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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16 Jun 2012
TL;DR: A new technique for extracting local features from images of architectural scenes, based on detecting and representing local symmetries, which can improve matching performance for this difficult task of matching challenging pairs of photos of urban scenes.
Abstract: We present a new technique for extracting local features from images of architectural scenes, based on detecting and representing local symmetries. These new features are motivated by the fact that local symmetries, at different scales, are a fundamental characteristic of many urban images, and are potentially more invariant to large appearance changes than lower-level features such as SIFT. Hence, we apply these features to the problem of matching challenging pairs of photos of urban scenes. Our features are based on simple measures of local bilateral and rotational symmetries computed using local image operations. These measures are used both for feature detection and for computing descriptors. We demonstrate our method on a challenging new dataset containing image pairs exhibiting a range of dramatic variations in lighting, age, and rendering style, and show that our features can improve matching performance for this difficult task.
166 citations
TL;DR: Applications of this work range from coherent remeshing of geometry with respect to the symmetries of a shape to geometric compression, intelligent mesh editing, and automatic instantiation.
Abstract: We propose an automatic method for finding symmetries of 3D shapes, that is, isometric transforms which leave a shape globally unchanged. These symmetries are deterministically found through the use of an intermediate quantity: the generalized moments. By examining the extrema and spherical harmonic coefficients of these moments, we recover the parameters of the symmetries of the shape. The computation for large composite models is made efficient by using this information in an incremental algorithm capable of recovering the symmetries of a whole shape using the symmetries of its subparts. Applications of this work range from coherent remeshing of geometry with respect to the symmetries of a shape to geometric compression, intelligent mesh editing, and automatic instantiation.
140 citations
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28 May 2002
TL;DR: A new reflective symmetry descriptor is introduced that represents a measure of reflective symmetry for an arbitrary 3D voxel model for all planes through the model's center of mass (even if they are not planes of symmetry).
Abstract: Computing reflective symmetries of 2D and 3D shapes is a classical problem in computer vision and computational geometry. Most prior work has focused on finding the main axes of symmetry, or determining that none exists. In this paper, we introduce a new reflective symmetry descriptor that represents a measure of reflective symmetry for an arbitrary 3D voxel model for all planes through the model's center of mass (even if they are not planes of symmetry). The main benefits of this new shape descriptor are that it is defined over a canonical parameterization (the sphere) and describes global properties of a 3D shape. Using Fourier methods, our algorithm computes the symmetry descriptor in O(N4 logN) time for an N × N × N voxel grid, and computes a multiresolution approximation in O(N3 logN) time. In our initial experiments, we have found the symmetry descriptor to be useful for registration, matching, and classification of shapes.
134 citations
07 Oct 2012
TL;DR: This work creates and makes publicly available a ground-truth dataset for symmetry detection in natural images, and uses supervised learning to learn how to combine these cues, and employs MIL to accommodate the unknown scale and orientation of the symmetric structures.
Abstract: In this work we propose a learning-based approach to symmetry detection in natural images. We focus on ribbon-like structures, i.e. contours marking local and approximate reflection symmetry and make three contributions to improve their detection. First, we create and make publicly available a ground-truth dataset for this task by building on the Berkeley Segmentation Dataset. Second, we extract features representing multiple complementary cues, such as grayscale structure, color, texture, and spectral clustering information. Third, we use supervised learning to learn how to combine these cues, and employ MIL to accommodate the unknown scale and orientation of the symmetric structures. We systematically evaluate the performance contribution of each individual component in our pipeline, and demonstrate that overall we consistently improve upon results obtained using existing alternatives.
114 citations
TL;DR: A new algorithm for detecting structural redundancy in geometric data sets that computes rigid symmetries, i.e., subsets of a surface model that reoccur several times within the model differing only by translation, rotation or mirroring.
Abstract: In this paper, we describe a new algorithm for detecting structural redundancy in geometric data sets. Our algorithm computes rigid symmetries, i.e., subsets of a surface model that reoccur several times within the model differing only by translation, rotation or mirroring. Our algorithm is based on matching locally coherent constellations of feature lines on the object surfaces. In comparison to previous work, the new algorithm is able to detect a large number of symmetric parts without restrictions to regular patterns or nested hierarchies. In addition, working on relevant features only leads to a strong reduction in memory and processing costs such that very large data sets can be handled. We apply the algorithm to a number of real world 3D scanner data sets, demonstrating high recognition rates for general patterns of symmetry.
109 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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