A new mesh simplification scheme that uses the face constriction process is presented. By introducing a statistical measure that can distinguish triangles having vertices of high local roughness from triangles in flat regions into our weight-ordering equation, along with other heuristics, our scheme can better preserve visually important features in the original mesh. To improve the shape quality of triangles, we adopt non-linear face area sensitivity in the weight ordering. Learning and feedback mechanism is also utilized to enhance user controllability. The computations are simple, making our scheme time-effective and easy to implement. In addition to comparing our scheme with other mesh simplification algorithms empirically, we compare their perform-ances by establishing a unifying ground among three basic simplification processes - decimate vertex, collapse edge, and constrict face. This unification allows us to analyze the intrinsic merits and demerits of simplification algorithms to help users make better selections.