Geometric Measures on Arbitrary Dimensional Digital Surfaces

Abstract

Abstract. This paper proposes a set of tools to analyse the geometry of multidimensional digital surfaces. Our approach is based on several works of digital topology and discrete geometry: representation of digital surfaces, bel adjacencies and digital surface tracking, 2D tangent computation by discrete line recognition, 3D normal estimation from slice contours. The main idea is to notice that each surface element is the crossing point of n − 1 discrete contours lying on the surface. Each of them can be seen as a 4-connected 2D contour. We combine the directions of the tangents extracted on each of these contours to compute the normal vector at the considered surface element. We then define the surface area from the normal field. The presented geometric estimators have been implemented in a framework able to represent subsets of n-dimensional spaces. As shown by our experiments, this generic implementation is also efficient.

Normals and Curvature Estimation for Digital Surfaces Based on Convolutions

“…It is however clear that the size of the mask should be set according to the resolution, as we do here. With the approach mentioned in the introduction, the best result in [10] is obtained on a sphere with radius 100 and an average error of 1.51 • (std. dev.…”

“…Roughly speaking, this property is what is called multigrid convergence in the literature. Several surface normal estimation methods have been proposed, among which we may cite [13,19,10] is past DGCI's, but no method has been proved to be convergent in that sense.…”

“…These methods have been extended to the 3D case with digital plane recognition [17]. Directional tangent estimation based on segments recognition was used in [19] to compute normal vectors on a digital surface and in [10] for the nD case. All of these methods are sensitive to noise.…”

Multi-Label Simple Points Definition for 3D Images Digital Deformable Model

“…Energy E s is based on a discrete area estimator proposed in [17]. The area estimation is not the same whether it is computed on one side or the other of the surface.…”

“…We indeed propose an original regularization term which uses a discrete geometric estimator for computing the area of each surfel. Its principle is to extract maximal digital straight segments to estimate the surfel normal, area being a byproduct [17]. We get therefore a digital equivalent of continuous active surfaces minimizing their area, which is also an 3D extension of discrete deformable boundaries [16].…”