Abstract Are simple strokes unique to the artist or designer who renders them? If so, can this idea be used to identify authorship or to classify artistic drawings? Also, could training methods be devised to develop particular styles? To answer these questions, we propose the Stroke Authorship Recognition (SAR) approach, a novel method that distinguishes the authorship of 2D digitized drawings. SAR converts a drawing into a histogram of stroke attributes that is discriminative of authorship. We provide extensive classification experiments on a large variety of data sets, which validate SAR's ability to distinguish unique authorship of artists and designers. We also demonstrate the usefulness of SAR in several applications including the detection of fraudulent sketches, the training and monitoring of artists in learning a particular new style and the first quantitative way to measure the quality of automatic sketch synthesis tools.

Non-negative matrix factorization (NMF) provides the advantage of parts-based data representation through additive only combinations. It has been widely adopted in areas like item recommending, text mining, data clustering, speech denoising, etc. In this paper, we provide an algorithm that allows the factorization to have linear or approximately linear constraints with respect to each factor. We prove that if the constraint function is linear, algorithms within our multiplicative framework will converge. This theory supports a large variety of equality and inequality constraints, and can facilitate application of NMF to a much larger domain. Taking the recommender system as an example, we demonstrate how a specialized weighted and constrained NMF algorithm can be developed to fit exactly for the problem, and the tests justify that our constraints improve the performance for both weighted and unweighted NMF algorithms under several different metrics. In particular, on the Movie lens data with 94% of items, the Constrained NMF improves recall rate 3% compared to SVD50 and 45% compared to SVD150, which were reported as the best two in the top-N metric.

Critical points of a vector field are key to their characterization. Their positions as well as their indexes are crucial for understanding vector fields. Considerable work exists in 2D, but less is available for 3D or higher dimensions. Geometric algebra is a derivative of Clifford algebra that not only enables a succinct definition of the index of a critical point in higher dimension; it also provides insight and computational pathways for calculating the index. We describe the problems in terms of geometric algebra and present an octree based solution using the algebra for finding critical points and their index in a 3D vector field.

Adaptively Sampled Distance Fields (ADFs) are a unifying representation of shape that integrate numerous concepts in computer graphics including the representation of geometry and volume data and a broad range of processing operations such as rendering, sculpting, level-of-detail management, surface offsetting, collision detection, and color gamut correction. Its structure is uncomplicated and direct, but is especially effective for quality reconstruction of complex shapes, e.g., artistic and organic forms, precision parts, volumes, high order functions, and fractals. We characterize one implementation of ADFs, illustrating its utility on two diverse applications: 1) artistic carving of fine detail, and 2) representing and rendering volume data and volumetric effects. Other applications are briefly presented.

Recently, there have been some developments in constructing infinitely smooth genus-n objects over a single domain using the so-called topological design method. Compared with existing geometric design methods, it provides many advantages: a simpler data structure, a compact data set and infinite continuity. Improvement in the user interface, however, is still needed for designing free-form genus-n objects. This research improves the topological design interface by incorporating a radial-basis scattered data interpolating function for designing infinitely smooth genus-n objects over a single domain. As part of this research, a constrained Delaunay triangulation over the given unstructured data on the single domain is activated. This allows visualization of a polygonized genus-n object in the object space as well as providing interactive data control behavior in the object space. This method is characterized by the unconstrained control points inside the boundaries and by the small-size data set needed to design a free-form genus-n object.

We present our results on the visualization of nonlinear vector field topology. The underlying mathematics is done in Clifford algebra, a system describing geometry by extending the usual vector space by a multiplication of vectors. We started with the observation that all known algorithms for vector field topology are based on piecewise linear or bilinear approximation, and that these methods destroy the local topology if nonlinear behavior is present. Our algorithm looks for such situations, chooses an appropriate polynomial approximation in these areas, and, finally, visualizes the topology. This overcomes the problem, and the algorithm is still very fast because we are using linear approximation outside these small but important areas. The paper contains a detailed description of the algorithm and a basic introduction to Clifford algebra.

Presents an algorithm for the visualization of vector field topology based on Clifford algebra. It allows the detection of higher-order singularities. This is accomplished by first analysing the possible critical points and then choosing a suitable polynomial approximation, because conventional methods based on piecewise linear or bilinear approximation do not allow higher-order critical points and destroy the topology in such cases. The algorithm is still very fast, because of using linear approximation outside the areas with several critical points.

Abstract All but the simplest of dynamical systems contain nonlinearities that play an important role in modeling and simulating physical systems. They create unpredictable (chaotic) behavior that is often hidden or neglected in traditional solutions. A simple dynamical system, the spherical pendulum, is introduced to illustrate issues, principles, and effects of chaos in dynamics. The spherical pendulum is a two degrees of freedom nonlinear system with a pivot point in space. The equations of motion for the pendulum are derived, simulated, and animated. A periodical perturbation is applied to the pivot point producing radically different behavior.

This paper briefly presents an efficient and intuitive 3D free-form deformation approach based on iterative affine transformations, a generalized de Casteljau algorithm, whereby the object warps along a Be´zier curve as its skeleton.

Topology is primal geometry. Our design philosophy embodies this principle. We report on a new surface &sign perspective based on a “marked” polygon for each object. The marked polygon captures the topology of the object surface. We construct multiply periodic mappings from polygon to sculptured surface. The mappings arise naturally from the topology and other design considerations. Hence we give a single domain global parameteriration for surfaces with handles. Examples demonstrate the design of sculptured objects and their ntanufimture.

A general method for rendering isosurfaces of multivariate rational and polynomial tensor products is described. The method is robust up to degree 15, handling singularities without introducing spurious rendering artifacts. The approach does not solve the problem of singularities in general, but it removes the problem from the rendering domain to the interpolation/approximation domain. It is based on finding real roots of a polynomial in Bernstein form. This makes it particularly suitable for parallel and pipelined processing. It is envisioned that the tensor products will be used as approximants or interpolants for empirical data or scalar fields. An interpolation scheme is given as an example.>

Rational tensor product surfaces, (Bézier, NURBS, Hermite, polynomial, etc.) are rendered in real-time by uniform faceting. The described methods are modular and can be balanced for optimal implementation on different hardware platforms. Discretization anomalies such as angularities, Mach banding, cracking etc. are avoided by tessellating the surface patches and segmenting the trimming curves based on the view.

To date, methods that blend solids, that is, B-rep or CSG models, with implicit functions require successive composition of the blending functions to handle an arbitrary solid model. The shape of the resulting surfaces depends upon the algebraic distances defined by these functions. To achieve meaningful shapes, previous methods have relied on blending functions that have a pseudo-Euclidean distance measure. These methods are abstracted, resulting in some general observations. Unfortunately, the functions used can exhibit unwanted discontinuities. A new method, the displacement form of blending, embeds the zero surface of the blending functions in a form for which algebraic distance is C 1 continuous in the entire domain of definition. Characteristics of the displacement form are demonstrated using the superelliptic blending functions. Intuitive and mathematical underpinnings are provided.

An object space method is given for interpolating between sampled and locally averaged signals, resulting in an antialiasing filter which provides a continuous transition from a sampled signal to its selectively dampened local averages. This method is applied to the three standard Euclidean dimensions and time, resulting in spatial and frame to frame coherence. The theory allows filtering of a variety of functions, including continuous and discrete representations of planar texture.

In 1965, Shepard introduced a general interpolation formula for arbitrarily spaced data over any finite dimensional Euclidean space. The deficiencies in his initial results have been corrected and later derivations have been widely applied to surface fitting problems in aviation design and geology.