Computers & Graphics (2019) Contents lists available at ScienceDirect Computers & Graphics journal homepage: www.elsevier.com/locate/cag A Comparison of Methods for Gradient Field Estimation on Simplicial Meshes Claudio Mancinelli a , Marco Livesu b , Enrico Puppo a a DIBRIS - University of Genoa (Italy) b IMATI - CNR, Genoa (Italy) ARTICLE INFO Article history: Received March 21, 2019 Keywords: Simplicial meshes, Gradient estimation, Discrete geometry ABSTRACT The estimation of the differential properties of a function sampled at the vertices of a discrete domain is at the basis of many applied sciences. In this paper, we focus on the computation of function gradients on triangle and tetrahedral meshes. We study one cell-based method (the standard the facto), plus three vertex-based methods. Com- parisons regard accuracy, ability to perform on different domain discretizations, and efficiency. We performed extensive tests and provide an in-depth analysis of our re- sults. Besides some common behaviour, we found that some methods perform better than others, considering both accuracy and efficiency. This directly translates to use- ful suggestions for the implementation of gradient estimators in research and industrial code. c  2019 Elsevier B.V. All rights reserved. 1. Introduction 1 Geometric meshes play a central role in a number of do- 2 mains, including computer graphics and scientific visualization, 3 mechanical, structural and electrical engineering, and computa- 4 tional methods in physics, chemistry and geology. Quite often, 5 a scalar field is sampled at the vertices of a geometric mesh, 6 which discretizes a given domain. Computational analysis of 7 such a field may require evaluating its gradient and, possibly, 8 tracing its integral curves. 9 In the geometry processing and FEM literature, a scalar field 10 is often extended from vertices to the interior of higher dimen- 11 sional cells by linear interpolation. Under this approach, a con- 12 stant gradient is associated to each cell with a straightforward 13 computation, thus providing the simplest form of evaluation 14 of the gradient field. Although sufficient for many applica- 15 tions, the piece-wise constant gradient has several limitations 16 and drawbacks: being not continuous, the gradient has diver- 17 gent covariant derivative at the interface between cells; singu- 18 larities are forced to lie just at vertices of the mesh; and integral 19 curves are discretised into polylines that travel parallel to each 20 other inside each element, so that their distribution does not de- 21 pend just on the field, but also on the orientation of edges and 22 vertex valences of the underlying mesh (see Figure 1). 23 As discussed in [1], discrete vector fields can be given per 24 face, per edge, or per vertex. Despite the entity to which the 25 gradient estimate is attached, the gradient field can be extended 26 to the whole mesh either in a piece-wise constant manner – i.e., 27 defining a local region surrounding each vertex/edge/face and 28 assuming the gradient to be constant within it, or via interpola- 29 tion – e.g., linearly inside each simplex. When vector fields are 30 extended to the whole mesh by linear interpolation, they can be 31 traced exactly inside each region [2, 3]. However, there exist 32 surprisingly few works dealing with the estimation and assess- 33 ment of piece-wise linear gradient fields. 34 In this work, we review common methods for estimating a 35 per-vertex gradient field linearly interpolated within each ele- 36 ment, and we compare their accuracy with respect to the stan- 37 dard method to compute a per-face constant gradient field. 38 Since our interest is to study the performances of the vertex- 39 based methods in every point of the domain, such comparisons 40 will be made considering not only the vertices of the mesh, as 41 shown in Sec.4.3. This choice is further motivated by the expec- 42 tation of applying these results to the tracing of integral curves. 43 We limit our study to the most common cases of planar and vol- 44 umetric complexes. Since all methods can be extended to arbi- 45