  Implicit 3D surface reconstruction of an object from its partial and noisy 3D
point cloud scan is the classical geometry processing and 3D computer vision
problem. In the literature, various 3D shape representations have been
developed, differing in memory efficiency and shape retrieval effectiveness,
such as volumetric, parametric, and implicit surfaces. Radial basis functions
provide memory-efficient parameterization of the implicit surface. However, we
show that training a neural network using the mean squared error between the
ground-truth implicit surface and the linear basis-based implicit surfaces does
not converge to the global solution. In this work, we propose locally supported
compact radial basis functions for a linear representation of the implicit
surface. This representation enables us to generate 3D shapes with arbitrary
topologies at any resolution due to their continuous nature. We then propose a
neural network architecture for learning the linear implicit shape
representation of the 3D surface of an object. We learn linear implicit shapes
within a supervised learning framework using ground truth Signed-Distance Field
(SDF) data for guidance. The classical strategies face difficulties in finding
linear implicit shapes from a given 3D point cloud due to numerical issues
(requires solving inverse of a large matrix) in basis and query point
selection. The proposed approach achieves better Chamfer distance and
comparable F-score than the state-of-the-art approach on the benchmark dataset.
We also show the effectiveness of the proposed approach by using it for the 3D
shape completion task.
