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Abstract: In this work we propose a new method for estimating the normal orientation of unorganized point clouds. Consistent assignment of normal orientation is a challenging task in the presence of sharp features, nearby surface sheets, noise, undersampling, and missing data. Existing approaches, which consider local geometric properties often fail when operating on such point clouds as local neighborhood measures inherently face issues of robustness. Our approach circumvents these issues by orienting normals based on globally smooth functions defined on point clouds with measures that depend only on single points. More specifically, we consider harmonic functions, or functions which lie in the kernel of the point cloud Laplace–Beltrami operator. Each harmonic function in the set is used to define a gradient field over the point cloud. The problem of normal orientation is then cast as an assignment of cross-product ordering between gradient fields. Global smoothness ensures a highly consistent orientation, rendering our method extremely robust in the presence of imperfect point clouds. [Copyright &y& Elsevier]