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

Average Curve of n Digital Curves

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DEM RECONSTRUCTION USING LIGHT FIELD AND BIDIRECTIONAL REFLECTANCE FUNCTION FROM MULTI-VIEW HIGH RESOLUTION SPATIAL IMAGES

This paper presents a method for dense DSM reconstruction from high resolution, mono sensor, passive imagery, spatial panchromatic image sequence. The interest of our approach is four-fold. Firstly, we extend the core of light field approaches using an explicit BRDF model from the Image Synthesis community which is more realistic than the Lambertian model. The chosen model is the Cook-Torrance BRDF which enables us to model rough surfaces with specular effects using specific material parameters. Secondly, we extend light field approaches for non-pinhole sensors and non-rectilinear motion by using a proper geometric transformation on the image sequence. Thirdly, we produce a 3D volume cost embodying all the tested possible heights and filter it using simple methods such as Volume Cost Filtering or variational optimal methods. We have tested our method on a Pleiades image sequence on various locations with dense urban buildings and report encouraging results with respect to classic multi-label methods such as MIC-MAC, or more recent pipelines such as S2P. Last but not least, our method also produces maps of material parameters on the estimated points, allowing us to simplify building classification or road extraction

Multi-scale Analysis of Discrete Contours for Unsupervised Noise Detection

International audienceBlurred segments [Debled06] were introduced in discrete geometry to address possible noise along discrete contours. The noise is not really detected but is rather canceled out by thickening digital straight segments. The thickness is tuned by a user and set globally for the contour, which requires both supervision and non-adaptive contour processing. To overcome this issue, we propose an original strategy to detect locally both the amount of noise and the meaningful scales of each point of a digital contour. Based on the asymptotic properties of maximal segments, it also detects curved and flat parts of the contour. From a given maximal observation scale, the proposed approach does not require any parameter tuning and is easy to implement. We demonstrate its effectiveness on several datasets. Its potential applications are numerous, ranging from geometric estimators to contour reconstruction

DOI: 10.1007/11919629 Convex shapes and convergence speed of discrete tangent estimators

Abstract. Discrete geometric estimators aim at estimating geometric characteristics of a shape with only its digitization as input data. Such an estimator is multigrid convergent when its estimates tend toward the geometric characteristics of the shape as the digitization step h tends toward 0. This paper studies the multigrid convergence of tangent estimators based on maximal digital straight segment recognition. We show that such estimators are multigrid convergent for some family of convex shapes and that their speed of convergence is on average O(h 2 3). Experiments confirm this result and suggest that the bound is tight.

What Does Digital Straightness Tell about Digital Convexity?

International audienceThe paper studies local convexity properties of parts of dig- ital boundaries. An online and linear-time algorithm is introduced for the decomposition of a digital boundary into convex and concave parts. In addition, other data are computed in the same time without any extra cost: the hull of each convex or concave part as well as the Bezout points of each edge of those hulls. The proposed algorithm involves well- understood algorithms: adding a point to the front or removing a point from the back of a digital straight segment and computing the set of maximal segments. The output of the algorithm is useful either for a polygonal representation of digital boundaries or for a segmentation into circular arcs