We present an analysis of weakly convex discontinuity adaptive (WCDA) models for regularizing three-dimensional (3D) quantitative microwave imaging. In particular, we are concerned with complex permittivity reconstructions from sparse measurements such that the reconstruction process is significantly accelerated. When dealing with such a highly underdetermined problem, it is crucial to employ regularization, relying in this case on prior knowledge about the structural properties of the underlying permittivity profile: we consider piecewise homogeneous objects. We present a numerical study on the choice of the potential function parameter for the Huber function and for two selected WCDA functions, one of which (the Leclerc-Cauchy-Lorentzian function) is designed to be more edge-preserving than the other (the Leclerc-Huber function). We evaluate the effect of reducing the number of (simulated) scattered field data on the reconstruction quality. Furthermore, reconstructions from subsampled single-frequency experimental data from the 3D Fresnel database illustrate the effectiveness of WCDA regularization.