Author

# Rajendra Nagar

Other affiliations: Indian Institute of Technology, Jodhpur

Bio: Rajendra Nagar is an academic researcher from Indian Institute of Technology Gandhinagar. The author has contributed to research in topic(s): Reflection symmetry & Symmetry (geometry). The author has an hindex of 6, co-authored 18 publication(s) receiving 80 citation(s). Previous affiliations of Rajendra Nagar include Indian Institute of Technology, Jodhpur.

##### Papers

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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.

15 citations

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TL;DR: This work detects the intrinsic reflective symmetry in triangle meshes where the authors have to find the intrinsically symmetric point for each point of the shape by establishing correspondences between functions defined on the shapes by extending the functional map framework and then recovering the point-to-point correspondences.

Abstract: In computer vision and graphics, various types of symmetries are extensively studied since symmetry present in objects is a fundamental cue for understanding the shape and the structure of objects. In this work, we detect the intrinsic reflective symmetry in triangle meshes where we have to find the intrinsically symmetric point for each point of the shape. We establish correspondences between functions defined on the shapes by extending the functional map framework and then recover the point-to-point correspondences. Previous approaches using the functional map for this task find the functional correspondences matrix by solving a non-linear optimization problem which makes them slow. In this work, we propose a closed form solution for this matrix which makes our approach faster. We find the closed-form solution based on our following results. If the given shape is intrinsically symmetric, then the shortest length geodesic between two intrinsically symmetric points is also intrinsically symmetric. If an eigenfunction of the Laplace-Beltrami operator for the given shape is an even (odd) function, then its restriction on the shortest length geodesic between two intrinsically symmetric points is also an even (odd) function. The sign of a low-frequency eigenfunction is the same on the neighboring points. Our method is invariant to the ordering of the eigenfunctions and has the least time complexity. We achieve the best performance on the SCAPE dataset and comparable performance with the state-of-the-art methods on the TOSCA dataset.

10 citations

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TL;DR: In this paper, the authors propose an algorithm to detect approximate reflection symmetry present in a set of volumetrically distributed points belonging to the same point set containing a distorted reflection symmetry pattern.

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 and 3-D reflection symmetry detection along with comparisons.

9 citations

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TL;DR: An algorithm to over-segment an image through the propagation of reflection symmetry evident at the pixel level to superpixel boundaries is proposed and an application called unsupervised symmetric object segmentation is presented to illustrate the effectiveness of the proposed approach.

Abstract: Over-segmentation of an image into superpixels has become an useful tool for solving various problems in computer vision. Reflection symmetry is quite prevalent in both natural and man-made objects. Existing algorithms for estimating superpixels do not preserve the reflection symmetry of an object which leads to different sizes and shapes of superpixels across the symmetry axis. In this work, we propose an algorithm to over-segment an image through the propagation of reflection symmetry evident at the pixel level to superpixel boundaries. In order to achieve this goal, we exploit the detection of a set of pairs of pixels which are mirror reflections of each other. We partition the image into superpixels while preserving this reflection symmetry information through an iterative algorithm. We compare the proposed method with state-of-the-art superpixel generation methods and show the effectiveness of the method in preserving the size and shape of superpixel boundaries across the reflection symmetry axes. We also present an application called unsupervised symmetric object segmentation to illustrate the effectiveness of the proposed approach.

8 citations

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TL;DR: A novel $k-symmetry clustering algorithm is proposed to minimize this energy function in order to efficiently find all the symmetry axes present in the given image.

Abstract: We propose an energy minimization approach to detect multiple reflection symmetry axes present in a given image representing fronto-parallel view of a scene. We perform local feature matching to detect the pairs of mirror symmetric points, and in order to formulate an energy function, we use the geometric characteristics of the symmetry axis. That is, it passes through the midpoint of line segment joining the two mirror symmetric points and is perpendicular to the vector joining two mirror symmetric points. We propose a novel $k$ - symmetry clustering algorithm to minimize this energy function in order to efficiently find all the symmetry axes present in the given image. We evaluate the proposed method on the standard datasets and show that we get comparable and better results than that of the state-of-the-art reflection symmetry detection methods.

7 citations

##### Cited by

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01 Jan 2006

2,669 citations

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TL;DR: In this paper, a simple and efficient method for refining maps or correspondences by iterative upsampling in the spectral domain that can be implemented in a few lines of code is presented.

Abstract: We present a simple and efficient method for refining maps or correspondences by iterative upsampling in the spectral domain that can be implemented in a few lines of code. Our main observation is that high quality maps can be obtained even if the input correspondences are noisy or are encoded by a small number of coefficients in a spectral basis. We show how this approach can be used in conjunction with existing initialization techniques across a range of application scenarios, including symmetry detection, map refinement across complete shapes, non-rigid partial shape matching and function transfer. In each application we demonstrate an improvement with respect to both the quality of the results and the computational speed compared to the best competing methods, with up to two orders of magnitude speed-up in some applications. We also demonstrate that our method is both robust to noisy input and is scalable with respect to shape complexity. Finally, we present a theoretical justification for our approach, shedding light on structural properties of functional maps.

67 citations

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Rutgers University

^{1}, Technical University of Berlin^{2}, Princeton University^{3}, Columbia University^{4}TL;DR: In this article, an automatic method that can be used to abstract high resolution images into very low resolution outputs with reduced color palettes in the style of pixel art is presented, simultaneously solving for a mapping of features and a reduced palette needed to construct the output image.

Abstract: We present an automatic method that can be used to abstract high resolution images into very low resolution outputs with reduced color palettes in the style of pixel art. Our method simultaneously solves for a mapping of features and a reduced palette needed to construct the output image. The results are an approximation to the results generated by pixel artists. We compare our method against the results of a naive process common to image manipulation programs, as well as the hand-crafted work of pixel artists. Through a formal user study and interviews with expert pixel artists we show that our results offer an improvement over the naive methods.

37 citations

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Pennsylvania State University

^{1}, Kyung Hee University^{2}, ETH Zurich^{3}, University of Toronto^{4}, Shanghai University^{5}TL;DR: This report provides a detailed summary of the evaluation methodology for each type of symmetry detection algorithm validated, and demonstrates and analyzes quantified detection results in terms of precision-recall curves and F-measures for all algorithms evaluated.

Abstract: Motivated by various new applications of computational symmetry in computer vision and in an effort to advance machine perception of symmetry in the wild, we organize the third international symmetry detection challenge at ICCV 2017, after the CVPR 2011/2013 symmetry detection competitions. Our goal is to gauge the progress in computational symmetry with continuous benchmarking of both new algorithms and datasets, as well as more polished validation methodology. Different from previous years, this time we expand our training/testing data sets to include 3D data, and establish the most comprehensive and largest annotated datasets for symmetry detection to date; we also expand the types of symmetries to include densely-distributed and medial-axis-like symmetries; furthermore, we establish a challenge-and-paper dual track mechanism where both algorithms and articles on symmetry-related research are solicited. In this report, we provide a detailed summary of our evaluation methodology for each type of symmetry detection algorithm validated. We demonstrate and analyze quantified detection results in terms of precision-recall curves and F-measures for all algorithms evaluated. We also offer a short survey of the paper-track submissions accepted for our 2017 symmetry challenge.

35 citations

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13 Jun 2010

21 citations