  Spatial relations between objects in an image have proved useful for
structural object recognition. Structural constraints can act as regularization
in neural network training, improving generalization capability with small
datasets. Several relations can be modeled as a morphological dilation of a
reference object with a structuring element representing the semantics of the
relation, from which the degree of satisfaction of the relation between another
object and the reference object can be derived. However, dilation is not
differentiable, requiring an approximation to be used in the context of
gradient-descent training of a network. We propose to approximate dilations
using convolutions based on a kernel equal to the structuring element. We show
that the proposed approximation, even if slightly less accurate than previous
approximations, is definitely faster to compute and therefore more suitable for
computationally intensive neural network applications.
