We present a variational method for subdivision surface reconstruction from a noisy dense mesh. A new set of subdivision rules with continuous sharpness control is introduced into Loop subdivision for better modeling subdivision surface features such as semi-sharp creases, creases, and corners. The key idea is to assign a sharpness value to each edge of the control mesh to continuously control the surface features. Based on the new subdivision rules, a variational model with $L_1$ norm is formulated to find the control mesh and the corresponding sharpness values of the subdivision surface that best fits the input mesh. An iterative solver based on the augmented Lagrangian method and particle swarm optimization is used to solve the resulting non-linear, non-differentiable optimization problem. Our experimental results show that our method can handle meshes well with sharp/semi-sharp features and noise.