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A linear method for segmentation of digital arcs

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A linear method for segmentation of digital arcs
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   apport  de recherche       I      S      S      N     0     2     4     9   -     6     3     9     9      I      S      R      N      I      N      R      I      A      /      R      R   -   -     0     0     0     1   -   -      F      R    +      E      N      G Thème COG INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE  A linear method for segmentation of digital arcs Thanh Phuong NGUYEN — Isabelle DEBLED-RENNESSON N° 0001 February 2010  Centre de recherche INRIA Nancy – Grand EstLORIA, Technopôle de Nancy-Brabois, Campus scientifique,615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy Téléphone : +33 3 83 59 30 00 — Télécopie : +33 3 83 27 83 19 A linear method for segmentation of digital arcs Thanh Phuong NGUYEN  ∗ , Isabelle DEBLED-RENNESSON  ∗ Th`eme COG — Syst`emes cognitifs´Equipes-Projets ADAGIoRapport de recherche n° 0001 — February 2010 — 16 pages Abstract:  A linear algorithm based on a discrete geometry approach is proposed forthe recognition of digital arcs and digital circles using a new representation of them. Byutilizing this representation, we transform the problem of digital arc recognition intothe problem of digital straight line recognition. We then develop a linear method forthe segmentation of digital arcs. In addition, the proposed method can naturally work with disconnected or possibly noisy curves thanks to the notion of blurred segment [1].This simple and robust method can be applied to detect the arcs in graphic documents,technical drawing images. The experimentation of our method gives good results. Key-words:  discrete circle, dicrete arc, blurred segment, discrete geometry This work is supported by ANR in the framework of GEODIB project, ANR-FI-071215-01-01 ∗ Email: { nguyentp,debled } @loria.fr  Segmentation en arcs en temps linaire R´esum´e :  Nous proposons un algorithme linaire reposant sur une approche de gomtriediscrte pour segmenter une courbe en arcs et cercles discrets. Cette mthode utiliseune reprsentation srcinale des arcs et cercles discrets. En utilisant cette reprsentation,nous transformons le problme de reconnaissance d’arcs discrets en un problme de re-connaissance de droites discrtes et nous en dduisons un algorithme de segmentation.Cet methode peut travailler sur des donnes dconnectes ou bien bruites grˆace la notionde segment flou [1]. Cette mthode simple et efficace peut ˆetre applique pour dtecter des arcs dans les documents techniques. L’exprimentation de notre mthode donne desbons rsultats. Mots-cl´es :  cercle discret, arc discret, segment flou, gomtrie discrte   A linear method for segmentation of digital arcs  3 1 Introduction The segmentation of discrete objects is an important topic in image analysis and dis-crete geometry. This is the first step for the decomposition of an image contour intoprimitives such as lines, circles. Many authors [2–5] have proposed algorithms for therecognition of digital straight lines.Beside the digital straight lines, the digital arcs and circles are also basic geometricobjects of which the recognition is an interesting topic. However, there are less workson this topic than that on discrete line recognition in the literature. Nakamura et al. [6]proposed a recursive algorithm for determining the center of a digital circle, but itscomplexity is exponential in the general case. Kim [7,8] proposed several results ondigital disks. The first result [7] detects if a set of grid points in a N   × N   image is adigital disk with complexity O ( n 3 ) . The second result [8] reduces this task to O ( n 2 ) .Based on the classical separating arc problem in computational geometry, Fisk [9] alsoproposed an algorithm for the recognition of a digital disk in O ( n 2 )  time. By using thesame approach, Kovalevsky [10] introduced a method for digital circle recognition in O ( n 2 log n )  time. Coeurjolly [11] transformed the problem of circle recognition intothe problem of search a 2D point that belongs to the intersection of  n 2 half-plane. Heused a preprocessing stage that utilizes arithmetic properties of digital curves to reducethe complexity to O ( n 4 / 3 log n ) . Sauer [12] presented a linear algorithm to decide if acurve is a digital circle. However, it is not appropriate to recognize a digital arc. Later,Damaschke [13] presented a linear algorithm for recognition of digital arcs based on alinear programming approach. He showed that this problem is equivalent to solve a setof  n inequalities in dimension 3 which can be done in linear time by using Megiddo’salgorithm [14]. Worring [15] introduced a digital circle segmentation method by usinga fixed size window process. Roussillon [16] proposed a linear algorithm of circlerecognition with one of three conditions: digital arcs have a fixed radius, boundarycurves contact a fixed point or a center belongs to a fixed straight line. But theseconstrains do not lead to an efficient method for digital circle recognition in the generalcase.We present in this paper a linear method for the recognition of digital circles or dig-ital arcs based on a discrete geometry approach. Firstly, a polygonalization is appliedin linear time on the input curve. Secondly, we utilize a transform proposed by Lateckiet al. [17] to represent the obtained polygon in a novel space called tangent space. Weshow that a sequence of chords of a circle will correspond to a sequence of colinearpoints in the tangent space. So the problem of arc/circle recognition can be regarded asthe problem of digital straight line recognition. Because the digitalization process of a digital circle causes distortion to straight lines in this space, we use the algorithm of blurred segment recognition [1] to detect the straight segments in this space.This paper is organized as follows. Section 2 recalls some definitions concerningdigital circles and blurred segments. The next section presents a technique to transforman arc into the tangent space and proposes some principal properties of the arc in thisrepresentation. Section 4 proposes a linear algorithm for the recognition of digital arcsor digital circles. Section 5 presents the upper bound of the approximation error of ourmethod. In section 6, we present a method for the segmentation of a curve into digitalarcs. Some experimentations are given in section 7. RR n° 0001
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