Minimum decomposition of a digital surface into digital plane segments is NP-hard

This paper deals with the complexity of the decomposition of a digital surface into digital plane segments (DPSs for short). We prove that the decision problem (does there exist a decomposition with less than λλ DPSs?) is NP-complete, and thus that the optimization problem (finding the minimum number of DPSs) is NP-hard. The proof is based on a polynomial reduction of any instance of the well-known 3-SAT problem to an instance of the digital surface decomposition problem. A geometric model for the 3-SAT problem is proposed.