Projects on analyzing sets of shapes: Supervised Part Correspondence, Unsupervised Co-Segmentation, Active Co-Analysis

Co-Hierarchical Analysis of Shape Structures

Oliver van Kaick1, Kai Xu2, Hao Zhang1, Yanzhen Wang2, Shuyang Sun1, Ariel Shamir3, Daniel Cohen-Or4,
1Simon Fraser University
2HPCL, National University of Defense Technology
3The Interdisciplinary Center
4Tel Aviv University
Structural co-hierarchical analysis of a set of velocipedes (bicycles, tricycles and four-cycles)

Figure: Structural co-hierarchical analysis of a set of velocipedes (bicycles, tricycles and four-cycles). The resulting co-hierarchy (center) is illustrated by a single sample shape from the set, where each node represents a part assembly. Two of the nodes (highlighted in blue and green) are expanded to show the insight gained by the analysis which relates parts with rather different geometries but similar functions.


We introduce an unsupervised co-hierarchical analysis of a set of shapes, aimed at discovering their hierarchical part structures and revealing relations between geometrically dissimilar yet functionally equivalent shape parts across the set. The core problem is that of representative co-selection. For each shape in the set, one representative hierarchy (tree) is selected from among many possible interpretations of the hierarchical structure of the shape. Collectively, the selected tree representatives maximize the within-cluster structural similarity among them. We develop an iterative algorithm for representative co-selection. At each step, a novel cluster-and-select scheme is applied to a set of candidate trees for all the shapes. The tree-to-tree distance for clustering caters to structural shape analysis by focusing on spatial arrangement of shape parts, rather than their geometric details. The final set of representative trees are unified to form a structural co-hierarchy. We demonstrate co-hierarchical analysis on families of man-made shapes exhibiting high degrees of geometric and finer-scale structural variabilities.



Figure: Consistent hierarchical segmentation results corresponding to structural co-hierarchies obtained for various sets.


PDF (17MB)
Supplementary material (16MB)

BibTex Reference

    author = {Oliver van Kaick and Kai Xu and Hao Zhang and Yanzhen Wang and Shuyang Sun and Ariel Shamir and Daniel Cohen-Or},
    title = {Co-Hierarchical Analysis of Shape Structures},
    journal = {ACM Trans. on Graphics (Proc. SIGGRAPH)},
    volume = {32},
    number = {4},
    pages = {69:1--69:10},
    year = 2013,

Presentation slides



Dataset and segmentation results available upon request.


We would like to thank all the reviewers for their comments and suggestions. This work is supported in part by the Natural Science and Engineering Research Council of Canada (grant no. 611370), NSFC (61202333), CPSF (2012M520392), and the Israel Science Foundation (grant no. 324/11).


GrUVi Project Page
Related projects:
Prior Knowledge for Part Correspondence
Unsupervised Co-Segmentation of a Set of Shapes via Descriptor-Space Spectral Clustering
Active Co-Analysis of a Set of Shapes