  Single-view 3D object reconstruction is a challenging fundamental problem in
computer vision, largely due to the morphological diversity of objects in the
natural world. In particular, high curvature regions are not always captured
effectively by methods trained using only set-based loss functions, resulting
in reconstructions short-circuiting the surface or cutting corners. In
particular, high curvature regions are not always captured effectively by
methods trained using only set-based loss functions, resulting in
reconstructions short-circuiting the surface or cutting corners. To address
this issue, we propose learning an image-conditioned mapping function from a
canonical sampling domain to a high dimensional space where the Euclidean
distance is equal to the geodesic distance on the object. The first three
dimensions of a mapped sample correspond to its 3D coordinates. The additional
lifted components contain information about the underlying geodesic structure.
Our results show that taking advantage of these learned lifted coordinates
yields better performance for estimating surface normals and generating
surfaces than using point cloud reconstructions alone. Further, we find that
this learned geodesic embedding space provides useful information for
applications such as unsupervised object decomposition.
