Section: New Results

Discrete geometry

Noisy curves

The recognition of digital objects, such as discrete lines and arcs, is an important topic in discrete geometry that has been subject of numerous works [27] , [45] . We got interested in the notion of ``noisy'' digital objects and in their detection. This problem has a direct application in image processing, in particular when existing geometrical shapes have to be interpreted in digital images.

Blurred segments

We introduced a new concept – fuzzy or blurred segments – that enables a flexible segmentation of discrete curves by taking into account a noise present in them.

A blurred segment is an 8-connected sequence of points that belong to an arithmetical discrete line of a given thickness. A parameter – the order of a blurred segment – controls the level of the allowed noise via the thickness of the discrete line bounding the blurred segment. Adding a point to a blurred segment amonts to compute the slope and the thickness of the new bounding discrete line. We showed that this computation can be done with a simple method. This led to an incremental and very efficient algorithm for splitting a discrete curve into blurred segments of fixed order. A paper on this subject was published in a special issue of Discrete Applied Mathematics [4] .

In collaboration with Fabien Feschet from LLAIC (Clermont-Ferrand), we also proposed new results on a restriction of the class of blurred segments in order to guarantee the optimality in the recognition process. This work was presented at the International Conference Discrete Geometry for Computer Imagery (DGCI) [13] .

Multi-order analysis

A possible application of the above approach occurs in the area of document analysis. In collaboration with Antoine Tabbone and Laurent Wendling of the QGAR team, we designed an algorithm for the polygonal approximation of noisy curves from a multi-order analysis algorithm. Due to the notion of blurred segment, this algorithm does not use fixed parameters and automatically provides a partitioning of a discrete curve into its meaningful parts.

This work was published this year [5] in the Electronic Letter on Computer Vision and Image Analysis (ELCVIA) .

Discrete curvature

We proposed a new notion of discrete tangent, relying on the definition of blurred segments and adapted to noisy curves. An algorithm permits to calculate the parameters of the tangent at each point of a discrete curve. From this algorithm, we can calculate several parameters as the normal vector or the curvature at all points of the considered curve [31] . These results have been improved in dimension 2 and extended to dimension 3 with the notion of 3D blurred segment in the framework of a DEA work [18] .

In bioinformatics, we are interested in the computation of the DNA curvature. Discrete models of the representation of the 3D structure of the DNA require the development of specific discrete geometry algorithms. We adapted to this problem the discrete curvature algorithms. The development of a software program is in progress.

Discrete convexity and concavity, polygonal decomposition of discrete sets

The study of convexity of a discrete region of the plane can be reduced to particular figures called hv-convex polyominoes. Previously, we developed a linear-time incremental algorithm to detect the convexity of such polyominoes [32] .

Several years ago, we established contacts with the University of Hamburg, namely with Professor Ulrich Eckart and his student Helene Reiter. In collaboration with Helene Reiter, we developed a linear-time algorithm for decomposition of the boundary of a plane digital object into convex and concave parts. Such a decomposition is very useful for describing the form of an object. The obtained algorithm uses properties of discrete straight lines for the convex case [32] , and extends them to the concave case. A paper describing this work was published by Kluwer in a book [6] .

Digital plane recognition

A naive digital plane with integer coefficients is defined as a subset of points verifying a double inequality hax+ by+ cz< h+ m a x {| a |, | b |, | c |} , where . Given a finite subset of , the problem is to determine whether or not there exists a naive digital plane containing it. This question is rather classical in the field of discrete geometry.

With Yan Gerard (LLAIC, Clermont-Ferrand) and Paul Zimmermann ( Spaces team), we proposed a new algorithm that solves this problem. The algorithm uses a strategy of optimization in a set of triangular facets (called triangles). A short program code (less than 300 lines) solving the problem is available on the Web (http://www.loria.fr/~debled/plane/ )and a paper describing this work is published to Discrete Applied Mathematics [7] .

We are now interested in the recognition of noisy discrete planes. This year, this topic was a subject of a DEA work done in our group [19] . In this work, techniques of discrete geometry were combined with those of computational geometry. A new definition was introduced: blurred discrete pieces of a plane. We have shown that the problem of recognition of these objects is equivalent to the one of computation of the thickness of a set of points in dimension 3. A corresponding algorithm was proposed. This study was presented at the Journées Informatique et Géométrie in October and is continued in the PhD work of L. Provot. The general goal of this work is to study the algorithms of recognition of pieces of naive discrete planes and their adaptation to the pieces of blurred planes in order to be used in the polyhedrisation of noisy discrete objects.

Discrete surface reconstruction from shading images

We introduced in [3] a discrete approach to the reconstruction of discrete surfaces from shading images. The main idea of this approach is to combine geometric information of the discrete surface to be reconstructed with photometric information. When only one light source oriented in viewer direction is used for the reconstruction, the method allows to add explicit constraints in order to reduce the concave/convex ambiguity.

Since this discrete approach does not use an analytical expression of the reflectance map (usually Lambertian), the reconstruction was applied with other reflectance map such as the specular Nayar's model. Furthermore, this method presents the advantage not to be limited to the uniform light source model. It can be easily extended with a non-distant light source.

Discrete surface smoothing

In collaboration with archeologists, we have been working on the problem of geometric smoothing and parameters extraction of discrete surfaces using the approach described in the previous section.

A statistical and geometrical method to smooth discrete surfaces was introduced in [14] . This method consists in smoothing the object surface by moving the center of each voxel to the unit cube according to the projection to the tangent plane. The tangent plane was estimated by a statistical estimation and by considering geometric constraints. The resulting surface representation allows us to get both smooth normal vectors of the surface and a smooth mesh while preserving the geometrical properties of the surface.

This work was recently accepted in an extended version in the special issue of DGCI in Computer & Graphics [8] .

Shape Modeling from Shading Design

Shading has a large impact to the human perception of 3D objects. Thus, in order to create or to deform a 3D object, it seems natural to manipulate its perceived shading. We proposed a new solution to implement this idea. Our approach is based on the ability of the user to coarsely draw a shading, under different lighting directions. With this intuitive process, the user can create or edit a height field (locally or globally), that will correspond to the drawn shading values. This approach is described in [15] and we are going to work on the extension of this approach in order to edit and create full 3D objects.