Black holes hold a tremendous discovery potential. In this paper the extent to which the Event Horizon Telescope and its next generation upgrade can resolve their structure is quantified. Black holes are characterized by a perfectly absorptive boundary, with a specific area determined by intrinsic parameters of the black hole. We use a general parametrization of spherically symmetric spacetimes describing deviations from this behavior, with parameters controlling the size of the central object and its interaction with light, in particular through a specular reflection coefficient G and an intrinsic luminosity measured as a fraction. of that of the accretion disc. This enables us to study exotic compact objects and compare them with black holes in a model-independent manner. We determine the image features associated with the existence of a surface in the presence of a geometrically thin and optically thick accretion disc, identifying requirements for very large baseline interferometry observations to be able to cast meaningful constraints on these parameters, in particular regarding angular resolution and image dynamic range. For face-on observations, constraints of order eta less than or similar to 10(-4), Gamma less than or similar to 10(-1) are possible with an enhanced Event Horizon Telescope array, imposing strong constraints on the nature of the central object.