable mapping between structurally different shapes. 1.6 Shape retrieval Figures 8 10 contain example queries and retrieval results of our method compared to others, on different sets from the test set of SHREC 2016 [Savva et al. 2016]. Figures contain example conditional queries and retrieval results of our method. References HUANG, Q., WANG, F., AND GUIBAS, L Functional map networks for analyzing and browsing large shape collections. ACM Trans. Graph. (SIGGRAPH) 33, 4, 36:1 11. KIM, V. G., LI, W., MITRA, N. J., CHAUDHURI, S., DIVERDI, S., AND FUNKHOUSER, T Learning part-based templates from large collections of 3d shapes. ACM Trans. Graph. (SIGGRAPH) 32, 4, 70:1 12. SAVVA, M., YU, F., SU, H., AONO, M., CHEN, B., COHEN- OR, D., DENG, W., SU, H., BAI, S., BAI, X., FISH, N., HAN, J., KALOGERAKIS, E., LEARNED-MILLER, E. G., LI, Y., LIAO, M., MAJI, S., TATSUMA, A., WANG, Y., ZHANG, N., AND ZHOU, Z Large-Scale 3D Shape Retrieval from ShapeNet Core55. In Eurographics Workshop on 3D Object Retrieval, The Eurographics Association, A. Ferreira, A. Giachetti, and D. Giorgi, Eds. SOLOMON, J., PEYRÉ, G., KIM, V., AND SRA, S Entropic metric alignment for correspondence problems. ACM Trans. Graph. (SIGGRAPH), to appear. WANG, Y., GONG, M., WANG, T., COHEN-OR, D., ZHANG, H., AND CHEN, B Projective analysis for 3D shape segmentation. ACM Trans. Graph. (SIGGRAPH Asia) 32, 6, 192:1 12. Figure 2: Comparison on Chair/Plane/Helicopter from the BHCP benchmark. Given groud-truth feature points on each shape, the curve of each method indicates the percentage of correspondences that were matched to within a certain error margin (in terms of Euclidean distance), indicated by the x-axis. We compare to Kim et al. [2013] and Huang et al. [2014] on two variations of our method. In OursDirect, we compute correspondences between every pair of shapes directly from their matching. In OursProp, we compute minimax propagation paths within our correspondence network, and then propagate correspondences through 3rd-party shapes.

3 (a) (b) (c) (d) (e) (f) (g) (h) Figure 4: Direct correspondence vs. correspondence propagation. The ground-truth feature points of the source chair are shown for two views in (a)+(b), and those of the target chair are shown in (c)+(d). The points on the target shape that correspond to the ground-truth points of the source chair are shown in (e)+(f), which are computed by our method using the direct approach. Corresponding points computed by our method using propagation are shown in (g)+(h). Figure 6: Segmentation propagation paths. We show the transference of a consistent semantic segmentation through the face-level correspondences obtained by our matching process. Starting from a manual segmentation given for the leftmost source shape in a sequence, we propagate the segmentation along the path towards the rightmost target shape. Parts colored in grey have no match to the preceding shape and remain unlabeled.

4 Query Method Top-5 retrievals * * (a) (b) (c) Figure 7: A correspondence between structurally distinct shapes computed with the method of Solomon et al. [2016]. We select four points on shape (a), and compute the regions with the highest correspondence probability in shape (b) (seen from another angle in (c)). Note that the resulting mapping is not semantically meaningful. * * Figure 8: Shape retrieval comparison (1 of 3). Top-5 retrievals given by our method (), compared to those given by the descriptor and the method of et al. [2015].

5 Query Method Top-5 retrievals Query Method Top-5 retrievals * * Figure 9: Shape retrieval comparison (2 of 3). Top-5 retrievals given by our method (), compared to those given by the descriptor and the method of et al. [2015]. Figure 10: Shape retrieval comparison (3 of 3). Top-5 retrievals given by our method (), compared to those given by the descriptor and the method of et al. [2015].

6 Query Condition Top-5 retrievals Query Condition Top-5 retrievals Figure 11: Directional shape retrieval (1 of 3). Examples of retrieval results for conditional queries, specifying one of two directions: simpler shapes / more complex shapes. The results are returned in the order of their structural similarity to the query. Figure 12: Directional shape retrieval (2 of 3). Examples of retrieval results for conditional queries, specifying one of two directions: simpler shapes / more complex shapes. The results are returned in the order of their structural similarity to the query.

7 Query Condition Top-5 retrievals Figure 13: Directional shape retrieval (3 of 3). Examples of retrieval results for conditional queries, specifying one of two directions: simpler shapes / more complex shapes. The results are returned in the order of their structural similarity to the query.

8 shape decomposition 1 matching 1 decomposition 2 matching 2 Figure 1: Decomposition and matching examples.

9 shape dec 1 match 1 dec 2 match 2 dec 3 match 3 dec 4 match 4 Figure 3: Direct correspondence vs. correspondence propagation. The source and target shapes are shown in the first two rows along with their decompositions and matchings. Note the missing and incorrect matches inferred as part of this direct matching. In the following pairs of rows, we feature pairs of shapes and their matchings, such that we start from the source shape, pass through two 3rd-party shapes, and end at the target shape. This propagation approach infers a better correspondence between the source and target, despite partial matches along the path.

10 (a) (b) (c) (d) (e) (f) Figure 5: Selected subtrees sampled from our correspondence networks. Given a source shape (the root at the bottom of each tree), we show shapes that can be reached by following a few neighbor connections in the network of correspondences.