Detecting Approximate Reflection Symmetry in a Point Set Using Optimization on Manifold
15 Mar 2019-IEEE Transactions on Signal Processing (Institute of Electrical and Electronics Engineers (IEEE))-Vol. 67, Iss: 6, pp 1582-1595
TL;DR: The robustness of the approach is shown by varying the amount of distortion in a perfect reflection symmetry pattern where the authors perturb each point by a different amount of perturbation, and the effectiveness of the method is demonstrated by applying it to the problem of 2-D and 3-D reflection symmetry detection.
Abstract: We propose an algorithm to detect approximate reflection symmetry present in a set of volumetrically distributed points belonging to $\mathbb {R}^d$ containing a distorted reflection symmetry pattern. We pose the problem of detecting approximate reflection symmetry as the problem of establishing correspondences between the points which are reflections of each other and we determine the reflection symmetry transformation. We formulate an optimization framework in which the problem of establishing the correspondences amounts to solving a linear assignment problem and the problem of determining the reflection symmetry transformation amounts to solving an optimization problem on a smooth Riemannian product manifold. The proposed approach estimates the symmetry from the geometry of the points and is descriptor independent. We evaluate the performance of the proposed approach on the standard benchmark dataset and achieve the state-of-the-art performance. We further show the robustness of our approach by varying the amount of distortion in a perfect reflection symmetry pattern where we perturb each point by a different amount of perturbation. We demonstrate the effectiveness of the method by applying it to the problem of 2-D (two-dimensional) and 3-D reflection symmetry detection along with comparisons.
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TL;DR: This work proposes a descriptor-free approach, in which, the problem of reflection symmetry detection as an optimization problem and provide a closed-form solution, and shows that the proposed method achieves state-of-the-art performance on the standard dataset.
17 citations
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TL;DR: R-PointHop as discussed by the authors determines a local reference frame (LRF) for every point using its nearest neighbors and finds its local attributes by point downsampling, neighborhood expansion, attribute construction and dimensionality reduction steps.
Abstract: Inspired by the recent PointHop classification method, an unsupervised 3D point cloud registration method, called R-PointHop, is proposed in this work. R-PointHop first determines a local reference frame (LRF) for every point using its nearest neighbors and finds its local attributes. Next, R-PointHop obtains local-to-global hierarchical features by point downsampling, neighborhood expansion, attribute construction and dimensionality reduction steps. Thus, we can build the correspondence of points in the hierarchical feature space using the nearest neighbor rule. Afterwards, a subset of salient points of good correspondence is selected to estimate the 3D transformation. The use of LRF allows for hierarchical features of points to be invariant with respect to rotation and translation, thus making R-PointHop more robust in building point correspondence even when rotation angles are large. Experiments are conducted on the ModelNet40 and the Stanford Bunny dataset, which demonstrate the effectiveness of R-PointHop on the 3D point cloud registration task. R-PointHop is a green and accurate solution since its model size and training time are smaller than those of deep learning methods by an order of magnitude while its registration errors are smaller. Our codes are available on GitHub.
16 citations
01 Jan 2022
TL;DR: R-PointHop as discussed by the authors determines a local reference frame (LRF) for every point using its nearest neighbors and finds local attributes by point downsampling, neighborhood expansion, attribute construction and dimensionality reduction steps.
Abstract: Inspired by the recent PointHop classification method, an unsupervised 3D point cloud registration method, called R-PointHop, is proposed in this work. R-PointHop first determines a local reference frame (LRF) for every point using its nearest neighbors and finds local attributes. Next, R-PointHop obtains local-to-global hierarchical features by point downsampling, neighborhood expansion, attribute construction and dimensionality reduction steps. Thus, point correspondences are built in hierarchical feature space using the nearest neighbor rule. Afterwards, a subset of salient points with good correspondence is selected to estimate the 3D transformation. The use of the LRF allows for invariance of the hierarchical features of points with respect to rotation and translation, thus making R-PointHop more robust at building point correspondence, even when the rotation angles are large. Experiments are conducted on the 3DMatch, ModelNet40, and Stanford Bunny datasets, which demonstrate the effectiveness of R-PointHop for 3D point cloud registration. R-PointHop's model size and training time are an order of magnitude smaller than those of deep learning methods, and its registration errors are smaller, making it a green and accurate solution. Our codes are available on GitHub.
11 citations
TL;DR: A novel image Retargeting approach which preserves the reflection symmetry present in the image during the image retargeting process and shows better preservation of symmetry axis, preservation of shape of the symmetric object, and quality of image retTargeting when compared to the existing methods.
9 citations
TL;DR: Wang et al. as mentioned in this paper presented two shape signature-based reflection symmetry detection methods with their theoretical underpinning and empirical evaluation, which can effectively deal with compound shapes which are challenging for traditional contour-based methods.
8 citations
References
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01 Oct 2017
TL;DR: Using the MSR-computed plane of symmetry, techniques for the optimal symmetric pairwise assignment between axon reconstructions are introduced and visualizations illustrating how neighborhood relationships between nearby axon pairs compare with the relationships between their mirror-reflected counterparts along the anteroposterior axis are provided.
Abstract: We demonstrate that the problem of fitting a plane of mirror symmetry to data in any Euclidian space can be reduced to the problem of registering two datasets. The exactness of the resulting solution depends entirely on the registration accuracy. This new Mirror Symmetry via Registration (MSR) framework involves (1) data reflection with respect to an arbitrary plane, (2) registration of original and reflected datasets, and (3) calculation of the eigenvector of eigenvalue -1 for the transformation matrix representing the reflection and registration mappings. To support MSR, we also introduce a novel 2D registration method based on random sample consensus of an ensemble of normalized cross-correlation matches. With this as its registration back-end, MSR achieves state-of-the-art performance for symmetry line detection in two independent 2D testing databases. We further demonstrate the generality of MSR by testing it on a database of 3D shapes with an iterative closest point registration back-end. We finally explore its applicability to examining symmetry in natural systems by assessing the degree of symmetry present in myelinated axon reconstructions from a larval zebrafish. Using the MSR-computed plane of symmetry, we introduce techniques for the optimal symmetric pairwise assignment between axon reconstructions and provide visualizations illustrating how neighborhood relationships between nearby axon pairs compare with the relationships between their mirror-reflected counterparts along the anteroposterior axis.
23 citations
"Detecting Approximate Reflection Sy..." refers methods or result in this paper
...first reflected the original point cloud about an arbitrary reflection plane and then used the ICP algorithm to align the original point cloud and the reflected point cloud [13]....
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...precision curves for the methods in [11]–[13], and...
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...But, the methods in [10], [11], [13], [14], and [15] do not establish correspondences....
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...plane detection, we compare the performance of our approach with the performance of the methods in [12], [13], and [11]....
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TL;DR: A convolutional approach to reflection symmetry detection in 2D, built on the products of complex-valued wavelet convolutions, that outperforms the best-performing algorithm on the CVPR 2013 Symmetry Detection Competition Database in the single-symmetry case.
23 citations
"Detecting Approximate Reflection Sy..." refers background in this paper
...12, we show the results on the datasets [61], [62], and [60]....
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TL;DR: It is proved that the group representation of each symmetry can be uniquely determined from a small number of symmetric pairs of points under certain conditions, where the number of pairs is equal to the maximum multiplicity of eigenvalues of the Laplace‐Beltrami operator.
Abstract: Global intrinsic symmetry detection of 3D shapes has received considerable attentions in recent years. However, unlike extrinsic symmetry that can be represented compactly as a combination of an orthogonal matrix and a translation vector, representing the global intrinsic symmetry itself is still challenging. Most previous works based on point‐to‐point representations of global intrinsic symmetries can only find reflectional symmetries, and are inadequate for describing the structure of a global intrinsic symmetry group. In this paper, we propose a novel group representation of global intrinsic symmetries, which describes each global intrinsic symmetry as a linear transformation of functional space on shapes. If the eigenfunctions of the Laplace‐Beltrami operator on shapes are chosen as the basis of functional space, the group representation has a block diagonal structure. We thus prove that the group representation of each symmetry can be uniquely determined from a small number of symmetric pairs of points under certain conditions, where the number of pairs is equal to the maximum multiplicity of eigenvalues of the Laplace‐Beltrami operator. Based on solid theoretical analysis, we propose an efficient global intrinsic symmetry detection method, which is the first one able to detect all reflectional and rotational global intrinsic symmetries with a clear group structure description. Experimental results demonstrate the effectiveness of our approach.
18 citations
"Detecting Approximate Reflection Sy..." refers background in this paper
...We refer the reader to some of the pioneering works for more details on this category ([19]–[27])....
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01 Jan 2008
TL;DR: In this paper, the authors presented a new symmetry detection algorithm for geometry represented as point clouds that is based on analyzing a graph of surface features, combining a general feature detection scheme with a RANSAC-based randomized subgraph searching algorithm in order to reliably detect reoccurring patterns of locally unique structures.
Abstract: 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
16 citations
TL;DR: This work proposes Hierarchical Intrinsic Symmetry Structure (HISS) to represent the structure of models and an automatic construction approach based on the skeleton of the model, and evaluates the representation and the algorithm with a set of experiments and applications.
5 citations