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When packets are transmitted over braided-paths in visual sensor networks some intermediate nodes may relay most part of the traffic, turning part or the entire network inactive when they run out of energy. Due to the nature of visual monitoring, source nodes may have different relevancies for the application, according to the desired monitoring tasks and the current sensors' poses and fields of view. Based on such relevancies, packets transmitted from different source nodes will have different relaying priorities, which are associated with energy thresholds. We propose an energy-efficient relay scheme where packets are forwarded to the next hop toward the sink only if the associated energy threshold is below the current energy level of the relaying node. Doing so, energy saving is achieved potentially prolonging the network lifetime, but with reduced impact to the overall monitoring quality of the application.