  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.
