Enhanced 3d shape analysis via information geometry

Three-dimensional point clouds are essential in computer graphics, computer vision, and robotics. Traditional metrics fail to capture global statistical structure, while existing KL divergence approximations for Gaussian mixture models (GMMs) are unbounded and numerically unstable. We establish an information-geometric framework where point clouds, represented as GMMs, form a statistical manifold. We introduce the modified symmetric KL (MSKL) divergence, defined via a generalized KL functional on square-root densities, and prove explicit upper and lower bounds guaranteeing finite, stable values. Experiments on MPI-FAUST and G-PCD datasets demonstrate MSKL outperforms traditional distances and existing KL approximations in shape discrimination tasks.