Registration of Point Cloud Data from a Geometric Optimization Perspective

Transcription

1 Eurographics Syposiu on Geoetry Processing (2004) R. Scopigno, D. Zorin, (Editors) Registration of Point Cloud Data fro a Geoetric Optiization Perspective Niloy J. Mitra, Natasha Gelfand, Helut Pottann, and Leonidas Guibas Coputer Graphics Laboratory, Stanford University Geoetric Modeling and Industrial Geoetry, Vienna University of Technology Abstract We propose a fraework for pairwise registration of shapes represented by point cloud data (PCD). We assue that the points are sapled fro a surface and forulate the proble of aligning two PCDs as a iniization of the squared distance between the underlying surfaces. Local quadratic approxiants of the squared distance function are used to develop a linear syste whose solution gives the best aligning rigid transfor for the given pair of point clouds. The rigid transfor is applied and the linear syste corresponding to the new orientation is build. This process is iterated until it converges. The point-to-point and the point-to-plane Iterated Closest Point (ICP) algoriths can be treated as special cases in this fraework. Our algorith can align PCDs even when they are placed far apart, and is experientally found to be ore stable than point-to-plane ICP. We analyze the convergence behavior of our algorith and of point-to-point and point-to-plane ICP under our proposed fraework, and derive bounds on their rate of convergence. We copare the stability and convergence properties of our algorith with other registration algoriths on a variety of scanned data. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Coputer Graphics]: Coputational Geoetry and Object Modelling 1. Introduction Registration plays an iportant role in 3D odel acquisition, object recognition, and geoetry processing. Given as input two shapes, often called the odel and the data, each in its own coordinate syste, the goal of registration is to find a transforation that optially positions that data with respect to the odel. In this paper, we consider the registration proble when both the odel and data inputs are given as point cloud data (PCD). This is a coon proble in 3D scanning, where ultiple views of an object need to be The authors wish to acknowledge support fro NSF grant CARGO , a Max-Planck-Institut Fellowship, and a Stanford Graduate Fellowship. Part of this research has been carried out within the Copetence Center Advanced Coputer Vision and has been funded by the Kplus progra. This work was also supported by the Austrian Science Fund under grant P16002-N05. brought into a coon coordinate syste, and in geoetry processing, where point cloud representations need to be aligned for applications such as texture transfer, orphing, or waterarking [CWPG04]. A popular ethod for aligning two point clouds is the Iterated Closest Point (ICP) algorith [BM92, CM91]. This algorith starts with two point clouds and an estiate of the aligning rigid body transfor. It then iteratively refines the transfor by alternating the steps of choosing corresponding points across the point clouds, and finding the best rotation and translation that iniizes an error etric based on the distance between the corresponding points. Despite a large aount of work on registration, convergence behavior of any registration algoriths, under different starting conditions, and error etrics, is poorly understood. Experientally, it has been shown that the rate of convergence of ICP heavily depends on the choice of the corresponding point-pairs, and the distance function that is being iniized [RL01]. Many enhanceents of ICP-style

2 algoriths for registration propose different error etrics, and point selection strategies, to iprove ICP s convergence behavior [GIRL03, Fit01, JH03, CLSB92, RL01]. Two distance etrics are coonly used in ICP and its variants. The point-to-point distance of Besl [BM92] uses the Eucledian distance between the corresponding points. This leads to an ICP algorith that converges slowly for certain types of input data and initial positions. Another error etric is the point-to-plane distance of Chen and Medioni [CM91], which uses the distance between a point and a planar approxiation of the surface at the corresponding point. When the initial position of the data is close to the odel, and when the input has relatively low noise, ICP with point-to-plane error etric has faster convergence than the point-to-point version. However, when the shapes start far away fro each other, or for noisy point clouds, point-toplane ICP tends to oscillate and fails to converge [GIRL03]. Another reason behind the slow convergence of registration algoriths based on ICP, is the local nature of the iniization. The only inforation used by the algorith is the point correspondences. As a result, the iniized error function only approxiates the squared distance between the two point clouds up to first order. In this paper, we propose an optiization fraework for studying registration algoriths. We pose registration between two point clouds as an optiization over the space of rigid transfors. We develop an objective function that is a second order approxiant to the squared distance between the odel and the data. Higher order inforation about the surfaces represented by the point clouds, such as local curvatures, are incorporated into the quadratic approxiant. Using such approxiant to the squared distance function, we develop a registration algorith. When the odel and the data PCDs are close, our algorith has a rate of convergence siilar to ICP with point-to-plane error etric. Moreover, our ethod has a stable behavior even when the initial displaceent is large. We also explain the convergence properties of the point-to-point and point-to-plane ICP variants, in ters of the accuracy of the distance function that they use during iniization. 2. Registration of Point Cloud Data Let P = {p 1,p 2,...,p n} and Q = {q 1,q 2,...,q } be two point clouds in IR d. The goal of the registration algorith is to find a rigid body transfor α coposed of a rotation atrix R and a translation vector t that best aligns the data PCD Q to atch the odel PCD P. Registration algoriths based on ICP work as follows. Given the initial position of the data with respect to the odel, the algorith chooses a set of k point pairs (p i,q i ) fro the odel and the data. The distance between the odel and data PCDs is approxiated by the su of distances between the point pairs. The algorith then searches for the rigid transfor that iniizes the residual distance, ε, between the odel and the transfored data: k ε(α) = d 2 (α(q i ),p i ), (1) where d can be point-to-point distance of Besl or point-toplane distance of Chen and Medioni. Notice, however, that the basic assuption is that the su of squared distances between pairs of points is a good approxiation for the distance between two PCDs. In the point cloud setting, we actually know that the odel and data PCDs are not arbitrary collections of points, but are sapled fro soe underlying surfaces Φ P and Φ Q. In this case, it is ore appropriate to iniize the distance fro the data PCD to the surface represented by the odel PCD. Pottann and Hofer showed that when the data and the odel are close, the point-to-plane distance is a good approxiation to the distance between a data point and the surface represented by the odel PCD. On the other hand, when the odel and the data are far apart, the point-to-point distance is a better choice [PLH02]. The goodness of a given error etric is deterined by two properties. First, we would like the error etric to accurately reflect the distance between a data point q and the surface represented by the odel PCD. Second, we would like the distance approxiation to be valid not just at a point q, but in a neighborhood around it. Both point-to-point and point-toplane error etrics are based only on first order inforation about the underlying input surfaces. As a result, they do not provide a good approxiation to the distance when we ove around in the neighborhood of a data point q. In this paper, we are concerned with developing a good approxiation to the distance function between two point clouds. In order to show theoretical bounds on the convergence behavior of registration algoriths based on our distance function, we pose the proble of registration of two point clouds as an optiization proble over the space of rigid transfors. This leads to the following optiization proble: we are searching for the best rigid transfor α = (R,t) that iniizes the error given by ε(α) = d 2 (Rq i + t,φ P ), (2) where R is the rotation atrix and t is the translation vector. The function d 2 (Rq i + t,φ P ) is the distance fro the transfored data point q i to the surface represented by the PCD P. Given the optiization proble of Equation 2, it is clear that the convergence behavior of any such optiization procedure depends on the accuracy of the function d 2. We call d 2 the squared distance function to the surface Φ P. Since the surface Φ P fro which the point odel was sapled is generally coplicated and unknown, a good approxiant to the squared distance function is required.

3 Contributions of the present paper We develop a quadratic approxiant to the squared distance function to the surface represented by a point cloud, and use this approxiant in a registration algorith. Our approxiant has the desired property of being valid not only at the query point where it is coputed, but also in a neighborhood around the query point. This property allows us to pose the registration proble in an optiization fraework, and use ethods such as Newton iteration, that depend on coputing accurate derivatives of the objective function. In our optiization fraework using the squared distance function, point-to-point and point-to-plane ICP variants are reduced to two special cases of the general iniization proble. Our distance function approxiates the squared distance fro a data query point to the surface represented by the odel PCD up to second order. We develop two ethods for coputing such local quadratic approxiants. The first, uses local curvature of the surface to incorporate second order inforation into the squared distance function. The second ethod, approxiates the global error landscape by locally fitting quadric patches to the squared distance function to the surface. The quadric patches are stored in a special octree like structure called the d2tree. For any point q, the registration algorith queries this special structure for the corresponding approxiant to the squared distance to the surface. Unlike coon ICP variants, the d2tree data structure allows us to perfor registration without explicitly using nearest neighbors for correspondence. Both of the above techniques incorporate inforation about the shape of the neighborhoods of the input surface into the error function. As a result we get better convergence behavior than purely local ethods of ICP. Using our distance function, we develop a registration algorith for point clouds that has fast convergence and is ore stable behavior than standard ICP variants. When two point clouds are close to each other, our algorith has quadratic convergence, siilar to point-to-plane ICP. However, unlike point-to-plane ICP, our algorith has ore stable convergence behavior and is less prone to oscillations when the initial distance between the odel and the data is large. 3. Registration using the squared distance function In this section, we assue the existence of a function d 2 that for any point x IR d, gives the squared distance to the odel PCD surface Φ P. Such a squared distance function defines the error landscape for our objective function as indicated by Equation 2. So this function d 2 is iportant for registration algoriths. Later in Section 4, we describe how to generate local quadratic approxiants F + of this function d 2 (x,φ P ). Assuing these approxiants are available, we now show how the registration proble can be solved in a least squared sense by a gradient descent search. Siply put, we try to place one point cloud in the squared distance field of the other, in order to iniize the placeent error. Given Φ P, we expect the points near its edial axis MA(Φ P ) to have bad quadratic approxiants, since locally, the squared distance function is not sooth. If we detect such points, we leave the out of our optiization procedure. We eploy an iterative schee to solve the nonlinear optiization proble over the coplex error landscape. At each stage, we solve for a rigid transfor coposed of a rotation R followed by a translation t. We use F + to solve for the rigid transfor α = (R,t) that brings the data PCD Q to the odel PCD P. We apply this transfor and repeat Registration in 2D We first explain the process in 2D. At any point x = [x y] IR 2, we assue the availability of an approxiant F + such that F + (x) d 2 (x,φ P ). Let F + be specified in the for F + (x) = Ax 2 + Bxy +Cy 2 + Dx + Ey + F, where A,B,C,D,E,F are the coefficients of the approxiant. More copactly, we can write F + in a quadratic for as F + (x) = [ x y 1 ] Q x [ x y 1 ] T, (3) where Q x is a syetric atrix that depends on x. Iportantly, the approxiant F + (x) is a valid locally around x. We denote any point q i of the data PCD Q by [x i y i ]. Let a atrix R corresponds to a rotation by angle θ around the origin. Our goal is to solve for a rigid transfor, which consists of R followed by a translation vector t = [t x t y], that iniizes F+ (Rq i + t). For sall θ, we can linearize the rotation by using sinθ θ and cosθ 1. So after each iteration step, we get, [x i y i ] [x i θy i + t x θx i + y i + t y]. If locally Q qi approxiately stays fixed, the residual error between the transfored data and the odel PCDs is given by ε(θ,t x,t y) = [ x i θy i +t x θx i + y i +t y 1 ]Q qi (4) [ x i θy i +t x θx i + y i +t y 1 ] T, where Q qi denotes the atrix representing the approxiant of the squared distance function to Φ P around the point q i. Our goal is to find values of θ, t x, and t y that iniizes this residual error. Setting the respective partial derivatives of the error ε to zero, we get the following linear syste I i J i K i θ J i 2A i B i t x = K i B i 2C i t y L i 2A i x i + B i y i + D i, (5) B i x i + 2C i y i + E i

4 where I i = 2(C i x 2 i B i x i y i + A i y 2 i ), J i = B i x i 2A i y i, K i = 2C i x i B i y i, L i = B i (x 2 i y 2 i ) + 2(C i A i )x i y i + E i x i D i y i and, A i,b i,c i,d i,e i,f i denotes the entries of the atrix Q qi. The transforation resulting fro solving Equation 5 is applied to Q. This copletes one iteration of our gradient descent process. Next we use the approxiants corresponding to the new positions of q i to get another linear syste whose solution is again applied to the data PCD. This process is iterated until the residual error falls below a pre-defined error threshold or a axiu nuber of iteration steps is reached. Since we do not ake any assuption about the initial alignent of P and Q, the rigid transfor coputed at any step, can be large. In such cases, we can only take sall steps in the direction of the transfor because its coputation is based on approxiants that are valid locally. This issue of applying a 1/η-fraction of a rigid transfor is an iportant proble and has been studied in depth in other places [Ale02]. We propose a siple way for coputing fractional transfors. In our notation, the coputed transfor vector [θ t x t y] denotes a rigid transfor coposed of a rotation atrix R followed by a translation vector t. This aps q to a point Rq + t. Let the fractional transfor be coposed of a rotation atrix R and a translation vector t. We define (R,t ) to be a 1/η fraction of (R,t) if the following relation holds, (..(R (R q + t ) + t )...η ties) Rq + t. (6) Fro this relation, we can get R = R 1/η, and t = (R I) 1 (R I)t where, I is the identity atrix. By R 1/η, we ean a rotation around the origin by angle θ/η. We defer the iportant issue of choosing a suitable value for η to Section Registration in 3D In this section, we extend the results fro the previous section to point cloud surface data in IR 3. For any point x = [x y z] IR 3, let the corresponding local quadratic approxiant F + be specified in the for F + (x) = Ax 2 + Bxy +Cy 2 + Dxz + Eyz + Fz 2 + Gx + Hy + Iz, where A through I are the coefficients of the quadratic approxiant. With slight abuse of notation, this equation can be written in a quadratic for as F + (x) = xq xx T, where x now denotes [x y z 1]. Once again, our goal is to find the rigid transfor which brings Q in best alignent with P. Let the rigid transfor be coposed of a rotation atrix, R, that is paraeterized by three angles (α,β,γ) in the X-Y-Z fixed angle orientation convention, followed by a translation vector t = [t x t y t z]. (7) Under sall otion, the rotation atrix can be linearized as, cosα sinα 0 cosβ 0 sinβ R = sinα cosα sinβ 0 cosβ cosγ sinγ 0 sinγ cosγ 1 α β α 1 γ β γ 1 Hence q Rq + t.. Now the registration proble reduces to finding values of α,β,γ,t x,t y,t z that iniize the residual error (8) ε(α,β,γ,t x,t y,t z) = (Rq i + t)q qi (Rq i + t) T. (9) This least square proble can be solved by setting the respective partial derivatives to zero. The resulting linear syste is given by [ ( Pi S i where, P i = S i = S T i )] [ α β γ tx t y t z ] T = R i U i V i W i, 2A i x i + B i y i + D i z i + G i B i x i + 2C i y i + E i z i + H i J i M i N i M i K i T i N i T i L i D i x i + E i y i + 2F i z i + I i, B i x i 2A i y i 2C i x i B i y i E i x i D i y i D i x i + 2A i z i E i x i + B i z i 2F i x i + D i z i D i y i B i z i E i y i 2C i z i 2F i y i E i z i 2A i B i D i R i = B i 2C i E i, D i E i 2F i J i = 2(C i x 2 i B i x i y i + A i y 2 i ), K i = 2(F i x 2 i D i x i z i + A i z 2 i ), L i = 2(F i y 2 i E i y i z i +C i z 2 i ), (10),

5 M i = E i x 2 i + D i x i y i + B i x i z i 2A i y i z i, N i = D i y 2 i + E i x i y i 2C i x i z i + B i y i z i, Let us look at two iportant special cases. T i = B i z 2 i 2F i x i y i + E i x i z i + D i y i z i, U i = B i (xi 2 y 2 For d = 0 we obtain i ) + 2(C i A i )x i y i + E i x i z i D i y i z i + H i x i G i y i, V i = D i (z 2 i xi 2 F d (x 1,x 2,x 3 ) = x3. 2 ) E i x i y i + 2(A i F i )x i z i + B i y i z i I i x i + G i z i, W i = E i (y 2 i z 2 i ) + D i x i y i B i x i z i + 2(F i C i )y i z i + I i y i H i z i, and A i through I i correspond to the entries of the atrix Q qi that represents the local quadratic approxiant around the point q i. As in the 2D case, whenever the coputed transfor (R,t) is large, we utilize a fractional transfor given by R = R 1/η and t = (R I) 1 (R I)t where, I denotes the identity atrix. A 1/η-fraction of the rotation atrix R can be coputed by the techniques proposed by Alexa [Ale02]. 4. Squared Distance Function Given a 3D point cloud P, we describe two ethods for constructing a quadratic approxiant F + to the squared distance function d 2 fro any point x IR 3 \ MA(Φ P ) to Φ P. At any point x, our goal is to construct an approxiant F + such that F + (x) d 2 (x,φ P ) is second order accurate. Points on the edial axis MA(Φ P ) have non-differentiable squared distance function and hence, their second order accurate approxiants do not exist. In the construction phase, we ensure that the approxiants are non-negative over IR 3, since F + is used as an objective function in a iniization process as shown in Section 3. In 2D, siilar approxiants can be easily coputed. Before we describe how to copute F + for a given PCD, we suarize a few basic results on the squared distance function of a surface as observed by Pottann and Hofer [PH03]. For each point on a given surface, we assue that the unit noral n along with the principal curvature directions e 1, e 2 are given. These three unit vectors cobine to for a local coordinate syste called the principal frae. At ubilical points, where the principal curvature directions are not well-defined, any two orthogonal unit vectors on the tangent plane ay be used as e 1, e 2. Let ρ i be the principal radius of curvature in the direction e i. The noral footpoint y denotes the closest point on the surface fro x. Let x 1,x 2,x 3 represent the coordinates of x in the principal frae at y. The signed distance fro x to its noral footprint is denoted by d. The sign of d is positive if x and the centers of the osculating circles at y, lie on the sae side of the surface around y. The second order Taylor approxiant [PH03] of the squared distance function to the surface at a point x can be expressed in the principal frae at y as F d (x) = F d (x 1,x 2,x 3 ) = d x1 2 + d x2 2 + x d ρ 1 d ρ 3. 2 (11) 2 We shall use δ j, j = 1,2, to denote d/(d ρ j ). Thus, if we are close to the surface, i.e. in its near-field, the squared distance function to the tangent plane at the noral footpoint, is a quadratic approxiant. For d = we obtain F d (x 1,x 2,x 3 ) = x x x 2 3, which is the squared distance fro x to its footpoint y. So the distances to noral footpoints are second order accurate if we are in the far-field of the surface. In order to use F d as an objective function for a iniization, we want the approxiant to be non-negative over IR 3. To this end, we replace δ j with { d/(d ρ j ) if d < 0, ˆδ j = 0 otherwise. The resulting approxiant F + is positive definite and is given by F + (x) = ˆδ 1 x ˆδ 2 x x 2 3. (12) This quadratic approxiant F + of d 2 is siply a weighted su of the squared distance functions x1 2,x2 2,x2 3 to three planes: the two principal planes and the tangent plane at the noral footpoint. Based on this observation, we transfor Equation 12 to the global coordinate syste as, F + (x) = ˆδ 1 ( e 1 (x y)) 2 + ˆδ 2 ( e 2 (x y)) 2 + ( n (x y)) 2. (13) We can now express this equation in the for given by Equation 7 to get values for the coefficients A through I. e 1 ρ 1 p n d y x Φ P Figure 1: A query point x IR 2 has a footpoint y IR 2 on the surface Φ P represented by a PCD P. We approxiate the footpoint by p, the closest point in P fro x. The principal frae at the footpoint is spanned by e 1 and n. The osculating circle to Φ P at p has a radius of curvature ρ 1. In the figure, the signed distance d fro x to the footpoint p is positive.

6 4.1. On-Deand Coputation Given a point x, in our first ethod for coputing a second order accurate squared distance field F + for a given PCD P, we perfor an on-deand coputation of Equation 13. For this ethod we first need to copute the noral footpoint of x to P. As an approxiation, we treat p, the closest point to x in P, as the noral footpoint. This point is found using an approxiate nearest neighbor data structure [AMN 98]. Figure 1 shows the scenario in 2D. When the P is a sparse sapling of Φ P, we can use the underlying oving least square (MLS) surface to get a better approxiation for the noral footpoint [AK04]. We further need to evaluate local curvatures at points of P in order to use Equation 13. These quantities are coputed in the preprocessing step of our algorith. At each point of a given PCD, we first deterine the principal frae using a local covariance analysis as detailed in [CP03, MNG04]. If the the underlying surface Φ P is regular, at each of point p of P, a local paraetrization exists. In the principal frae at p, we estiate the local surface by least square fitting a quadratic function of the for ax 2 + bxy + cy 2 + dx + ey to the neighboring points in P. Once we estiate the coefficients a through e, we can use facts fro differential geoetry to get the Gaussian curvature K and the ean curvature H using K = 4ac b 2 (1 + d 2 + e 2 ) 2 (14) H = a(1 + e2 ) bde + c(1 + d 2 ) (1 + d 2 + e 2 ) 2. Finally we evalute the principal radii of curvature ρ i as 1/(H ± H 2 K). The correctness of these estiates depends on the sapling density of the given PCD and on the easureent noise. Further, the neighborhood size used for the least square fits can be adapted to the local shape [MNG04]. In low noise scenarios, when the local estiates of the differential properties can be reliably coputed, the approxiants F + given by this ethod are good Quadratic Approxiants using d2tree Our second ethod for coputing approxiate quadratic approxiants involves least square fitting of quadratic patches to a sapled squared distance function. For a given PCD, these quadratic patches are pre-coputed and stored in a special data structure called the d2tree[lpz03]. Given any point x, in this ethod we do a point location in the cells of the d2tree and return the quadratic approxiant stored in the corresponding cell. Siply put, the d2tree is an octree-like (quad-tree in 2D) data structure, where each cell stores a quadratic fit to the squared distance function, correct to soe axiu error Figure 2: d2tree can be used in 2D (left) and in 3D (right) to store quadratic approxiants of the squared distance fields correct to soe error threshold. The axiu nuber of levels and the error threshold, which are paraeters used during the construction of this quad-tree like data-structure, deterine the size of the cells. In the 2D case, we overlay the Voronoi diagra of the PCD on top of the d2tree, to illustrate that sall cells are created around the edial axis. threshold. The approxiants are stored in the for as given in Equation 7 (Equation 3 in 2D). Details of a top-down construction of d2tree can be found in [LPZ03]. Here we describe a botto-up construction, which is coputationally ore efficient. As a first step, a sapled squared distance field is build for an input PCD by sweeping the space starting fro the PCD P and propagating the squared distance inforation [LPZ03]. Depending on the nuber of levels, which is an input to the algorith, the space is divided into sallest allowable cells (see Figure 2). In each cell, a quadric patch, that best fits the sapled squared distance field, is coputed. The fitting error and the atrices used to copute the coefficients of the fit are saved in each cell. At the next level, the neighboring cells (four in 2D and eight in 3D) are erged to for a larger quadric patch, only if the resulting fitting error is below the given error threshold. The larger quadric patches can be efficiently fitted by re-using the atrices stored in the saller cells. The quadratic atrix stored in any cell is ade positive sei-definite during construction. The axiu nuber of levels of the tree and the error threshold are the required paraeters for the construction of this data structure. Notice that there exists a tradeoff between the size of the cells and the accuracy of the quadratic approxiants. Unlike the on-deand ethod for coputing quadratic approxiants described before, the d2tree approach does not need estiates of the local curvature or any nearest neighbor structure. Quadratic approxiants coputed by d2tree, iplicitly learn the local curvature inforation by fitting quadrics to the sapled squared distance field. We find this ethod to be robust to noisy or under-sapled PCD. Given a query point x, coputing F + (x) siply involves a point location in this d2tree structure, and does not rec The Eurographics Association 2004.

7 x z Figure 3: Funnel of Convergence for aligning a bunny with itself. The bunny is rotated (around the y-axis) and translated (along the x-z plane) to generate different initial positions for the data PCD. The figure in the iddle denotes the sapling pattern used to get the initial positions. The rotation angle is sapled at 10 intervals, while the axiu radial translation of the bunny is around 5 the height of the bunny. Regions in black denote convergence to the correct solution. The convergence funnel of the point-to-plane ICP (left) is found to be quite narrow and unstable. Under siilar conditions, our on-deand algorith (right) is found to have a significantly broader, and uch ore stable convergence funnel. The shape of the convergence funnel corresponding to our d2tree based approach is siilar. quire any explicit correspondence between points of the input PCDs Point-to-point and point-to-plane ICP error etrics as special cases of quadratic approxiant In our fraework, the standard ICP algoriths can be reduced to special cases by selecting suitable approxiants to the squared distance function. Basic point-to-point ICP uses squared distance to the closest point as its approxiant, i.e. F + (x) = x y 2 while the Chen Medioni point-to-plane ICP uses F + (x) = ( n (x y)) 2 as the quadratic approxiant (Equation 7). In the for given by Equation 13, pointto-point ICP has ˆδ i = 1, and point-to-plane ICP has ˆδ i = 0. However, as pointed out in Section 4, such approxiants are second order accurate only in the far field and near field of the PCD, respectively, and hence neither of these algoriths is well-behaved for all relative placeents of the odel and data PCDs. 5. Convergence Issues In this section, we discuss the convergence behavior of point-to-point and point-to-plane ICP algoriths, and then give bounds on the convergence rates for our algorith. In contrast to ICP algoriths, our schee uses second order accurate square distance approxiants at all point in space, and hence, exhibits better convergence properties. Experientally, the point-to-point ICP algorith converges linearly. In a recent result, Pottann has provided theoretical justification for this behavior [Pot04]. We define a low residual proble as one where the data shape fits the odel shape well, and a zero residual proble as one where the fit is exact. For a low residual proble, when the iniizer is approached tangentially, point-to-point ICP has a very slow convergence[pot04, RL01]. We recall that the Chen-Medioni approach iteratively iniizes the su of squared distances to the tangent planes at the noral footpoints of the current data point locations. This iplies a gradient descent in the error landscape, where the squared distance to the tangent plane is used to define the objective function. Fro an optiization perspective, this process corresponds to Gauss Newton iteration. For a zero residual proble, and a sufficiently good initial position, this algorith converges quadratically [Pot04]. In practice, the Chen-Medioni ethod also works well for low residual probles. Notice that in the near-field, the squared distance to the tangent plane is a second order accurate approxiant to d 2. So point-to-plane ICP perfors uch better for fine registration than the point-to-point ICP algorith. However, there is no reason to expect convergence when the two PCDs are sufficiently apart in the transfor space, and in practice, this is found to be true. For low residue probles, our algorith also exhibits quadratic convergence, which eans that the error reduction is of the for ε + Cε 2 c (15) where, C (0,1) denotes the convergence constant, ε c denotes the residual error in the current step, and ε + denotes the residual error after application of the coputed rigid transfor. For each point q i, our algorith coputes a second order approxiant F + to the squared distance function of the surface Φ P represented by PCD P. Using these approxiants, we derive the best aligning transfor for

8 Algorith 1 Brings a data PCD Q in alignent to a odel PCD P using on-deand coputation or d2tree based approach or point-to-point ICP or point-to-plane ICP. 1: if using on-deand ethod then 2: Build an approxiate nearest neighbor structure for P. 3: Pre-copute principal frae and radii of curvature ρ j, j = 1,2, at each point p i P as described in Section : else if using d2tree ethod then 5: Build d2tree with a suitable error threshold. (see Section 4.2) 6: end if 7: count MAXCOUNT 8: repeat 9: for each point q i Q do 10: Copute F + (q i ) using ethod described in Section 4.1 for on-deand approach. For d2tree based approach refer to Section 4.2. For point-to-point or point-to-plane ICP refer to Section : end for 12: Using F + (q i ), build and solve the linear syste given in Section 3. 13: if Arijo condition not satisfied then 14: Take 1/η fraction of the coputed rigid transfor (see Section 3). Value of η chosen via line-search to satisfy Arijo rule (see Section 5). 15: end if 16: if (residual error < ERRORTHRESHOLD) then 17: break 18: end if 19: count count 1 20: until count 0 Q by following a gradient descent with Newton iteration steps [Kel99] in the rigid transfor group (see Section 3). We continue until the residual error falls below a pre-defined threshold or a axiu nuber of iteration steps has been reached. Since the presented ethod is a Newton algorith, it converges quadratically [Kel99, Pot04]. As entioned in Section 3, if the residue ε is large, we apply only 1/η fraction of the coputed transfor to prevent oscillations, or even divergence. Various line search strategies exist for choosing good values for η [Kel99]. In our ipleentation, we used the Arijo condition [Kel99] to select η. This results in a daped Gauss Newton algorith. It is well known in optiization, that algoriths which uses the Arijo rule converge linearly. Hence, to ensure faster convergence for large residue probles, it ay be better to select a quadratic approxiant of the otion, instead of a linear one [Pot04]. Our gradient descent based optiization can get stuck at a local iniu. We bound the axiu residual error for a given PCD pair, and use it to detect a local iniu. A point cloud P, sapled fro a surface Φ P, is said to be sapled r-dense, if any sphere with radius r centered on Φ P contains at least one saple point in P [HDD 92]. Suppose that the odel PCD P is an r P -dense sapling of Φ P. Further, assue the easureent noise only perturbs any point by a axiu aount of σ P and σ Q respectively for the given PCDs P, Q. Under this restrictive sapling odel, when Φ Q represents a subset of Φ P, a final residual atching error ε greater than M(r P + σ P + σ Q ) 2 indicates the algorith has been stuck at a local inia during the search process. When such a situation happens, we ay randoly perturb Q to a new orientation, and try to align the two PCDs starting fro that position. To further study this global convergence property, we define the funnel of convergence for a registration algorith, as the set of all initial poses of a PCD Q, which can be successfully aligned with P, using the given algorith. Notice that the funnel only easures global convergence and not speed. A broad funnel indicates that the algorith can successfully handle a wide range of initial positions. An algorith is said to have a stable funnel if the convergence zones, in the transfor space, are clustered and not arbitrarily distributed. A stable funnel is desirable, since this can enable a systeatic way of generating positions for rando re-starts, using soe branch and bound approach. Experientally, we observe that our algorith has a broader and ore stable funnel of convergence as copared to the point-to-plane ICP variant. This can be explained by the fact, that our algorith akes use of higher order surface properties. 6. Results We test our algorith on a variety of data sets with different aounts of noise, and copare its perforance against point-to-point and point-to-plane ICP algoriths. A brief suary of our registration fraework is given in Algorith 1. We copare the perforance of approaches based on the choice of the approxiant F + of the squared distance function at any point x: 1. on-deand coputation of quadratic approxiant (Section 4.1) 2. quadratic approxiant using d2tree (Section 4.2) 3. squared point-to-point distance (point-to-point ICP) 4. squared distance to the tangent point at the footpoint of x (point-to-plane ICP) In our ipleentation, we test for Arijo condition to ensure stability of the algoriths. On the bunny odel, which consists of 50,282 points, we copare the convergence funnel of point-to-plane ICP and that of our algorith based on on-deand coputation. A copy of the sae PCD is rotated around the y-axis and translated to different positions along the x-z plane. Figure 3

9 point point ICP on deand sq. dist. approx. d2tree sq. dist. approx. point plane ICP 3.5 x point point ICP on deand sq. dist. approx. d2tree sq. dist. approx. point plane ICP 3.5 x d2tree sq. dist. approx. 8e 5 1e 5 7e 4 1e 3 Residual Error Residual Error Residual Error Error threshold for construction of d2tree Iteration Count Iteration Count Iteration Count Figure 4: Plots of residual error vs iteration count for bunny PCD. When the odel and data PCDs, both corrupted with noise, start far apart in the transfor space, the point-to-plane ICP fails to converge to the correct solution (left). However, algoriths using any of the other square distance approxiants do converge, with the d2tree based approach converging fastest. Middle: For good initial position and sall residual proble (the two PCDs align well), the point-to-point ICP algorith has a slow convergence, while optiization based on any of the other squared distance approxiants, converges quadratically. The figure to the right shows the effect of changing the error threshold value used for constructing the d2tree. As the threshold is increased, a larger neighborhood of the squared distance function is captured by each of the cells of the tree and hence, the algorith converges faster. However, for sufficiently high error threshold values, the distance approxiants get too crude, and the ethod starts to deteriorate. shows that the convergence funnel of point-to-plane ICP is quite narrow when the initial displaceent is large. Under siilar conditions, our algorith is found to have a uch broader convergence funnel. Our convergence funnel is also ore stable. Experientally, the initial translation is found to have little effect on the convergence of our algorith. Next we copare the convergence rates for the four variants listed before. For both on-deand and d2tree based approaches, the pre-coputation tie depends on the size n of the odel PCD. For point-to-point, point-to-plane and ondeand coputation, at each iteration, F + for a point x can be coputed in O(1) tie after the nearest neighbor query has been answered. For d2tree, the nearest neighbor query is replaced by point location in the d2tree cells. The solution of the linear syste involves an inversion of a 6 6 atrix. Since the aount of work in each iteration step for any of the algoriths is roughly sae, we siply count the nuber of iterations for coparing speed. In Figure 4, we plot the residual error vs iteration count for four approaches. In the presence of noise and for large residues, point-to-plane ICP often fails to converge. In such noisy scenarios, since the estiates of local principal radii are bad, our on-deand algorith is found to be arginally worse than point-to-point ICP. The d2tree based ethod still converges fast, since the cell-sizes autoatically get adjusted during their construction phase, to partially average out the effect of noise. However, in low residual cases, for reasons explained in Section 5, all algoriths except for point-to-point ICP converge quadratically. The threshold value used for constructing the d2tree is also varied. As the threshold is increased, a larger neighborhood of the squared distance function is captured by each of the cells of the tree and hence, the algorith converges faster. However, for sufficiently high error threshold values, the distance approxiants get too crude, and the ethod starts to deteriorate. Our algorith is able to handle the case when the data PCD is a subset of the odel PCD. We take a partial scan of the bunny consisting of 17,600 points. This scan is fro scanned data and is corrupted with easureent noise. We used the on-deand algorith to atch the partial scan to the coplete bunny odel PCD. The starting arrangeent and the final atch are shown in Figure 5. Finally we test the robustness of our approach in presence of noise and varying sapling density.we try to align a part (consisting of 14,519 points) of a ball-joint with the socket of a hip-bone represented by 132, 538 points. Note the sapling density and sapling pattern are vastly different across the two odels. The ball-joint is uch densely sapled copared to the hip-bone. Even in this case, for Figure 5: Partial Match: A partial scan of the bunny (shown in purple) is registered to the bunny, the odel PCD. The initial arrangeent of the PCDs is shown to the left. Our algorith found the correct atch (iddle,right) in six iterations.

10 of 3d shapes. In IEEE Transactions on PAMI (1992), vol. 14, pp , 2 [CLSB92] CHAMPLEBOUX G., LAVALLEE S., SZELISKI R., BRUNIE L.: Fro accurate range iaging sensor calibration to accurate odel-based 3d object localization. In IEEE Conference of CVPR (1992), pp Figure 6: Partial Match: The goal is to fit a part (shown in purple) of the ball-joint to the hip-bone (shown in green). The starting arrangeent (left) and the final alignent (right) are shown. The whole ball-joint is shown to help the reader judge the correctness of the atch. Our algorith is robust enough to handle varying sapling density and noise in the given PCDs. reasonable starting positions, we got a good final alignent (see Figure 6). The whole ball-joint is shown just to illustrate the goodness of the alignent. We anually selected a part of the ball-joint to satisfy our constraint that Φ Q represents a subset of Φ P. 7. Conclusion and Future Work We have developed a fraework for pairwise registration of point cloud data. In this fraework, registration is treated as a distance iniization between the surfaces represented by the PCDs. We develop quadratic approxiants of the squared distance function to a point cloud and use the approxiants to perfor a iniization. Since our approxiants are second order accurate, we can use the for a Gauss-Newton optiization. As a result, copared to other coonly used registration algoriths, our algorith is stabler and has a faster convergence rate. Extending the fraework to copute partial atches will be very useful. Finally, we plan to extend our fraework to solve the proble of siultaneously aligning ore than two PCDs. References [AK04] [Ale02] AMENTA N., KIL Y.: Defining point-set surfaces. In ACM SIGGRAPH (to appear 2004). 6 ALEXA M.: Linear cobination of transforations. In Coputer graphics and interactive techniques (2002), pp , 5 [AMN 98] ARYA S., MOUNT D. M., NETANYAHU N. S., SIL- VERMAN R., WU A. Y.: An optial algorith for approxiate nearest neighbor searching. In Journal of the ACM (1998), vol. 45, pp [BM92] BESL P. J., MCKAY N. D.: A ethod for registration [CM91] CHEN Y., MEDIONI G.: Object odeling by registration of ultiple range iages. In IEEE Conference on Robotics and Autoation (1991). 1, 2 [CP03] CAZALS F., POUGET M.: Estiating differential quantities using polynoial fitting of osculating jets. In Eurographics/ACM SIGGRAPH syposiu on Geoetry processing (2003), pp [CWPG04] COTTING D., WEYRICH T., PAULY M., GROSS M.: Robust waterarking of point-sapled geoetry. In Shape Modeling International (2004). 1 [Fit01] FITZGIBBON A. W.: Robust registration of 2d and 3d point sets. In British Machine Vision Conference (2001). 2 [GIRL03] GELFAND N., IKEMOTO L., RUSINKIEWICZ S., LEVOY M.: Geoetrically stable sapling for the ICP algorith. In 3D Digital Iaging and Modeling (2003). 2 [HDD 92] HOPPE H., DEROSE T., DUCHAMP T., MCDONALD J., STUETZLE W.: Surface reconstruction fro unorganized points. In ACM SIGGRAPH (1992), pp [JH03] JOST T., HUGLI H.: A ulti-resolution ICP with heuristic closest point search for fast and robust 3d registration of range iages. In 3D Digital Iaging and Modeling (2003). 2 [Kel99] [LPZ03] [MNG04] [PH03] KELLEY. C. T.: Iterative Methods for Optiization. SIAM, Philadelphia, LEOPOLDSEDER S., POTTMANN H., ZHAO H.: The d 2 -tree: A hierarchical representation of the squared distance function. In Tech.Rep. 101, Institut of Geoetry, Vienna University of Technology (2003). 6 MITRA N. J., NGUYEN A., GUIBAS L.: Estiating surface norals in noisy point cloud data. In International Journal of Coputational Geoetry and Applications (to appear 2004). 6 POTTMANN H., HOFER M.: Geoetry of the squared distance function to curves and surfaces. In Visualization and Matheatics III (2003), pp [PLH02] POTTMANN H., LEOPOLDSEDER S., HOFER M.: Registration without ICP. In Technical Report 91, Institut für Geoetrie, TU Wien (2002). 2 [Pot04] [RL01] POTTMANN H.: Geoetry and convergence analysis of registration algoriths. In Tech.Rep. 117, Geoetry Preprint Series, Vienna University of Technology (2004). 7, 8 RUSINKIEWICZ S., LEVOY M.: Efficient variants of the ICP algorith. In 3D Digital Iaging and Modeling (2001). 1, 2, 7