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1、Eurographics Symposium on Geometry Processing (2006) Konrad Polthier, Alla Sheffer (Editors) Poisson Surface Reconstruction Michael Kazhdan1, Matthew Bolitho1and Hugues Hoppe2 1Johns Hopkins University, Baltimore MD, USA 2Microsoft Research, Redmond WA, USA Abstract We show that surface reconstructi
2、on from oriented points can be cast as a spatial Poisson problem. This Poisson formulation considers all the points at once, without resorting to heuristic spatial partitioning or blending, and is therefore highly resilient to data noise. Unlike radial basis function schemes, our Poisson approach al
3、lows a hierarchy of locally supported basis functions, and therefore the solution reduces to a well conditioned sparse linear system. We describe a spatially adaptive multiscale algorithm whose time and space complexities are pro- portional to the size of the reconstructed model. Experimenting with
4、publicly available scan data, we demonstrate reconstruction of surfaces with greater detail than previously achievable. 1. Introduction Reconstructing 3D surfaces from point samples is a well studied problem in computer graphics. It allows fi tting of scanned data, fi lling of surface holes, and rem
5、eshing of ex- isting models. We provide a novel approach that expresses surface reconstruction as the solution to a Poisson equation. Like much previous work (Section 2), we approach the problem of surface reconstruction using an implicit function framework. Specifi cally, like Kaz05 we compute a 3D
6、 in- dicator function (defi ned as 1 at points inside the model, and 0 at points outside), and then obtain the reconstructed surface by extracting an appropriate isosurface. Our key insight is that there is an integral relationship be- tween oriented points sampled from the surface of a model and th
7、e indicator function of the model. Specifi cally, the gra- dient of the indicator function is a vector fi eld that is zero almost everywhere (since the indicator function is constant almost everywhere), except at points near the surface, where it is equal to the inward surface normal. Thus, the orie
8、nted point samples can be viewed as samples of the gradient of the models indicator function (Figure 1). The problem of computing the indicator function thus re- ducestoinvertingthegradientoperator,i.e.fi ndingthescalar function whose gradient best approximates a vector fi eld V defi ned by the samp
9、les, i.e. minkVk. If we apply the divergence operator, this variational problem transforms into a standard Poisson problem: compute the scalar func- 1 1 1 0 0 ?M 0 0 0 0 0 1 1 1 0 Indicator function ?M Indicator gradient 00 0 0 0 0 Surface ?M Oriented points V ? Figure 1:Intuitive illustration of Po
10、isson reconstruction in 2D. tionwhose Laplacian (divergence of gradient) equals the divergence of the vector fi eldV, = V. We will make these defi nitions precise in Sections 3 and 4. Formulating surface reconstruction as a Poisson problem offers a number of advantages. Many implicit surface fi ttin
11、g methods segment the data into regions for local fi tting, and further combine these local approximations using blending functions. In contrast, Poisson reconstruction is a global so- lution that considers all the data at once, without resorting to heuristic partitioning or blending. Thus, like rad
12、ial basis function (RBF) approaches, Poisson reconstruction creates very smooth surfaces that robustly approximate noisy data. But, whereas ideal RBFs are globally supported and non- decaying, the Poisson problem admits a hierarchy of locally supported functions, and therefore its solution reduces t
13、o a well-conditioned sparse linear system. c? The Eurographics Association 2006. 模型的有向点集可以 看作是指示函数的梯 度的采样; 很多的隐式曲面重建方都会将 数据分成局部数据拟合。 启发式的分块. Kazhdan et al. / Poisson Surface Reconstruction Moreover, in many implicit fi tting schemes, the value of the implicit function is constrained only near the sa
14、m- ple points, and consequently the reconstruction may con- tain spurious surface sheets away from these samples. Typ- ically this problem is attenuated by introducing auxiliary “off-surface” points (e.g. CBC01, OBA03). With Pois- son surface reconstruction, such surface sheets seldom arise because
15、the gradient of the implicit function is constrained at all spatial points. In particular it is constrained to zero away from the samples. Poisson systems are well known for their resilience in the presence of imperfect data. For instance, “gradient domain” manipulation algorithms (e.g. FLW02) inten
16、tionally mod- ify the gradient data such that it no longer corresponds to any real potential fi eld, and rely on a Poisson system to recover the globally best-fi tting model. There has been broad interdisciplinary research on solv- ing Poisson problems and many effi cient and robust methods have been developed. One particular aspect of our problem instance is that an accurate solution to the Poisson equation is only necessary near the reconstructed surface. This a
