This thesis proposes novel reconstruction schemes for low frequency near-field electromagnetic imaging of high-contrast conductivity distributions enclosed inside shielded regions, using Maxwell's equations in 3D. Our focus lies in the estimation of conductivity value or shape enclosed in shielded regions from electromagnetic data measured externally, for one low frequency. We are interested in regions which are roughly equivalent in size to small rooms or medium-sized containers, though the reconstruction schemes proposed here can easily be adjusted to imaging situations at larger or smaller scale. The novel regularization techniques proposed here are based on a sparsity promoting regularization scheme on the one hand, and level set based shape evolution techniques on the other. For estimating the conductivity profile enclosed inside these shielded regions, we introduce a sparsity regularization scheme and compare its result to the shape-based schemes developed here and a traditional L2-based approach. In the shape-based regime, we introduce single and color level set regularization schemes which are designed to reconstruct binary and multi-phase material respectively. Alongside color level set regularization, we introduce a topological perturbation scheme which is designed to avoid a certain type of local minima that is characteristic to simultaneous multi-value shape-based reconstruction. In each reconstruction scheme, Landweber-Kaczmarz iterations are employed for the optimization process, with suitable tailor-made line search techniques designed accordingly. In our numerical simulations, we perform 3D reconstructions from noisy simulated data and compare the results with those obtained from a standard L2-based approach. Our results suggest, in the applications considered here, that the proposed novel schemes are able to yield significantly improved reconstructions when compared with traditional techniques. We end with using convolutional neural networks to classify electromagnetic images that result from the reconstruction schemes.