The mean curvature has been shown a proper regularization in various ill-posed inverse problems in signal processing. Traditional solvers are based on either gradient descent methods or Euler Lagrange Equation. However, it is not clear if this mean curvature regularization term itself is convex or not. In this paper, we first prove that the mean curvature regularization is convex if the dimension of imaging domain is not
larger than seven. With this convexity, all optimization methods lead to the same global optimal solution. Based on this convexity and Bernstein theorem, we propose an efficient filter solver, which can implicitly minimize the mean curvature.Our experiments show that this filter is at least two orders of magnitude faster than traditional solvers.