SUMMARY Muography is an imaging tool based on the attenuation of cosmic muons for observing density anomalies associated with large objects, such as underground caves or fractured zones. Tomography based on muography measurements, that is, 3-D reconstruction of density distribution from 2-D muon flux maps, brings along special challenges. The detector field of view covering must be as balanced as possible, considering the muon flux drop at high zenith angles and the detector placement possibilities. The inversion from directional muon fluxes to a 3-D density map is usually underdetermined (more voxels than measurements). Therefore, the solution of the inversion can be unstable due to partial coverage. The instability can be solved by geologically relevant Bayesian constraints. However, the Bayesian principle results in parameter bias and artefacts. In this work, linearized (density-length based) inversion is applied by formulating the constraints associated with inversion to ensure the stability of parameter fitting. After testing the procedure on synthetic examples, an actual high-quality muography measurement data set from seven positions is used as input for the inversion. The resulting tomographic imaging provides details on the complicated internal structures of karstic fracture zone. The existence of low density zones in the imaged space was verified by samples from core drills, which consist of altered dolomite powder within the intact high density dolomite.