This paper presents a novel image smoothing approach using a space-filling curve as the reduced domain to perform separation of edges and details. This structure-aware smoothing effect is achieved by modulating local extrema after empirical mode decomposition; it is highly effective and efficient since it is implemented on a one-dimensional curve instead of a two-dimensional image grid. To overcome edge staircase-like artifacts caused by a neighborhood deficiency in domain reduction, we next use a joint contrast-based filter to consolidate edge structures in image smoothing. The adoption of dimensional reduction makes our smoothing approach distinct for two reasons. First, overall structure-awareness is improved as more extrema are exploited to locate the salient edges and details. Second, envelope computation for local extrema is made much fast by using explicit interpolants on the curve. Moreover, our approach is simple and very easy to implement in practice. Experimental results demonstrate the merit of our approach, which outperforms previous state-of-the-art methods, for a variety of image processing tasks.