This paper introduces a mathematical formulation of the problem of detection and characterization of shallowly buried landmines (more generally, buried objects) using passive thermal infrared technique. The problem consists of two steps. In the first step, referred to as thermal modeling which aims at predicting the soil temperature provided by the thermal properties of the soil and the buried objects, a parabolic partial differential equation based model is formulated. The proposed model is validated using experimental data. For solving the model, a splitting finite difference scheme is used. In the second step, referred to as inverse problem setting for landmine detection, the forward thermal model and acquired infrared images are used to detect the presence of buried objects and characterize them based on the estimation of their thermal and geometrical properties. Mathematically, this inverse problem is stated as the estimation of a piecewise constant coefficient of the heat transfer equation. To reduce the ill-posedness of this problem, which is due to the lack of spatial information in the measured data, we make use of a parametrization of the coefficient which needs only a small number of unknowns. The problem is then solved by gradient-based optimization methods. Numerical results both validate the proposed thermal model and illustrate the performance of the suggested algorithm for the inverse problem.