The peculiar combined capability of nonvolatile resistance switching memories to store and process data within a common nanoscale physical medium allows to implement disruptive mem-computing paradigms in hybrid circuits leveraging the compatibility of CMOS and memristive technologies. This may pave the way toward the future development of minaturized, lightweight, ultra-dense, high-speed and low-power universal memcomputers with sensing functionality on board. Since the availability of technical products of this kind would respond to the current demands of the Internet-of-Things industry, it is timely to investigate the functionalities and limitations of memristive memcomputing structures, such as those arranged in cellular bio-inspired architectures. The adoption of memristors in circuit design brings new life to nonlinear system theory, since standard analysis and synthesis techniques from linear system theory are no longer applicable for the investigation of highly-nonlinear electronic systems. This paper demonstrates how the use of standard and novel concepts from nonlinear system theory allow to design a Memristor Cellular Nonlinear Network for the execution of pixel-wise logical boolean functions on binary images, and the concurrent storage of input or output data into the memristive memory bank, providing clear evidence for the truly mem-computing character of its memristor-centered signal processing paradigm.