Lane-Riesenfeld's subdivision schemes provide a versatile tool for generating smooth and detailed curves and surfaces, making them valuable assets in computer graphics and geometric modeling. These schemes offer flexibility in controlling the regularity of the limit function by adjusting some parameters. In this paper, we propose a nonlinear modification of these algorithms for piecewise smooth functions. The new algorithm converge to piecewise smooth limit functions maintaining the high regularity of Lane-Riesenfeld's subdivision schemes in smooth zones. The practical results presented show a very good numerical behavior of the proposed algorithms.

Twitter has become an important tool for communication and marketing. Topic model algorithms meant to characterize the discourse of online conversations and identify relevant audiences do not perform well for this task, despite their widespread usage. This paper proposes an iterative topic model, Gamma Filtration, and a social network-based method, Simmelian Filtration, to amplify tweet-topic probability signal and reduce noise. We demonstrate the method on a novel data set collected of European Racially and Ethnically Motivated Violent Extremist (REMVE) networks on Twitter. We find that Simmelian Filtering is most successful at reducing noise as measured by perplexity. This improves our ability to detect and monitor core conversations of a community that is disseminating propaganda to increase online extremism.

Handwriting is considered a unique "fingerprint" that characterizes a scribe (it is even used as evidence in modern forensics). In paleography (the study of ancient writing), it is presumed that each writer has a one prototype for each letter in the alphabet. Commonly, for ancient inscriptions, letters are organized into paleographic tables (where the rows are the alphabet letters, and the columns represent the examined inscriptions). These tables play a significant role in dating inscriptions based on their resemblance to columns in the table. In this paper, we argue that each scribe "fingerprint" is not represented by a single character prototype, but in fact by a distribution of characters. We introduce a framework for automatically identifying the writer style and constructing paleographic tables based on character histograms. Subsequently, we propose a method for comparing short documents utilizing letter distribution. We demonstrate the validity of the methods on two handwritten datasets: Modern and Ancient Hebrew pertaining to the First Temple period. Our methodology on the ancient dataset enables us to provide additional evidence concerning the level of literacy in the kingdom of Judah ca. 600 BCE.

We present a new framework for real-time shape deformation with local shape preservation and volume control. Given a 3D object, in any form, one would like to manipulate the object using convenient handles, so that the resulting shape is a natural variation of the given object. It is also important that the deformation is controlled, thereby enabling localized changes that do not influence nearby branches. For example, given a horse model, a movement of one of its hooves should not affect the other hooves. Another goal is the minimization of local shape distortion throughout the object. The first ingredient of our method is the use of interior radial basis functions (IRBF), where the functions are radial with respect to interior distances within the object. The second important ingredient is the reduction of local distortions by minimizing the distortion of a set of spheres placed within the object. Our method achieves the goals of convenient shape manipulation and local influence property, and improves the latest state-of-the-art cage-based methods by replacing the cage with the more flexible IRBF centers. The latter enables extra flexibility and fully automated construction, as well as simpler formulation. We also suggest the IRBF interpolation method that can extend any surface mapping to the whole subspace in a shape-aware manner.

We introduce Green Coordinates for closed polyhedral cages. The coordinates are motivated by Green's third integral identity and respect both the vertices position and faces orientation of the cage. We show that Green Coordinates lead to space deformations with a shape-preserving property. In particular, in 2D they induce conformal mappings, and extend naturally to quasi-conformal mappings in 3D. In both cases we derive closed-form expressions for the coordinates, yielding a simple and fast algorithm for cage-based space deformation. We compare the performance of Green Coordinates with those of Mean Value Coordinates and Harmonic Coordinates and show that the advantage of the shape-preserving property is not achieved at the expense of speed or simplicity. We also show that the new coordinates extend the mapping in a natural analytic manner to the exterior of the cage, allowing the employment of partial cages.

We introduce a Locally Optimal Projection operator (LOP) for surface approximation from point-set data. The operator is parameterization free, in the sense that it does not rely on estimating a local normal, fitting a local plane, or using any other local parametric representation. Therefore, it can deal with noisy data which clutters the orientation of the points. The method performs well in cases of ambiguous orientation, e.g., if two folds of a surface lie near each other, and other cases of complex geometry in which methods based upon local plane fitting may fail. Although defined by a global minimization problem, the method is effectively local, and it provides a second order approximation to smooth surfaces. Hence allowing good surface approximation without using any explicit or implicit approximation space. Furthermore, we show that LOP is highly robust to noise and outliers and demonstrate its effectiveness by applying it to raw scanned data of complex shapes.

This article introduces a method for mesh editing that is aimed at preserving shape and volume. We present two new developments: The first is a minimization of a functional expressing a geometric distance measure between two isometric surfaces. The second is a local volume analysis linking the volume of an object to its surface curvature. Our method is based upon the moving frames representation of meshes. Applying a rotation field to the moving frames defines an isometry. Given rotational constraints, the mesh is deformed by an optimal isometry defined by minimizing the distance measure between original and deformed meshes. The resulting isometry nicely preserves the surface details, but when large rotations are applied, the volumetric behavior of the model may be unsatisfactory. Using the local volume analysis, we define a scalar field by which we scale the moving frames. Scaled and rotated moving frames restore volumetric properties of the original mesh, while properly maintaining the surface details. Our results show that even extreme deformations can be applied to meshes, with only minimal distortion of surface details and object volume.

We introduce a rigid motion invariant mesh representation based on discrete forms defined on the mesh. The reconstruction of mesh geometry from this representation requires solving two sparse linear systems that arise from the discrete forms: the first system defines the relationship between local frames on the mesh, and the second encodes the position of the vertices via the local frames. The reconstructed geometry is unique up to a rigid transformation of the mesh. We define surface editing operations by placing user-defined constraints on the local frames and the vertex positions. These constraints are incorporated in the two linear reconstruction systems, and their solution produces a deformed surface geometry that preserves the local differential properties in the least-squares sense. Linear combination of shapes expressed with our representation enables linear shape interpolation that correctly handles rotations. We demonstrate the effectiveness of the new representation with various detail-preserving editing operators and shape morphing.

One of the main challenges in editing a mesh is to retain the visual appearance of the surface after applying various modifications. In this paper we advocate the use of linear differential coordinates as means to preserve the high-frequency detail of the surface. The differential coordinates represent the details and are defined by a linear transformation of the mesh vertices. This allows the reconstruction of the edited surface by solving a linear system that satisfies the reconstruction of the local details in least squares sense. Since the differential coordinates are defined in a global coordinate system they are not rotation-invariant. To compensate for that, we rotate them to agree with the rotation of an approximated local frame. We show that the linear least squares system can be solved fast enough to guarantee interactive response time thanks to a precomputed factorization of the coefficient matrix. We demonstrate that our approach enables to edit complex detailed meshes while keeping the shape of the details in their natural orientation.

We advocate the use of point sets to represent shapes. We provide a definition of a smooth manifold surface from a set of points close to the original surface. The definition is based on local maps from differential geometry, which are approximated by the method of moving least squares (MLS). The computation of points on the surface is local, which results in an out-of-core technique that can handle any point set. We show that the approximation error is bounded and present tools to increase or decrease the density of the points, thus allowing an adjustment of the spacing among the points to control the error. To display the point set surface, we introduce a novel point rendering technique. The idea is to evaluate the local maps according to the image resolution. This results in high quality shading effects and smooth silhouettes at interactive frame rates.

We advocate the use of point sets to represent shapes. We provide a definition of a smooth manifold surface from a set of points close to the original surface. The definition is based on local maps from differential geometry, which are approximated by the method of moving least squares (MLS). We present tools to increase or decrease the density of the points, thus, allowing an adjustment of the spacing among the points to control the fidelity of the representation. To display the point set surface, we introduce a novel point rendering technique. The idea is to evaluate the local maps according to the image resolution. This results in high quality shading effects and smooth silhouettes at interactive frame rates.

We present an object-space morphing technique that blends the interiors of given two- or three-dimensional shapes rather than their boundaries. The morph is rigid in the sense that local volumes are least-distorting as they vary from their source to target configurations. Given a boundary vertex correspondence, the source and target shapes are decomposed into isomorphic simplicial complexes. For the simplicial complexes, we find a closed-form expression allocating the paths of both boundary and interior vertices from source to target locations as a function of time. Key points are the identification of the optimal simplex morphing and the appropriate definition of an error functional whose minimization defines the paths of the vertices. Each pair of corresponding simplices defines an affine transformation, which is factored into a rotation and a stretching transformation. These local transformations are naturally interpolated over time and serve as the basis for composing a global coherent least-distorting transformation.

Given two or more objects of general topology, intermediate objects are constructed by a distance field metamorphosis. In the presented method the interpolation of the distance field is guided by a warp function controlled by a set of corresponding anchor points. Some rules for defining a smooth least-distorting warp function are given. To reduce the distortion of the intermediate shapes, the warp function is decomposed into a rigid rotational part and an elastic part. The distance field interpolation method is modified so that the interpolation is done in correlation with the warp function. The method provides the animator with a technique that can be used to create a set of models forming a smooth transition between pairs of a given sequence of keyframe models. The advantage of the new approach is that it is capable of morphing between objects having a different topological genus where no correspondence between the geometric primitives of the models needs to be established. The desired correspondence is defined by an animator in terms of a relatively small number of anchor points

Given (n-1)-dimensional parallel cross-sections of an n-dimensional body, one would like to reconstruct the n-dimensional body. The method based on Distance Field Interpolation (DFI) gives a robust solution to this problem in its ability to deal with any topology in any dimension. Still this method may give undesired solutions to the problem if the changes from one cross-section to the next are significant relative to the size of the details in the cross-sections. We consider the problem of solid reconstruction from contours, which can also be considered as a contour blending or contour morphing problem, where the third dimension is time. The method presented is based on interpolation of the distance field, guided by a warp function which is controlled by a set of corresponding anchor points. Some rules for defining a smooth least-distorting warp function are given. To reduce the distortion of the intermediate shapes, the warp function is decomposed into a rigid rotational part and an elastic part. The distance field interpolation method is modified so that the interpolation is guided by the warp function. The advantage of the new approach is that it is capable of blending between contours having different topological genus, and no correspondence between the geometric primitives should be established. The desired general correspondence is defined by the user in terms of a relatively small number of anchor points.