The MLS surface [Levin 2003], used for modeling and rendering with point clouds, was originally defined algorithmically as the output of a particular meshless construction. We give a new explicit definition in terms of the critical points of an energy function on lines determined by a vector field. This definition reveals connections to research in computer vision and computational topology. Variants of the MLS surface can be created by varying the vector field and the energy function. As an example, we define a similar surface determined by a cloud of surfels (points equipped with normals), rather than points. We also observe that some procedures described in the literature to take points in space onto the MLS surface fail to do so, and we describe a simple iterative procedure which does. the relationship of extremal surfaces and implicit surfaces. As we discuss in Section 5, there is an implicit surface containing every extremal surface, including the MLS surface. This can be quite useful, particularly for defining normals precisely. 1

This paper describes a method for building interpolating or approximating implicit surfaces from polygonal data. The user can choose to generate a surface that exactly interpolates the polygons, or a surface that approximates the input by smoothing away features smaller than some user-specified size. The implicit functions are represented using a moving least-squares formulation with constraints integrated over the polygons. The paper also presents an improved method for enforcing normal constraints and an iterative procedure for ensuring that the implicit surface tightly encloses the input vertices.

We present a method for modeling and animating a wide spectrum of volumetric objects, with material properties anywhere in the range from stiff elastic to highly plastic. Both the volume and the surface representation are point based, which allows arbitrarily large deviations form the original shape. In contrast to previous point based elasticity in computer graphics, our physical model is derived from continuum mechanics, which allows the specification of common material properties such as Young's Modulus and Poisson's Ratio. In each step

Point sets become an increasingly popular shape representation. Most shape processing and rendering tasks require the approximation of a continuous surface from the point data. We present a surface approximation that is motivated by an efficient iterative ray intersection computation. On each point on a ray, a local normal direction is estimated as the direction of smallest weighted co-variances of the points. The normal direction is used to build a local polynomial approximation to the surface, which is then intersected with the ray. The distance to the polynomials essentially defines a distance field, whose zero-set is computed by repeated ray intersection. Requiring the distance field to be smooth leads to an intuitive and natural sampling criterion, namely, that normals derived from the weighted co-variances are well defined in a tubular neighborhood of the surface. For certain, well-chosen weight functions we can show that well-sampled surfaces lead to smooth distance fields with non-zero gradients and, thus, the surface is a continuously differentiable manifold. We detail spatial data structures and efficient algorithms to compute ray-surface intersections for fast ray casting and ray tracing of the surface.

In recent years point-based geometry has gained increasing attention as an alternative surface representation, both for efficient rendering and for flexible geometry processing of highly complex 3D-models. Point sampled objects do neither have to store nor to maintain globally consistent topological information. Therefore they are more flexible compared to triangle meshes when it comes to handling highly complex or dynamically changing shapes. In this paper, we make an attempt to give an overview of the various point-based methods that have been proposed over the last years. In particular we review and evaluate different shape representations, geometric algorithms, and rendering methods which use points as a universal graphics primitive.

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Point set surfaces are a smooth manifold surface approximation from a set of sample points. The surface definition is based on a projection operation that constructs local polynomial approximations and respects a minimum feature size. We present techniques for ray tracing point set surfaces. For the computation of ray-surface intersection the properties of the projection operation are exploited: The surface is enclosed by a union of minimum feature size spheres. A ray is intersected with the spheres first and inside the spheres with local polynomial approximations. Our results show that 2--3 projections are sufficient to accurately intersect a ray with the surface.

Levin’s MLS projection operator allows defining a surface from a set of points and represents a versatile procedure to generate points on this surface. Practical problems of MLS surfaces are a complicated non-linear optimization to compute a tangent frame and the (commonly overlooked) fact that the normal to this tangent frame is not the surface normal. An alternative definition of Point Set Surfaces, inspired by the MLS projection, is the implicit surface version of Adamson & Alexa. We use this surface definition to show how to compute exact surface normals and present simple, efficient projection operators. The exact normal computation also allows computing orthogonal projections. Categories and Subject Descriptors (according to ACM CCS): G.1.2 [Numerical Analysis]: Approximation of surfaces and contours I.3.5 [Computer Graphics]: Curve, surface, solid, and object representations

Recent research on point-based surface representations suggests that point sets may be a viable alternative to parametric surface representations in applications where the topological constraints of a parameterization are unwieldy or inefficient. Particle systems offer a mechanism for controlling point samples and distributing them according to needs of the application.Furthermore, particle systems can serve as a surface representation in their own right, or to augment implicit functions, allowing for both efficient rendering and control of implicit function parameters. The state of the art in surface sampling particle systems, however, presents some shortcomings. First, most of these systems have many parameters that interact with some complexity, making it difficult for users to tune the system to meet specific requirements. Furthermore, these systems do not lend themselves to spatially adaptive sampling schemes, which are essential for efficient, accurate representations of complex surfaces. In this paper we present a new class of energy functions for distributing particles on implicit surfaces and a corresponding set of numerical techniques. These techniques provide stable, scalable, efficient, and controllable mechanisms for distributing particles that sample implicit surfaces within a locally adaptive framework. 1.

Figure 1: Illustration of the central features of our algebraic MLS framework. From left to right: efficient handling of very complex point sets, fast mean curvature evaluation and shading, significantly increased stability in regions of high curvature, sharp features with controlled sharpness. Sample positions are partly highlighted. In this paper we present a new Point Set Surface (PSS) definition based on moving least squares (MLS) fitting of algebraic spheres. Our surface representation can be expressed by either a projection procedure or in implicit form. The central advantages of our approach compared to existing planar MLS include significantly improved stability of the projection under low sampling rates and in the presence of high curvature. The method can approximate or interpolate the input point set and naturally handles planar point clouds. In addition, our approach provides a reliable estimate of the mean curvature of the surface at no additional cost and allows for the robust handling of sharp features and boundaries. It processes a simple point set as input, but can also take significant advantage of surface normals to improve robustness, quality and performance. We also present an novel normal estimation procedure which exploits the properties of the spherical fit for both direction estimation and orientation propagation. Very efficient computational procedures enable us to compute the algebraic sphere fitting with up to 40 million points per second on latest generation GPUs.