A two-level approach to implicit surface modeling with compactly supported radial basis functions

📝 Abstract

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.

💡 Analysis

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.

📄 Content

DRAFT

1 Surface Reconstruction with Higher-Order Smoothness Rongjiang Pan1, Vaclav Skala2 1School of Computer Science and Technology, Shandong University, Jinan, China, 250100, panrj@sdu.edu.cn 2Centre of Computer Graphics and Data Visualization, Department of Computer Science and Engineering, University of West Bohemia, Plzen, Czech Republic Abstract This work proposes a method to reconstruct surfaces with higher-order smoothness from noisy 3D measurements. The reconstructed surface is implicitly represented by the zero level-set of a continuous valued embedding function. The key idea is to find a function whose higher-order derivatives are regularized and whose gradient is best aligned with a vector field defined by the input point set. In contrast to methods based on the first-order variation of the function that are biased towards the constant functions and treat the extraction of the isosurface without aliasing artifacts as an afterthought, we impose higher-order smoothness directly on the embedding function. After solving a convex optimization problem with a multi-scale iterative scheme, a triangulated surface can be extracted using the marching cubes algorithm. We demonstrated the proposed method on several data sets obtained from raw laser-scanners and multi-view stereo approaches. Experimental results confirm that our approach allows us to reconstruct smooth surfaces from points in the presence of noise, outliers, large missing parts and very coarse orientation information. Keywords: Surface reconstruction, Higher-order smoothness, Convex optimization 1 Introduction Reconstructing three-dimensional digital models from real world objects is one of the major research topics in computer graphics as well as in computer vision. The majority of the developed geometric acquisition techniques, such as active and passive range sensing, usually measure a large number of 3D points. However, the discrete points are not useful for many practical applications although point-based geometry representation has been proposed [1]. Thus, reconstructing watertight surfaces from a set of sparse points is becoming a common step in the acquisition process. The problem has been researched extensively and many techniques have been developed over the past two decades [2]-[11]. However, surface reconstruction remains a difficult and, in general, an ill-posed problem since noise and outliers often contaminate the scanned data. Moreover, due to inaccessibility during scanning and some material properties, there will be cases where points are missing or incomplete.
To cope with most of the deficiencies, energy-based methods, which combine the quality of fit to data with surface regularization, are particularly appropriate for robustly constructing surfaces from sampled point sets. Recently, global optimization frameworks, e.g. graph-cut [12] and convex relaxation techniques [13], have been applied to the surface fitting problem, where the surfaces are represented implicitly by the binary-valued indicator functions. The binary volume techniques focus on segmenting a voxel as the interior or the exterior of the underlying shape. Once the function is computed, a triangulated surface model can be efficiently recovered using an isosurface extraction algorithm such as marching cubes [14]. Nevertheless, the isosurfaces often suffer from aliasing artifacts and require post-processing to achieve smooth surfaces [15]. In this paper, we propose to impose higher-order smoothness directly on a continuous-valued embedding function. Moreover, instead of measuring the distance between the surface and the given noisy data points, we wish to compute the function whose gradient is best aligned with an estimated coarse normal field. As a result, the surface reconstruction problem is formulated as a convex optimization, whose minimum yields higher quality surfaces. Computationally, the function is discretized on a regular 3D grid and constructed by solving a large sparse linear system using a multi-scale iterative scheme. The paper is organized as follows. We give some related work in the next section. Section 3 presents our energy formulation and the implementation details are provided in Section 4. In Section 5, we show some experimental results and a brief summary is concluded in Section 6. 2 Related Work In the presence of noise and inhomogeneous sample density, most popular approaches for surface reconstruction fit continuous valued or characteristic (inside-outside) functions to the input point set and then Surface Reconstruction with higher-order smoothness, The Visual Computer, on-line, ISSN 0178-2789, Springer, 2011

DRAFT

2 extract the reconstructed surface as an appropriate isosurface of this function. The pioneered work by Hoppe [2] defines the implicit function as the signed distance to the tangent plane of the closest input point. Signed distance can also be merged toget

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