David Coeurjolly

A comparative evaluation of length estimators of digital curves

This paper compares previously published length estimators in image analysis having digitized curves as input. The evaluation uses multigrid convergence (theoretical results and measured speed of convergence) and further measures as criteria. This paper also suggests a new gradient-based method for length estimation, and combines a previously proposed length estimator for straight segments with a polygonalization method.

Generalizations of angular radial transform for 2D and 3D shape retrieval

The angular radial transform (ART) is a moment-based image description method adopted in MPEG-7 as a 2D region-based shape descriptor. This paper proposes generalizations of the ART to describe two-dimensional images and three-dimensional models. First, we propose an 2D extension, called GART, which allows applying ART to images while insuring robustness to all possible rotations and to perspective deformations. Then, we generalize the ART to index 3D models. This new 3D shape descriptor, so called 3D ART, has the same properties that the original transform: robustness to rotation, translation, noise and scaling while keeping a compact size and a good retrieval cost. The size of the descriptor is an essential evaluation parameter on which depends the response time of a content-based retrieval system. For both generalizations, many experiments were made on large databases and have shown, that GART outperforms ART in accuracy at the cost of speed, and that 3D ART outperforms the spherical harmonics shape descriptor (Vranic, D.V., Saupe, D., 2002. Description of 3D-shape using a complex function on the sphere, in: IEEE International Conference on Multimedia and Expo (ICME 2002), Lausanne, Switzerland, 2002, pp. 177-180; Funkhouser, T., Min, P. Kazhdan, M., Chen, J., Halderman, A., Dobkin, D., Jacobs, D., 2003. A search engine for 3D models. ACM Trans. Graphics 22(1), 83-105) in speed at the cost of accuracy.

Discrete bisector function and Euclidean skeleton in 2D and 3D

We propose a new definition and an algorithm for the discrete bisector function, which is an important tool for analyzing and filtering Euclidean skeletons. We also introduce a new thinning algorithm which produces homotopic discrete Euclidean skeletons. These algorithms, which are valid both in 2D and 3D, are integrated in a skeletonization method which is based on exact transformations, allows the filtering of skeletons, and is computationally efficient.

Optimal Separable Algorithms to Compute the Reverse Euclidean Distance Transformation and Discrete Medial Axis in Arbitrary Dimension

In binary images, the distance transformation (DT) and the geometrical skeleton extraction are classic tools for shape analysis. In this paper, we present time optimal algorithms to solve the reverse Euclidean distance transformation and the reversible medial axis extraction problems for d-dimensional images. We also present a d-dimensional medial axis filtering process that allows us to control the quality of the reconstructed shape

Digital planarity-A review

Digital planarity is defined by digitizing Euclidean planes in the three-dimensional digital space of voxels; voxels are given either in the grid-point or the grid-cube model. The paper summarizes results (also including most of the proofs) about different aspects of digital planarity, such as supporting or separating Euclidean planes, characterizations in arithmetic geometry, periodicity, connectivity, and algorithmic solutions. The paper provides a uniform presentation, which further extends and details a recent book chapter in [R. Klette, A. Rosenfeld, Digital Geometry-Geometric Methods for Digital Picture Analysis, Morgan Kaufmann, San Francisco, 2004].

Discrete Curvature Based on Osculating Circle Estimation

In this paper, we make an overview of the existing algorithms concerning the discrete curvature estimation. We extend the Worring and Smeulders [WS93] classification to new algorithms and we present a new and purely discrete algorithm based on discrete osculating circle estimation.

Adaptive estimation of normals and surface area for discrete 3-D objects: application to snow binary data from X-ray tomography

Estimating the normal vector field on the boundary of discrete three-dimensional objects is essential for rendering and image measurement problems. Most of the existing algorithms do not provide an accurate determination of the normal vector field for shapes that present edges. Here, we propose a new and simple computational method in order to obtain accurate results on all types of shapes, whatever their local convexity degree. The presented method is based on the gradient vector field analysis of the object distance map. This vector field is adaptively filtered around each surface voxel using angle and symmetry criteria so that as many relevant contributions as possible are accounted for. This optimizes the smoothing of digitization effects while preserving relevant details of the processed numerical object. Thanks to the precise normal field obtained, a projection method can be proposed to immediately derive the surface area from a raw discrete object. An empirical justification of the validity of such an algorithm in the continuous limit is also provided. Some results on simulated data and snow images from X-ray tomography are presented, compared to the Marching Cubes and Convex Hull results, and discussed.

On digital plane preimage structure

In digital geometry, digital straightness is an important concept both for practical motivations and theoretical interests. Concerning the digital straightness in dimension 2, many digital straight line characterizations exist and the digital straight segment preimage is well known. In this article, we investigate the preimage associated to digital planes. More precisely, we present first structure theorems that describe the preimage of a digital plane. Furthermore, we present a bound on the number of preimage faces under some given hypotheses.

2D and 3D visibility in discrete geometry: an application to discrete geodesic paths

In this article, we present a discrete definition of the classical visibility in computational geometry based on digital straight fines. We present efficient algorithms to compute the set of pixels in a non-convex domain that are visible from a source pixel. Based on these definitions, we define discrete geodesic paths in discrete domain with obstacles. This allows us to introduce a new geodesic metric in discrete geometry.

Multigrid convergence and surface area estimation

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.