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 present an algorithm for multi-scale partial intrinsic symmetry detection over 2D and 3D shapes, where the scale of a symmetric region is defined by intrinsic distances between symmetric points over the region. To identify prominent symmetric regions which overlap and vary in form and scale, we decouple scale extraction and symmetry extraction by performing two levels of clustering. First, significant symmetry scales are identified by clustering sample point pairs from an input shape. Since different point pairs can share a common point, shape regions covered by points in different scale clusters can overlap. We introduce the symmetry scale matrix (SSM), where each entry estimates the likelihood two point pairs belong to symmetries at the same scale. The pair-to-pair symmetry affinity is computed based on a pair signature which encodes scales. We perform spectral clustering using the SSM to obtain the scale clusters. Then for all points belonging to the same scale cluster, we perform the second-level spectral clustering, based on a novel point-to-point symmetry affinity measure, to extract partial symmetries at that scale. We demonstrate our algorithm on complex shapes possessing rich symmetries at multiple scales. We introduce the Symmetry Factored Embedding (SFE) and the Symmetry Factored Distance (SFD) as new tools to analyze and represent symmetries in a point set. The SFE provides new coordinates in which symmetry is "factored out," and the SFD is the Euclidean distance in that space. These constructions characterize the space of symmetric correspondences between points -- i.e., orbits. A key observation is that a set of points in the same orbit appears as a clique in a correspondence graph induced by pairwise similarities. As a result, the problem of finding approximate and partial symmetries in a point set reduces to the problem of measuring connectedness in the correspondence graph, a well-studied problem for which spectral methods provide a robust solution. We provide methods for computing the SFE and SFD for extrinsic global symmetries and then extend them to consider partial extrinsic and intrinsic cases. During experiments with difficult examples, we find that the proposed methods can characterize symmetries in inputs with noise, missing data, non-rigid deformations, and complex symmetries, without a priori knowledge of the symmetry group. As such, we believe that it provides a useful tool for automatic shape analysis in applications such as segmentation and stationary point detection. An unsupervised procedure to local symmetry detection in natural images is proposed.Achieved via a Hough-style voting approach made at different scales and coordinate spaces.A final denoising scheme aims at removing noise and spurious local symmetries.Experiments have been conducted on the recent extension of the Berkeley Segmentation Dataset.The proposed method performs well compared to the state-of-the-art algorithms. Display Omitted In this paper, we propose a new unsupervised and simple approach to local symmetry detection of ribbon-like structure in natural images. The proposed model consists in quantifying the presence of a partial medial axis segment, existing between each pair of (preliminary detected) line segments delineating the boundary of two textured regions, by a set of heuristics related both to the geometrical structure of each pair of line segments and its ability to locally delimit a homogeneous texture region in the image. This semi-local approach is finally embedded in a two-step algorithm with an amplification step, via a Hough-style voting approach achieved at different scales and coordinate spaces which aims at determining the dominant local symmetries present in the image and a final denoising step, via an averaging procedure, which aims at removing noise and spurious local symmetries. The experiments, reported in this paper and conducted on the recent extension of the Berkeley Segmentation Dataset for the local symmetry detection task, demonstrate that the proposed symmetry detector performs well compared to the best existing state-of-the-art algorithms recently proposed in the literature. Local reflection symmetry detection in nature images is a quite important but challenging task in computer vision. The main obstacle is both the scales and the orientations of symmetric structure are unknown. The multiple instance learning (MIL) framework sheds lights onto this task owing to its capability to well accommodate the unknown scales and orientations of the symmetric structures. However, to differentiate symmetry vs non-symmetry remains to face extreme confusions caused by clutters scenes and ambiguous object structures. In this paper, we propose a novel multiple instance learning framework for local reflection symmetry detection, named multiple instance subspace learning (MISL), which instead learns a group of models respectively on well partitioned subspaces. To obtain such subspaces, we propose an efficient dividing strategy under MIL setting, named partial random projection tree (PRPT), by taking advantage of the fact that each sample (bag) is represented by the proposed symmetry features computed at specific scale and orientation combinations (instances). Encouraging experimental results on two datasets demonstrate that the proposed local reflection symmetry detection method outperforms current state-of-the-arts. HighlightsWe perform clustering on samples represented by multiple instances.We learn a group of MIL classifiers on subspaces.We report state-of-the-arts results on the symmetry detection benchmark.