Piecewise smooth reconstruction of normal vector field on digital data

Abstract

a) (b) (c) (d) Figure 1: Piecewise smooth reconstruction of a normal vector field on a digital shape, normal vectors are represented through the flat shading of faces according to illumination (top-left: perfect digitization / down-right: noisy digitization): (a) input digital shape V , (b) input normal vector field g obtained with digital integral invariant (II) method [CLL14] with r = 3, (c) output normal vector field u and (d) sharpfeatures v superposed in red. Perfect and noisy digitization results are obtained using the same parameters. Our approach smoothes the input normal vector field except across sharp features, which are precisely delineated, thin, and consistent with the smoothing. AbstractWe propose a novel method to regularize a normal vector field defined on a digital surface (boundary of a set of voxels). When the digital surface is a digitization of a piecewise smooth manifold, our method localizes sharp features (edges) while regularizing the input normal vector field at the same time. It relies on the optimisation of a variant of the Ambrosio-Tortorelli functional, originally defined for denoising and contour extraction in image processing [AT90]. We reformulate this functional to digital surface processing thanks to discrete calculus operators. Experiments show that the output normal field is very robust to digitization artifacts or noise, and also fairly independent of the sampling resolution. The method allows the user to choose independently the amount of smoothing and the length of the set of discontinuities. Sharp and vanishing features are correctly delineated even on extremely damaged data. Finally, our method can be used to enhance considerably the output of state-ofthe-art normal field estimators like Voronoi Covariance Measure [MOG11] or Randomized Hough Transform [BM12].

Research on Biomimetic Models and Nanomechanical Behaviour of Membranous Wings of Chinese Bee Apis cerana cerana Fabricius

“…The point cloud data should be smoothed in order to eliminate or reduce the effects of noise, that is to say, let low-frequency data pass through and high-frequency noise are intercepted. The quality of a 3-dimensional model reestablished would be enhanced by data smoothing [16]. …”

Combining voxel and normal predictions for multi-view 3D sketching

“…To do so, we first locate the voxels de-33 lineating the object's boundary, and re-project the normal maps 34 on the resulting surface to obtain a distribution of candidate nor-35 mals for each surface element. We then solve for the smoothest 36 normal field that best agrees with these observations [5]. Finally, 37 we optimize the surface elements to best align with this normal 38 field [6].…”

“…In the first stage, we project the normal predictions onto the 64 surface of the volumetric prediction, and complement this in-65 formation with normals estimated directly from the voxel grid. 66 We then solve for the piecewise-smooth normal field that is most 67 consistent with all these candidate normals, such that sharp sur-68 face discontinuities automatically emerge at their most likely lo-69 cations [5]. In the second stage, we optimize the surface of the 70 voxel grid such that it respects the normal field resulting from 71 the first stage, while staying close to the initial predicted voxel 72 geometry [6].…”

“…For each surface element, we now have a unique normal vector 94 n as well as an estimate of its standard deviation σ n . We obtain 95 our final piecewise-smooth normal field n * by minimizing a dis-1 crete variant of the Ambrosio-Tortorelli energy [5].…”

“…for some parameters α, λ, ε ∈ R. Note that the scalar function v 3 tends to be close to 0 along sharp features and close to 1 else-4 where. 5 The first term ensures that the output normal n * is close to the 6 inputn. The second term encourages n * to be smooth where there 7 is no discontinuity.…”