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The spatial relationship between different objects plays an important role in defining the context of scenes. Most previous 3D classification and retrieval methods take into account either the individual geometry of the objects or simple relationships between them such as the contacts or adjacencies. In this paper we propose a new method for the classification and retrieval of 3D objects based on the Interaction Bisector Surface (IBS), a subset of the Voronoi diagram defined between objects. The IBS is a sophisticated representation that describes topological relationships such as whether an object is wrapped in, linked to or tangled with others, as well as geometric relationships such as the distance between objects. We propose a hierarchical framework to index scenes by examining both the topological structure and the geometric attributes of the IBS. The topology-based indexing can compare spatial relations without being severely affected by local geometric details of the object. Geometric attributes can also be applied in comparing the precise way in which the objects are interacting with one another. Experimental results show that our method is effective at relationship classification and content-based relationship retrieval.

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Highlights• We propose a feature-based 2D+3D multimodal facial expression recognition method.• It is fully automatic benefit from a large set of automatically detected landmarks.• The complementarities between 2D and 3D features are comprehensively demonstrated.• Our method achieves the best accuracy on the BU-3DFE database so far.• A good generalization ability is shown on the Bosphorus database. AbstractWe present a fully automatic multimodal 2D + 3D feature-based facial expression recognition approach and demonstrate its performance on the BU-3DFE database. Our approach combines multi-order gradient-based local texture and shape descriptors in order to achieve efficiency and robustness. First, a large set of fiducial facial landmarks of 2D face images along with their 3D face scans are localized using a novel algorithm namely incremental Parallel Cascade of Linear Regression (iPar-CLR). Then, a novel Histogram of Second Order Gradients (HSOG) based local image descriptor in conjunction with the widely used first-order gradient based SIFT descriptor are used to describe the local texture around each 2D landmark. Similarly, the local geometry around each 3D landmark is described by two novel local shape descriptors constructed using the first-order and the second-order surface differential geometry quantities, i.e., Histogram of mesh Gradients (meshHOG) and Histogram of mesh Shape index (curvature quantization, meshHOS). Fi- * Corresponding author A C C E P T E D M A N U S C R I P Tnally, the Support Vector Machine (SVM) based recognition results of all 2D and 3D descriptors are fused at both feature-level and score-level to further improve the accuracy. Comprehensive experimental results demonstrate that there exist impressive complementary characteristics between the 2D and 3D descriptors. We use the BU-3DFE benchmark to compare our approach to the state-of-the-art ones. Our multimodal feature-based approach outperforms the others by achieving an average recognition accuracy of 86.32%. Moreover, a good generalization ability is shown on the Bosphorus database.

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