Properties of Gauss digitized sets and digital surface integration

International audienceThis paper presents new topological and geometrical properties of Gauss digitizations of Euclidean shapes, most of them holding in arbitrary dimension $d$. We focus on $r$-regular shapes sampled by Gauss digitization at gridstep $h$. The digitized boundary is shown to be close to the Euclidean boundary in the Hausdorff sense, the minimum distance $\frac{\sqrt{d}}{2}h$ being achieved by the projection map $\xi$ induced by the Euclidean distance. Although it is known that Gauss digitized boundaries may not be manifold when $d \ge 3$, we show that non-manifoldness may only occur in places where the normal vector is almost aligned with some digitization axis, and the limit angle decreases with $h$. We then have a closer look at the projection of the digitized boundary onto the continuous boundary by $\xi$. We show that the size of its non-injective part tends to zero with $h$. This leads us to study the classical digital surface integration scheme, which allocates a measure to each surface element that is proportional to the cosine of the angle between an estimated normal vector and the trivial surface element normal vector. We show that digital integration is convergent whenever the normal estimator is multigrid convergent, and we explicit the convergence speed. Since convergent estimators are now available in the litterature, digital integration provides a convergent measure for digitized objects

Robust Geometry Estimation using the Generalized Voronoi Covariance Measure

The Voronoi Covariance Measure of a compact set K of R^d is a tensor-valued measure that encodes geometric information on K and which is known to be resilient to Hausdorff noise but sensitive to outliers. In this article, we generalize this notion to any distance-like function delta and define the delta-VCM. We show that the delta-VCM is resilient to Hausdorff noise and to outliers, thus providing a tool to estimate robustly normals from a point cloud approximation. We present experiments showing the robustness of our approach for normal and curvature estimation and sharp feature detection

Fully Deformable 3D Digital Partition Model with Topological Control

International audienceWe propose a purely discrete deformable partition model for segmenting 3D images. Its main ability is to maintain the topology of the partition during the minimization process. To do so, our main contribution is a new definition of multi-label simple points (ML simple point) that is easily computable. An ML simple point can be relabeled without modifying the overall topology of the partition. The definition is based on intervoxel properties, and uses the notion of collapse on cubical complexes. This work is an extension of a former restricted definition [DupasAl09] that prohibits the move of intersections of boundary surfaces. A deformation process is carried out with a greedy energy minimization algorithm. A discrete area estimator is used to approach at best standard regularizers classically used in continuous energy minimizing methods. We illustrate the potential of our approach with the segmentation of 3D medical images with known expected topology

Combining Topological Maps, Multi-Label Simple Points, and Minimum-Length Polygons for Efficient Digital Partition Model

International audienceDeformable models have shown great potential for image segmentation. They include discrete models whose combinatorial formulation leads to efficient and sometimes optimal minimization algorithms. In this paper, we propose a new discrete framework to deform any partition while preserving its topology. We show how to combine the use of multi-label simple points, topological maps and minimum-length polygons in order to implement an efficient digital deformable partition model. Our experimental results illustrate the potential of our framework for segmenting images, since it allows the mixing of region-based, contour-based and regularization energies, while keeping the overall image structure

Meaningful Thickness Detection on Polygonal Curve

International audienceThe notion of meaningful scale was recently introduced to detect the amount of noise present along a digital contour. It relies on the asymptotic properties of the maximal digital straight segment primitive. Even though very useful, the method is restricted to digital contour data and is not able to process other types of geometric data like disconnected set of points. In this work, we propose a solution to overcome this limitation. It exploits another primitive called the Blurred Segment which controls the straight segment recognition precision of disconnected sets of points. The resulting noise detection provides precise results and is also simpler to implement. A first application of contour smoothing demonstrates the efficiency of the proposed method. The algorithms can also be tested online

Curvature based corner detector for discrete, noisy and multi-scale contours

International audienceEstimating curvature on digital shapes is known to be a difficult problem even in high resolution images 10,19. Moreover the presence of noise contributes to the insta- bility of the estimators and limits their use in many computer vision applications like corner detection. Several recent curvature estimators 16,13,15, which come from the dis- crete geometry community, can now process damaged data and integrate the amount of noise in their analysis. In this paper, we propose a comparative evaluation of these estimators, testing their accuracy, efficiency, and robustness with respect to several type of degradations. We further compare the best one with the visual curvature proposed by Liu et al. 14, a recently published method from the computer vision community. We finally propose a novel corner detector, which is based on curvature estimation, and we provide a comprehensive set of experiments to compare it with many other classical cor- ner detectors. Our study shows that this corner detector has most of the time a better behavior than the others, while requiring only one parameter to take into account the noise level. It is also promising for multi-scale shape description