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Surface editing operations commonly require geometric details of the surface to be preserved as much as possible. We argue that geometric detail is an intrinsic property of a surface and that, consequently, surface editing is best performed by operating over an intrinsic surface representation. We provide such a representation of a surface, based on the Laplacian of the mesh, by encoding each vertex relative to its neighborhood. The Laplacian of the mesh is enhanced to be invariant to locally linearized rigid transformations and scaling. Based on this Laplacian representation, we develop useful editing operations: interactive free-form deformation in a region of interest based on the transformation of a handle, transfer and mixing of geometric details between two surfaces, and transplanting of a partial surface mesh onto another surface. The main computation involved in all operations is the solution of a sparse linear system, which can be done at interactive rates. We demonstrate the effectiveness of our approach in several examples, showing that the editing operations change the shape while respecting the structural geometric detail.

Free-form deformation (FFD) is a powerful modeling tool, but controlling the shape of an object under complex deformations is often difficult. The interface to FFD in most conventional systems simply represents the underlying mathematics directly; users describe deformations by manipulating control points. The difficulty in controlling shape precisely is largely due to the control points being extraneous to the object; the deformed object does not follow the control points exactly. In addition, the number of degrees of freedom presented to the user can be overwhelming. We present a method that allows a user to control a free-form deformation of an object by manipulating the object directly, leading to better control of the deformation and a more intuitive interface. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling - Curve, Surface, Solid, and Object Representations; I.3.6 [Computer Graphics ]: Methodology and Techniques - Interaction Techniques. Ad...

Traditionally, shape transformation using implicit functions is performed in two distinct steps: 1) creating two implicit functions, and 2) interpolating between these two functions. We present a new shape transformation method that combines these two tasks into a single step. We create a transformation between two N- dimensional objects by casting this as a scattered data interpolation problem in N + 1 dimensions. For the case of 2D shapes, we place all of our data constraints within two planes, one for each shape. These planes are placed parallel to one another in 3D. Zero-valued constraints specify the locations of shape boundaries and positivevalued constraints are placed along the normal direction in towards the center of the shape. We then invoke a variational interpolation technique (the 3D generalization of thin-plate interpolation), and this yields a single implicit function in 3D. Intermediate shapes are simply the zero-valued contours of 2D slices through this 3D function. Shape transformation between 3D shapes can be performed similarly by solving a 4D interpolation problem. To our knowledge, ours is the first shape transformation method to unify the tasks of implicit function creation and interpolation. The transformations produced by this method appear smooth and natural, even between objects of differing topologies. If desired, one or more additional shapes may be introduced that influence the intermediate shapes in a sequence. Our method can also reconstruct surfaces from multiple slices that are not restricted to being parallel to one another.

Figure 1: Objects created with our system. (a) boolean operations with scanned geometry, (b) an Octopus modeled by deforming and extruding a sphere, (c) a design study for a Siggraph coffee mug created by boolean operations, free-form deformation and displacement mapping. We present a versatile and complete free-form shape modeling framework for point-sampled geometry. By combining unstructured point clouds with the implicit surface definition of the moving least squares approximation, we obtain a hybrid geometry representation that allows us to exploit the advantages of implicit and parametric surface models. Based on this representation we introduce a shape modeling system that enables the designer to perform large constrained deformations as well as boolean operations on arbitrarily shaped objects. Due to minimum consistency requirements, point-sampled surfaces can easily be re-structured on the fly to support extreme geometric deformations during interactive editing. In addition, we show that strict topology control is possible and sharp features can be generated and preserved on point-sampled objects. We demonstrate the effectiveness of our system on a large set of input models, including noisy range scans, irregular point clouds, and sparsely as well as densely sampled models.

This paper presents a survey of the work done in modeling deformable objects within the computer graphics research community. The research has a long history and a wide variety of approaches have been used. This paper organizes the diversity of research by the technique used rather than by the application, although applications are discussed throughout. This paper presents some purely geometric approaches for modeling deformable objects, but focuses on physically based approaches. In the latter category are mass-spring models, nite element models, approximate continuum models, and low degree of freedom models. Special emphasis is placed on nite element models, which o er the greatest accuracy, but have seen limited use in computer graphics. The paper also suggests important areas for future research. 1

In this paper, we introduce a novel approach to mesh editing with the Poisson equation as the theoretical foundation. The most distinctive feature of this approach is that it modifies the original mesh geometry implicitly through gradient field manipulation. Our approach can produce desirable and pleasing results for both global and local editing operations, such as deformation, object merging, and smoothing. With the help from a few novel interactive tools, these operations can be performed conveniently with a small amount of user interaction. Our technique has three key components, a basic mesh solver based on the Poisson equation, a gradient field manipulation scheme using local transforms, and a generalized boundary condition representation based on local frames. Experimental results indicate that our framework can outperform previous related mesh editing techniques.

Finding effective interactive deformation techniques for complex geometric objects continues to be a challenging problem in modeling and animation. We present an approach that is inspired by armatures used by sculptors, in which wire curves give definition to an object and shape its deformable features. We also introduce domain curves that define the domain of deformation about an object. A wire together with a collection of domain curves provide a new basis for an implicit modeling primitive. Wires directly reflect object geometry, and as such they provide a coarse geometric representation of an object that can be created through sketching. Furthermore, the aggregate deformation from several wires is easy to define. We show that a single wire is an appealing direct manipulation deformation technique; we demonstrate that the combination of wires and domain curves provide a new way to outline the shape of an implicit volume in space; and we describe techniques for the aggregation of deformations resulting from multiple wires, domain curves and their interaction with each other and other deformation techniques. The power of our approach is illustrated using applications of animating figures with flexible articulations, modeling wrinkled surfaces and stitching geometry together.

Abstract This paper presents a new method for determining the minimal non-rigid deformation between two 3-D surfaces, such as those which describe anatomical structures in 3-D medical images. Although we match surfaces, we represent the deformation as a volumetric transformation. Our method performs a least squares minimization of the distance between the two surfaces of interest. To quickly and accurately compute dislances between points on the two surfaces, we use a precomputed distance map represented using an octree spline whose resolution increases near the surface. To quickly and robustly compute the deformation, we use a volumetric spline to model the deformation function. We present experimental results on both synthetic and real 3-D surfaces. Introduction The matching of 3-D anatomical surfaces between anatomical atlases and patient data, or between different patient data sets, is an important element of 3-D medical image analysis and quantification. Matching between atlases and patient data enables more accurate and reliable segmentation and the functional labeling of medical images, as well as multimodality data registration and integration. In computer vision, this problem corresponds to finding the non-rigid deformation between two surfaces, with applications to model-based object recognition and deformable object tracking. In previous work [1 ], we developed a fast and accurate technique for determining the rigid transformation between two surfaces, and also between a 3-D surface and its 2-D projections. In this paper, we extend our technique to recover smooth non-rigid deformations between 3-D surfaces. Our approach is based on describing the deformation as a warping of the space containing one of the surfaces. In particular, we use a multiresolution warp or displacement field based on concepts from free-form deformations [2]. Our approach enables us to locally adapt the resolution of the deformation field to bring the two surfaces into registration, while maintaining smoothness and avoiding unnecessary computation. The result is a rapid and efficient registration algorithm which does not require the extraction of features from the

Figure 1: Original horse model with enclosing triangle control mesh shown in black (a). Several deformations generated using our 3D mean value coordinates applied to a modified control mesh (b,c,d). Constructing a function that interpolates a set of values defined at vertices of a mesh is a fundamental operation in computer graphics. Such an interpolant has many uses in applications such as shading, parameterization and deformation. For closed polygons, mean value coordinates have been proven to be an excellent method for constructing such an interpolant. In this paper, we generalize mean value coordinates from closed 2D polygons to closed triangular meshes. Given such a mesh P, we show that these coordinates are continuous everywhere and smooth on the interior of P. The coordinates are linear on the triangles of P and can reproduce linear functions on the interior of P. To illustrate their usefulness, we conclude by considering several interesting applications including constructing volumetric textures and surface deformation.