Under the bandwidth-limited network, the memory event-triggered based set-membership filtering issue is investigated for a class of two-dimensional (2-D) systems subject to unknown-but-bounded (UBB) noise and censored measurement, aiming to guarantee filtering performance and conserve the network resources. First, a 2-D system is formulated using the Fornasini-Marchesini II (F-M II) local state-space model. The Tobit model and a saturation strategy are then used to characterize and handle censored measurements. Then, introducing historical measurement information, a novel memory event-triggered mechanism is proposed to determine whether the data can be transmitted, thereby reducing the frequency of data transmission. Based on this, a memory event-triggered set-membership filter is designed. By means of the 2-D mathematical induction and the S-procedure, sufficient conditions are derived to ensure the existence of the desired set-membership filter. Furthermore, the filter gain is obtained in terms of the solutions to a series of convex optimization problems such that the filtering error remains within the minimum ellipsoidal bound. Finally, the practicality and superiority of the proposed filtering scheme are demonstrated with taking a heat exchange process as a case study.