A Quantitative Evaluation of Surface Normal Estimation in Point Clouds

Transcription

1 A Quanttatve Evaluaton of Surface Normal Estmaton n Pont Clouds Krzysztof Jordan 1 and Phlppos Mordoha 1 Abstract We revst a well-studed problem n the analyss of range data: surface normal estmaton for a set of unorganzed ponts. Surface normal estmaton has been well-studed ntally due to ts theoretcal appeal and more recently due to ts many practcal applcatons. The latter cover several aspects of range data analyss from plane or surface fttng to segmentaton, object detecton and scene analyss. Followng the vast majorty of the lterature, we also focus our attenton on technques that operate n small neghborhoods around the pont whose normal s to be estmated. We pay close attenton to aspects of the mplementaton, such as the use of weghts and normalzaton, that have not been studed n detal n the past. We perform quanttatve evaluaton on a dverse set of pont clouds derved from 3D meshes, whch allows us to obtan accurate ground truth. I. INTRODUCTION Pont clouds are ncreasngly becomng more prevalent as a form of data. They are drectly acqured by range sensors such as LIDAR or depth cameras based on structured lght. Whle range mages can be analyzed relyng partally on mage processng technques, f more than one range mage s avalable, these technques become napplcable n general. Pont clouds are often drectly suffcent for tasks such as obstacle avodance, but requre further processng for hgher level tasks. The most common estmaton on pont cloud nputs s per-pont normal estmaton, snce surface normals are useful for the segmentaton of range data [1], [2] and for computng shape descrptors [3], [4], [5], [6], [7]. Whle surface normal estmaton s typcally treated as a tool that does not requre any attenton n terms of algorthm desgn, smlar to a mean flter for nstance, normals are often estmated sub-optmally even by state of the art systems. The dfferences n accuracy due to these choces may be small, and the mpact on overall system performance may also be small, but there s no reason to nclude a sub-optmal estmator n a system. It s possble that an ncrease n error of a few degrees n surface normal estmaton wll not affect the results of a sophstcated segmentaton algorthm, but why should one use a worse estmator when better optons are avalable at the same or smlar computatonal cost? Along the lnes of most surface normal estmaton methods from pont clouds [8], [9], all algorthms n ths paper operate n local neghborhoods around each pont. They am at estmatng the normal at a pont based on mnmzng the fttng error of planar or quadratc surfaces passng through 1 Department of Computer Scence, Stevens Insttute of Technology, Hoboken, NJ 07030, USA {kjordan1, Phlppos.Mordoha}@stevens.edu Ths work was supported n part by a Google Research Award. each pont of the neghborhood, or by estmatng a 2D subspace that s tangent at the pont of nterest based on parwse pont relatonshps. Even though at least three ponts would be requred to nfer a plane, the vector from each pont of the neghborhood to the anchor pont of the plane must le n the desred plane. Thus, parwse operatons provde constrants that can be aggregated to lead to complete solutons. Technques that operate on trplets of ponts have also been reported n the lterature, but are out of scope of ths paper because they ether requre Delaunay trangulaton as pre-processng or they consder all combnatons of three ponts and are computatonally expensve. In ths paper, we extend a recent study by Klasng et al. [10] n two ways: by augmentng the set of methods evaluated and by performng the evaluaton on a much larger, dverse set of 3D shapes. We keep four of the methods proposed n [10] (Secton III-A) and augment the set by consderng three modfcatons: () usng the reference pont, nstead of the neghborhood mean, as the anchor pont of the plane, () normalzng the vectors formed by ponts n the neghborhood and the anchor pont, and () applyng weghts to the contrbutons of neghborng ponts that decay wth dstance. Fg. 1. Screenshots of the armadllo, grl, arplane and glasses models that are used as nputs n our experments. The models are dsplayed as meshes, but only the vertces are actually used n the computaton.

2 We evaluate all methods on a set of 40 3D models made avalable onlne by Dutagac et al. [11] (Fg. 1). The models are n the form of meshes, whch allow the computaton of surface normals wth much hgher accuracy than what s possble usng just the pont cloud as nput. Most of the surfaces have regons of hgh curvature and are not just smooth planar or quadratc patches. Only the pont clouds are provded as nput to the varous method n the experments and the accuracy of the estmated surface normals s compared to the ground truth. As expected, we observe non-trval errors even wthout addtonal nose. Ths loss of accuracy s nevtable due to the absence of connectvty nformaton. Whle ths may appear as an unfar comparson, the estmaton of normals of the orgnal contnuous surfaces, whch are approxmated by the pont clouds, s precsely the problem we tackle. II. RELATED WORK The prevalng approach for estmatng surface normals from unorganzed pont clouds was ntroduced n the classc paper by Hoppe et al. [8] and s based on total least squares (TLS) mnmzaton. Ths approach has been adopted by numerous authors wthout modfcatons. Among these publcatons, the paper by Mtra et al. [9] contans an n depth analyss of the effects of neghborhood sze, curvature, samplng densty, and nose n normal estmaton usng the TLS approach. We study dfferent aspects of the computaton, focusng on the objectve functon tself, but refer readers to [9] for a rgorous analyss of the TLS approach ncludng error bounds on the estmated normals. An early comparatve study on surface normal estmaton for range mages was publshed by Wang et al. [12]. Whle the methods evaluated and the authors observaton reman relevant, we adopt the notaton and termnology of the more recent work by Klasng et al. [10]. Detals on the latter can be found n Secton III and are not repeated here. Badno et al. [13] also evaluated dfferent technques surface normal estmaton from ponts recently, but ther focus was on range mages explotng the nherent neghborhood structure to further accelerate computaton. Tombar et al. [7] also solve the standard total least squares formulaton, but apply weghts the lnearly decay wth dstance on the contrbuton of each neghborng pont. For effcency, they chose to not compute the centrod of the regon, centerng the computaton on the reference pont tself. We evaluate both of these modfcatons to the TLS formulaton, but we use Gaussan nstead of lnear weghts. III. METHODS In ths secton, we present the methods that are evaluated n Secton IV. We begn wth the methods presented by Klasng et al. [10] and proceed by ntroducng the modfcatons that are the focus of our study. We adopt the notaton of [10] to allow drect comparsons and only consder methods they classfy as optmzaton-based, as opposed to averagng whch requre Delaunay trangulaton of the pont cloud. The nput pont cloud s a set of n ponts P = {p 1, p 2,..., p n }, p R 3. We are nterested n the estmaton of the surface normal for a pont n the pont cloud, whch we refer to as the reference pont. The reference pont s denoted by p = [p x, p y, p z] T and ts normal by n = [n x, n y, n z ] T. Only ponts n the neghborhood of p, whch s denoted by Q = {q 1, q 2,..., q k }, q j P, q j p, are used for the estmaton. The number of ponts k ncluded n the neghborhood s a key parameter for all methods. Standard nearest neghbor searches based on k-d trees are used to fnd the neghbors of each pont. Followng [10], we defne the data matrx, the neghbor matrx and the augmented neghbor matrx as follows: P = [p 1, p 2,..., p n ] T (1) Q = [q 1, q 2,..., q k ] T (2) Q + = [p, q 1, q 2,..., q k ] T. (3) The augmented neghbor matrx Q + contans the reference pont p n addton to ts neghbors. A. Objectve Functons for Normal Estmaton As Klasng et al. [10] remark, seemngly dfferent crtera, such as mnmzng the fttng error of a local plane passng through Q + or maxmzng the angle between the surface normal at p and the vectors formed by p and ts neghbors, result n very smlar objectve functons to be optmzed. In all cases, the desred soluton s the sngular vector assocated wth the mnmal sngular value of an approprate matrx. PlaneSVD: Ths method [12] fts a local plane S (x, y, z) = n x x + n y y + n z z + d to the ponts n Q +. The objectve functon s: J 1 (b ) = [Q + 1 k+1 ] b 2, (4) where 1 k+1 s column vector of k+1 ones and b = [n T d] T. The mnmzer of (4) s the sngular vector of [Q + 1 k+1 ] assocated wth the smallest sngular value. The frst three elements of ths vector are normalzed and assgned to n. PlanePCA: PlaneSVD mnmzes the fttng error of the local plane n the negborhood of p. A dfferent crteron s to fnd the drecton of mnmum varance n the neghborhood Q +. If the ponts n Q+ formed a perfect plane, the drecton of mnmum varance would be the normal to ths plane and the varance would be 0. In the presence of nose or non-planar surfaces, the drecton of mnmum varance s the best approxmaton to the normal. To obtan the objectve functon (TLS) we subtract the emprcal mean of the data matrx [8], [14], [9]: J 2 (n ) = [Q + Q + ] n 2, (5) where Q + s a matrx contanng the mean vector q + = 1 k+1 (p + k j=1 q j) n every row. The mnmzer of (5) s the sngular vector of [Q + Q + ] assocated wth the smallest sngular value. The name PlanePCA s justfed because ths computaton s equvalent to takng the prncpal component wth the smallest varance after performng Prncpal

3 Component Analyss (PCA) on Q +. On the other hand, n PlaneSVD the data are not centered by subtractng the mean. Ths s also equvalent to formng the emprcal covarance matrx of the ponts n Q +, computng ts egendecomposton and settng n equal to the egenvector assocated wth the mnmum egenvalue [15], [12]. VectorSVD: An alternatve formulaton for nferrng the surface normal at p s to seek the vector that maxmzes the angle between tself and the vectors from p to q j, whch should be n the two-dmensonal tangent subspace of p. The dfference wth the prevous methods s that VectorSVD forces the estmated plane to pass through the reference pont p. The maxmzaton of angles can be formulated as the mnmzaton of the nner products between n and the vectors from p to q j. J 3 (n ) = (q 1 p ) T (q 2 p ) T... (q k p ) T n 2. (6) The sngular vector assocated wth the mnmum sngular value of the matrx above s the mnmzer of ths objectve and the desred normal. Followng Klasng et al. [10], we omt the VectorPCA method due to ts smlarty to PlanePCA. QuadSVD: Ths s the only quadratc method evaluated n ths paper. QuadSVD [16], [17] assumes that the surface s composed of small quadratc patches, nstead of small planar patches as all other methods n ths paper do. Specfcally, a surface n the form of S (x, y, z) = c 1 x 2 + c 2 y 2 + c 3 z 2 + c 4 xy + c 5 xz + c 6 yz + c 7 x + c 8 y + c 9 z + c 10 s ftted to to the ponts n Q +. The objectve functon s: J 4 (c ) = R c 2, (7) where c s the vector of coeffcents and each row of R contans the lnear and quadratc terms correspondng to a pont n Q +, that s, each row of R contans: [qjx 2 q2 jy q2 jx q jxq jxy q jx q jz q jy q jz q jx q jy q jz 1] wth the frst row correspondng to p. The fnal normal n s computed by evaluatng the gradent of S at p. B. Modfcatons In ths secton, we ntroduce three modfcatons that are sometmes appled n the lterature to mprove the robustness of surface normal estmaton. To the best of our knowledge, however, a quanttatve analyss of ther effects and a set of recommendatons for practtoners have not been publshed before. We begn wth an equvalent formulaton of VectorSVD, but not centered on the reference pont, as the baselne here. For each neghborhood Q, we form a 3 3 square matrx M as follows: M = q j Q (q j q + )(q j q + )T, (8) where q + s the mean of the neghborhood ncludng the reference pont. The egenvector of M correspondng the mnmum egenvalue mnmzes the followng crteron and s taken as the normal at p. J 5 (n ) = M n 2. (9) Anchorng on the reference pont: A varaton that has already appeared n the methods of Secton III-A s the use of ether the neghborhood mean or the reference pont tself as the anchor pont through whch the plane s assumed to pass. Whle n many cases, especally for large values of k, ths choce should not make a large dfference, t s worth nvestgatng the advantages and dsadvantages of ether choce. The conventonal TLS approach [8], [15], [9], termed PlanePCA here, uses the neghborhood mean as the anchor, whle VectorSVD uses the reference pont as the anchor. Conceptually, f the data are assumed nose-free, usng the reference pont should lead to more precse estmates. Conversely, f the data are assumed to be corrupted by nose, the neghborhood mean, may be a better proxy for the anchor pont than p. In terms of processng tme, avodng the computaton of the neghborhood mean saves a number of operatons that s lnear n k. M (R) q j Q (q j p )(q j p ) T. (10) Normalzed vectors: An observaton that can be made from (8) and all precedng objectve functons s that vectors formed usng neghbors closer to the anchor pont are shorter than vector formed usng the furthest ponts n Q. Ths s not a desrable property n general, snce t volates the assumpton that the surface s only locally planar. The second modfcaton of (8) we examne s by normalzng the vectors n the outer products that form M : M (N) + (q j q )(q j q + )T q q j q + j Q 2 2, (11) n case the plane s anchored at the neghborhood mean, or M (NR) (q j p )(q j p )T q j p 2, (12) q j Q 2 n case the plane s anchored at the reference pont. As a result of normalzaton, all the matrces contrbutng to the sum are rank-1 wth ther non-zero egenvalue equal to 1. Weghted contrbutons: For smlar reasons to the above modfcaton and n order to reduce the effects of ponts that were ncluded n Q despte beng far from p, several authors [18], [7] have appled weghts to the outer products as they are summed to form M. The weghts decrease wth dstance to make the effects of dstant ponts smaller. M (W) q j Q e q j q σ 2 (q j q + )(q j q + )T. (13)

4 The dstances used n the weght computaton above are for the case the plane s anchored at the neghborhood mean. The computaton of M (WR) s analogous when the plane s anchored on p. In ths paper, we defne the weghts as shown n (13), but other alternatves exst. For example, Tombar et al. [7] choose a lnear decay wth dstance. In most cases, an addtonal parameter s requred for specfyng the weghts. We set σ equal to the average dstance from each pont n the pont cloud to ts k th neghbor. σ remans fxed for a pont cloud and assgns non-trval weghts to all ponts n most neghborhoods, as long as they are not outlers, very far from the anchor pont. Makng σ a functon of the dstance to a certan nearest neghbor allows us to use the same procedure for settng t, regardless of scale and number of ponts n the nput pont cloud. We dd not attempt to quantfy the effects of σ n ths paper. Notaton: Havng three bnary choces leads to a total of eght methods. We use N for normalzaton, W for weghts and R for reference pont to denote whch modfcatons are actve. For example, the method wth all modfcatons actve s denoted by NW R and computes the normal as the egenvector correspondng to the mnmum egenvalue of the followng matrx. M (NWR) e q j p 2 2 (q j p )(q j p ) T 2σ 2 q j p 2. q j Q 2 (14) Note that all methods n Secton III-B operate n neghborhoods that have not been augmented by the reference pont p, except for the computaton of the neghborhood mean. We made ths choce because we do not assume that p q + les on local plane. When none of the modfcatons are actve, the method s denoted by base and s smlar to the PlanePCA algorthm up to the excluson of the reference pont from the neghborhood. R s dentcal to VectorSVD, snce p s used as the anchor. IV. EXPERIMENTAL RESULTS In ths secton, we descrbe the data and ground truth generaton for our experments followed by quanttatve results. A. Expermental Setup We performed experments on 40 3D meshes, made avalable onlne by Dutagac et al. [11]. Some examples are shown n Fg. 1. These are more challengng than the data used by prevous studes [10], [13] that conssted of smooth surfaces wth fewer dscontnutes. The average number of vertces per model s 8,548, for a grand total of 341,909 vertces. Each pont cloud s centered and scaled so that the dagonal of ts boundng box s equal to one unt of dstance. It should be ponted out that the vertces are not unformly sampled. Instead, they are denser n areas wth more detals. We decded aganst re-samplng the meshes, snce non-unform densty s a common challenge for normal (a) Teddy, σ n = 0.6% (b) Gargoyle, σ n = 0.3% (c) Ant, σ n = 0.6% (d) Glasses, σ n = 0.3% Fg. 2. Screenshots of meshes corrupted by nose wth σ n gven as a fracton of the dagonal of the boundng box of the entre mesh. Only the vertces are used for surface normal estmaton. estmaton n practce. We only use the mesh connectvty nformaton to generate ground truth, but provde only the pont clouds as nput to the normal estmaton methods of Secton III. Ground truth normal estmaton: We explot mesh connectvty nformaton to compute the ground truth as the weghted average of the surface normals of all trangles ncdent at a vertex. We choose to wegh the normal of each trangle accordng to the angle under whch the trangle s ncdent at the vertex of nterest [19]. (Weghng the normals accordng to the areas of the trangles [20], [19] produces estmates wthn 1 of the angle-weghted ones on average. Ether weghtng scheme could have been used.) We estmated surface normals for all vertces n all 40 meshes wth k rangng from 10 to 50 n steps of 10. We report the average error per pont n degrees, dvdng by the total number of ponts. Snce we are only nterested n surface normal estmaton accuracy over a large set of ponts, ths error metrc s more relevant than averagng the error per mesh and then averagng the per-mesh errors. Addtve zero-mean Gaussan nose was added to the ponts. We set the standard devaton of the nose σ n to 0.15%, 0.3%, 0.45% and 0.6% of the dagonal and repeat the experments as above. Increasng σ n even further resulted n extremely nosy models. Some examples of the effects of nose can be seen n Fg. 2. The ground truth normals of the nose-free meshes were used as ground truth for these experments as well. Otherwse, the experments would be equvalent to the nose-free case on deformed nputs. B. Quanttatve Results We begn by evaluatng the effects of the three modfcatons presented n Secton III-B, before comparng the most

5 effectve among them to the methods of Secton III-A. Fgure 3 shows the average orentaton error per pont n degrees as a functon of k for the nose-free case, as well as for addtve nose wth σ n = 0.3%. TABLE I AVERAGE ORIENTATION ERROR PER POINT IN DEGREES OVER ALL 3D MODELS FOR k = 10. R on R off NW N W base TABLE II AVERAGE ORIENTATION ERROR PER POINT IN DEGREES OVER ALL 3D MODELS FOR k = 10. (a) Nose-free nputs N on N off WR W R base (b) Addtve nose σ n = 0.3% Fg. 3. Average orentaton error per pont n degrees over all 3D models as a functon of k for the methods of Secton III-B. The lowest overall error s acheved by N on the nose-free data, whle NW R shows greater stablty as k ncreases. NW R shows smlar behavor on nosy data as well. The lowest error s acheved by N for k 20 for all values of σ n. Effects of anchor pont selecton: One of the most consstent fndngs among all combnatons of nose, normalzaton and use of weghts s that anchorng the plane on the neghborhood mean leads to hgher accuracy. Table I shows the average error n degrees for the nose-free case comparng methods wth dfferent choces for the anchor pont sde by sde. The mddle column shows error values when the local planes are anchored at the reference pont (R on) and the rght column the same errors when the planes are anchored on the neghborhood mean (R off). Results for all nose levels are very smlar and have been omtted. Effects of normalzaton: A second consstent fndng n all our experments s that normalzng the vectors used n the formulaton of the objectve functons s benefcal snce t does not gve more weght to dstant ponts. Table II shows a smlar comparson wth that of Table I focusng on the effects of settng N on or off. The results below are for the nose free case. Effects of weghts: The concluson from these experments s less clear. The use of weghts mproves accuracy when the reference pont s used as anchor. On the other hand, t s detrmental when the neghborhood mean s used as anchor. As nose levels ncrease, the use of weghts has a slght postve mpact on accuracy. It s possble that adaptng σ n (13) approprately may reveal more postve effects, but our concluson from ths study s that normalzaton s more effectve than the use of weghts n reducng the effects of dstant neghbors and t also does not requre addtonal parameters. Comparson of all methods: Fgure 4 shows the average orentaton error over all 3D models for PlaneSVD, PlanePCA, VectorSVD, QuadSVD, NW R and N. We chose N as the top performng among the modfcatons we evaluated above and NW R due to ts more stable behavor wth respect to varatons n k. The lowest error s acheved by N on the nose-free data for k = 10, whle NW R shows greater stablty as k ncreases. (There s no beneft from consderng more neghbors snce the data s nose-free.) When nose wth σ n = 0.3% s added, N stll acheves the smallest error (8.04 ) for k = 20. PlanePCA and PlaneSVD are vrtually ndstngushable from N for all values of k. Increasng σ n to 0.6% does not alter the results sgnfcantly. PlanePCA acheves the smallest orentaton error at k = 30 by a very small margn over PlaneSVD and N. VectorSVD, or equvalently R, s clearly worse than the other methods. QuadSVD does not work well based on a mnmal sample of 10 neghbors and keeps mprovng as k grows wthout comng partcularly close to the other methods. NW R appears to be more senstve to nose then the other methods. Fgure 5 contans vsualzatons of varous models wth dfferent levels of addtve nose generated by the base method. In general, dscernng the per-pont dfferences among dfferent methods s essentally mpossble, so we do not provde vsualzatons of results from other methods. The vertces n these models have been color-coded accordng

6 (a) Nose-free nputs (b) Addtve nose σn = 0.3% (c) Addtve nose σn = 0.6% Fg. 4. Average orentaton error per pont n degrees over all 3D models as a functon of k for PlaneSVD, PlanePCA, VectorSVD, QuadSVD, N W R and N. See text for an analyss of the results. PlaneSVD and PlanePCA overlap almost perfectly and cannot be dstngushed n the plots. to the errors of the estmated normals. Fgure 6 shows the average orentaton error for k = 20 as a functon of σn. N and PlanePCA appear to be more robust to addtve nose, whle QuadSVD and N W R show a steep ncrease n error. V. CONCLUSIONS We presented an evaluaton of surface normal estmaton methods applcable to pont clouds that s extensve both n terms of the number of methods beng evaluated and also n terms of the sze of the valdaton set. The use of 40 3D meshes of complex geometry poses challenges to these algorthms and exposes potental weaknesses. These challenges are due to the presence of detals (large curvature) (a) Gargoyle, σn = 0.0% (b) Gargoyle, σn = 0.3% (c) Arplane, σn = 0.6% (d) Glasses, σn = 0.3% Fg. 5. Screenshots of meshes color-coded accordng to surface normal orentaton error. All models were generated by the base method. The color codng s wth respect to the mean error for the specfc mesh. Vertces wthn 25% of the mean are colored blue, vertces wth error larger than 125% of the mean are colored red and vertces wth error less than 75% of the mean are colored green. The color of the nteror ponts n each trangle s the result of barycentrc blendng of the vertex colors producng shades of purple or cyan. As before, σn s gven as a fracton of the dagonal of the boundng box of the mesh. on the surfaces, and due to the non-unform densty of the ponts, snce densty s proportonal to the local level of detal. Moreover, we perturbed the vertces by addng zeromean Gaussan nose to them. Our results shows that the use of the reference pont, the pont for whch the normal s estmated, as the anchor through whch the plane must pass s detrmental for accuracy. Ths s due to the non-sotropy of the neghborhoods and, n some experments, the presence of strong addtve nose that make the reference pont unsutable for ths role. Usng the neghborhood mean has proven to be more effectve. As a result, VectorSVD s nferor to the other methods presented n [10]. A second clear concluson from our work s that all vectors used to construct the matrces whose spectral analyss produces the sought-after normals should be normalzed. Whle the use of weghts dd not seem to be benefcal n our expermental setup (non-sotropc neghborhoods and addtve nose), t has shown to be very effectve when the data are corrupted by outlers [21], [22], [23]. We plan to perform further analyss on data corrupted by both addtve and outler nose n our future work. It s well known that PlaneSVD may suffer from numercal nstablty. Ths behavor s not observed on our data because the pont clouds are centered and scaled so that the dagonal of ther boundng box s equal to one, but practtoners should be aware of ths drawback. Our experments do not support fttng quadratc surfaces

7 Fg. 6. Average orentaton error per pont n degrees over all 3D models as a functon of σ n for PlaneSVD, PlanePCA, VectorSVD, QuadSVD, NW R and N. The number of neghbors s fxed at 20 for all nose levels. PlaneSVD and PlanePCA overlap almost perfectly and cannot be dstngushed n the plot. to the data. QuadSVD does not perform partcularly well and comes at a much hgher computatonal cost due to the need to perform SVD on a matrx for each pont, n contrast to most of the other methods that operate on 3 3 scatter matrces. Our fnal recommendaton s the use of the optmal total least squares soluton, termed PlanePCA n [10], modfed to use normalzed vectors. Normalzaton, n general, sgnfcantly mproves the numercal stablty and accuracy of TLS optmzaton. See, for example, the work of Hartley [24]. REFERENCES [1] D. Anguelov, B. Taskarf, V. Chatalbashev, D. Koller, D. Gupta, G. Hetz, and A. Ng, Dscrmnatve learnng of markov random felds for segmentaton of 3d scan data, n IEEE. Conf. on Computer Vson and Pattern Recognton, vol. 2, 2005, pp [2] A. Golovnsky, V. Km, and T. Funkhouser, Shape-based recognton of 3d pont clouds n urban envronments, n Int. Conf. on Computer Vson, [3] A. E. Johnson and M. Hebert, Usng spn mages for effcent object recognton n cluttered 3d scenes, IEEE Trans. on Pattern Analyss and Machne Intellgence, vol. 21, no. 5, pp , [4] A. Frome, D. Huber, R. Kollur, T. Bulow, and J. Malk, Recognzng objects n range data usng regonal pont descrptors, n European Conf. on Computer Vson, 2004, pp. Vol III: [5] A. Patterson, P. Mordoha, and K. Danlds, Object detecton from large-scale 3D datasets usng bottom-up and top-down descrptors, n European Conf. on Computer Vson, 2008, pp [6] R. B. Rusu, N. Blodow, and M. Beetz, Fast pont feature hstograms (fpfh) for 3d regstraton, n IEEE Internatonal Conference on Robotcs and Automaton (ICRA). IEEE, 2009, pp [7] F. Tombar, S. Salt, and L. D Stefano, Unque sgnatures of hstograms for local surface descrpton, n European Conf. on Computer Vson. Sprnger, 2010, pp [8] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, Surface reconstructon from unorganzed ponts, ACM SIGGRAPH, pp , [9] N. J. Mtra, A. Nguyen, and L. Gubas, Estmatng surface normals n nosy pont cloud data, Internatonal Journal of Computatonal Geometry and Applcatons, vol. 14, no. 4 5, pp , [10] K. Klasng, D. Althoff, D. Wollherr, and M. Buss, Comparson of surface normal estmaton methods for range sensng applcatons, n IEEE Internatonal Conference on Robotcs and Automaton (ICRA). IEEE, 2009, pp [11] H. Dutagac, C. Cheung, and A. Godl, Evaluaton of 3D nterest pont detecton technques va human-generated ground truth, The Vsual Computer, vol. 28, pp , [12] C. Wang, H. Tanahash, H. Hrayu, Y. Nwa, and K. Yamamoto, Comparson of local plane fttng methods for range data, n IEEE. Conf. on Computer Vson and Pattern Recognton, vol. 1, 2001, pp. I 663 I 669. [13] H. Badno, D. Huber, Y. Park, and T. Kanade, Fast and accurate computaton of surface normals from range mages, n IEEE Internatonal Conference on Robotcs and Automaton (ICRA), 2011, pp [14] M. Gop, S. Krshnan, and C. T. Slva, Surface reconstructon based on lower dmensonal localzed delaunay trangulaton, n Computer Graphcs Forum, vol. 19, no. 3, 2000, pp [15] K. Kanatan, Statstcal Optmzaton for Geometrc Computaton: Theory and Practce. Elsever Scence, [16] W. Sun, C. Bradley, Y. Zhang, and H. T. Loh, Cloud data modellng employng a unfed, non-redundant trangular mesh, Computer-Aded Desgn, vol. 33, no. 2, pp , [17] D. OuYang and H.-Y. Feng, On the normal vector estmaton for pont cloud data from smooth surfaces, Computer-Aded Desgn, vol. 37, no. 10, pp , [18] G. Medon, M. S. Lee, and C. K. Tang, A Computatonal Framework for Segmentaton and Groupng. Elsever, New York, NY, [19] S. Jn, R. R. Lews, and D. West, A comparson of algorthms for vertex normal computaton, The Vsual Computer, vol. 21, no. 1-2, pp , [20] G. Taubn, Estmatng the tensor of curvature of a surface from a polyhedral approxmaton, n Int. Conf. on Computer Vson, 1995, pp [21] G. Guy and G. Medon, Inference of surfaces, 3d curves, and junctons from sparse, nosy, 3d data, IEEE Trans. on Pattern Analyss and Machne Intellgence, vol. 19, no. 11, pp , [22] W. Tong, C. Tang, P. Mordoha, and G. Medon, Frst order augmentaton to tensor votng for boundary nference and multscale analyss n 3d, IEEE Trans. on Pattern Analyss and Machne Intellgence, vol. 26, no. 5, pp , May [23] M. Lu, F. Pomerleau, F. Colas, and R. Segwart, Normal estmaton for pontcloud usng gpu based sparse tensor votng, n IEEE Internatonal Conference on Robotcs and Bommetcs (ROBIO), 2012, pp [24] R. I. Hartley, In defense of the eght-pont algorthm, IEEE Trans. on Pattern Analyss and Machne Intellgence, vol. 19, no. 6, pp , 1997.