We propose a framework for intensity-based registration of images by linear transformations, based on a discrete Markov random field (MRF) formulation. Here, the challenge arises from the fact that optimizing the energy associated with this problem requires a high-order MRF model. Currently, methods for optimizing such high-order models are less general, easy to use, and efficient, than methods for the popular second-order models. Therefore, we propose an approximation to the original energy by an MRF with tractable second-order terms. The approximation at a certain point p in the parameter space is the normalized sum of evaluations of the original energy at projections of p to two-dimensional subspaces. We demonstrate the quality of the proposed approximation by computing the correlation with the original energy, and show that registration can be performed by discrete optimization of the approximated energy in an iteration loop. A search space refinement strategy is employed over iterations to achieve sub-pixel accuracy, while keeping the number of labels small for efficiency. The proposed framework can encode any similarity measure is robust to the settings of the internal parameters, and allows an intuitive control of the parameter ranges. We demonstrate the applicability of the framework by intensity-based registration, and 2D-3D registration of medical images. The evaluation is performed by random studies and real registration tasks. The tests indicate increased robustness and precision compared to corresponding standard optimization of the original energy, and demonstrate robustness to noise. Finally, the proposed framework allows the transfer of advances in MRF optimization to linear registration problems.