Previous joint/guided filters directly transfer the structural information in the reference image to the target one. In this paper, we first analyze its major drawback – that is, there may be completely different edges in the two images. Simply passing all patterns to the target could introduce significant errors. To address this issue, we propose the concept of mutual-structure, which refers to the structural information that is contained in both images and thus can be safely enhanced by joint filtering, and an untraditional objective function that can be efficiently optimized to yield mutual structure. Our method results in necessary and important edge preserving, which greatly benefits depth completion, optical flow estimation, image enhancement, stereo matching, to name a few.