Many image completion methods are based on a low-rank approximation of the underlying image using matrix or tensor decomposition models. In this study, we assume that the image to be completed is represented by a multi-way array and can be approximated by a conical hull of subtensors in the observation space. If an observed tensor is near-separable along at least one mode, the extreme rays, represented by the selected subtensors, can be found by analyzing the corresponding convex hull. Following this assumption, we propose a geometric algorithm to address a low-rank image completion problem. The extreme rays are extracted with a segmented convex-hull algorithm that is suitable for performing noise-resistant non-negative tensor factorization. The coefficients of a conical combination of such rays are estimated using Douglas–Rachford splitting combined with the rank-two update least-squares algorithm. The proposed algorithm was applied to incomplete RGB images and a hyperspectral 3D array with a large number of randomly missing entries. Experiments confirm its good performance with respect to other well-known image completion methods.

许多图像完成方法基于使用矩阵或张量分解模型的基础图像的低秩逼近。在这项研究中，我们假设要完成的图像由多路阵列表示，并且可以由观察空间中的锥形张量的锥壳近似。如果观察到的张量沿至少一种模式几乎是可分离的，则可以通过分析相应的凸包来找到由选定的次张量表示的极端射线。根据这个假设，我们提出了一种几何算法来解决低秩图像完成问题。使用分段凸包算法提取极端射线，该算法适合执行抗噪声的非负张量分解。此类射线的圆锥形组合的系数是使用Douglas–Rachford分裂与二级更新最小二乘算法组合估算的。所提出的算法被应用于不完整的RGB图像和具有大量随机丢失条目的高光谱3D阵列。实验证实了其相对于其他众所周知的图像完成方法的良好性能。