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.

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We propose an algorithm to detect approximate reflection symmetry present in a set of volumetrically distributed points belonging to 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.

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Over-segmentation of an image into superpixels has become a useful tool for solving various problems in image processing and computer vision. Reflection symmetry is quite prevalent in both natural and man-made objects and is an essential cue in understanding and grouping the objects in natural scenes. 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 first find the reflection symmetry in the image and represent it by a set of pairs of pixels which are mirror reflections of each other. We partition the image into superpixels while preserving this reflection symmetry through an iterative algorithm. We compare the proposed method with state-of-the-art superpixel generation methods and show the effectiveness in preserving the size and shape of superpixel boundaries across the reflection symmetry axes. We also present two applications, symmetry axes detection and unsupervised symmetric object segmentation, to illustrate the effectiveness of the proposed approach.

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