Maximal digital straight segments and convergence of discrete geometric estimators

Discrete geometric estimators approach geometric quantities on digitized shapes without any knowledge of the continuous shape. A classical yet difficult problem is to show that an estimator asymptotically converges toward the true geometric quantity as the resolution increases. We study here the convergence of local estimators based on Digital Straight Segment (DSS) recognition. It is closely linked to the asymptotic growth of maximal DSS, for which we show bounds both about their number and sizes. These results not only give better insights about digitized curves but indicate that curvature estimators based on local DSS recognition are not likely to converge. We indeed invalidate an hypothesis which was essential in the only known convergence theorem of a discrete curvature estimator. The proof involves results from arithmetic properties of digital lines, digital convexity, combinatorics, continued fractions and random polytopes

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

Curvature Estimation along Noisy Digital Contours by Approximate Global Optimization

International audienceIn this paper we introduce a new curvature estimator along digital contours, that we called Global Min-Curvature estimator (GMC). As opposed to previous curvature estimators, it considers all the possible shapes that are digitized as this contour, and selects the most probable one with a global optimization approach. The GMC estimator exploits the geometric properties of digital contours by using local bounds on tangent directions defined by the maximal digital straight segments. The estimator is then adapted to noisy contours by replacing maximal segments with maximal blurred digital straight segments. Experiments on perfect and damaged digital contours are performed and in both cases, comparisons with other existing methods are presented

Алгоритм и программа распознавания контуров изображений как последовательности отрезков цифровых прямых

В данной работе рассматриваются алгоритм распознавания отрезков цифровых прямых в контурах бинарных изображений и программная реализация алгоритма. Использование этого алгоритма при обработке изображений приведет к более естественному и экономному по сравнению с известными способами описанию изображений. Рассматриваемый алгоритм и программная реализация могут быть использованы также и для описания контуров при обработке полутоновых и цветных изображений.У даній роботі наводиться алгоритм розпізнавання відрізків цифрових прямих у контурах бінарних зображень, а також програмна реалізація алгоритму. Використання цього алгоритму для оброблення зображень призведе до того, що опис зображень буде більш натуральним та економним порівняно з відомими засобами кодування зображень. Запропоновані алгоритм і програмна реалізація можуть застосовуватись для кодування контурів при обробленні напівтонових та кольорових зображень.In the given work the algorithm of the recognition of the digital direct line segment in contours of the binary images and the software implementation of the algorithm is considered. Utilization of this algorithm to process the images will result to more natural and economical description in comparison with known ways of the description of the images. The considered algorithm and the software implementation can be used also for the description of contours when processing the half-tone and colour images

Multidimensional cell lists for investigating 3-manifolds

AbstractThe paper presents a new method of investigating topological properties of three-dimensional manifolds by means of computers. Manifolds are represented as block complexes. The paper contains definitions and a theorem necessary to transfer some basic knowledge of the classical topology to finite topological spaces. The method is based on subdividing the given set into blocks of cells in such a way that a k-dimensional block be homeomorphic to a k-dimensional ball. The block structure is described by the data structure known as “cell list” which is generalized here for the multidimensional case. Results of computer experiments are presented

The complexity of tangent words

In a previous paper, we described the set of words that appear in the coding of smooth (resp. analytic) curves at arbitrary small scale. The aim of this paper is to compute the complexity of those languages.Comment: In Proceedings WORDS 2011, arXiv:1108.341

On Christoffel and standard words and their derivatives

We introduce and study natural derivatives for Christoffel and finite standard words, as well as for characteristic Sturmian words. These derivatives, which are realized as inverse images under suitable morphisms, preserve the aforementioned classes of words. In the case of Christoffel words, the morphisms involved map $a$ to $a^{k+1}b$ (resp.,~$ab^{k}$) and $b$ to $a^{k}b$ (resp.,~$ab^{k+1}$) for a suitable $k>0$. As long as derivatives are longer than one letter, higher-order derivatives are naturally obtained. We define the depth of a Christoffel or standard word as the smallest order for which the derivative is a single letter. We give several combinatorial and arithmetic descriptions of the depth, and (tight) lower and upper bounds for it.Comment: 28 pages. Final version, to appear in TC