Topology-Agnostic Mesh Reconstruction of Deformable Objects from Sparse Touch

Estimating the full shape of a deformable object is especially challenging when vision is unavailable: in the dark, inside an opaque bag, behind the manipulating hand, or under heavy self-occlusion. Touch is the natural sensor in these settings, but touches are sparse and local. We present a single topology-agnostic estimator that reconstructs the full mesh of a deformable object from only a few touches and no vision, using one permutation-invariant cross-attention architecture that handles a 1D rope, a 2D cloth, and a 3D volumetric soft body. The learned estimator reduces reconstruction error by roughly two-thirds relative to non-learned geometric mesh completion and a Gaussian-process surface baseline, and it outperforms a simpler global-pool set encoder, with the gap growing as more touches are observed. We then show that the estimator's deep-ensemble uncertainty can be used to learn where to touch next, which lowers error further and beats both random touching and a Gaussian-process active baseline at sparse budgets. This gain is modest on average but grows with self-occlusion and on the error tail. When vision is also available, where to touch barely matters, motivating the vision-free setting we study.