Humans take advantage of real world symmetries for various tasks, yet capturing their superb symmetry perception mechanism into a computational model remains elusive. Encouraged by a new discovery (CVPR 2016) demonstrating extremely high inter-person accuracy of human perceived symmetries in the wild, we have created the first deep-learning neural network for reflection and rotation symmetry detection (Sym-NET), trained on photos from MS-COCO (Common Object in COntext) dataset with nearly 11K symmetry-labels from more than 400 human observers. We employ novel methods to convert discrete human labels into symmetry heatmaps, capture symmetry densely in an image and quantitatively evaluate Sym-NET against multiple existing computer vision algorithms. Using the symmetry competition testsets from CVPR 2013 and unseen MS-COCO photos, Sym-NET comes out as the winner with significantly superior performance over all other competitors. Beyond mathematically well-defined symmetries on a plane, Sym-NET demonstrates abilities to identify viewpoint-varied 3D symmetries, partially occluded symmetrical objects and symmetries at a semantic level.
Lee and Liu @cite_11 @cite_32 @cite_58 have developed an algorithm to detect (1) the center of rotation, (2) the number of folds, (3) type of symmetry group (dihedral cyclic O(2)), and (4) the region of support. The first step of their algorithm is rotation symmetry center detection where they use a novel frieze expansion to transform the image at each pixel location into polar coordinates and then search for translation symmetry. The second step applies a Discrete Fourier Transform (DFT) on the frieze expansion to determine (2)-(4) listed above. In our work, for rotation symmetries we only focus on detecting rotation symmetry centers.
@cite_58 @cite_32 @cite_11
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We present a novel and effective algorithm for affinely skewed rotation symmetry group detection from real-world images. We define a complete skewed rotation symmetry detection problem as discovering five independent properties of a skewed rotation symmetry group: 1) the center of rotation, 2) the affine deformation, 3) the type of the symmetry group, 4) the cardinality of the symmetry group, and 5) the supporting region of the symmetry group in the image. We propose a frieze-expansion (FE) method that transforms rotation symmetry group detection into a simple, 1D translation symmetry detection problem. We define and construct a pair of rotational symmetry saliency maps, complemented by a local feature method. Frequency analysis, using Discrete Fourier Transform (DFT), is applied to the frieze-expansion patterns (FEPs) to uncover the types (cyclic, dihedral, and O(2)), the cardinalities, and the corresponding supporting regions, concentric or otherwise, of multiple rotation symmetry groups in an image. The phase information of the FEP is used to rectify affinely skewed rotation symmetry groups. Our result advances the state of the art in symmetry detection by offering a unique combination of region-based, feature-based, and frequency-based approaches. Experimental results on 170 synthetic and natural images demonstrate superior performance of our rotation symmetry detection algorithm over existing methods. We present a novel and effective algorithm for rotation symmetry group detection from real-world images. We propose a frieze-expansion method that transforms rotation symmetry group detection into a simple translation symmetry detection problem. We define and construct a dense symmetry strength map from a given image, and search for potential rotational symmetry centers automatically. Frequency analysis, using discrete Fourier transform (DFT), is applied to the frieze-expansion patterns to uncover the types and the cardinality of multiple rotation symmetry groups in an image, concentric or otherwise. Furthermore, our detection algorithm can discriminate discrete versus continuous and cyclic versus dihedral symmetry groups, and identify the corresponding supporting regions in the image. Experimental results on over 80 synthetic and natural images demonstrate superior performance of our rotation detection algorithm in accuracy and in speed over the state of the art rotation detection algorithms. Abstract The purpose of this paper is to detect and characterize geometric rosettes which are among many different star-like motifs used in the most common Islamic Geometric Patterns. Based on symmetry group theory, a geometric rosette can be characterized by its center; order and group of symmetry. It is easy to observe that a rosette is formed by a central star surrounded by two types of shapes (mid-cells and outer-cells) called “ furmah” by Moroccan artisans. The center of symmetry is the center of the concentric circles circum-circling respectively the central star, mid-cells and outer-cells. The order of the rosette is connected with the number of mid-cells. To detect the center of symmetry in a rosette image, we propose an original method based on genetic algorithm that extracts the circle passing by the maximum pixels belonging to the binary rosette image. The center and the radius of the detected circle correspond respectively to the symmetry center of the rosette and the radius of its supporting concentric region. To determine the type and the order of symmetry, Frequency analysis using Discrete Fourier Transform is applied to the Frieze-expansion pattern. This later is obtained by applying the Frieze-Expansion technique to the extracted supporting concentric region. Experimental results prove that the proposed algorithm for rotational symmetry detection is rather successful because it manages to localize the real center of symmetry of any rosette pattern. The key point of this algorithm lies at the same time in its simplicity and its efficiency. We present an algorithm that detects rotational and reflectional symmetries of two-dimensional objects. Both symmetry types are effectively detected and analyzed using the angular correlation (AC), which measures the correlation between images in the angular direction. The AC is accurately computed using the pseudopolar Fourier transform, which rapidly computes the Fourier transform of an image on a near-polar grid. We prove that the AC of symmetric images is a periodic signal whose frequency is related to the order of the symmetry. This frequency is recovered via spectrum estimation, which is a proven technique in signal processing with a variety of efficient solutions. We also provide a novel approach for finding the center of symmetry and demonstrate the applicability of our scheme to the analysis of real images A method for efficient detection of the dominant local reflectional symmetry in grey level images is described. The general approach is to define a local measure of reflectional symmetry that transforms the symmetry detection problem to an optimization problem, and obtain the symmetric regions by an efficient global optimization algorithm. The symmetry of a 1D function can be measured in the frequency domain as the fraction of its energy that resides in symmetric Fourier basis functions. This approach is extended to two dimensions. Locality can be formally treated in terms of the Gabor decomposition and implemented via soft windowing. The resulting measure is a complicated multimodal function of the location of the center of the supporting region, its size, and the orientation of the symmetry axis. A new probabilistic generic algorithm is applied to the determination of the global maximum of the reflectional symmetry function. Less than one thousand evaluations of the local symmetry measure are typically needed in order to locate the dominant symmetry in natural, wildlife test images. The main challenge in Super Resolution (SR) is to discover the mapping between the low-and high-resolution manifolds of image patches, a complex ill-posed problem which has recently been addressed through piecewise linear regression with promising results. In this paper we present a novel regression-based SR algorithm that benefits from an extended knowledge of the structure of both manifolds. We propose a transform that collapses the 16 variations induced from the dihedral group of transforms (i.e. rotations, vertical and horizontal reflections) and antipodality (i.e. diametrically opposed points in the unitary sphere) into a single primitive. The key idea of our transform is to study the different dihedral elements as a group of symmetries within the high-dimensional manifold. We obtain the respective set of mirror-symmetry axes by means of a frequency analysis of the dihedral elements, and we use them to collapse the redundant variability through a modified symmetry distance. The experimental validation of our algorithm shows the effectiveness of our approach, which obtains competitive quality with a dictionary of as little as 32 atoms (reducing other methods' dictionaries by at least a factor of 32) and further pushing the state-of-the-art with a 1024 atoms dictionary. A novel and efficient method is presented for grouping feature points on the basis of their underlying symmetry and characterising the symmetries present in an image. We show how symmetric pairs of features can be efficiently detected, how the symmetry bonding each pair is extracted and evaluated, and how these can be grouped into symmetric constellations that specify the dominant symmetries present in the image. Symmetries over all orientations and radii are considered simultaneously, and the method is able to detect local or global symmetries, locate symmetric figures in complex backgrounds, detect bilateral or rotational symmetry, and detect multiple incidences of symmetry. We present an algorithm for detecting multiple rotational symmetries in natural images. Given an image, its gradient magnitude field is computed, and information from the gradients is spread using a diffusion process in the form of a gradient vector flow (GVF) field. We construct a graph whose nodes correspond to pixels in tire image, connecting points that are likely to be rotated versions of one another The n-cycles present in tire graph are made to vote for C sub n symmetries, their votes being weighted by the errors in transformation between GVF in the neighborhood of the voting points, and the irregularity of the n-sided polygons formed by the voters. The votes are accumulated at tire centroids of possible rotational symmetries, generating a confidence map for each order of symmetry. We tested the method with several natural images. Natural images often exhibit symmetries that should be taken into account when editing them. In this paper we present Nautilus --- a method for automatically identifying symmetric regions in an image along with their corresponding symmetry transformations. We compute dense local similarity symmetry transformations using a novel variant of the Generalised PatchMatch algorithm that uses Metropolis-Hastings sampling. We combine and refine these local symmetries using an extended Lucas-Kanade algorithm to compute regional transformations and their spatial extents. Our approach produces dense estimates of complex symmetries that are combinations of translation, rotation, scale, and reflection under perspective distortion. This enables a number of automatic symmetry-aware image editing applications including inpainting, rectification, beautification, and segmentation, and we demonstrate state-of-the-art applications for each of them. The goal of our work is to develop an algorithm for automatic and robust detection of global intrinsic symmetries in 3D surface meshes. Our approach is based on two core observations. First, symmetry invariant point sets can be detected robustly using critical points of the Average Geodesic Distance (AGD) function. Second, intrinsic symmetries are self-isometries of surfaces and as such are contained in the low dimensional group of Mobius transformations. Based on these observations, we propose an algorithm that: 1) generates a set of symmetric points by detecting critical points of the AGD function, 2) enumerates small subsets of those feature points to generate candidate Mobius transformations,and 3) selects among those candidate Mobius transformations the one(s) that best map the surface onto itself. The main advantages of this algorithm stem from the stability of the AGD in predicting potential symmetric point features and the low dimensionality of the Mobius group for enumerating potential self-mappings. During experiments with a benchmark set of meshes augmented with human-specified symmetric correspondences, we find that the algorithm is able to find intrinsic symmetries for a wide variety of object types with moderate deviations from perfect symmetry. Symmetry is an important cue for many applications, including object alignment, recognition, and segmentation. In this paper, we describe a planar reflective symmetry transform (PRST) that captures a continuous measure of the reflectional symmetry of a shape with respect to all possible planes. This transform combines and extends previous work that has focused on global symmetries with respect to the center of mass in 3D meshes and local symmetries with respect to points in 2D images. We provide an efficient Monte Carlo sampling algorithm for computing the transform for surfaces and show that it is stable under common transformations. We also provide an iterative refinement algorithm to find local maxima of the transform precisely. We use the transform to define two new geometric properties, center of symmetry and principal symmetry axes, and show that they are useful for aligning objects in a canonical coordinate system. Finally, we demonstrate that the symmetry transform is useful for several applications in computer graphics, including shape matching, segmentation of meshes into parts, and automatic viewpoint selection. Symmetry is a pervasive phenomenon presenting itself in all forms and scales in natural and manmade environments. Its detection plays an essential role at all levels of human as well as machine perception. The recent resurging interest in computational symmetry for computer vision and computer graphics applications has motivated us to conduct a US NSF funded symmetry detection algorithm competition as a workshop affiliated with the Computer Vision and Pattern Recognition (CVPR) Conference, 2013. This competition sets a more complete benchmark for computer vision symmetry detection algorithms. In this report we explain the evaluation metric and the automatic execution of the evaluation workflow. We also present and analyze the algorithms submitted, and show their results on three test sets of real world images depicting reflection, rotation and translation symmetries respectively. This competition establishes a performance baseline for future work on symmetry detection. In previous work on "transformed mixtures of Gaussians" and "transformed hidden Markov models", we showed how the EM algorithm in a discrete latent variable model can be used to jointly normalize data (e.g., center images, pitch-normalize spectrograms) and learn a mixture model of the normalized data. The only input to the algorithm is the data, a list of possible transformations, and the number of clusters to find. The main criticism of this work was that the exhaustive computation of the posterior probabilities over transformations would make scaling up to large feature vectors and large sets of transformations intractable. Here, we describe how a tremendous speed-up is acheived through the use of a variational technique for decoupling transformations, and a fast Fourier transform method for computing posterior probabilities. For N × N images, learning C clusters under N rotations, N scales, N x-translations and N y-translations takes only (C + 2 log N)N2 scalar operations per iteration. In contrast, the original algorithm takes CN6 operations to account for these transformations. We give results on learning a 4-component mixture model from a video sequence with frames of size 320×240. The model accounts for 360 rotations and 76,800 translations. Each iteration of EM takes only 10 seconds per frame in MATLAB, which is over 5 million times faster than the original algorithm. Symmetry, as one of the key components of Gestalt theory, provides an important mid-level cue that serves as input to higher visual processes such as segmentation. In this work, we propose a complete approach that links the detection of curved reflection symmetries to produce symmetry-constrained segments of structures regions in real images with clutter. For curved reflection symmetry detection, we leverage on patch-based symmetric features to train a Structured Random Forest classifier that detects multiscaled curved symmetries in 2D images. Next, using these curved symmetries, we modulate a novel symmetry-constrained foreground-background segmentation by their symmetry scores so that we enforce global symmetrical consistency in the final segmentation. This is achieved by imposing a pairwise symmetry prior that encourages symmetric pixels to have the same labels over a MRF-based representation of the input image edges, and the final segmentation is obtained via graph-cuts. Experimental results over four publicly available datasets containing annotated symmetric structures: 1) SYMMAX-300 [38], 2) BSD-Parts, 3) Weizmann Horse (both from [18]) and 4) NY-roads [35] demonstrate the approach's applicability to different environments with state-of-the-art performance. PatchMatch is a fast algorithm for computing dense approximate nearest neighbor correspondences between patches of two image regions [1]. This paper generalizes PatchMatch in three ways: (1) to find k nearest neighbors, as opposed to just one, (2) to search across scales and rotations, in addition to just translations, and (3) to match using arbitrary descriptors and distances, not just sum-of-squared-differences on patch colors. In addition, we offer new search and parallelization strategies that further accelerate the method, and we show performance improvements over standard kd-tree techniques across a variety of inputs. In contrast to many previous matching algorithms, which for efficiency reasons have restricted matching to sparse interest points, or spatially proximate matches, our algorithm can efficiently find global, dense matches, even while matching across all scales and rotations. This is especially useful for computer vision applications, where our algorithm can be used as an efficient general-purpose component. We explore a variety of vision applications: denoising, finding forgeries by detecting cloned regions, symmetry detection, and object detection. In this article we outline a method that automatically transforms an Euclidean ornament into a hyperbolic one. The necessary steps are pattern recognition, symmetry detection, extraction of a Euclidean fundamental region, conformal deformation to a hyperbolic fundamental region and tessellation of the hyperbolic plane with this patch. Each of these steps has its own mathematical subtleties that are discussed in this article. In particular, it is discussed which hyperbolic symmetry groups are suitable generalizations of Euclidean wallpaper groups. Furthermore it is shown how one can take advantage of methods from discrete differential geometry in order to perform the conformal deformation of the fundamental region. Finally it is demonstrated how a reverse pixel lookup strategy can be used to obtain hyperbolic images with optimal resolution. We present a spectral approach for detecting and analyzing rotational and reflectional symmetries in n-dimensions. Our main contribution is the derivation of a symmetry detection and analysis scheme for sets of points IRn and its extension to image analysis by way of local features. Each object is represented by a set of points S ∈ IRn, where the symmetry is manifested by the multiple self-alignments of S . The alignment problem is formulated as a quadratic binary optimization problem, with an efficient solution via spectral relaxation. For symmetric objects, this results in a multiplicity of eigenvalues whose corresponding eigenvectors allow the detection and analysis of both types of symmetry. We improve the scheme's robustness by incorporating geometrical constraints into the spectral analysis. Our approach is experimentally verified by applying it to 2D and 3D synthetic objects as well as real images. This paper introduces an algorithm for the registration of rotated and translated volumes using the three-dimensional (3-D) pseudopolar Fourier transform, which accurately computes the Fourier transform of the registered volumes on a near-spherical 3-D domain without using interpolation. We propose a three-step procedure. The first step estimates the rotation axis. The second step computes the planar rotation relative to the rotation axis. The third step recovers the translational displacement. The rotation estimation is based on Euler's theorem, which allows one to represent a 3-D rotation as a planar rotation around a 3-D rotation axis. This axis is accurately recovered by the 3-D pseudopolar Fourier transform using radial integrations. The residual planar rotation is computed by an extension of the angular difference function to cylindrical motion. Experimental results show that the algorithm is accurate and robust to noise 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. Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalisation of the vector-field method that allows the use of higher-order symmetry operators. In particular, our method makes use of the second-order Carter operator, which is a hidden symmetry in the sense that it does not correspond to a Killing symmetry of the spacetime.