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We propose an unsupervised real-time dense depth completion from a sparse depth map guided by a single image. Our method generates a smooth depth map while preserving discontinuity between different objects. Our key idea is a Binary Anisotropic Diffusion Tensor (B-ADT) which can completely eliminate smoothness constraint at intended positions and directions by applying it to variational regularization. We also propose an Image-guided Nearest Neighbor Search (IGNNS) to derive a piecewise constant depth map which is used for B-ADT derivation and in the data term of the variational energy. Our experiments show that our method can outperform previous unsupervised and semi-supervised depth completion methods in terms of accuracy. Moreover, since our resulting depth map preserves the discontinuity between objects, the result can be converted to a visually plausible point cloud. This is remarkable since previous methods generate unnatural surface-like artifacts between discontinuous objects.