The asynchronous development between the observation capability and the transition capability results in that an original image data (OID) formed by one-time observation cannot be completely transmitted in one transmit chance between the EOS and GS (named as a visible time window, VTW). It needs to segment the OID to several segmented image data (SID) and then transmits them in several VTWs, which enriches the extension of satellite image data downlink scheduling problem (SIDSP). We define the novel SIDSP as satellite image data downlink scheduling problem with family attribute (SIDSPWFA), in which some big OID is segmented by a fast segmentation operator first, and all SID and other no-segmented OID is transmitted in the second step. Two optimization objectives, the image data transmission failure rate (FR) and the segmentation times (ST), are then designed to formalize SIDSPWFA as a bi-objective discrete optimization model. Furthermore, a bi-stage differential evolutionary algorithm(DE+NSGA-II) is developed holding several bi-stage operators. Extensive simulation instances show the efficiency of models, strategies, algorithms and operators is analyzed in detail.