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.
The exists a class of techniques that exploits the redundancy in repeating structures to robustly detect symmetries @cite_13 ; @cite_27 ; @cite_1 ; @cite_7 ; @cite_9 . The transformation space voting method proposed by @cite_30 is extended in @cite_7 by fitting parameters of a transformation generator to optimally register the clusters in transformation space. @cite_5 and @cite_13 have taken a similar approach using subgraph matching of feature points and feature lines, respectively, to establish potential correspondences between repeated structures. This is followed by an iterative closest points (ICP) algorithm to simultaneously grow corresponding regions and refine matches over all detected patterns, allowing the detection of repeated patterns even in noisy data @cite_7 , but at the cost of requiring a-priori knowledge of the commutative group expected in the data. Also, the non-linear local optimization procedure within the ICP algorithm could cause it to get trapped in a local minimum if the initialization is not good enough.
@cite_30 @cite_7 @cite_9 @cite_1 @cite_27 @cite_5 @cite_13
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"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. We introduce a computational framework for discovering regular or repeated geometric structures in 3D shapes. We describe and classify possible regular structures and present an effective algorithm for detecting such repeated geometric patterns in point- or meshbased models. Our method assumes no prior knowledge of the geometry or spatial location of the individual elements that define the pattern. Structure discovery is made possible by a careful analysis of pairwise similarity transformations that reveals prominent lattice structures in a suitable model of transformation space. We introduce an optimization method for detecting such uniform grids specifically designed to deal with outliers and missing elements. This yields a robust algorithm that successfully discovers complex regular structures amidst clutter, noise, and missing geometry. The accuracy of the extracted generating transformations is further improved using a novel simultaneous registration method in the spatial domain. We demonstrate the effectiveness of our algorithm on a variety of examples and show applications to compression, model repair, and geometry synthesis. In this paper, we present a new geometry compression technique particularly suitable for 3D mesh models of engineering class ‐ architectural models, machine plants, factories, etc. We observe that such models have a number of repeating features at various levels of granularity. In most of the widely available models in this class, the geometric description of such features are also repeated. A major distinctive aspect of our compression technique is that repeating geometric features in the geometric model are automatically discovered and then compactly encoded. The proposed method discovers repetition first at the connected component level and then at the subcomponent level across components and also at the aggregate component level. The results from a straight forward implementation tried on large mesh models downloaded from the net are extremely encouraging. Decorative reliefs are widely used for e.g. packaging and porcelain design. In periodic reliefs, the relief repeats a pattern, for example all the way around an underlying surface of revolution. Reverseengineering of existing reliefs allows them to be re-applied to different base surfaces; we show here how to segment a single repeat unit of a periodic relief starting from a scanned triangle mesh. We first briefly review how we segment the relief from the background surface using our previous work. The rest of the paper then concentrates on how we extract a single repeat unit from the relief. To do so, the user provides two points on one relief boundary which are in approximate correspondence on consecutive repeats of the relief. We first refine the relative locations of these points, and then determine a third corresponding point using relief boundary information. These are used to determine three initial cutting planes across the relief. Then surface registration strategies are utilised to refine the correspondence between adjacent repeat units. Finally, we refine the exact locations of the cutting planes by considering only surface information close to the cutting planes. This allows a repeat unit of the periodic relief to be extracted. We demonstrate that our algorithm is successful and practical, using various real scanned models: user input can be quite imprecise, and we can cope with hand-made reliefs in which the pattern units are only approximately copies of each other. Approximate geometric models, e.g. as created by reverse engineering, describe the approximate shape of an object, but do not record the underlying design intent. Automatically inferring geometric aspects of the design intent, represented by feature trees and geometric constraints, enhances the utility of such models for downstream tasks. One approach to design intent detection in such models is to decompose them into regularity features. Geometric regularities such as symmetries may then be sought in each regularity feature, and subsequently be combined into a global, consistent description of the model's geometric design intent. This paper describes a systematic approach for finding such regularity features based on recovering broken symmetries in the model. The output is a tree of regularity features for subsequent use in regularity detection and selection. Experimental results are given to demonstrate the operation and efficiency of the algorithm. Symmetry detection aims at discovering redundancy in the form of reoccurring structures in geometric objects. In this paper, we present a new symmetry detection algorithm for geometry represented as point clouds that is based on analyzing a graph of surface features. We combine a general feature detection scheme with a RANSAC-based randomized subgraph searching algorithm in order to reliably detect reoccurring patterns of locally unique structures. A subsequent segmentation step based on a simultaneous region growing variant of the ICP algorithm is applied to verify that the actual point cloud data supports the pattern detected in the feature graphs. We apply our algorithm to synthetic and real-world 3D scanner data sets, demonstrating robust symmetry detection results in the presence of scanning artifacts and noise. The modular and flexible nature of the graph-based detection scheme allows for easy generalizations of the algorithm, which we demonstrate by applying the same technique to other data modalities such as images or triangle meshes.