Bernard_A_Solution_for_2015_CVPR_paper

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[ToG13]Poisson Surface Reconstruction

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 an indicator function χrather than a parametric function.This leads to 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 .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.。

机器学习领域的知名人物和论文

机器学习领域的知名人物和论文机器学习作为人工智能领域的重要分支及研究方向，不断涌现出许多杰出的知名人物以及具有重要影响力的论文。

这些人物和论文在推动机器学习技术发展和应用方面起到了重要的作用。

本文将介绍几位机器学习领域的知名人物以及他们的重要论文，带领读者了解机器学习领域的发展脉络和重要思想。

1. Andrew Ng（吴恩达）在机器学习领域，Andrew Ng无疑是一个家喻户晓的人物。

他是斯坦福大学的教授，并且曾经是谷歌的首席科学家。

他的重要贡献之一是创建了Coursera上非常著名的机器学习课程，该课程使得机器学习技术的学习变得更加便捷和可普及。

他的学术研究涉及深度学习、神经网络以及数据挖掘等领域。

他的论文《Deep Learning》被广泛引用，对深度学习领域的发展起到了重要推动作用。

2. Geoffrey Hinton（杰弗里·辛顿）Geoffrey Hinton被誉为“深度学习之父”，他是深度学习领域的杰出研究者和学者。

他的重要贡献之一是开发了BP（Backpropagation）算法，该算法为神经网络的训练提供了有效的方法。

他还提出了“Dropout”技术，通过随机丢弃一些神经元的方式来防止神经网络的过拟合问题。

他的论文《Deep Neural Networks for Acoustic Modeling in Speech Recognition》对语音识别等领域产生了巨大的影响。

3. Yoshua BengioYoshua Bengio是加拿大蒙特利尔大学教授，也是深度学习领域的重要人物之一。

他在深度学习领域的贡献源远流长。

他的论文《Learning Deep Architectures for AI》介绍了深度学习的概念和技术，并提出了一种深度置信网络（Deep Belief Networks）的训练方法。

这篇论文的发表引发了深度学习的研究和应用的热潮。

4. Ian GoodfellowIan Goodfellow是深度学习领域的年轻研究者，其主要贡献是提出了生成对抗网络（GAN）的概念。

基于人工免疫网络的联想记忆器

止的条件。当实际应用模型有时间条件约束时，这种限制形式很
一
人工免疫网络（Ｎ）Ａ１已经用于解决优化问题】同时，，通过
研究人工免疫机制，就可以利用这些方法创建～种新型的联想记忆器。这种机制用于定义在ｒ维空间上的点群。点群表示上述ｌ
超平面。这拌一个点群中最佳点等价于吸引子。
所设计的记忆器学习和记忆阶段流程图如图１图２，所示。需要指出的是，在学习阶段首先必须定义形式空间。假设被记忆摸型是一个用ｎ维双极性向量表示的Ｐ模型集合。则形式空间是一个超立方体，它的中心在轴心，并且边长为２势为的初始群体，是在整个形式空间产生的。对群体中每一个个体Ｉ，在那些表示群体中的个体中，通过调换一个随机选择的基因位来确定候选个体。
生物免疫系统的自修复机制。利用这些特性，在实数空间和海明
空间中都可以产生新的最优方法。本文中，于这些最优方法简基
述了人工免疫阿错联想记忆器模型的设计思想和实现方法下
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方面称为系统动态性。也就是控制一组确定的淋巴细胞繁殖
和相应的免疫蛋白的数量增减的微分方程。另一方面称为系统的后动态性。这是一种算法。谈算法控制从群体免疫细胞中除去一定的克隆细胞，同时控制来自骨髓产生的新淋巴细胞池中的新繁殖细胞的补充机制。免疫网络的学习和自适应特性来自于

外文翻译---基于模糊逻辑技术图像上边缘检测

译文二：1基于模糊逻辑技术图像上边缘检测[2]摘要：模糊技术是经营者为了模拟在数学水平的代偿行为过程的决策或主观评价而引入的。

下面介绍经营商已经完成了的计算机视觉应用。

本文提出了一种基于模糊逻辑推理战略为基础的新方法，它被建议使用在没有确定阈值的数字图像边缘检测上。

这种方法首先将用3⨯3的浮点二进制矩阵将图像分割成几个区域。

边缘像素被映射到一个属性值与彼此不同的范围。

该方法的鲁棒性所得到的不同拍摄图像将与线性Sobel运算所得到的图像相比较。

并且该方法给出了直线的线条平滑度、平直度和弧形线条的良好弧度这些永久的效果。

同时角位可以更清晰并且可以更容易的定义。

关键词：模糊逻辑，边缘检测，图像处理，电脑视觉，机械的部位，测量1.引言在过去的几十年里，对计算机视觉系统的兴趣，研究和发展已经增长了不少。

如今，它们出现在各个生活领域，从停车场，街道和商场各角落的监测系统到主要食品生产的分类和质量控制系统。

因此，引进自动化的视觉检测和测量系统是有必要的，特别是二维机械对象[1,8]。

部分原因是由于那些每天产生的数字图像大幅度的增加(比如，从X光片到卫星影像)，并且对于这样图片的自动处理有增加的需求[9,10,11]。

因此，现在的许多应用例如对医学图像进行计算机辅助诊断，将遥感图像分割和分类成土地类别(比如，对麦田，非法大麻种植园的鉴定，以及对作物生长的估计判断)，光学字符识别，闭环控制，基于目录检索的多媒体应用，电影产业上的图像处理，汽车车牌的详细记录的鉴定，以及许多工业检测任务(比如，纺织品，钢材，平板玻璃等的缺陷检测)。

历史上的许多数据已经被生成图像，以帮助人们分析(相比较于数字表之类的，图像显然容易理解多了)[12]。

所以这鼓励了数字分析技术在数据处理方面的使用。

此外，由于人类善于理解图像，基于图像的分析法在算法发展上提供了一些帮助(比如，它鼓励几何分析)，并且也有助于非正式确认的结果。

虽然计算机视觉可以被总结为一个自动(或半自动)分析图像的系统，一些变化也是可能的[9,13]。

傅里叶望远术的实验室验证系统

傅里叶望远术的实验室验证系统
董洪舟，吴健，刘艺，张炎
（电子科技大学光电信息学院，成都６０５１０４）
摘要：本文介绍了傅里叶望远术成像的基本原理，为验证傅里叶望远术成像原理，在实验室中建了四光束的傅搭
里叶望远术验证成像系统，对灰度透射式目标进行成像验证，利用ＬｂＩＷ软件完成了实验中相关控制软件、ａＶＥ检测软件和信号处理软件程序设计。通过形成不同空间频率的干涉条纹提取目标的频谱值，利用相位闭合技术，
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收稿日期：２１６１；收到修改稿日期：２１一９００卜０— １０ｉ０— ５
０引言
随着航空航天技术的日益发展，间目空标的高分辨率成像技术成为一个重要研究方向¨ Ｊ其中傅里叶，
望远术是一种具有较大发展潜力的成像技术。它的成像原理是利用直线干涉条纹场扫描目标来获取携带目
标频谱信息的回波信号，并通过相位闭合技术消除光束初始相位和大气湍流带来的相位畸变，最终还原出

工业自动化技术讲稿5人工神经网络与神经网络控制课件

人工神经网络与神经网络控制
胆道疾病病人护理化工企业本质安全理论实践及方法内科护理学呼吸系统总论概论脾胃病常见症状及治疗经验偏瘫截瘫康复训练手册偏执性精神障碍品管圈实践
（2）神经网络控制应用
基于神经网络的特点，神经网络在控制系统中可充当系统模型、直接做控制器或提供优化算法。它已被应用到自动控制领域的各个方面，包括系统建模与辨识、优化设计、预测控制、最优化控制、自适应控制、容错控制、模糊控制、专家控制和学习控制等等。
人工神经网络与神经网络控制
胆道疾病病人护理化工企业本质安全理论实践及方法内科护理学呼吸系统总论概论脾胃病常见症状及治疗经验偏瘫截瘫康复训练手册偏执性精神障碍品管圈实践
（1）神经网络控制概念
神经网络控制是研究和利用人脑的某些结构机理以及人的知识和经验对系统的控制。利用神经网络，可以把控制问题看成模式识别问题．被识别的模式是映射成“行为”信号的“变化”信号。神经控制最显著的特点是具有学习能力。它是通过不断修正神经元之间的连接权值，并离散存储在连接网络来实现的。它对非线性系统和难以建模的系统的的控制具有良好效果。
1943年，生理学家W.S.McCulloch和数学家 W.A.Pitts提出了神经元M-P模型。
1958年，计算机学家Frank Rosenblatt提出了 “感知器”（Perceptron）。
人工神经网络与神经网络控制
胆道疾病病人护理化工企业本质安全理论实践及方法内科护理学呼吸系统总论概论脾胃病常见症状及治疗经验偏瘫截瘫康复训练手册偏执性精神障碍品管圈实践
4神经网络的工作方式及其特点
学习期

System, Method, and Computer Program Product for A

专利名称：System, Method, and Computer ProgramProduct for Accessing Manipulating RemoteDatasets发明人：Bei Gu,Kenneth Howard,Eric J. Kaplan,PeterChirlian,Aleksandr Shukhat申请号：US13567841申请日：20120806公开号：US20130031050A1公开日：20130131专利内容由知识产权出版社提供专利附图：摘要：A method, system and computer program product for creating a report on thebasis of a plurality of remote datasets includes an intelligence server, one or more tree servers and one or more databases. Each tree server creates one or more segments, or slices, of a report, using information that resides on the tree server. Slices are aggregated into a tree structure, and the tree structure is converted into a report. The intelligence server receives updates from the tree servers. The tree servers and the intelligence server remain in communication for the purposes of passing update messages. The update messages are received and processed at the intelligence--server in a manner that facilitates synchronization with the contributing tree servers and provides live updates to the user,申请人：Bei Gu,Kenneth Howard,Eric J. Kaplan,Peter Chirlian,Aleksandr Shukhat地址：Hillsborough NJ US,Stony Point NY US,Allendale NJ US,Basking Ridge NJUS,Milburn NJ US国籍：US,US,US,US,US更多信息请下载全文后查看。

基于遗传算法的RBF神经网络在铂电阻温度传感器非线性补偿中的应用

基于遗传算法的ＲＦ神经网络在铂电阻Ｂ温度传感器非线性补偿中的应用
董玲娇
（温州职业技术学院电气电子工程系，浙江温州３５３）２０５
摘要：针对铂电阻温度传感器在实际应用中存在非线性问题，提出了基于遗传算法优化径向基函数（Ｂ）ＲＦ神经网络实
第２８卷第１期２１００年２月
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现其非线性补偿的方法。分析了非线性补偿原理。设计了ＲＦ神经网络补偿器，Ｂ并引入遗传算法优化神经网络结构和
参数。实验结果表明，所提出的铂电阻温度传感器非线性补偿方法是实用和可行的。图４表１参１０关键词：控制技术；温度非线性补偿；径向基函数神经网络；遗传算法

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1. Introduction
The alignment of a set of objects by means of transformations plays an important role in the ﬁeld of computer vision and recognition. For instance, for the creation of statistical shape models (SSMs) [5] training shapes are initially aligned for removing pose differences in order to only model shape variability. The most common way of shape representation is by encoding each shape as a point-cloud. In order to be able to process a set of shapes it is necessary that correspondences
A Solution for Multi-Alignment by Transformation Synchronisation
Florian Bernard1,2,3 Johan Thunberg2 Andreas Husch1,2,3
1 2
Peter Gemmar3 Jorge Goncalves2
the set of all pairwise transformations in such a way that they globally exhibit transitive consistency. Experiments demonstrate the effectiveness of this method in denoising noisy pairwise transformations. Furthermore, using this novel method the GPA is solved in an unbiased manner in closed-form, i.e. non-iterative. Transformation synchronisation is applied to solve the GPA with missing data as well as with wrong correspondence assignments and results in superior performance compared to existing methods. Our main contribution is a generalisation of the techniques presented by Singer et al. [3, 9, 10, 17], who have introduced a method for minimising global self-consistency errors between pairwise orthogonal transformations based on eigenvalue decomposition and semideﬁnite programming. With permutation transformations being a subset of orthogonal transformations, in [14] the authors demonstrate that the method by Singer et al. is also able to effectively synchronise permutation transformations for globally consistent matchings. In our case, rather than considering the special case of orthogonal matrices, we present a synchronisation method for invertible linear transformations. Furthermore, it is demonstrated how this method can be applied for the synchronisation of similarity, euclidean and rigid transformations, which are of special interest for the groupwise alignment of shapes. Whilst the proposed synchronisation method is applicable in many other ﬁelds where noisy pairwise transformations are to be denoised (e.g. groupwise image registration or multi-view registration), in this paper GPA is used as illustrating example.
between all shapes are established. Whilst there is a vast amount of research in the ﬁeld of shape correspondences (for an overview see [11, 18]), in this paper we focus on the alignment of shapes and we assume that correspondences have already been established. The alignment of two objects by removing location, scale and rotation is known as Absolute Orientation Problem (AOP) [13] or Procrustes Analysis [8]. For the AOP there are various closed-form solutions, among them methods based on singular value decomposition (SVD) [1, 16]; based on eigenvalue decomposition [13]; based on unit quaternions [12] or based on dual quaternions [19]. A comparison of these methods [6] has revealed that the accuracy and the robustness of all methods are comparable. The alignment of more than two objects is known as Generalised Procrustes Analysis (GPA). Whilst a computationally expensive global solution for GPA in two and three dimensions has been presented in [15], the most common way for solving the GPA is to align the objects with a reference object. However, ﬁxing any of the objects as reference induces a bias. An unbiased alternative is to align all objects with the adaptive mean object as reference. An iterative algorithm then alternatingly updates the reference object and estimates the transformations aligning the objects. The iterative nature of these methods constitutes a problem if the relative transformation between any pair of objects is noisy. This is for example the case if data is missing, correspondences are wrong or if the transformations are observed by independent sensors (e.g. non-communicating robots observe each other). Noisy relative transformations can be characterised by transitive inconsistency, i.e. transforming A to B and B to C might lead to a different result than transforming A directly to C . This paper presents a novel method for synchronising
{johan.thunberg,jorge.goncalves}@uni.lu
3
Trier University of Applied Sciences, Germany
p.gemmar@hochschule-Hale Waihona Puke rier.deAbstract