91 research outputs found

    Analysis of the Weight Function for Implicit Moving Least Squares Techniques

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    In this thesis, I analyze the weight functions used in moving least squares (MLS) methods to construct implicit surfaces that interpolate or approximate polygon soup. I found that one previous method that presented an analytic solution to the integrated moving least squares method has issues with degeneracies because they changed the weight functions to decrease too slowly. Inspired by their method, I derived a bound for the choice of weight function for implicit moving least squares (IMLS) methods to avoid these degeneracies in two-dimensions and in three-dimensions. Based on this bound, I give a theoretical proof of the correctness of the moving least squares interpolation and approximation scheme with weight function used in Shen et al. when used on closed polyhedrons. Further, previous IMLS implicit surface reconstruction algorithms that ll holes and gaps create surfaces with obvious bulges due to an intrinsic property of MLS. I propose a generalized IMLS method using a Gaussian distribution function to re-weight each polygon, making nearer polygons dominate and reducing the bulges on holes and gaps

    BSP-fields: An Exact Representation of Polygonal Objects by Differentiable Scalar Fields Based on Binary Space Partitioning

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    The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions to the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of the function-based modeling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples

    An Exact Representation of Polygonal Objects by C1-continuous Scalar Fields Based on Binary Space Partitioning

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    The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions to the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of function-based modelling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples

    Construction of Implicit Surfaces from Point Clouds Using a Feature-based Approach

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    Adequate Inner Bound for Geometric Modeling with Compact Field Function

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    International audienceRecent advances in implicit surface modeling now provide highly controllable blending effects. These effects rely on the field functions of R3→R\mathbb{R}^3 \rightarrow \mathbb{R} in which the implicit surfaces are defined. In these fields, there is an outside part in which blending is defined and an inside part. The implicit surface is the interface between these two parts. As recent operators often focus on blending, most efforts have been made on the outer part of field functions and little attention has been paid on the inner part. Yet, the inner fields are important as soon as difference and intersection operators are used. This makes its quality as crucial as the quality of the outside. In this paper, we analyze these shortcomings, and deduce new constraints on field functions such that differences and intersections can be seamlessly applied without introducing discontinuities or field distortions. In particular, we show how to adapt state of the art gradient-based union and blending operators to our new constraints. Our approach enables a precise control of the shape of both the inner or outer field boundaries. We also introduce a new set of asymmetric operators tailored for the modeling of fine details while preserving the integrity of the resulting fields

    Implicit Surface Modelling as an Eigenvalue Problem

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    We discuss the problem of fitting an implicit shape model to a set of points sampled from a co-dimension one manifold of arbitrary topology. The method solves a non-convex optimisation problem in the embedding function that defines the implicit by way of its zero level set. By assuming that the solution is a mixture of radial basis functions of varying widths we attain the globally optimal solution by way of an equivalent eigenvalue problem, without using or constructing as an intermediate step the normal vectors of the manifold at each data point. We demonstrate the system on two and three dimensional data, with examples of missing data interpolation and set operations on the resultant shapes
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