  Boundaries are among the primary visual cues used by human and computer
vision systems. One of the key problems in boundary detection is the label
representation, which typically leads to class imbalance and, as a consequence,
to thick boundaries that require non-differential post-processing steps to be
thinned. In this paper, we re-interpret boundaries as 1-D surfaces and
formulate a one-to-one vector transform function that allows for training of
boundary prediction completely avoiding the class imbalance issue.
Specifically, we define the boundary representation at any point as the unit
vector pointing to the closest boundary surface. Our problem formulation leads
to the estimation of direction as well as richer contextual information of the
boundary, and, if desired, the availability of zero-pixel thin boundaries also
at training time. Our method uses no hyper-parameter in the training loss and a
fixed stable hyper-parameter at inference. We provide theoretical
justification/discussions of the vector transform representation. We evaluate
the proposed loss method using a standard architecture and show the excellent
performance over other losses and representations on several datasets. Code is
available at https://github.com/edomel/BoundaryVT.
