Revisiting Digital Straight Segment Recognition

This paper presents new results about digital straight segments, their recognition and related properties. They come from the study of the arithmetically based recognition algorithm proposed by I. Debled-Rennesson and J.-P. Reveill\`es in 1995 [Debled95]. We indeed exhibit the relations describing the possible changes in the parameters of the digital straight segment under investigation. This description is achieved by considering new parameters on digital segments: instead of their arithmetic description, we examine the parameters related to their combinatoric description. As a result we have a better understanding of their evolution during recognition and analytical formulas to compute them. We also show how this evolution can be projected onto the Stern-Brocot tree. These new relations have interesting consequences on the geometry of digital curves. We show how they can for instance be used to bound the slope difference between consecutive maximal segments

Minimal Decomposition of a Digital Surface into Digital Plane Segments Is NP-Hard

Abstract. This paper deals with the complexity of the decomposition of a digital surface into digital plane segments (DPS for short). We prove that the decision problem (does there exist a decomposition with less than k DPS?) is NP-complete, and thus that the optimisation problem (finding the minimal number of DPS) is NP-hard. The proof is based on a polynomial reduction of any instance of the well-known 3-SAT problem to an instance of the digital surface decomposition problem. A geometric model for the 3-SAT problem is proposed.

Unsupervised Polygonal Reconstruction of Noisy Contours by a Discrete Irregular Approach

International audienceIn this paper, we present an original algorithm to build a polygonal reconstruction of noisy digital contours. For this purpose, we first improve an algorithm devoted to the vectorization of discrete irregular isothetic objects. Afterwards we propose to use it to define a reconstruction process of noisy digital contours. More precisely, we use a local noise detector, introduced by Kerautret and Lachaud in IWCIA 2009, that builds a multi-scale representation of the digital contour, which is composed of pixels of various size depending of the local amount of noise. Finally, we compare our approach with previous works, by con- sidering the Hausdorff distance and the error on tangent orientations of the computed line segments to the original perfect contour. Thanks to both synthetic and real noisy objects, we show that our approach has interesting performance, and could be applied in document analysis systems

Digital hyperplane fitting

International audienceThis paper addresses the hyperplane fitting problem of discrete points in any dimension (i.e. in Z d). For that purpose, we consider a digital model of hyperplane, namely digital hyperplane, and present a combinatorial approach to find the optimal solution of the fitting problem. This method consists in computing all possible digital hyperplanes from a set S of n points, then an exhaustive search enables us to find the optimal hyperplane that best fits S. The method has, however, a high complexity of O(n d), and thus can not be applied for big datasets. To overcome this limitation, we propose another method relying on the Delaunay triangulation of S. By not generating and verifying all possible digital hyperplanes but only those from the elements of the triangula-tion, this leads to a lower complexity of O(n d 2 +1). Experiments in 2D, 3D and 4D are shown to illustrate the efficiency of the proposed method

Algorithms for Fast Digital Straight Segments Union

International audienceGiven two Digital Straight Segments (DSS for short) of known minimal characteristics, we investigate the union of these DSSs: is it still a DSS ? If yes, what are its minimal characteristics ? We show that the problem is actually easy and can be solved in, at worst, logarithmic time using a state-of-the-art algorithm. We moreover propose a new algorithm of logarithmic worst-case complexity based on arithmetical properties. But when the two DSSs are connected, the time complexity of this algorithm is lowered to O(1) and experiments show that it outperforms the state-of-the art one in any case

Segmenting simplified surface skeletons

A novel method for segmenting simplified skeletons of 3D shapes is presented. The so-called simplified Y-network is computed, defining boundaries between 2D sheets of the simplified 3D skeleton, which we take as our skeleton segments. We compute the simplified Y-network using a robust importance measure which has been proved useful for simplifying complex 3D skeleton manifolds. We present a voxel-based algorithm and show results on complex real-world objects, including ones containing large amounts of boundary noise

Reversible Polygonalization of a 3D Planar Discrete Curve: Application on Discrete Surfaces.

International audienceReversible polyhedral modelling of discrete objects is an important issue to handle those objects. We propose a new algorithm to compute a polygonal face from a discrete planar face (a set of voxels belonging to a discrete plane). This transformation is reversible, i.e. the digitization of this polygon is exactly the discrete face. We show how a set of polygons modelling exactly a discrete surface can be computed thanks to this algorithm

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. Key words: digital plane preimage, digital straight line, dual transformation.