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In this paper we propose a novel variational formulation for arbitrary surface reconstruction from 3D scattered points. An implicit surface is adopted for its more advantages, such as continuity and differentiability, easy estimation of points inside or outside the shape, and convenient implementation for these complicated set operations. The presented new energy functional in this paper considers more factors on properties of the scattered points (including distances and normal vectors), smoothness and the constraint of the signed distance function. The gradient flow which minimizes the total energy functional updates and drives the motion of the zero level set interface to the desired surface. The derived formulations were applied to certain 3D surface reconstruction with good results.