BendFields

Abstract

and ADRIEN BOUSSEAU INRIADesigners frequently draw curvature lines to convey bending of smooth surfaces in concept sketches. We present a method to extrapolate curvature lines in a rough concept sketch, recovering the intended 3D curvature field and surface normal at each pixel of the sketch. This 3D information allows to enrich the sketch with 3D-looking shading and texturing.We first introduce the concept of regularized curvature lines that model the lines designers draw over curved surfaces, encompassing curvature lines and their extension as geodesics over flat or umbilical regions. We build on this concept to define the orthogonal cross field that assigns two regularized curvature lines to each point of a 3D surface. Our algorithm first estimates the projection of this cross field in the drawing, which is nonorthogonal due to foreshortening. We formulate this estimation as a scattered interpolation of the strokes drawn in the sketch, which makes our method robust to sketchy lines that are typical for design sketches. Our interpolation relies on a novel smoothness energy that we derive from our definition of regularized curvature lines. Optimizing this energy subject to the stroke constraints produces a dense nonorthogonal 2D cross field which we then lift to 3D by imposing orthogonality. Thus, one central concept of our approach is the generalization of existing cross field algorithms to the nonorthogonal case.We demonstrate our algorithm on a variety of concept sketches with various levels of sketchiness. We also compare our approach with existing work that takes clean vector drawings as input. Fig. 1. Designers commonly draw curvature lines to emphasize surface bending in concept sketches ((a), red lines). We extrapolate these lines in a bitmap to form a dense cross field, from which we estimate the 3D curvature directions and the surface normal at each pixel ((b)(c)). We use this information to compute shading, greatly enhancing the 3D look of the sketch (d). Note that this sketch is composed of several layers to represent the main body, handle, and nozzle of the vacuum cleaner.

Vectorization of Line Drawings via Polyvector Fields

“…3. The target application for Bendfields [Iarussi et al 2015] leads to di erent frame field design assumptions (a) that are unsuitable for our type of vectorization. Compare with our result (b).…”

“…Frame elds originally were proposed for guiding anisotropic quad meshing via inversion-free mesh parameterization [Panozzo et al 2014]. Since then, frame elds have found additional applications, such as inferring 3D normals from a 2D sketch [Iarussi et al 2015] and recovery of damaged historical documents [Pal et al 2016]. Our work is driven by the frame eld synthesis and interpolation tool set developed in [Diamanti et al 2015;Panozzo et al 2014].…”

“…Our work is driven by the frame eld synthesis and interpolation tool set developed in [Diamanti et al 2015;Panozzo et al 2014]. Namely, we use their de nition and representation of a PolyVector eld, as described in Section 3.1. e BendFields algorithm proposed by Iarussi et al [2015] inspired some aspects of our approach. While their algorithm is targeted to 3D surface reconstruction from curvature lines, they initially generate a frame eld aligned to directions in a bitmap image.…”

“…Optimizing for a (u, ) pair per pixel induces challenging issues involving labeling and sign; for example, this representation in the BendFields algorithm requires the use of a mixed-integer solver [Iarussi et al 2015]. To avoid this complexity, we instead optimize for a (c 0 , c 2 ) pair per pixel, which has no sign or ordering ambiguity.…”

“…Frame elds a ach two pairs of vectors {±u, ± } to each point on the plane. ey have been recently used to generate anisotropic quadrilateral meshes and to estimate 3D normals from 2D sketches [Iarussi et al 2015;Panozzo et al 2014]. Although frame elds are natural for tracking the orientations of curves meeting at sharp junctions, to our knowledge they never have been applied to image vectorization.…”