Backbone extraction is a fundamental process in network science that reduces complexity while preserving essential structural and functional properties. Many real-world networks exhibit a component structure, where dense intra-community connections coexist with sparse but crucial inter-community links. To address this multiscale organization, we propose a Bi-Scale Filtering Framework, which first decomposes the network into local (intra-community) and global (inter-community) components before applying tailored backbone extraction methods. Specifically, we use the Disparity Filter to preserve locally significant edges within communities and the High Salience Skeleton to retain globally important connections. The resulting backbones are then merged into a unified, bi-scale structure that balances local detail and large-scale connectivity. Using a real-world air transportation network, we demonstrate that this approach preserves multiscale structures more effectively than single-scale methods. Using the component structure, Bi-Scale Filtering offers a more faithful representation of network backbones, making it a promising tool for complex network analysis.