A comprehensive framework for detection and characterization of overlapping intrinsic symmetry over 3D shapes is proposed. To identify prominent symmetric regions which overlap in space and vary in form, the proposed framework is decoupled into a Correspondence Space Voting procedure followed by a Transformation Space Mapping procedure. In the correspondence space voting procedure, significant symmetries are first detected by identifying surface point pairs on the input shape that exhibit local similarity in terms of their intrinsic geometry while simultaneously maintaining an intrinsic distance structure at a global level. Since different point pairs can share a common point, the detected symmetric shape regions can potentially overlap. To this end, a global intrinsic distance-based voting technique is employed to ensure the inclusion of only those point pairs that exhibit significant symmetry. In the transformation space mapping procedure, the Functional Map framework is employed to generate the final map of symmetries between point pairs. The transformation space mapping procedure ensures the retrieval of the underlying dense correspondence map throughout the 3D shape that follows a particular symmetry. Additionally, the formulation of a novel cost matrix enables the inner product to succesfully indicate the complexity of the underlying symmetry transformation. The proposed transformation space mapping procedure is shown to result in the formulation of a semi-metric symmetry space where each point in the space represents a specific symmetry transformation and the distance between points represents the complexity between the corresponding transformations. Experimental results show that the proposed framework can successfully process complex 3D shapes that possess rich symmetries.
Several recent approaches to detect approximate and partial extrinsic symmetries have focused on algorithms that cluster votes for symmetries in a parameterized “transformation space” @cite_24 ; @cite_30 ; @cite_31 ; @cite_19 ]. @cite_30 generate “votes” in a transformation space to align pairs of similar points and then cluster them in a space spanned by the parameters of the potential symmetry transformations. Regardless of how good the shape descriptors are, the aforementioned methods are not effective at finding correspondences between points in complex symmetry orbits that are spread across multiple distinct clusters in the transformation space. Since the dimensionality of the transformation space increases with the complexity of the symmetry, the voting procedure in transformation space becomes increasing intractable when dealing with complex symmetries.
@cite_24 @cite_19 @cite_31 @cite_30
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We propose a randomized method for the detection of symmetry in polyhedra without assuming the predetermination of the centroids of the objects. Using a voting process, which is the main concept of the Hough transform in image processing, we transform the geometric computation for symmetry detection based on graph theory, to the peak detection problem in a voting space in the context of the Hough transform. In this paper, a simple and yet robust Hough transform algorithm is proposed to detect and analyze reflectional symmetry and skew-symmetry (reflectional symmetry under parallel projection). It is applicable to shapes that contain global, local and slightly deformed reflectional skew-symmetries under the presence of noise and occlusion. "Symmetry is a complexity-reducing concept [...]; seek it every-where." - Alan J. PerlisMany natural and man-made objects exhibit significant symmetries or contain repeated substructures. This paper presents a new algorithm that processes geometric models and efficiently discovers and extracts a compact representation of their Euclidean symmetries. These symmetries can be partial, approximate, or both. The method is based on matching simple local shape signatures in pairs and using these matches to accumulate evidence for symmetries in an appropriate transformation space. A clustering stage extracts potential significant symmetries of the object, followed by a verification step. Based on a statistical sampling analysis, we provide theoretical guarantees on the success rate of our algorithm. The extracted symmetry graph representation captures important high-level information about the structure of a geometric model which in turn enables a large set of further processing operations, including shape compression, segmentation, consistent editing, symmetrization, indexing for retrieval, etc.