International audience

Abstract The estimation of differential quantities on oriented point cloud is a classical step for many geometry processing tasks in computer graphics and vision. Even if many solutions exist to estimate such quantities, they usually fail at satisfying both a stable estimation with theoretical guarantee, and the efficiency of the associated algorithm. Relying on the notion of corrected curvature measures [LRT22, LRTC20] designed for surfaces, the method introduced in this paper meets both requirements. Given a point of interest and a few nearest neighbours, our method estimates the whole curvature tensor information by generating random triangles within these neighbours and normalising the corrected curvature measures by the corrected area measure. We provide a stability theorem showing that our pointwise curvatures are accurate and convergent, provided the noise in position and normal information has a variance smaller than the radius of neighbourhood. Experiments and comparisons with the state‐of‐the‐art confirm that our approach is more accurate and much faster than alternatives. The method is fully parallelizable, requires only one nearest neighbour request per point of computation, and is trivial to implement.

Voxel based modeling is a very attractive way to represent complex multi-material objects. Beside artistic choices of pixel/voxel arts, representing objects as voxels allows efficient and dynamic interactions with the scene. For geometry processing purposes, many applications in material sciences, medical imaging or numerical simulation rely on a regular partitioning of the space with labeled voxels. In this article, we consider a variational approach to reconstruct interfaces in multi-labeled digital images. This approach efficiently produces piecewise smooth quadrangulated surfaces with some theoretical stability guarantee. Non-manifold parts at intersecting interfaces are handled naturally by our model. We illustrate the strength of our tool for digital surface regularization, as well as voxel art regularization by transferring colorimetric information to regularized quads and computing isotropic geodesic on digital surfaces.

Abstract A consistent and yet practically accurate definition of curvature onto polyhedral meshes remains an open problem. We propose a new framework to define curvature measures, based on the Corrected Normal Current, which generalizes the normal cycle: it uncouples the positional information of the polyhedral mesh from its geometric normal vector field, and the user can freely choose the corrected normal vector field at vertices for curvature computations. A smooth surface is then built in the Grassmannian ℝ3 × 𝕊2 by simply interpolating the given normal vector field. Curvature measures are then computed using the usual Lipschitz–Killing forms, and we provide closed‐form formulas per triangle. We prove a stability result with respect to perturbations of positions and normals. Our approach provides a natural scale‐space for all curvature estimations, where the scale is given by the radius of the measuring ball. We show on experiments how this method outperforms state‐of‐the‐art methods on clean and noisy data, and even achieves pointwise convergence on difficult polyhedral meshes like digital surfaces. The framework is also well suited to curvature computations using normal map information.

Abstract The Mumford‐Shah functional approximates a function by a piecewise smooth function. Its versatility makes it ideal for tasks such as image segmentation or restoration, and it is now a widespread tool of image processing. Recent work has started to investigate its use for mesh segmentation and feature lines detection, but we take the stance that the power of this functional could reach far beyond these tasks and integrate the everyday mesh processing toolbox. In this paper, we discretize an Ambrosio‐Tortorelli approximation via a Discrete Exterior Calculus formulation. We show that, combined with a new shape optimization routine, several mesh processing problems can be readily tackled within the same framework. In particular, we illustrate applications in mesh denoising, normal map embossing, mesh inpainting and mesh segmentation.

Essential image processing and analysis tasks, such as image segmentation, simplification and denoising, can be conducted in a unified way by minimizing the Mumford-Shah (MS) functional. Although seductive, this minimization is in practice difficult because it requires to jointly define a sharp set of contours and a smooth version of the initial image. For this reason, various relaxations of the original formulations have been proposed, together with optimisation methods. Among these, the Ambrosio-Tortorelli (AT) parametric functional is of particular interest, because minimizers of AT can be shown to converge to a minimizer of MS. However this convergence is difficult to achieve numerically using standard finite difference schemes. Indeed, with AT, discontinuities need to be represented explicitly rather than implicitly. In this work, we propose to formulate AT using the full framework of Discrete Calculus (DC), which is able to sharply represent discontinuities thanks to a more sophisticated topological framework. We present our proposed formulation, its resolution, and results on synthetic and real images. We show that we are indeed able to represent sharp discontinuities and as a result significantly better stability to noise, compared with finite difference schemes.

This paper proposes an original method for extracting the centerline of 3D objects given only partial mesh scans as input data. Its principle relies on the construction of a normal vector accumulation map build by casting digital rays from input vertices. This map is then pruned according to a confidence voting rule: confidence in a point increases if this point has maximal votes along a ray. Points with high confidence accurately delineate the centerline of the object. The resulting centerline is robust enough to allow the reconstruction of the associated graph by a simple morphological processing of the confidence and a geodesic tracking. The overall process is unsupervised and only depends on a user-chosen maximal object radius. Experiments show a good behavior on standard mesh scans. Moreover, the proposed method is not only competitive with state-of-the-art methods on perfect data, but appears to be much more reliable on imperfect or damaged data, like holes, partial scans, noise, and scans from only one direction.

Abstract We 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‐of‐the‐art normal field estimators like Voronoi Covariance Measure [ MOG11 ] or Randomized Hough Transform [ BM12 ].

The Voronoi covariance measure (VCM) of a compact set $K$ of $\mathbb{R}^d$ is a tensor-valued measure that encodes geometrical information on $K$ and which is known to be resilient to Hausdorff noise but sensitive to outliers. In this paper, we generalize this notion to any distance-like function $\delta$ and define the $\delta$-VCM. Combining the VCM with the distance to a measure and also with the witnessed-$k$-distance, we get a provably good tool for normal estimation that is resilient to Hausdorff noise and to outliers. We present experiments showing the robustness of our approach for normal and curvature estimation and sharp feature detection.

The automatic detection of noisy or damaged parts along digital contours is a difficult problem since it is hard to distinguish between information and perturbation without further a priori hypotheses. However, solving this issue has a great impact on numerous applications, including image segmentation, geometric estimators, contour reconstruction, shape matching, or image edition. We propose an original strategy to detect what the relevant scales are at which each point of the digital contours should be considered. It relies on theoretical results of asymptotic discrete geometry. A direct consequence is the automatic detection of the noisy or damaged parts of the contour, together with its quantitative evaluation (or noise level). Apart from a given maximal observation scale, the proposed approach does not require any parameter tuning and is easy to implement. We demonstrate its effectiveness on several datasets. We present different direct applications of this local measure to contour smoothing and geometric estimators whose algorithms initially required a noise/scale parameter to tune: They show the pertinence of the proposed measure for digital shape analysis and reconstruction.

A theory for the multiscale analysis of digital shapes would be very interesting for the pattern recognition community, giving a digital equivalent of the continuous scale-space theory. We focus here on providing analytical formulae of the multiresolution of Digital Straight Segments (DSS), which is a fundamental tool for describing digital shape contours.

This work highlights the relation existing between isosurfaces of an image with a given threshold (as classically computed by a marching-cubes algorithm) and digital surfaces of this thresholded image. The first step has been to extend the classical adjacency relation defined between elements of a digital surface. It turns out that the induced surface graphs are 2D combinatorial manifolds without boundary which can easily be mapped into closed and orientable surfaces in R/sup 3/. Hence, digital surfaces can be processed to compute corresponding isosurfaces; the converse is also true.