Multigrid Convergence and Surface Area Estimation

  title={Multigrid Convergence and Surface Area Estimation},
  author={David Coeurjolly and Fr{\'e}d{\'e}ric Flin and Olivier Teytaud and Laure Tougne},
  booktitle={Theoretical Foundations of Computer Vision},
Surface area of discrete objects is an important feature for model-based image analysis. In this article, we present a theoretical framework in order to prove multigrid convergence of surface area estimators based on discrete normal vector field integration. The paper details an algorithm which is optimal in time and multigrid convergent to estimate the surface area and a very efficient algorithm based on a local but adaptive computation. 

On Multigrid Convergence of Local Algorithms for Intrinsic Volumes

  • A. Svane
  • Computer Science
    Journal of Mathematical Imaging and Vision
  • 2013
It is shown that local algorithms for intrinsic volumes other than volume are not multigrid convergent on the class of convex polytopes, and for convex particles in 2D with a lower bound on the interior angles, a multigride convergent local algorithm for the Euler characteristic is constructed.

3 D Object Digitization : Volume and Surface Area Estimation

Good estimators should be multigrid convergent, i.e. the error goes to zero with increasing sampling density, and such estimators both for volume and for surface area estimation based on simple counting of voxels are given.

Multigrid Convergence of Discrete Geometric Estimators

Global geometric estimators of area, length, moments, as well as local geometric estimator of tangent and curvature are presented, and their multigrid convergence is studied, a fundamental property which guarantees that the estimation tends toward the exact one as the sampling resolution gets finer and finer.

Asymptotic variance of grey-scale surface area estimators

Segmentation of Noisy Discrete Surfaces

A flexible approach considering arithmetic discrete planes with a variable width is used to avoid the oversegmentation that might happen when classical segmentation algorithms based on regular discrete planes are used to decompose the surface of the object.

Geometric Feature Estimators for Noisy Discrete Surfaces

We present in this paper robust geometric feature estimators on the border of a possibly noisy discrete object. We introduce the notion of patch centered at a point of this border. Thanks to a width

Surface area estimation of digitized 3D objects using weighted local configurations

On the Computations of Specific Surface Area and Specific Grain Contact Area from Snow 3 D Images

Estimating the Specific Surface Area (SSA) of snow and firn using three-dimensional (3D) images is now widely used in the snow and ice community. However, little information is available about

3D noisy discrete objects: Segmentation and application to smoothing



Surface area estimation for digitized regular solids

The paper summarizes work on global polyhedrization techniques with experimental results pointing towards correct multigrid convergence and the class of general ellipsoids is suggested to be a test set for such multigrids convergence studies.

Digital Planar Segment Based Polyhedrization for Surface Area Estimation

The paper estimates the surface area using another global technique called DPS (Digital Planar Segment) algorithm, based on the projection of these DPSes into Euclidean planes, which shows a tendency to be multigrid convergent.

Multigrid Convergence of Calculated Features in Image Analysis

  • R. KletteJ. Zunic
  • Computer Science, Mathematics
    Journal of Mathematical Imaging and Vision
  • 2004
Estimates of worst-case bounds for quantization errors in calculating features such as moments, moment based features, or perimeters in image analysis and probability-theoretical estimates of error bounds (e.g. standard deviations) for such digital approximations are provided.

Segmentation and Length Estimation of 3D Discrete Curves

An arithmetical definition of 3-D discrete lines as well as an efficient construction algorithm and a proof of the multigrid convergence of length estimators is presented.

Normal Computation for Discrete Surfaces in 3D Space

A new method to compute normals on discrete surfaces in object space by considering one of these subsets in a small neighbourhood of a surface point enables the surface normal to be derived from this set.

Fast computation of the normal vector field of the surface of a 3-D discrete object

A fast computational technique to compute the normal vector field of a discrete object at a given scale, whose time cost is proportional to the number of surfels at and little dependent on the scale is presented.

3D Discrete Normal Vectors

A new method for the calculation of normal vectors to a digital object that relies on discrete geometry theories : the recognition of discrete straight lines and tangential lines in dimension 2 is introduced.

On Approximation of Jordan Surfaces in 3D

A polyhedral approximation of closed Jordan surfaces is described, based on the notion of a relative convex hull in a polyhedrally bounded compact set obtained by gridding technique, which can be applied also to approximation of surfaces of functions.

Digital and Image Geometry

This paper relates this approach to digital topology with several other approaches to Digital Topology appeared in literature through a deep analysis of the axioms involved in the definition of digital space.

Surface area estimation of digitized planes

A method for estimating surface area is developed for three-dimensional binary objects. This method is based on assigning surface area weights to the surface volume-elements (voxels) of a binary