Co-Hierarchical Analysis of Shape Structures Oliver van Kaick 1,4 Kai Xu 2 Hao Zhang 1 Yanzhen Wang 2 Shuyang Sun 1 Ariel Shamir 3 Daniel Cohen-Or 4 4 Tel Aviv University 1 Simon Fraser University 3 The Interdisciplinary Center 2 HPCL, Nat. Univ. of Defense Tech.
Shape segmentation 2 Analysis of sets of shapes Joint segmentation Huang et al Co-segmentation Sidi et al Active co-analysis Wang et al. 2012
Shape segmentation 3 Segmentation: a flat representation
Part hierarchy 4 Hierarchy: a higher-level organization of shape parts
Applications of hierarchies 5 Use the hierarchy for various tasks Structure-aware shape editing [Wang et al. 2011] Hierarchical segmentation
Part hierarchies 6 Extraction of hierarchies from individual or pairs of shapes Symmetry hierarchy Wang et al Geometry structuring Martinet 2007 Part recombination Jain et al. 2012
Co-hierarchical analysis 7 Our goal: Extraction of a unified (binary) hierarchy Through an unsupervised co-analysis of the set
Co-hierarchical analysis 8 A unified explanation of the structures Top-down to account for the structural variability
Co-hierarchical analysis 9 The co-hierarchy of a set of velocipedes Capturing the functionality of the parts
Co-hierarchical analysis 10 The co-hierarchy of a set of velocipedes Capturing the functionality of the parts
Co-hierarchical analysis 11 The co-hierarchy of a set of velocipedes Capturing the functionality of the parts
Challenge of co-hierarchical analysis 12 Shapes can have many possible hierarchies We need to select one hierarchy per shape …
Challenge of co-hierarchical analysis 13 There can be geometric variability in the set We need to compare the shape structures
Challenge of co-hierarchical analysis 14 There can also be much structural variability We need to account for that
Challenge of co-hierarchical analysis 15 Cluster-and-select scheme: clustering, representative selection, and resampling
Overview 16
Overview 17
Sampling the space of hierarchies We sample the space by sampling the splits Difficult to define a generic splitting criterion Criterion: balance of volume, compactness of parts, normalized cut? We resort to random sampling We sample splits in a top-down manner 18
Tree-to-tree distance 19 Tree-to-tree distance: structural differences
Node distance 20 Transformation between bounding boxes Bounding boxes focus on the structural similarity
Shape distance 21 Shape distance: distance between hierarchies
Cluster-and-select motivation 22 Representative selection
Cluster-and-select 23 Minimal illustrative example with four shapes
Cluster-and-select 24 Multiple possible hierarchies per shape
Cluster-and-select 25 Sampling of hierarchies
Cluster-and-select 26 Multi-instance clustering
Cluster-and-select 27 Representative selection
Cluster-and-select 28 Traditional clustering: maximize similarity within clusters and dissimilarity between clusters
Cluster-and-select 29 Our problem: maximize similarity within clusters and similarity between clusters
Cluster-and-select 30 Samples maximize the similarity within clusters
Cluster-and-select 31 Also maximize the similarity between clusters
Cluster-and-select 32 Resampling of hierarchies
Cluster-and-select 33 Resampling of hierarchies
Cluster-and-select 34 Repeat the process: clustering, selection
Cluster-and-select 35 Representative movement
Results of co-hierarchical analysis 36 The co-hierarchies are shown as a hierarchical segmentation
Results of co-hierarchical analysis 37 Hierarchical segmentation results
Results of co-hierarchical analysis 38 Hierarchical segmentation results
Results of co-hierarchical analysis 39 Hierarchical segmentation results
Results of co-hierarchical analysis 40 Consistency of the co-hierarchy [Wang et al. 2011] Ours
Results of co-hierarchical analysis 41 Cluster-and-select on a mixed set of shapes
Summary of contributions Co-hierarchical analysis of sets of shapes Structure-driven shape analysis – To deal with geometric variability Hierarchical analysis – To deal with structural variability A novel cluster-and-select scheme – To account for both variability and similarity The structural co-hierarchy representation – Unifies the learned structures 42
Limitations and future work Co-hierarchical analysis: only a first step More sophisticated node and tree distances Initial random sampling of trees Integrate segmentation and hierarchical analysis Multi-class co-hierarchies Which hierarchy should be selected? 43
44 Co-Hierarchical Analysis of Shape Structures Project page: Thank you for your attention!
Appendix 45
Tree-to-tree distance 46 Node distance Recursive children distance NiNi NjNj
Tree-to-tree distance 47 NiNi NjNj Node distance Recursive children distance
Results of co-hierarchical analysis 48 Hierarchical segmentation results
Results of co-hierarchical analysis 49 Hierarchical segmentation results: deeper levels
Results of co-hierarchical analysis 50 Improvements shown by the cluster-and-select