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Approximate Implicitization of Space Curves
"... Abstract The process of implicitization generates an implicit representation of a curve or surface from a given parametric one. This process is potentially interesting for applications in Computer Aided Design, where the robustness and efficiency of intersection algorithm can be improved by simultan ..."
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Abstract The process of implicitization generates an implicit representation of a curve or surface from a given parametric one. This process is potentially interesting for applications in Computer Aided Design, where the robustness and efficiency of intersection algorithm can be improved by simultaneously considering implicit and parametric representations. This paper gives an brief survey of the existing techniques for approximate implicitization of hyper surfaces. In addition it describes a framework for the approximate implicitization of space curves.
Robust Fitting of Implicitly Defined Surfaces using GaussNewtontype techniques
, 2009
"... We describe GaussNewton type methods for fitting implicitly defined curves and surfaces to given unorganized data points. The methods are suitable not only for leastsquares approximation, but they can also deal with general error functions, such as approximations to the ℓ1 or ℓ∞ norm of the vector ..."
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We describe GaussNewton type methods for fitting implicitly defined curves and surfaces to given unorganized data points. The methods are suitable not only for leastsquares approximation, but they can also deal with general error functions, such as approximations to the ℓ1 or ℓ∞ norm of the vector of residuals. Two different definitions of the residuals will be discussed, which lead to two different classes of methods: direct methods and data–based ones. In addition we discuss the continuous versions of the methods, which furnish geometric interpretations as evolution processes. It is shown that the data–based methods – which are less costly, as they work without the computation of the closest points – can efficiently deal with error functions that are adapted to noisy and uncertain data. In addition, we observe that the interpretation as evolution process allows to deal with the issues of regularization and with additional constraints.
Industrial Geometry INDUSTRIAL GEOMETRY Computer Vision Modeling and 3D Object Reconstruction by Implicitly Defined Surfaces with Sharp Features Modeling and 3D Object Reconstruction by Implicitly Defined Surfaces with Sharp Features
"... Abstract We propose a new method for describing sharp features (i.e., edges and vertices) of implicitly defined surfaces. We consider an initial implicitly defined surface, which is represented as the zero set of a C 1 smooth scalar field with nonvanishing gradients. In order to represent sharp ed ..."
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Abstract We propose a new method for describing sharp features (i.e., edges and vertices) of implicitly defined surfaces. We consider an initial implicitly defined surface, which is represented as the zero set of a C 1 smooth scalar field with nonvanishing gradients. In order to represent sharp edges and vertices, this surface is augmented by adding a new type of implicit representations, which are called edge descriptors and vertex descriptors. They are defined with the help of the distance field of edge curves. In our implementation, we use circular splines to describe these edge curves, since they support a fast and noniterative closest point computation. After adding the edge and vertex descriptors to the initial scalar field, the zero set of the augmented function contains the sharp features. We apply the new representation to surface modelling by implicitly defined surfaces with sharp features and to object reconstruction. In the latter case we describe an algorithm for detecting the sharp curves and vertices of a shape which is given by an unorganized point cloud, which are then approximated by circular splines, in order to define the edge and vertex descriptors.
Piecewise rational manifold surfaces with sharp features
"... We present a construction of a piecewise rational freeform surface of arbitrary topological genus which may contain sharp features: creases, corners or cusps. The surface is automatically generated from a given closed triangular mesh. Some of the edges are tagged as sharp ones, defining the featu ..."
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We present a construction of a piecewise rational freeform surface of arbitrary topological genus which may contain sharp features: creases, corners or cusps. The surface is automatically generated from a given closed triangular mesh. Some of the edges are tagged as sharp ones, defining the features on the surface. The surface is C s smooth, for an arbitrary value of s, except for the sharp features defined by the user. Our method is based on the manifold construction and follows the blending approach.
Modeling and 3D Object Reconstruction by Implicitly Defined Surfaces with Sharp Features
, 2009
"... We propose a new method for describing sharp features (i.e., edges and vertices) of implicitly defined surfaces. We consider an initial implicitly defined surface, which is represented as the zero set of a C 1 smooth scalar field with non–vanishing gradients. In order to represent sharp edges and ve ..."
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We propose a new method for describing sharp features (i.e., edges and vertices) of implicitly defined surfaces. We consider an initial implicitly defined surface, which is represented as the zero set of a C 1 smooth scalar field with non–vanishing gradients. In order to represent sharp edges and vertices, this surface is augmented by adding new types of implicit representations, called edge descriptors and vertex descriptors. They are defined with the help of the distance field of edge curves. In our implementation, we use circular splines to describe these edge curves, since they support a fast and noniterative closest point computation. After adding the edge and vertex descriptors to the initial scalar field, the zero set of the augmented function contains the sharp features. We apply the new representation to surface modelling by implicitly defined surfaces with sharp features and to object reconstruction. In the latter case we describe an algorithm for detecting the sharp curves and vertices of a shape which is given by an unorganized point cloud, which are then approximated by circular splines, in order to define the edge and vertex descriptors.