Discrete spherical means of directional derivatives and Veronese maps

We describe and study geometric properties of discrete circular and spherical means of directional derivatives of functions, as well as discrete approximations of higher order differential operators. For an arbitrary dimension we present a general construction for obtaining discrete spherical means of directional derivatives. The construction is based on using the Minkowski's existence theorem and Veronese maps. Approximating the directional derivatives by appropriate finite differences allows one to obtain finite difference operators with good rotation invariance properties. In particular, we use discrete circular and spherical means to derive discrete approximations of various linear and nonlinear first- and second-order differential operators, including discrete Laplacians. A practical potential of our approach is demonstrated by considering applications to nonlinear filtering of digital images and surface curvature estimation

Surface patch reconstruction by touching

This thesis studies the reconstruction of unknown curved surfaces in 3D through contour tracking. The implementation involves a 2-axis joystick sensor and a 4-DOF Adept robot. The joystick\u27s force sensing is combined with the Adept\u27s high positional accuracy to yield precise contact measurements.;A surface patch in 3D can be rebuilt by tracking along three concurrent curves on the surface. These data curves lie in different planes and are acquired via planar contour tracking. The Darboux frame at the curve intersection is first estimated to reflect the local geometry. Then polynomial fitting is carried out in this frame. Minimization of the total (absolute) Gaussian curvature of the surface fit effectively prevents unnecessary folding otherwise expected to result from the use of touching data. Experiments have demonstrated high accuracy of reconstruction

Image segmentation with adaptive region growing based on a polynomial surface model

A new method for segmenting intensity images into smooth surface segments is presented. The main idea is to divide the image into flat, planar, convex, concave, and saddle patches that coincide as well as possible with meaningful object features in the image. Therefore, we propose an adaptive region growing algorithm based on low-degree polynomial fitting. The algorithm uses a new adaptive thresholding technique with the L∞ fitting cost as a segmentation criterion. The polynomial degree and the fitting error are automatically adapted during the region growing process. The main contribution is that the algorithm detects outliers and edges, distinguishes between strong and smooth intensity transitions and finds surface segments that are bent in a certain way. As a result, the surface segments corresponding to meaningful object features and the contours separating the surface segments coincide with real-image object edges. Moreover, the curvature-based surface shape information facilitates many tasks in image analysis, such as object recognition performed on the polynomial representation. The polynomial representation provides good image approximation while preserving all the necessary details of the objects in the reconstructed images. The method outperforms existing techniques when segmenting images of objects with diffuse reflecting surfaces

Generalized ellipsoids and anisotropic filtering for segmentation improvement in 3D medical imaging

Deformable models have demonstrated to be very useful techniques for image segmentation. However, they present several weak points. Two of the main problems with deformable models are the following: (1) results are often dependent on the initial model location, and (2) the generation of image potentials is very sensitive to noise. Modeling and preprocessing methods presented in this paper contribute to solve these problems. We propose an initialization tool to obtain a good approximation to global shape and location of a given object into a 3D image. We also introduce a novel technique for corner preserving anisotropic diffusion filtering to improve contrast and corner measures. This is useful for both guiding initialization (global shape) and subsequent deformation for fine tuning (local shape).This work was supported by the Spanish Government and the Xunta de Galicia by projects TIC2000-0399-C02-02 and PGIDT99PXI20606B, respectively.2005-04-01S

3D Reconstruction Using High Resolution Implicit Surface Representations and Memory Management Strategies

La disponibilité de capteurs de numérisation 3D rapides et précis a permis de capturer de très grands ensembles de points à la surface de différents objets qui véhiculent la géométrie des objets. La métrologie appliquée consiste en l'application de mesures dans différents domaines tels que le contrôle qualité, l'inspection, la conception de produits et la rétroingénierie. Une fois que le nuage de points 3D non organisés couvrant toute la surface de l'objet a été capturé, un modèle de la surface doit être construit si des mesures métrologiques doivent être effectuées sur l'objet. Dans la reconstruction 3D en temps réel, à l'aide de scanners 3D portables, une représentation de surface implicite très efficace est le cadre de champ vectoriel, qui suppose que la surface est approchée par un plan dans chaque voxel. Le champ vectoriel contient la normale à la surface et la matrice de covariance des points tombant à l'intérieur d'un voxel. L'approche globale proposée dans ce projet est basée sur le cadre Vector Field. Le principal problème abordé dans ce projet est la résolution de l'incrément de consommation de mémoire et la précision du modèle reconstruit dans le champ vectoriel. Ce tte approche effectue une sélection objective de la taille optimale des voxels dans le cadre de champ vectoriel pour maintenir la consommation de mémoire aussi faible que possible et toujours obtenir un modèle précis de la surface. De plus, un ajustement d e surface d'ordre élevé est utilisé pour augmenter la précision du modèle. Étant donné que notre approche ne nécessite aucune paramétrisation ni calcul complexe, et qu'au lieu de travailler avec chaque point, nous travaillons avec des voxels dans le champ vectoriel, cela réduit la complexité du calcul.The availability of fast and accurate 3D scanning sensors has made it possible to capture very large sets of points at the surface of different objects that convey the geometry of the objects. A pplied metrology consists in the application of measurements in different fields such as quality control, inspection, product design and reverse engineering. Once the cloud of unorganized 3D points covering the entire surface of the object has been capture d, a model of the surface must be built if metrologic measurements are to be performed on the object. In realtime 3D reconstruction, using handheld 3D scanners a very efficient implicit surface representation is the Vector Field framework, which assumes that the surface is approximated by a plane in each voxel. The vector field contains the normal to the surface and the covariance matrix of the points falling inside a voxel. The proposed global approach in this project is based on the Vector Field framew ork. The main problem addressed in this project is solving the memory consumption increment and the accuracy of the reconstructed model in the vector field. This approach performs an objective selection of the optimal voxels size in the vector field frame work to keep the memory consumption as low as possible and still achieve an accurate model of the surface. Moreover, a highorder surface fitting is used to increase the accuracy of the model. Since our approach do not require any parametrization and compl ex calculation, and instead of working with each point we are working with voxels in the vector field, then it reduces the computational complexity