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Magnetic detection electrical impedance tomography (MDEIT) is an imaging modality that aims to compute the cross-sectional distribution of the conductivity of a volume from the magnetic flux density surrounding the object. Owing to the Biot-Savart law, the MDEIT inverse problem is inherently ill-conditioned making image reconstruction highly susceptible to the effects of noise and numerical errors. Appropriate priors or penalties are needed to facilitate reconstruction and to restrict the search space to a specific solution set. The images have the sparsity property and sharp variations. Consequently, this paper presents an approach involving a combination of the L¹ and total variation norm penalties, the former to suppress spurious background signals and enforce sparsity and the latter to preserve local smoothness and piecewise constancy in the MDEIT reconstructed images. The primal dual-interior point method (PD-IPM) for minimizing the joint L¹--TV penalty was used in the paper. The method was validated by using MDEIT simulated data and experimental data in comparison with the performances of the L², L¹ and total variation norm penalty-based approaches. The results show that the joint L¹--TV regularized algorithm preserves sparsity property, local smoothness and piecewise constancy, leading to improvements in the localization of the reconstructed images in MDEIT. [ABSTRACT FROM AUTHOR]