In this paper we propose a new mem-computing image processing architecture, called Memristor Cellular Nonlinear Network, which leverages the unique capability of nonvolatile memristors to compute and store data in the same physical nano-scale locations. Adopting a bistable-like memristor in place for the linear resistor in the standard realization of a cell of the nonlinear dynamic array, the resulting network is capable to process information by exploiting the time evolution of the voltages across the memristors as well as to store/retrieve results into/ from the memristances. This attractive feature, absent in a standard Cellular Nonlinear Network, may pave the way towards the future development of a new generation of visual processors with unprecedented spatial resolution.