Multi-dimensional visual data completion via weighted hybrid graph-Laplacian

Graph Laplacian has been extensively studied as a priori to assist in visual completion of partially observed entries. However, a simple treatment using first-order operation may be suboptimal since the inherent smooth property within the visual data is ill-considered. To relieve this issue, in this paper, we propose an improved priori, namely weighted hybrid graph Laplacian (WHGL), to better characterize the spatially hierarchical knowledge hidden in the multi-dimensional data. On the whole, two individual layers are presented to encode the structured properties of the given data. In the first layer, the underlying adjacency matrix of the local and nonlocal relationships within the data is composed to capture the pixel-level structure information. Global smoothness is considered as the second layer to reveal the piecewise correlations of different visual elements. Mathematically, we generalize the graph Laplacian in the first-and second-order to construct the respective layers. Finally, a re-weighting strategy is used to mitigate the discontinuity of the reconstructed data under extreme cases, e.g. 95% voxels missing. With an initial step borrowed from some off-the-shelf methods, our model can be solved under limited iterations by the Preconditioned Conjugate Gradients algorithm. The experimental results demonstrate the superiority of the proposed method.