Reconstruction of inhomogeneous dielectric objects from microwave scattering by means of quantitative microwave tomography is a nonlinear, ill-posed inverse problem. In this paper, we employ the Huber function as a robust regularization approach for this challenging problem. The resulting reconstructions both in 2D and 3D from sparse data points for piecewise constant objects are encouraging. The reconstructions of more complex permittivity profiles from breast phantom data indicate potential for use in biomedical imaging.