泊松表面重建1
泊松重建算法 -回复
泊松重建算法-回复泊松重建算法（Poisson Reconstruction Algorithm）是一种基于泊松方程的三维重建方法，在计算机图形学领域得到了广泛的应用。
该算法可以根据已知的边界条件进行三维模型的恢复和重建，具有高效、稳定且能够处理非规则边界的优点。
本文将详细介绍泊松重建算法的原理、步骤和应用。
一、算法原理泊松重建算法的原理基于泊松方程，它利用离散化的数据计算出一个逼近真实模型的解。
泊松方程可以描述出一些物理现象的平衡状态，因此适用于模型恢复和重建。
离散化是泊松重建算法中的关键步骤，它将三维模型分割成一系列小的网格单元。
每个网格单元上都有一个未知变量，表示模型在该位置上的属性值。
通过求解离散泊松方程，可以得到这些未知变量的数值解，从而实现模型的重建。
二、算法步骤1. 数据采集和预处理：泊松重建算法需要输入一组离散化的数据作为重建的依据。
这些数据可以通过激光扫描、相机拍摄等方式获取。
在将数据输入到算法之前，需要对其进行预处理，包括去噪、对齐等操作。
2. 网格生成：通过将离散化的数据划分为一系列小的网格单元，构建出一个三维网格。
网格的大小和形状可以根据实际情况进行调整。
3. 边界条件设置：在重建过程中，需要给出一些已知的边界条件来约束模型的恢复。
这些边界条件可以是模型的外形边界、表面法向等信息。
4. 泊松方程求解：将离散化后的泊松方程转化为一个线性方程组，通过求解该方程组得到未知变量的数值解。
这个过程一般通过迭代方法进行，直到达到一定的收敛准则为止。
5. 后处理：通过求解泊松方程得到的数值解并不能直接作为重建的结果，还需要进行一些后处理的步骤。
例如，可以进行模型的平滑和细化操作来改善模型的质量。
三、算法应用泊松重建算法在计算机图形学领域有着广泛的应用。
一方面，它可以用于三维模型的重建和恢复，尤其适用于非规则形状的模型。
例如，在医学图像处理中，可以使用泊松重建算法从断层扫描图像中恢复出人体器官的三维形状。
meshlab泊松重建参数设置
meshlab泊松重建参数设置MeshLab是一款免费的三维模型处理软件，它可以进行各种三维模型的处理和编辑。
其中，泊松重建是MeshLab中的一个重要功能，它可以将点云数据转换为三角网格模型。
在进行泊松重建时，参数设置是非常重要的，下面我们来详细介绍一下MeshLab泊松重建参数设置。
1. 输入点云数据在进行泊松重建之前，首先需要导入点云数据。
在MeshLab中，可以通过“文件”菜单中的“导入”命令来导入点云数据。
在导入点云数据时，需要注意点云数据的格式，MeshLab支持多种点云数据格式，如PLY、OBJ、XYZ等。
2. 设置泊松重建参数在导入点云数据后，可以通过“滤波器”菜单中的“重建”命令来进行泊松重建。
在进行泊松重建时，需要设置一些参数，包括：（1）采样率采样率是指对点云数据进行采样的比例，采样率越高，生成的三角网格模型的精度越高，但是计算时间也会增加。
一般来说，采样率可以设置为0.1到0.5之间。
（2）深度深度是指泊松重建算法中的迭代次数，深度越大，生成的三角网格模型的精度越高，但是计算时间也会增加。
一般来说，深度可以设置为8到12之间。
（3）八叉树深度八叉树深度是指在泊松重建算法中使用的八叉树的深度，八叉树深度越大，生成的三角网格模型的精度越高，但是计算时间也会增加。
一般来说，八叉树深度可以设置为6到10之间。
（4）最小点数最小点数是指在进行泊松重建时，每个八叉树节点中最少需要包含的点数，最小点数越小，生成的三角网格模型的精度越高，但是计算时间也会增加。
一般来说，最小点数可以设置为10到50之间。
（5）平滑系数平滑系数是指在进行泊松重建时，对法向量进行平滑的程度，平滑系数越大，生成的三角网格模型的平滑度越高，但是可能会导致一些细节丢失。
一般来说，平滑系数可以设置为0.1到0.5之间。
3. 进行泊松重建在设置好泊松重建参数后，可以点击“应用”按钮来进行泊松重建。
在进行泊松重建时，需要注意点云数据的大小和计算机的性能，如果点云数据过大或计算机性能不足，可能会导致计算时间过长或程序崩溃。
泊松表面重建源码
泊松表面重建源码泊松表面重建是一种用于三维形状重建的算法，可以根据离散的点云数据生成连续的曲面模型。
本文将介绍泊松表面重建的原理和实现，并讨论其在计算机图形学和计算机视觉领域的应用。
一、引言在计算机图形学和计算机视觉领域，三维形状重建是一个重要的研究方向。
它涉及将从传感器获取的离散点云数据转换为连续的曲面模型，以便进行后续分析和处理。
泊松表面重建算法就是其中一种常用的方法。
二、泊松表面重建原理泊松表面重建算法基于泊松方程的解，通过求解离散点云数据的泊松方程，得到一个光滑的曲面模型。
具体而言，泊松方程可以表示为：∇²f = ρ其中，f是待求解的函数，表示曲面的高度；∇²是拉普拉斯算子，用于描述曲面的光滑程度；ρ是离散点云数据的密度函数，表示点云数据在空间中的分布情况。
泊松表面重建算法的核心思想是将离散点云数据的密度函数ρ估计为一个常数，然后通过求解泊松方程得到曲面模型。
具体的步骤如下：1. 对离散点云数据进行网格化，将点云数据转换为一个有限元网格；2. 估计密度函数ρ为常数，常数的选择可以根据点云数据的特点进行调整；3. 构造泊松方程的离散形式，将其转化为一个线性方程组；4. 求解线性方程组，得到曲面的高度函数f；5. 根据高度函数f生成曲面模型。
三、泊松表面重建实现泊松表面重建算法的实现可以使用各种编程语言和计算工具进行，例如C++、Python和MATLAB等。
下面以Python为例，介绍一种简单的实现方法。
需要导入相关的库和模块，例如numpy和scipy等。
然后，读取离散点云数据，可以从本地文件中读取，也可以通过传感器获取。
接下来，对点云数据进行网格化，将其转换为一个有限元网格。
可以使用Delaunay三角剖分算法或其他网格生成算法。
然后，估计密度函数ρ为常数。
常数的选择可以根据点云数据的特点进行调整，一般可以选择为点云数据的平均密度。
接着，构造泊松方程的离散形式。
可以使用有限元法或其他离散方法来进行离散化，将其转化为一个线性方程组。
基于屏幕空间的泊松表面重建
Screened Poisson Surface ReconstructionMICHAEL KAZHDANJohns Hopkins UniversityandHUGUES HOPPEMicrosoft ResearchPoisson surface reconstruction creates watertight surfaces from oriented point sets.In this work we extend the technique to explicitly incorporate the points as interpolation constraints.The extension can be interpreted as a generalization of the underlying mathematical framework to a screened Poisson equation.In contrast to other image and geometry processing techniques,the screening term is defined over a sparse set of points rather than over the full domain.We show that these sparse constraints can nonetheless be integrated efficiently.Because the modified linear system retains the samefinite-element discretization,the sparsity structure is unchanged,and the system can still be solved using a multigrid approach. Moreover we present several algorithmic improvements that together reduce the time complexity of the solver to linear in the number of points, thereby enabling faster,higher-quality surface reconstructions.Categories and Subject Descriptors:I.3.5[Computer Graphics]:Compu-tational Geometry and Object ModelingAdditional Key Words and Phrases:screened Poisson equation,adaptive octree,finite elements,surfacefittingACM Reference Format:Kazhdan,M.,and Hoppe,H.Screened Poisson surface reconstruction. ACM Trans.Graph.NN,N,Article NN(Month YYYY),PP pages.DOI=10.1145/XXXXXXX.YYYYYYY/10.1145/XXXXXXX.YYYYYYY1.INTRODUCTIONPoisson surface reconstruction[Kazhdan et al.2006]is a well known technique for creating watertight surfaces from oriented point samples acquired with3D range scanners.The technique is resilient to noisy data and misregistration artifacts.However, as noted by several researchers,it suffers from a tendency to over-smooth the data[Alliez et al.2007;Manson et al.2008; Calakli and Taubin2011;Berger et al.2011;Digne et al.2011].In this work,we explore modifying the Poisson reconstruc-tion algorithm to incorporate positional constraints.This mod-ification is inspired by the recent reconstruction technique of Calakli and Taubin[2011].It also relates to recent work in im-age and geometry processing[Nehab et al.2005;Bhat et al.2008; Chuang and Kazhdan2011],in which a datafidelity term is used to“screen”the associated Poisson equation.In our surface recon-struction context,this screening term corresponds to a soft con-straint that encourages the reconstructed isosurface to pass through the input points.The approach we propose differs from the traditional screened Poisson formulation in that the position and gradient constraints are defined over different domain types.Whereas gradients are constrained over the full3D space,positional constraints are introduced only over the input points,which lie near a2D manifold. We show how these two types of constraints can be efficiently integrated,so that we can leverage the original multigrid structure to solve the linear system without incurring a significant overhead in space or time.To demonstrate the benefits of screening,Figure1compares results of the traditional Poisson surface reconstruction and the screened Poisson formulation on a subset of11.4M points from the scan of Michelangelo’s David[Levoy et al.2000].Both reconstructions are computed over a spatial octree of depth10,corresponding to an effective voxel resolution of10243.Screening generates a model that better captures the input data(as visualized by the surface cross-sections overlaid with the projection of nearby samples), even though both reconstructions have similar complexity(6.8M and6.9M triangles respectively)and required similar processing time(230and272seconds respectively,without parallelization).1 Another contribution of our work is to modify both the octree structure and the multigrid implementation to reduce the time complexity of solving the Poisson system from log-linear to linear in the number of input points.Moreover we show that hierarchical point clustering enables screened Poisson reconstruction to attain this same linear complexity.2.RELA TED WORKReconstructing surfaces from scanned points is an important and extensively studied problem in computer graphics.The numerous approaches can be broadly categorized as follows. Combinatorial Algorithms.Many schemes form a triangula-tion using a subset of the input points[Cazals and Giesen2006]. Space is often discretized using a tetrahedralization or a voxel grid,and the resulting elements are partitioned into inside and outside regions using an analysis of cells[Amenta et al.2001; Boissonnat and Oudot2005;Podolak and Rusinkiewicz2005], eigenvector computation[Kolluri et al.2004],or graph cut [Labatut et al.2009;Hornung and Kobbelt2006].Implicit Functions.In the presence of sampling noise,a common approach is tofit the points using the zero set of an implicit func-tion,such as a sum of radial bases[Carr et al.2001]or piecewise polynomial functions[Ohtake et al.2005;Nagai et al.2009].Many techniques estimate a signed-distance function[Hoppe et al.1992; 1The performance of the unscreened solver is measured using our imple-mentation with screening weight set to zero.The implementation of the original Poisson reconstruction runs in412seconds.ACM Transactions on Graphics,V ol.VV,No.N,Article XXX,Publication date:Month YYYY.2•M.Kazhdan and H.HoppeFig.1:Reconstruction of the David head ‡,comparing traditional Poisson surface reconstruction (left)and screened Poisson surface reconstruction which incorporates point constraints (center).The rightmost diagram plots pixel depth (z )values along the colored segments together with the positions of nearby samples.The introduction of point constraints significantly improves fit accuracy,sharpening the reconstruction without amplifying noise.Bajaj et al.1995;Curless and Levoy 1996].If the input points are unoriented,an important step is to correctly infer the sign of the resulting distance field [Mullen et al.2010].Our work extends Poisson surface reconstruction [Kazhdan et al.2006],in which the implicit function corresponds to the model’s indicator function χ.The function χis often defined to have value 1inside and value 0outside the model.To simplify the derivations,inthis paper we define χto be 12inside and −12outside,so that its zero isosurface passes near the points.The function χis solved using a Laplacian system discretized over a multiresolution B-spline basis,as reviewed in Section 3.Alliez et al.[2007]form a Laplacian system over a tetrahedral-ization,and constrain the solution’s biharmonic energy;the de-sired function is obtained as the solution to an eigenvector prob-lem.Manson et al.[2008]represent the indicator function χusing a wavelet basis,and efficiently compute the basis coefficients using simple local sums over an adapted octree.Calakli and Taubin [2011]optimize a signed-distance function to have value zero at the points,have derivatives that agree with the point normals,and minimize a Hessian smoothness norm.The resulting optimization involves a bilaplacian operator,which requires estimating derivatives of higher order than in the Laplacian.The reconstructed surfaces are shown to have good accuracy,strongly suggesting the importance of explicitly fitting the points within the optimization.This motivated us to explore whether a Laplacian system could be extended in this respect,and also be compatible with a multigrid solver.Screened Poisson Surface Fitting.The method of Nehab et al.[2005],which simultaneously fits position and normal constraints,may also be viewed as the solution of a screened Poisson equation.The fitting algorithm assumes that a 2D parametric domain (i.e.,a plane or triangle mesh)is already established.The position and derivative constraints are both defined over this 2D domain.In contrast,in Poisson surface reconstruction the 2D domain manifold is initially unknown,and therefore the goal is to infer anindicator function χrather than a parametric function.This leadsto a hybrid problem with derivative (Laplacian)constraints defined densely over 3D and position constraints defined sparsely on the set of points sampled near the unknown 2D manifold.3.REVIEW OF POISSON SURFACE RECONSTRUCTIONThe approach of Poisson surface reconstruction is based on the observation that the (inward pointing)normal field of the boundary of a solid can be interpreted as the gradient of the solid’s indicator function.Thus,given a set of oriented points sampling the boundary,a watertight mesh can be obtained by (1)transforming the oriented point samples into a continuous vector field in 3D,(2)finding a scalar function whose gradients best match the vector field,and (3)extracting the appropriate isosurface.Because our work focuses primarily on the second step,we review it here in more detail.Scalar Function Fitting.Given a vector field V :R 3→R 3,thegoal is to solve for the scalar function χ:R 3→R minimizing:E (χ)=∇χ(p )− V (p ) 2d p .(1)Using the Euler-Lagrange formulation,the minimum is obtainedby solving the Poisson equation:∆χ=∇· V .System Discretization.The Galerkin formulation is used totransform this into a finite-dimensional system [Fletcher 1984].First,a basis {B 1,...,B N }:R 3→R is chosen,namely a collection of trivariate (usually triquadratic)B-spline functions.With respect to this basis,the discretization becomes:∆χ,B i [0,1]3= ∇· V ,B i [0,1]31≤i ≤Nwhere ·,· [0,1]3is the standard inner-product on the space of(scalar-and vector-valued)functions defined on the unit cube:F ,G [0,1]3=[0,1]3F (p )·G (p )d p , U , V [0,1]3=[0,1]3U (p ), V (p ) d p .Since the solution is itself expressed in terms of the basis functions:χ(p )=N∑i =1x i B i (p ),ACM Transactions on Graphics,V ol.VV ,No.N,Article XXX,Publication date:Month YYYY .1.离散化->连续2.找个常量函数最佳拟合这些这些向量域;3.抽取等值面这里已经将离散的有向点转化为了连续的向量域表示;点集合的最初的思考Screened Poisson Surface Reconstruction•3finding the coefficients{x i}of the solution reduces to solving the linear system Ax=b where:A i j= ∇B i,∇B j [0,1]3and b i= V,∇B i [0,1]3.(2) The basis functions{B1,...,B N}are chosen to be compactly supported,so most pairs of functions do not have overlapping support,and thus the matrix A is sparse.Because the solution is expected to be smooth away from the input samples,the linear system is discretized byfirst adapting an octree to the input samples and then associating an(appropriately scaled and translated)trivariate B-spline function to each octree node. This provides high-resolution detail in the vicinity of the surface while reducing the overall dimensionality of the system.System Solution.Given the hierarchy defined by an octree of depth D,a multigrid approach is used to solve the linear system. The basis functions are partitioned according to the depths of their associated nodes and,for each depth d,a linear system A d x d=b d is defined using the corresponding B-splines{B d1,...,B d Nd},such thatχ(p)=∑D d=0∑i x d i B d i(p).Because the octree-selected B-spline functions do not form a complete grid at each depth,it is generally not possible to prolong the solution x d at depth d into the solution x d+1at depth d+1. (The B-spline associated with a given node is a sum of B-spline functions associated not only with its own child nodes,but also with child nodes of its neighbors.)Instead,the constraints at depth d+1are adjusted to account for the part of the solution already realized at coarser depths.Pseudocode for a cascadic solver,where the solution is only relaxed on the up-stroke of the V-cycle,is given in Algorithm1.Algorithm1:Cascadic Poisson Solver1For d∈{0,...,D}Iterate from coarse tofine2For d ∈{0,...,d−1}Remove the constraints3b d=b d−A dd x d met at coarser depths4Relax A d x d=b d Adjust the system at depth dHere,A dd is the N d×N d matrix used to transform solution coefficients at depth d into constraints at depth d:A dd i j= ∇B d i,∇B d j [0,1]3.Note that,by definition,A d=A dd.Isosurface Extraction.Solving the Poisson equation,one obtains a functionχthat approximates the indicator function.Ideally,the function’s zero level-set should therefore correspond to the desired surface.In practice however,the functionχcan differ from the true indicator function due to several sources of error:—The point sampling may be noisy,possibly containing outliers.—The Galerkin discretization is only an approximation of the continuous problem.—The point sampling density is approximated during octree construction.To mitigate these errors,in[Kazhdan et al.2006]the implicit function is adjusted by globally subtracting the average value of the function at the input samples.4.INCORPORA TING POINT CONSTRAINTSThe original Poisson surface reconstruction algorithm adjusts the implicit function using a single global offset such that its average value at all points is zero.However,the presence of errors can cause the implicit function to drift so that no global offset is satisfactory. Instead,we seek to explicitly interpolate the points.Given the set of input points P with weights w:P→R≥0,we add to the energy of Equation1a term that penalizes the function’s deviation from zero at the samples:E(χ)=V(p)−∇χ(p) 2d p+α·Area(P)∑p∈P∑p∈Pw(p)χ2(p)(3)whereαis a weight that trades off the importance offitting the gradients andfitting the values,and Area(P)is the area of the reconstructed surface,estimated by computing the local sampling density as in[Kazhdan et al.2006].In our implementation,we set the per-sample weights w(p)=1,although one can also use confidence values if these are available.The energy can be expressed concisely asE(χ)= V−∇χ, V−∇χ [0,1]3+α χ,χ (w,P)(4)where ·,· (w,P)is the bilinear,symmetric,positive,semi-definite form on the space of functions in the unit-cube,obtained by taking the weighted sum of function values:F,G (w,P)=Area(P)∑p∈P w(p)∑p∈Pw(p)·F(p)·G(p).4.1Interpretation as a Screened Poisson EquationThe energy in Equation4combines a gradient constraint integrated over the spatial domain with a value constraint summed at discrete points.As shown in the appendix,its minimization can be interpreted as a screened Poisson equation(∆−α˜I)χ=∇· V with an appropriately defined operator˜I.4.2DiscretizationWe apply a discretization similar to that in Section3to the minimization of the energy in Equation4.The coefficients of the solutionχwith respect to the basis{B1,...,B N}are again obtained by solving a linear system of the form Ax=b.The right-hand-side b is unchanged because the constrained value at the sample points is zero.Matrix A now includes the point constraints:A i j= ∇B i,∇B j [0,1]3+α B i,B j (w,P).(5) Note that incorporating the point constraints does not change the sparsity of matrix A because B i(p)·B j(p)is nonzero only if the supports of the two functions overlap,in which case the Poisson equation has already introduced a nonzero entry in the matrix.As in Section3,we solve this linear system using a cascadic multigrid algorithm–iterating over the octree depths from coarsest tofinest,adjusting the constraints,and relaxing the system.Similar to Equation5,the matrix used to transform a solution at depth d to a constraint at depth d is expressed as:A dd i j= ∇B d i,∇B d j [0,1]3+α B d i,B d j (w,P).ACM Transactions on Graphics,V ol.VV,No.N,Article XXX,Publication date:Month YYYY.4•M.Kazhdan and H.HoppeFig.2:Visualizations of the reconstructed implicit function along a planar slice through the cow ‡(shown in blue on the left),for the original Poisson solver,and for the screened Poisson solver without and with scale-independent screening.This operator adjusts the constraint b d (line 3of Algorithm 1)not only by removing the Poisson constraints met at coarser resolutions,but also by modifying the constrained values at points where the coarser solution does not evaluate to zero.4.3Scale-Independent ScreeningTo balance the two energy terms in Equation 3,it is desirable to adjust the screening parameter αsuch that (1)the reconstructed surface shape is invariant under scaling of the input points with respect to the solver domain,and (2)the prolongation of a solution at a coarse depth is an accurate estimate of the solution at a finer depth in the cascadic multigrid approach.We achieve both these goals by adjusting the relative weighting of position and gradient constraints across the different octree depths.Noting that the magnitude of the gradient constraint scales with resolution,we double the weight of the interpolation constraint with each depth:A ddi j = ∇B d i ,∇B dj [0,1]3+2d α B d i ,B dj (w ,P ).The adaptive weight of 2d is chosen to keep the Laplacian and screening constraints around the surface in balance.To see this,assume that the points are locally planar,and consider the row of the system matrix corresponding to an octree node overlapping the points.The coefficients of the system in that row are the sum of Laplacian and screening terms.If we consider the rows corresponding to the child nodes that overlap the surface,we find that the contribution from the Laplacian constraints scales by a factor of 1/2while the contribution from the screening term scales by a factor of 1/4.2Thus,scaling the screening weights by a factor of two with each resolution keeps the two terms in balance.Figure 2shows the benefit of scale-independent screening in reconstructing a cow model.The leftmost image shows a plane passing through the bounding cube of the cow,and the images to the right show the values of the computed indicator function along that plane,for different implementations of the solver.As the figure shows,the unscreened Poisson solver provides a good approximation of the indicator functions,with values inside (resp.outside)the surface approximately 1/2(resp.-1/2).However,applying the same solver to the screened Poisson equation (second from right)provides a solution that is only correct near the input samples and returns to zero near the faces of the bounding cube,2Forthe Laplacian term,the Laplacian scales by a factor of 4with refinement,and volumetric integrals scale by a factor of 1/8.For the screening term,area integrals scale by a factor of 1/4.potentially resulting in spurious surface sheets away from the surface.It is only with scale-independent screening (right)that we obtain a high-quality solution to the screened Poisson ing this resolution adaptive weighting,our system has the property that the reconstruction obtained by solving at depth D is identical to the reconstruction that would be obtained by scaling the point set by 1/2and solving at depth D +1.To see this,we consider the two energies that guide the reconstruc-tion,E V (χ)measuring the extent to which the gradients of the so-lution match the prescribed vector field,and E (w ,P )(χ)measuring the extent to which the solution meets the screening constraint:E V (χ)=V (p )−∇χ(p )2d p E (w ,P )(χ)=Area (P )∑p ∈P w (p )∑p ∈Pw (p )χ2(p ).Scaling by 1/2,we obtain a new point set (˜w ,˜P)with positions scaled by 1/2,unchanged weights,˜w (p )=w (2p ),and scaled area,Area (˜P )=Area (P )/4;a new scalar field,˜χ(p )=χ(2p );and a new vector field,˜ V (p )=2 V (2p ).Computing the correspondingenergies,we get:E ˜ V (˜χ)=1E V(χ)and E (˜w ,˜P )(˜χ)=1E (w ,P )(χ).Thus,scaling the screening weight by a factor of two with eachsuccessive depth ensures that the sum of energies is unchanged (up to multiplication by a constant)so the minimizer remains the same.4.4Boundary ConditionsIn order to define the linear system,it is necessary to define the behavior of the function space along the boundary of the integration domain.In the original Poisson reconstruction the authors imposed Dirichlet boundary conditions,forcing the implicit function to havea value of −12along the boundary.In the present work we extend the implementation to support Neumann boundary conditions as well,forcing the normal derivative to be zero along the boundary.In principle these two boundary conditions are equivalent for watertight surfaces,since the indicator function has a constant negative value outside the model.However,in the presence of missing data we find Neumann constraints to be less restrictive because they only require that the implicit function have zero derivative across the boundary of the integration domain,a property that is compatible with the gradient constraint since the guiding vector field V is set to zero away from the samples.(Note that when the surface does cross the boundary of the domain,the Neumann boundary constraints create a bias to crossing the domain boundary orthogonally.)Figure 3shows the practical implications of this choice when reconstructing the Angel model,which was only scanned from the front.The left image shows the original point set and the reconstructions using Dirichlet and Neumann boundary conditions are shown to the right.As the figure shows,imposing Dirichlet constraints creates a water-tight surface that closes off before reaching the boundary while using Neumann constraints allows the surface to extend out to the boundary of the domain.ACM Transactions on Graphics,V ol.VV ,No.N,Article XXX,Publication date:Month YYYY .Screened Poisson Surface Reconstruction•5Fig.3:Reconstructions of the Angel point set‡(left)using Dirichlet(center) and Neumann(right)boundary conditions.Similar results can be seen at the bases of the models in Figures1 and4a,with the original Poisson reconstructions obtained using Dirichlet constraints and the screened reconstructions obtained using Neumann constraints.5.IMPROVED ALGORITHMIC COMPLEXITYIn this section we discuss the efficiency of our reconstruction al-gorithm.We begin by analyzing the complexity of the algorithm described above.Then,we present two algorithmic improvements. Thefirst describes how hierarchical clustering can be used to re-duce the screening overhead at coarser resolutions.The second ap-plies to both the unscreened and screened solver implementations, showing that the asymptotic time complexity in both cases can be reduced to be linear in the number of input points.5.1Efficiency of basic solverLet us begin by analyzing the computational complexity of the unscreened and screened solvers.We assume that the points P are evenly distributed over a surface,so that the depth of the adapted octree is D=O(log|P|)and the number of octree nodes at depth d is O(4d).We also note that the number of nonzero entries in matrix A dd is O(4d),since the matrix has O(4d)rows and each row has at most53nonzero entries.(Since we use second-order B-splines, basis functions are supported within their one-ring neighborhoods and the support of two functions will overlap only if one is within the two-ring neighborhood of the other.)Assuming that the matrices A dd have already been computed,the computational complexity for the different steps in Algorithm1is: Step3:O(4d)–since A dd has O(4d)nonzero entries.Step4:O(4d)–since A d has O(4d)nonzero entries and the number of relaxation steps performed is constant.Steps2-3:∑d−1d =0O(4d)=O(4d·d).Steps2-4:O(4d·d+4d)=O(4d·d).Steps1-4:∑D d=0O(4d·d)=O(4D·D)=O(|P|·log|P|). There still remains the computation of matrices A dd .For the unscreened solver,the complexity of computing A dd is O(4d),since each entry can be computed in constant time.Thus, the overall time complexity remains O(|P|·log|P|).For the screened solver,the complexity of computing A dd is O(|P|)since defining the coefficients requires accumulating the screening contribution from each of the points,and each point contributes to a constant number of rows.Thus,the overall time complexity is dominated by the cost of evaluating the coefficients of A dd which is:D∑d=0d−1∑d =0O(|P|)=O(|P|·D2)=O(|P|·log2|P|).5.2Hierarchical Clustering of Point ConstraintsOurfirst modification is based on the observation that since the basis functions at coarser resolutions are smooth,it is unnecessary to constrain them at the precise sample locations.Instead,we cluster the weighted points as in[Rusinkiewicz and Levoy2000]. Specifically,for each depth d,we define(w d,P d)where p i∈P d is the weighted average position of the points falling into octree node i at depth d,and w d(p i)is the sum of the associated weights.3 If all input points have weight w(p)=1,then w d(p i)is simply the number of points falling into node i.This alters the computation of the system matrix coefficients:A dd i j= ∇B d i,∇B d j [0,1]3+2dα B d i,B d j (w d,P d).Note that since d>d ,the value B d i,B d j (w d,P d)is obtained by summing over points stored with thefiner resolution.In particular,the complexity of computing A dd for the screened solver becomes O(|P d|)=O(4d),which is the same as that of the unscreened solver,and both implementations now have an overall time complexity of O(|P|·log|P|).On typical examples,hierarchical clustering reduces execution time by a factor of almost two,and the reconstructed surface is visually indistinguishable.5.3Conforming OctreesTo account for the adaptivity of the octree,Algorithm1subtracts off the constraints met at all coarser resolutions before relaxing at a given depth(steps2-3),resulting in an algorithm with log-linear time complexity.We obtain an implementation with linear complexity by forcing the octree to be conforming.Specifically, we define two octree cells to be mutually visible if the supports of their associated B-splines overlap,and we require that if a cell at depth d is in the octree,then all visible cells at depth d−1must also be in the tree.Making the tree conforming requires the addition of new nodes at coarser depths,but this still results in O(4d)nodes at depth d.While the conforming octree does not satisfy the condition that a coarser solution can be prolonged into afiner one,it has the property that the solution obtained at depths{0,...,d−1}that is visible to a node at depth d can be expressed entirely in terms of the coefficients at depth d−ing an accumulation vector to store the visible part of the solution,we obtain the linear-time implementation in Algorithm2.3Note that the weight w d(p)is unrelated to the screening weight2d introduced in Section4.3for scale-independent screening.ACM Transactions on Graphics,V ol.VV,No.N,Article XXX,Publication date:Month YYYY.6•M.Kazhdan and H.HoppeHere,P d d−1is the B-spline prolongation operator,expressing a solution at depth d−1in terms of coefficients at depth d.The number of nonzero entries in P d d−1is O(4d),since each column has at most43nonzero entries,so steps2-5of Algorithm2all have complexity O(4d).Thus,the overall complexity of both the unscreened and screened solvers becomes O(|P|).Algorithm2:Conforming Cascadic Poisson Solver1For d∈{0,...,D}Iterate from coarse tofine.2ˆx d−1=P d−1d−2ˆx d−2Upsample coarseraccumulation vector.3ˆx d−1=ˆx d−1+x d−1Add in coarser solution.4b d=b d−A d d−1ˆx d−1Remove constraintsmet at coarser depths.5Relax A d x d=b d Adjust the system at depth d.5.4Implementation DetailsThe algorithm is implemented in C++,using OpenMP for multi-threaded parallelization.We use a conjugate-gradient solver to re-lax the system at each multigrid level.With the exception of the octree construction,most of the operations involved in the Poisson reconstruction can be categorized as operations that either“accu-mulate”or“distribute”information[Bolitho et al.2007,2009].The former do not introduce write-on-write conflicts and are trivial to parallelize.The latter only involve linear operations,and are par-allelized using a standard map-reduce approach:in the map phase we create a duplicate copy of the data for each thread to distribute values into,and in the reduce phase we merge the copies by taking their sum.6.RESULTSWe evaluate the algorithm(Screened)by comparing its accuracy and computational efficiency with several prior methods:the original Poisson reconstruction of Kazhdan et al.[2006](Poisson), the Wavelet reconstruction of Manson et al.[2008](Wavelet),and the Smooth Signed Distance reconstruction of Calakli and Taubin [2011](SSD).For the new algorithm,we set the screening weight toα=4and use Neumann boundary conditions in all experiments.(Numerical results obtained using Dirichlet boundaries were indistinguishable.) For the prior methods,we set algorithmic parameters to values recommended by the authors,using Haar Wavelets in the Wavelet reconstruction and setting the value/normal/Hessian weights to 1/1/0.25in the SSD reconstruction.For Poisson,SSD,and Screened we set the“samples-per-node”parameter to1and the “bounding-box-scale”parameter to1.1.(For Wavelet the bounding box scale is hard-coded at1and there is no parameter to adjust the sampling density.)6.1AccuracyWe run three different types of experiments.Real Scanner Data.To evaluate the accuracy of the different reconstruction algorithms on real-world data,we gathered several scanned datasets:the Awakening(10M points),the Stanford Bunny (0.2M points),the David(11M points),the Lucy(1.0M points), and the Neptune(2.4M points).For each dataset,we randomly partitioned the points into two equal-sized subsets:input points for the reconstruction algorithms,and validation points to measure point-to-reconstruction distances.Figure4a shows reconstructions results for the Neptune and David models at depth10.It also shows surface cross-sections overlaid with the validation points in their vicinity.These images reveal that the Poisson reconstruction(far left),and to a lesser extent the SSD reconstruction(center left),over-smooth the data,while the Wavelet reconstruction(center left)has apparent derivative discontinuities.In contrast,our screened Poisson approach(far right)provides a reconstruction that faithfullyfits the samples without introducing noise.Figure4b shows quantitative results across all datasets,in the form of RMS errors,measured using the distances from the validation points to the reconstructed surface.(We also computed the maximum error,but found that its sensitivity to individual outlier points made it an unreliable and unindicative statistic.)As thefigure indicates,the Screened Poisson reconstruction(blue)is always more accurate than both the original Poisson reconstruction algorithm(red)and the Wavelet reconstruction(purple),and generates reconstruction whose RMS errors are comparable to or smaller than those of the SSD reconstruction(green).Clean Uniformly Sampled Data.To evaluate reconstruction accuracy on clean data,we used the approach of Osada et al.[2001] to generate oriented point sets by uniformly sampling the surfaces of the Fandisk,Armadillo Man,Dragon,and Raptor models.For each model,we generated datasets of100K and1M points and reconstructed surfaces from each point set using the four different reconstruction algorithms.As an example,Figure5a shows the reconstructions of the fandisk and raptor models using1M point samples at depth10.Despite the lack of noise in the input data,the Wavelet reconstruction has spurious high-frequency detail.Focusing on the sharp edges in the model,we also observe that the screened Poisson reconstruction introduces less smoothing,providing a reconstruction that is truer to the original data than either the original Poisson or the SSD reconstructions.Figure5b plots RMS errors across all models,measured bidirec-tionally between the original surface and the reconstructed surface using the Metro tool[Cignoni and Scopigno1998].As in the case of real scanner data,screened Poisson reconstruction always out-performs the original Poisson and Wavelet reconstructions,and is comparable to or better than the SSD reconstruction. Reconstruction Benchmark.We use the benchmark of Berger et al.[2011]to evaluate the accuracy of the algorithms under different simulations of scanner error,including nonuniform sampling,noise,and misalignment.The dataset consists of mul-tiple virtual scans of implicit surfaces representing the Anchor, Dancing Children,Daratech,Gargoyle,and Quasimodo models. As an example,Figure6a visualizes the error in the reconstructions of the anchor model from a virtual scan consisting of210K points (demarked with a dashed rectangle in Figure6b)at depth9.The error is visualized using a red-green-blue scale,with red signifyingACM Transactions on Graphics,V ol.VV,No.N,Article XXX,Publication date:Month YYYY.。
表面重建算法
表面重建算法概述表面重建算法是计算机图形学中的一个重要研究领域，其主要目的是从点云数据中生成连续、光滑的曲面模型。
表面重建算法应用广泛，如三维扫描、医学成像、地形建模等领域。
本文将介绍表面重建算法的基本原理、分类以及常用算法。
基本原理表面重建算法的基本原理是从离散的点云数据中生成连续、光滑的曲面模型。
点云数据通常由三维扫描仪或激光雷达等设备获取。
对于一个给定的点云，表面重建算法需要确定每个点在曲面上的位置和法向量。
分类表面重建算法可以分为两类：基于网格和基于隐式函数。
1. 基于网格基于网格的表面重建算法将点云转换为一个三角网格，然后通过对网格进行平滑处理来生成曲面模型。
其中最常用的方法是Poisson重构算法。
Poisson重构算法基于Poisson方程，该方程描述了曲面上任意一点处梯度向量与曲面法向量之间的关系。
该算法首先计算每个点在曲面上的法向量，然后通过对点云进行重采样得到一个规则的网格，最后利用Poisson方程求解得到曲面模型。
2. 基于隐式函数基于隐式函数的表面重建算法将点云转换为一个隐式函数，然后通过等值面提取算法生成曲面模型。
其中最常用的方法是Moving Least Squares (MLS)算法。
MLS算法首先对点云进行平滑处理，然后对每个点构建一个局部加权多项式函数。
该函数表示了该点附近的曲面形状，然后通过等值面提取算法生成曲面模型。
常用算法1. Marching Cubes算法Marching Cubes算法是一种基于网格的表面重建算法。
该算法将三维空间划分为一系列小立方体，并在每个立方体中确定等值面的位置和拓扑结构。
最终将所有立方体中的等值面拼接起来形成曲面模型。
2. Poisson重构算法Poisson重构算法是一种基于网格的表面重建算法。
该算法首先计算每个点在曲面上的法向量，然后通过对点云进行重采样得到一个规则的网格，最后利用Poisson方程求解得到曲面模型。
3. MLS算法MLS算法是一种基于隐式函数的表面重建算法。
meshlab泊松重建代码 -回复
meshlab泊松重建代码-回复MeshLab是一个免费的开源三维网格处理软件，它提供了一系列强大的工具和算法，方便用户对三维模型进行编辑和转换。
其中一个重要的功能是泊松重建，可以通过点云数据重建出光滑的三维表面模型。
本文将介绍MeshLab中泊松重建的代码实现步骤。
首先，我们需要准备好点云数据作为输入。
点云数据表示了三维模型表面的一系列离散的点，可以通过激光扫描、三维扫描仪等方式获取。
在本文中，我们将使用一个简单的点云数据来演示泊松重建的过程。
第一步，打开MeshLab软件。
在软件的菜单栏中选择"文件"，然后点击"导入"来加载点云数据。
选择点云文件并点击"打开"按钮，点云数据将会被加载到软件的视图窗口中。
第二步，点击菜单栏中的"滤镜"，然后选择"重建"子菜单中的"泊松表面重建"。
弹出的对话框中将显示出一些参数选项，我们可以根据需要进行调整。
第三步，选择泊松重建算法的类型。
在对话框中的"泊松表面重建"下拉菜单中，有两种不同的算法可供选择："增值法"和"加权法"。
两种算法都可以进行泊松重建，但在不同的情况下性能可能会有所不同。
第四步，确定重建后的模型的细节水平。
在对话框中的"详细度"滑块上拖动滑块可以调整模型的细节水平。
向左拖动滑块会降低模型的细节水平，向右拖动滑块会增加模型的细节水平。
根据点云数据的密度和所需的细节水平，我们可以选择合适的值。
第五步，调整其他参数。
在对话框中还有其他一些参数可以调整，如"平滑程度"、"多面体包围球半径"等。
这些参数可以根据需求进行调整，以获得更好的重建效果。
第六步，点击对话框中的"应用"按钮，开始进行泊松重建。
心内膜表面几何模型三维重建算法研究
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【计算机应用研究】_数据采集系统_期刊发文热词逐年推荐_20140723
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科研热词 远程监控 设备管理 设备点检 计算机辅助审计 能量采集 编码 纹理获取 纹理图像 监管 深度图像 桥梁结构 曲面重建 无线传感器网络 数据库技术 数据库 数据传输 数据优先级 数据 故障诊断 审计方法 列车脱轨 分布式 入侵检测 光纤光栅传感器 信息安全 传输延时 传感器 代理技术 代理
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科研热词 推荐指数 马尔可夫链 1 预测与健康管理 1 预测 1 非线性校正 1 附加动量变学习速率法 1 路况监控 1 自组织网络 1 脉搏 1 网关矩阵排布技术 1 系统实现 1 简化路网模型 1 立交桥 1 移动医疗 1 环道 1 灰色系统 1 灰色神经网络模型 1 泊松表面重建 1 污染物排放量 1 模型修正 1 曲线拟合 1 智能手机 1 智能农业温室 1 故障预测 1 改进灰色神经网络 1 心内膜三维标测 1 应变式称重传感器 1 实时修正 1 农业物联网软件设计模型 1 云计算服务 1 三维重建 1 android 1
possion表面重建 向量场散度
英文回答：Postsion surface reconstruction is an important graphic reconstruction method that can recreate a smooth curve based on known cloud data. A key step in the reconstruction of the possion surface is to calculate the dispersion of vector fields. Vector field is a vector function defined at any point in space, which can be used to describe fluid motion, force field distribution, etc. The dispersion of the vector field is a measure of the difference between the source and the flow of the vector field, thus helping us to understand the characteristics of the fluid or force field. This technology is important under current policy for scientific research and engineering practice and helps to refine the approach and policy.possion表面重建是一种重要的图形重建方法，它可以根据已知的点云数据来重建出一个平滑的曲面。
在possion表面重建中，一个关键的步骤就是计算向量场的散度。
向量场是一个在空间中任意点上都有定义的向量函数，它可以用来描述流体的运动、力场的分布等。
泊松曲面重建法
泊松曲面重建法
泊松曲面重建法是一种三维模型重建方法，其基本思想是通过求解泊松方程实现曲面重建。
该方法通过将点云数据转换成一个能量函数，并利用泊松方程的边界条件来求解曲面。
泊松曲面重建法具有较好的鲁棒性和适应性，能够处理噪声点、不完整数据以及非连通点云等情况。
同时，该方法能够提供具有平滑性和逼近真实模型的曲面重建结果。
泊松曲面重建法的实现过程包括三个主要步骤：点云预处理、泊松方程求解和曲面重建。
在点云预处理中，需要进行数据清洗、采样和法向量计算等操作。
在泊松方程求解中，需要构建泊松方程矩阵，并通过求解线性方程组得到泊松曲面。
在曲面重建中，需要对泊松曲面进行拓扑操作和曲面平滑处理，以得到最终的三维模型。
泊松曲面重建法在计算机视觉、计算机图形学、医学图像处理等领域得到广泛应用，尤其在三维重建、虚拟现实、数字化建筑等方面具有重要意义。
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泊松重建算法 -回复
泊松重建算法-回复什么是泊松重建算法？泊松重建算法是一种用于图像处理和计算机图形学中的技术，其主要目标是将一个稀疏的、噪声污染的输入图像重建得到一个更加清晰、准确的图像。
该算法主要应用于填充缺失的图像区域、去除图像中的噪声以及图像增强等方面。
泊松重建算法的原理和步骤是什么？泊松重建算法的实现主要通过以下几个步骤来完成：1. 数据的准备：首先，需要将输入的图像数据进行预处理，包括提取感兴趣的图像区域，去除噪声，并将图像转化为适合处理的数据结构。
2. 建立泊松方程：泊松重建算法的核心是建立和求解泊松方程。
泊松方程是一个偏微分方程，用于描述物理现象和图像上的无边界问题。
在泊松重建算法中，泊松方程被用于传递信息，以填充缺失的图像区域或去除噪声。
3. 图像修复：通过求解泊松方程，可以得到缺失图像区域或受损图像的修复结果。
此时，泊松重建算法将根据已知的图像信息和边界条件，推导出缺失区域的像素值。
修复过程中，泊松方程提供了一个平滑约束，以确保修复结果具有良好的视觉效果。
4. 迭代求解：泊松重建算法通常需要进行迭代求解，以逐渐逼近最优解。
在每次迭代中，算法将根据当前的修复结果和泊松方程重新计算下一轮迭代的像素值。
迭代次数的选择取决于输入图像的复杂程度和修复的要求。
5. 后处理：在泊松重建算法完成修复后，还需要进行一些后处理步骤，以进一步提升修复结果的质量。
这些后处理步骤可以包括去除锯齿、增强对比度、应用滤波器等操作，以使修复后的图像更加真实和自然。
泊松重建算法的优缺点是什么？泊松重建算法有如下的优点：1. 结果准确：该算法基于泊松方程的物理模型，可以对缺失区域进行精确的重建，使得修复后的图像与原始图像尽量接近。
2. 平滑效果好：泊松重建算法通过泊松方程的平滑约束，可以有效地去除图像噪声，使修复后的图像更加平滑，具有良好的视觉效果。
3. 参数少，易于调整：泊松重建算法的参数较少，通常只需要调整迭代次数和平滑约束系数即可，易于使用和调试。
泊松表面重建
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Copyright 2008 C. Gotsman & M. Kazhdan Technion, Johns Hopkins Univ.
Poisson Surface Reconstruction
Motivation
Scanning often gives only local connectivity
Surface Reconstruction
Generate a mesh from a set of surface samples
Digital Geometry Processing Poisson Mesh Reconstruction
Motivation
In many domains, scanners are used to obtain virtual representations of 3D shapes
/digitalhammurabi/ /projects/mich/
泊松曲面重建
泊松曲面重建引言泊松曲面重建是一种常用的三维几何重建方法，可以根据一组离散的点云数据还原出原始的曲面模型。
该方法利用了曲面上的泊松方程及调和函数的性质，通过求解泊松方程来实现对曲面的还原。
本文将介绍泊松曲面重建方法的原理、流程和应用。
方法原理泊松曲面重建的核心原理是基于泊松方程的调和函数性质。
对于给定的离散点云数据，我们可以构造一个离散的拉普拉斯算子，该算子可以用来近似表示原始曲面上的调和函数。
泊松方程可以描述该调和函数与给定边界条件之间的关系，通过求解泊松方程，可以得到原始曲面的近似表示。
具体而言，泊松曲面重建的方法流程如下：1.数据预处理：对于给定的点云数据，首先需要进行预处理，包括去噪、采样等操作。
去噪可以通过滤波等方法实现，采样可以选择合适的点云密度，以保证重建结果的精度和效果。
2.构造拉普拉斯算子：构造一个离散的拉普拉斯算子，用于描述给定离散点云数据上的调和函数。
常用的方法是使用固定半径邻域，对每个点的邻域进行加权平均，将原始点云数据转化为一个稀疏的线性方程组。
3.求解泊松方程：在给定的边界条件下，求解离散泊松方程，得到曲面上的调和函数的近似表示。
这里可以使用迭代法或矩阵分解等方法进行求解。
4.曲面重建：根据求解得到的调和函数近似表示，通过插值等方法还原出原始曲面的离散模型。
可以使用三角化方法将曲面表示为三角网格模型，也可以使用其他方法进行曲面重建。
应用领域泊松曲面重建在三维重建、计算机图形学和计算机视觉等领域有着广泛的应用。
三维重建泊松曲面重建可以被应用于三维重建中，将离散的点云数据转化为连续的曲面模型。
三维重建在计算机辅助设计、虚拟现实、医学影像处理等领域都有着广泛的应用，泊松曲面重建方法能够提供高质量的曲面重建结果。
计算机图形学在计算机图形学中，泊松曲面重建可以用于建模和渲染。
通过将离散的点云数据转化为曲面模型，可以实现更精确和真实的模拟和渲染效果。
泊松曲面重建方法能够提供高度平滑的曲面表示，为计算机图形学中的建模和渲染任务提供了可靠的基础。
泊松表面重建.
泊松表面重建摘要：我们展示了对有向点集的表面重建可以转化为一个空间泊松问题。
这种泊松公式化表达的方法同时考虑了所有的点，而不借助于启发式的空间分割或合并，于是对数据的噪声有很大的抵抗性。
不像径向基函数的方案，我们的泊松方法允许对局部基函数划分层次结构，从而使问题的解缩减为一个良态的稀疏线性系统。
我们描述了一个空间自适应的多尺度算法，其时间和空间复杂度正比于重建模型的大小。
使用公共提供的扫描数据进行实验，在重建的表面，我们的方法比先前的方法显示出更详细的细节。
1、 引言：由点样本重建三维表面在计算机图形学中是一个热门研究问题。
它允许对扫描数据的拟合，对表面空洞的填充，和对现有模型的重新构网。
我们提出了一种重要的方法，把表面重建问题表示为泊松方程的解。
跟许多先前的工作一样(参见第2部分)，我们使用隐式函数框架来处理表面重建问题。
特别地，像[K a z 05]我们计算了一个三维指示函数 (在模型内部的点定义为1，外部的点定义为0)，然后可以通过提取合适的等值面获得重建的表面。
我们的核心观点是从模型表面采样的有向点集和模型的指示函数之间有一个内在关系。
特别地，指示函数的梯度是一个几乎在任何地方都是零的向量场(由于指示函数在几乎任何地方都是恒定不变的)，除了模型表面附近的点，在这些地方指示函数的梯度等于模型表面的内法线。
这样，有向点样本可视为模型的指示函数梯度的样本(如图1)。
图1 二维泊松重建的直观图例计算指示函数的问题因此简化为梯度算子的反算，即找到标量函数~χ，使其梯度最佳逼近样本定义的向量场V u r ，即，，如果我们使用散度算子，那么这个变化的问题转化为标准的泊松问题：计算标量函数~χ，它的拉普拉斯算子(梯度的散度)等于向量场V u r的散度，在第3、4部分我们将会对上式作精确的定义。
把表面重建问题表达成泊松问题拥有许多优点。
很多对隐式表面进行拟合的方法把数据分割到不同区域以进行局部拟合，然后使用合成函数进一步合并这些局部拟合结果。
泊松重建法向量判定-概述说明以及解释
泊松重建法向量判定-概述说明以及解释1.引言1.1 概述概述部分的内容：引言部分旨在介绍本篇长文的主要内容和目的。
本文将重点讨论泊松重建法和法向量判定的相关问题。
泊松重建法是一种基于离散点云数据的重建方法，其通过对点云的采样和插值，利用泊松方程对缺失的部分进行补全，从而实现对物体的三维重建。
本文将首先对泊松重建法进行详细介绍，包括原理、算法流程和应用领域等方面。
同时，将重点讨论泊松重建法存在的一些问题和挑战，并提出相应的解决办法。
其次，本文还将深入探讨法向量判定的重要性和应用场景。
法向量是表征物体表面方向和形状的重要属性，对于三维模型的后续处理、渲染和分析具有重要作用。
然而，在点云数据中获取准确的法向量信息是一项具有挑战性的任务，在实际应用中存在着一些困难和问题。
因此，本文将介绍目前常用的法向量计算方法，并对其优缺点进行分析和比较。
最后，本文将对泊松重建法和法向量判定的研究进行总结，并展望未来的研究方向和发展趋势。
希望通过本篇长文的阐述，能够对读者深入了解泊松重建法和法向量判定的理论基础和实际应用有所帮助。
1.2 文章结构文章结构部分的内容可以包括以下内容：文章结构部分旨在向读者介绍本文的组织结构和各个章节的主要内容。
本文采用如下的文章结构：第一部分是引言部分。
在这一部分中，我们将概述本文的背景和目的，为读者提供一个整体的了解。
第二部分是正文部分，主要包括泊松重建法和法向量判定两个小节。
在泊松重建法的部分，我们将详细介绍泊松重建法的原理和算法。
我们将首先解释什么是泊松重建法，它被广泛用于三维重建和几何处理中。
然后，我们将逐步介绍泊松重建法的基本步骤和关键技术。
最后，我们将讨论一些常见的应用场景和问题，并提供相应的解决方案。
在法向量判定的部分，我们将介绍法向量的概念和在几何处理中的重要性。
我们将解释什么是法向量判定，以及它在模型重建、曲面渲染等方面的应用。
然后，我们将介绍一些常用的法向量判定算法和技术，并讨论它们的优缺点以及适用场景。
泊松曲面重建原理
泊松曲面重建原理一、前言泊松曲面重建是一种三维点云重建技术，可以将无序的点云数据转换为平滑的曲面模型。
在计算机视觉和计算机图形学领域中得到广泛应用。
本文将详细介绍泊松曲面重建的原理。
二、点云数据点云数据是由大量离散的三维坐标点组成的数据集合，通常来自于激光雷达、摄像头或其他测量设备。
点云数据可以表示物体表面的三维形状信息，但由于其离散性和噪声干扰等问题，很难直接进行可视化或分析。
三、网格模型网格模型是一种基于三角形片元构成的连续曲面模型，可以通过有限数量的顶点和连接它们的边来描述物体表面。
网格模型具有可视化效果好、易于处理等优点，在计算机图形学和计算机辅助设计等领域得到广泛应用。
四、泊松方程泊松方程是一个偏微分方程，用于描述在给定边界条件下的物理场问题。
在数学上，它可以用拉普拉斯算子表示为∇²u=f(x,y,z)，其中u是要求解的函数，f(x,y,z)是已知的边界条件。
在计算机图形学中，泊松方程可以用于曲面重建和图像处理等领域。
五、泊松曲面重建泊松曲面重建是一种基于泊松方程的曲面重建方法，其基本思想是将点云数据看作泊松方程的边界条件，并通过求解泊松方程得到平滑的曲面模型。
具体步骤如下：1. 采样：对点云数据进行采样，以减少数据量和噪声干扰。
2. 重心插值：将采样后的点云数据转换为三角形网格模型，并使用重心插值法计算每个三角形片元上的顶点坐标。
3. 泊松方程求解：将三角形网格模型看作泊松方程的边界条件，通过求解泊松方程得到平滑的曲面模型。
4. 网格优化：对生成的曲面模型进行网格优化，以提高拓扑结构和减少三角形数量。
5. 后处理：对优化后的曲面模型进行后处理，如纹理映射、光照计算等操作。
六、优缺点分析泊松曲面重建方法具有以下优点：1. 可以处理大规模的点云数据，适用于复杂的三维形状重建。
2. 生成的曲面模型平滑、连续，具有较好的几何特征和拓扑结构。
3. 可以处理噪声干扰和不完整数据，具有较好的鲁棒性。
三维模型表面重构算法
三维模型表面重构算法
三维模型表面重构算法是一种用于从点云数据生成三维表面模型的算法。
以下是几种常见的三维模型表面重构算法：
1. Poisson表面重建算法：该算法通过最小化表面能量函数来重建三维表面。
它使用迭代优化技术，不断优化表面形状，直到达到收敛为止。
该算法可以生成高质量的三维表面，但计算复杂度较高。
2. Ball Pivoting算法：该算法通过旋转一个球体并检测球体与点云数据的交点来重建三维表面。
它使用迭代方式不断优化表面形状，最终生成三维表面模型。
该算法计算效率较高，但需要手动选择球体半径参数。
3. Marching Cubes算法：该算法是一种基于体素的表面重建算法，它通过在三维数据场中遍历体素并提取表面三角形来重建三维表面。
该算法计算效率较高，但生成的表面模型质量较低。
4. Poisson-based Marching Cubes算法：该算法是Marching Cubes算法和Poisson表面重建算法的结合，它使用Marching Cubes算法提取体素表面三角形，然后使用Poisson 表面重建算法对三角形进行优化处理，最终生成高质量的三维表面模型。
这些算法各有优缺点，应根据具体情况选择合适的算法来重建三维表面模型。
pcl库中的泊松曲面重建算法
pcl库中的泊松曲面重建算法
PCL（点云库）中的泊松曲面重建算法是一种用于从点云数据中重建光滑曲面的方法。
泊松曲面重建算法基于泊松方程，通过将点云数据转换为隐式函数表示的曲面来实现重建。
这种方法适用于从离散的点云数据中生成平滑的曲面模型，常用于三维重建、计算机辅助设计和医学图像处理等领域。
泊松曲面重建算法的原理是通过将点云数据转换为泊松方程的右侧项，然后解泊松方程来得到曲面的隐式表示。
该算法首先对点云数据进行表面重建，然后利用泊松方程的性质进行平滑处理，最终得到光滑的曲面模型。
在PCL库中，泊松曲面重建算法通常通过PoissonReconstruction类来实现。
该类提供了对点云数据进行曲面重建的功能，用户可以通过设置一些参数来调整重建的精度和平滑度。
在使用该算法时，需要注意点云数据的质量和密度对重建结果的影响，以及合适的参数设置对最终曲面模型的影响。
除了PCL库中的泊松曲面重建算法，还有其他一些开源库和软件也提供了类似的功能，如Open3D、MeshLab等，它们也可以用于
进行泊松曲面重建。
总的来说，泊松曲面重建算法是一种常用的从点云数据中重建光滑曲面的方法，它在PCL库中得到了很好的实现和应用。
通过合理设置参数和处理点云数据，可以得到高质量的曲面重建结果，满足不同应用场景的需求。
泊松表面重建
Eurographics Symposium on Geometry Processing(2006)Konrad Polthier,Alla Sheffer(Editors)Poisson Surface ReconstructionMichael Kazhdan1,Matthew Bolitho1and Hugues Hoppe21Johns Hopkins University,Baltimore MD,USA2Microsoft Research,Redmond WA,USAAbstractWe show that surface reconstruction 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 allows 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 publicly available scan data,we demonstrate reconstruction of surfaces with greater detail than previously achievable.1.IntroductionReconstructing3D surfaces from point samples is a well studied problem in computer graphics.It allowsfitting of scanned data,filling of surface holes,and remeshing of ex-isting models.We provide a novel approach that expresses surface reconstruction as the solution to a Poisson equation. Like much previous work(Section2),we approach the problem of surface reconstruction using an implicit function framework.Specifically,like[Kaz05]we compute a3D in-dicator functionχ(defined as1at points inside the model, and0at 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 the indicator function of the model.Specifically,the gra-dient of the indicator function is a vectorfield 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 oriented point samples can be viewed as samples of the gradient of the model’s indicator function(Figure1).The problem of computing the indicator function thus re-duces to inverting the gradient operator,i.e.finding the scalar functionχwhose gradient best approximates a vectorfield V defined by the samples,i.e.minχ ∇χ− V .If we applythe divergence operator,this variational problem transforms into a standard Poisson problem:compute the scalar func-111F M00111Indicator functionF MIndicator gradient Surfacew M Oriented pointsVGFigure1:Intuitive illustration of Poisson reconstruction in2D.tionχwhose Laplacian(divergence of gradient)equals the divergence of the vectorfield V,∆χ≡∇·∇χ=∇· V.We will make these definitions precise in Sections3and4. Formulating surface reconstruction as a Poisson problem offers a number of advantages.Many implicit surfacefitting methods segment the data into regions for localfitting,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 radial 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 to a well-conditioned sparse linear system.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。
泊松表面重建1
实验 结果 (shíyàn)
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10。实验场景包括牛牛实验室，和图书馆。为度深树
实验室稀疏点云
共十九页
表面重建结果
实验 结果 (shíyàn)
图书馆稀疏(xīshū) 点云
表面重建结果
共十九页
总结(zǒngjié)
例图观直的(nàij ɡnòhz)建重松泊维二
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隐函数(hánshù)
创建一个指示函数来表示表面(biǎomiàn)模型
Mp 1 0
i i
fpM fpM
0பைடு நூலகம்
0
1
1 0
0 1
0
指示函数
M
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构造 指示函数 (gòuzào)
如何(rúhé)构造指示函数？
原始点集
指示函数
M
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构造 泊松方程 (gòuzào)
（3）基于隐式曲面重建法； （4）基于统计学重建法。
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三维场景表面 重建 (biǎomiàn)
整完到得难很也景场的富丰不理纹，理处散扩过经使即。均不布分、疏稀、大声噪，云点间空维三的成生像图由
洞孔充填理合、据数声噪滤过、构结扑拓的面表断推要需(ǎfgnāf)法方建重，在存的题问些这于由。云点密稠的 上以决解地效有以可法算建重面表松泊的出提等Michael Kazhdan而，等络网型模分划、据数样采整调、
。建重面表行进云点的出建重像图对现实来法算建重面表松泊用采此因。题问
东北大学信息楼
共十九页
体育馆
泊松表面 重建 (biǎomiàn)
隐函数 方法 (hánshù)
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东北大学信息楼
体育馆
泊松表面重建
隐函数方法
计算了一个三维指示函 数 ( 在模型内部的点定 义为1，外部的点定义 为0) 提取合适的等值面获得 重建的表面
<0
0
>0
泊松表面重建
二维泊松重建的直观图例
隐函数
创建一个指示函数来表示表面模型
0 0 1 1 0 1 0
1 if p M M p 0 if p M
Thank You
算法实现
输入有向点云数据:
建立八叉树空间 计算向量场 解泊松方程求指示函数 提取等值面
实现: 适应的八叉树空间
实现: 向量场
实现：解泊松方程
min V
构造稀疏矩阵解泊松方程求解指示函数
V
实现: 表面提取
为了获得重建表面 ，首先需要选择一个等值，然后通过计算指示函数提 取对应的等值面。选择等值使得提取的等值面近似逼近输入的样本点的 位置。方案是首先通过在样本点的位置估计 ，然后使用平均值来提取等 值面：
0
指示函数
M
构造指示函数
如何构造指示函数？
原始点集
指示函数
M
构造泊松方程
由已知有向点云估计向量场 V 求解函数 ，它的梯度最接近 向量场 : V
min V
应用散度算子,我们可以将这变成一个泊松 问题: V V
泊松表面重建
表面重建
由点样本重建三维表面
表面重建方法
散乱点云的网格重建方法发展到现在，主要有如 下几个研究方向: （1）基于Delaunay重建法； (2）区域扩张重建法； （3）基于隐式曲面重建法； （4）基于统计学重建法。
ቤተ መጻሕፍቲ ባይዱ
三维场景表面重建
由图像生成的三维空间点云，噪声大、稀疏、分布不均。即使经过扩散处理，纹理不 丰富的场景也很难得到完整的稠密点云。由于这些问题的存在，重建方法需要推断表 面的拓扑结构、过滤噪声数据、合理填充孔洞、调整采样数据、划分模型网络等，而 Michael Kazhdan等提出的泊松表面重建算法可以有效地解决以上问题。因此采用泊松 表面重建算法来实现对图像重建出的点云进行表面重建。
实验结果
在稀疏点云重建的基础上采用泊松表面重建算法重建场景的表面，取 八叉树深度为10。实验场景包括牛牛实验室，和图书馆。
实验室稀疏点云
表面重建结果
实验结果
图书馆稀疏点云
表面重建结果
总结
1，由实验结果可知泊松表面重建可以很好地解决基于图像生成的三 维空间点云，噪声大、稀疏、分布不均等问题，可以满足不对模型精 度过高要求应用的需求。 2，只是对部分场景做了实验，还没完成大规模的场景表面重建。