Shape preserving surface reconstruction using locally anisotropic radial basis function interpolants (original) (raw)
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Reconstructing surfaces using anisotropic basis functions
2001
Point sets obtained from computer vision techniques are often noisy and non-uniform. We present a new method of surface reconstruction that can handle such data sets using anisotropic basis functions. Our reconstruction algorithm draws upon the work in variational implicit surfaces for constructing smooth and seamless 3D surfaces. Implicit functions are often formulated as a sum of weighted basis functions that are radially symmetric. Using radially symmetric basis functions inherently assumes, however, that the surface to be reconstructed is, everywhere, locally symmetric. Such an assumption is true only at planar regions, and hence, reconstruction using isotropic basis is insufficient to recover objects that exhibit sharp features. We preserve sharp features using anisotropic basis that allow the surface to vary locally. The reconstructed surface is sharper along edges and at corner points. We determine the direction of anisotropy at a point by performing principal component analysis of the data points in a small neighborhood. The resulting field of principle directions across the surface is smoothed through tensor filtering. We have applied the anisotropic basis functions to reconstruct surfaces from noisy synthetic 3D data and from real range data obtained from space carving.
Quasi-interpolation for surface reconstruction from scattered data with radial basis function
Computer Aided Geometric Design, 2012
Radial Basis Function (RBF) has been used in surface reconstruction methods to interpolate or approximate scattered data points, which involves solving a large linear system. The linear systems for determining coefficients of RBF may be ill-conditioned when processing a large point set, which leads to unstable numerical results. We introduce a quasiinterpolation framework based on compactly supported RBF to solve this problem. In this framework, implicit surfaces can be reconstructed without solving a large linear system. With the help of an adaptive space partitioning technique, our approach is robust and can successfully reconstruct surfaces on non-uniform and noisy point sets. Moreover, as the computation of quasi-interpolation is localized, it can be easily parallelized on multi-core CPUs.
Proceedings Shape Modeling Applications, 2004.
Implicit surfaces can be constructed from scattered surface points using radial basis functions (RBFs) to interpolate the surface's embedding function. Many researchers have used thin-plate spline RBFs for this because of their desirable smoothness properties. Others have used compactly supported RBFs, leading to a sparse matrix solution with lower computational complexity and better conditioning. However, the limited radius of support introduces a free parameter that leads to varying solutions as well as varying computational requirements: a larger radius of support leads to smoother and more accurate solutions but requires more computation. This paper presents an empirical analysis of this radius of support. The results using compactly supported RBFs are compared for varying model sizes and radii of support, exploring the relationship between data density and the accuracy of the interpolated surface.
ACM SIGGRAPH 2005 Courses on - SIGGRAPH '05, 2005
We describe algebraic methods for creating implicit surfaces using linear combinations of radial basis interpolants to form complex models from scattered surface points. Shapes with arbitrary topology are easily represented without the usual interpolation or aliasing errors arising from discrete sampling. These methods were first applied to implicit surfaces by Savchenko, et al. and later developed independently by Turk and O'Brien as a means of performing shape interpolation. Earlier approaches were limited as a modeling mechanism because of the order of the computational complexity involved. We explore and extend these implicit interpolating methods to make them suitable for systems of large numbers of scattered surface points by using compactly supported radial basis interpolants. The use of compactly supported elements generates a sparse solution space, reducing the computational complexity and making the technique practical for large models. The local nature of compactly supported radial basis functions permits the use of computational techniques and data structures such as k-d trees for spatial subdivision, promoting fast solvers and methods to divide and conquer many of the subproblems associated with these methods. Moreover, the representation of complex models permits the exploration of diverse surface geometry. This reduction in computational complexity enables the application of these methods to the study of shape properties of large complex shapes.
Univariate interpolation by Curvature-based scaled radial basis functions
2022
Choosing the scale or shape parameter of radial basis functions (RBFs) is a well-documented but still an open problem in kernel-based methods. It is common to tune shape parameter according to the applications, and it plays a crucial role both for the accuracy and stability of the method. In this paper, we first devise an explicit relation between the shape parameter of univariate RBFs and their curvature at each point. This leads to characterizing RBFs to scalable and unscalable ones. We prove that all scalable univariate RBFs lie in the 1 c2 -class which means that their curvature at the point xj is proportional to 1 c2 j , where cj is the corresponding spatially variable shape parameter at xj . Some of the most commonly used RBFs are then characterized and classified accordingly to their curvature. Then the fundamental theory of plane curves helps us recover univariate functions from scattered data, by enforcing the exact and approximate solutions have the same curvature at the p...
A PRACTICAL USE OF RADIAL BASIS FUNCTIONS INTERPOLATION AND APPROXIMATION
Interpolation and approximation methods are used across many fields. Standard interpolation and approximation methods rely on "ordering" that actually means tessellation in-dimensional space in general, like sorting, triangulation, generating of tetrahedral meshes etc. Tessellation algorithms are quite complex in-dimensional case. On the other hand, interpolation and approximation can be made using meshfree (meshless) techniques using Radial Basis Function (RBF). The RBF interpolation and approximation methods lead generally to a solution of linear system of equations. However, a similar approach can be taken for a reconstruction of a surface of scanned objects, etc. In this case this leads to a linear system of homogeneous equations, when a different approach has to be taken. In this paper we describe novel approaches based on RBFs for data interpolation and approximation generally in d-dimensional space. We will show properties and differences of "global" and "Compactly Supported RBF (CSRBF)", run-time and memory complexities. As the RBF interpolation and approximation naturally offer smoothness, we will analyze such properties as well as approaches how to decrease computational expenses. The proposed meshless interpolation and approximation will be demonstrated on different problems, e.g. inpainting removal, restoration of corrupted images with high percentage of corrupted pixels, digital terrain interpolation and approximation for GIS applications and methods for decreasing computational complexity.
Surface Reconstruction Based on Compactly Supported Radial Basis Functions
Geometric Modeling: Techniques, Applications, Systems and Tools, 2004
In this chapter the use of compactly-supported radial basis functions for surface reconstruction is described. To solve the problem of reconstruction or volume data generation specially designed software is employed. Time performance of the algorithm is investigated. Thanks to the efficient octree algorithm used in this study, the resulting matrix is a band diagonal matrix that reduces computational costs.
The regularizing properties of anisotropic radial basis functions
2007
In the present work we consider the problem of interpolating scattered data using radial basis functions (RBF). In general, it is well known that this leads to a discrete linear inverse problem that needs to be regularized in order to provide a meaningful solution. The work focuses on a metric-regularization approach, based on a new class of RBF, called anisotropic RBF. The work provides theoretical justifications for the regularization approach and it considers a suitable proposal for the metric, supporting it by numerical examples.
A two-level approach to implicit surface modeling with compactly supported radial basis functions
Engineering with Computers, 2010
We describe a two-level method for computing a function whose zero-level set is the surface reconstructed from given points scattered over the surface and associated with surface normal vectors. The function is defined as a linear combination of compactly supported radial basis functions (CSRBFs). The method preserves the simplicity and efficiency of implicit surface interpolation with CSRBFs and the reconstructed implicit surface owns the attributes, which are previously only associated with globally supported or globally regularized radial basis functions, such as exhibiting less extra zero-level sets, suitable for inside and outside tests. First, in the coarse scale approximation, we choose basis function centers on a grid that covers the enlarged bounding box of the given point set and compute their signed distances to the underlying surface using local quadratic approximations of the nearest surface points. Then a fitting to the residual errors on the surface points and additional off-surface points is performed with fine scale basis functions. The final function is the sum of the two intermediate functions and is a good approximation of the signed distance field to the surface in the bounding box. Examples of surface reconstruction and set operations between shapes are provided.