Digital Surface Regularization by Normal Vector Field Alignment

Abstract

Digital objects and digital surfaces are isothetic structures per se. Such surfaces are thus not adapted to direct visualization with isothetic quads, or to many geometry processing methods. We propose a new regularization technique to construct a piecewise smooth quadrangulated surface from a digital surface. More formally we propose a variational formulation which efficiently regularizes digital surface vertices while complying with a prescribed, eventually anisotropic, input normal vector field estimated on the digital structure. Beside visualization purposes, such regularized surface can then be used in any geometry processing tasks which operates on triangular or quadrangular meshes (e.g. compression, texturing, anisotropic smoothing, feature extraction).

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

“…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]. We evaluate our approach on the dataset of Delanoy 39 et al [1], on which we recover smoother surfaces with sharper 40 discontinuities.…”

“…We first use ex-3 isting deep neural networks [1, 24] to predict a volumetric re-4 construction of the shape, along with one normal map per sketch 5 (Figure 1b). We then project the normal maps on the surface of 6 the volumetric reconstruction and combine this information with 7 the initial surface normal to obtain a distribution of normals for 8 each surface element (Figure 1c,d). While the normals coming 9 from different sources are mostly consistent, some parts of the 10 shape exhibit significant ambiguity due to erroneous predictions 11 and misalignment between the input sketches and the volumet- 12 ric reconstruction.…”

“…Therefore in the next step of our approach 13 we reconstruct a piecewise-smooth normal field by a variational estimate. Finally, we regularize the initial surface such that its 19 quads and edges align with this normal field [6], resulting in a 20 piecewise-smooth object that follows the overall shape of the vol- that this type of architecture performs well on the task of nor-49 mal prediction from sketches. We base our implementation on 50 Pix2Pix [24], from which we remove the discriminator network 51 for simplicity.…”

“…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]. We begin by thresholding the volumetric prediction to obtain 75 a binary voxel grid.…”

“…Equipped with a piecewise-smooth normal field n * , we finally 29 reconstruct a regularized surface whose quads are as close to or-30 thogonal to the prescribed normals as possible. We achieve this 31 goal using the variational model proposed in [6]. As illustrated 32 in Figure 2, this surface reconstruction guided by our piecewise-33 smooth normal vector field effectively aligns quad edges with 34 sharp surface discontinuities.…”