Denoising Manifold and Non-Manifold Point Clouds
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- Erika Newman
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1 Denosng Manfold and Non-Manfold Pont Clouds Ranjth Unnkrshnan Martal Hebert Carnege Mellon Unversty, Pttsburgh, PA 23 Abstract The fathful reconstructon of 3-D models from rregular and nosy pont samples s a task central to many applcatons of computer vson and graphcs. We present an approach to denosng that naturally handles ntersectons of manfolds, thus preservng hgh-frequency detals wthout oversmoothng. Ths s accomplshed through the use of a modfed locally weghted regresson algorthm that models a neghborhood of ponts as an mplct product of lnear subspaces. By posng the problem as one of energy mnmzaton subject to constrants on the coeffcents of a hgher order polynomal, we can also ncorporate ansotropc error models approprate for data acqured wth a range sensor. We demonstrate the effectveness of our approach through some prelmnary results n denosng synthetc data n 2-D and 3-D domans. Introducton Surface reconstructon from unorganzed pont samples s a challengng problem relevant to several applcatons, such as the dgtzaton of archtectural stes for creatng vrtual envronments, reverse-engneerng of CAD models from probed postons, remote sensng and geospatal analyss. Improvements n scanner technology have made t possble to acqure dense sets of ponts, and have fueled the need for algorthms that are robust to nose nherent n the samplng process. In several domans, partcularly those nvolvng man-made objects, the underlyng geometry conssts of surfaces that are only pece-wse smooth. Such objects possess sharp features such as corners and edges whch are created when these smooth surfaces ntersect. The reconstructon of these sharp features s partcularly challengng as nose and sharp features are nherently ambguous, and physcal lmtatons n scanner resoluton prevent proper samplng of such hgh-frequency features. Ths paper proposes a denosng technque to accurately reconstruct ntersectons of manfolds from rregular pont samples. The technque can correctly account for the ansotropc nature of sensng errors n the sampled data under the assumpton that a nose model for the sensor used to acqure the ponts s avalable. The method does not assume pror avalablty of connectvty nformaton, and avods computng surface normals or meshes at ntermedate steps. Prepared through collaboratve partcpaton n the Robotcs Consortum sponsored by the U.S Army Research Laboratory under the Collaboratve Technology Allance Program, Cooperatve Agreement DAAD
2 Nosy nput. σ = 0.0 Denosed output wth HEIV estmate Denosed output usng degenerate estmate (a) (b) (c) Fgure : Example of denosng a toy dataset by global fttng of an mplct degenerate polynomal (a) Input data consstng of ponts from two ntersectng lne segments corrupted wth unform Gaussan nose of std. devaton σ = 0. (b) Denosed data usng an mplct quadratc ft wth the HEIV estmator []. Note that the sharp feature formed by the ntersecton s not preserved. (c) Denosed output after mposng degeneracy constrants on ft coeffcents fxes ths problem.. Related Work There have been several proposed approaches to recover geometry from nosy pont samples. They may be coarsely categorzed as based on computatonal geometry, local regresson, or mplct functon fttng. In general, past approaches have often made smplfyng assumptons about the data due to the ll-posed nature of the problem. () Methods based on classcal regresson typcally assume that the geometry can be treated locally as a smooth surface, whch s clearly a problem at surface ntersectons. (2) Most approaches assume the nose n the data to be sotropc and homogeneous, perhaps because they often lead to convenent closed-form analytcal expressons. However, nose s almost always hghly drectonal and dependent on the dstance of the pont to the sensor. Ths s, for example, the case wth laser range scanners. Ignorng the ansotropy n the nose model typcally results n a systematc bas n the surface reconstructon []. (3) Some methods assume the relable avalablty of addtonal nformaton about the geometry, such as connectvty nformaton (meshes) and surfaces normals, and try to produce estmates of geometry that agree wth ths nformaton. However, the estmaton of both these quanttes s errorprone. Estmaton of dfferental quanttes lke surface normals and tangents s dffcult n the presence of nose even for relatvely smooth surfaces [7, 0], and s of course not even well-defned at ntersectons. Several methods based on computatonal geometry have been developed and rgorously analyzed n the lterature [3]. Many algorthms n ths category come wth theoretcal guarantees of accuracy n the reconstructon but ther applcablty s largely restrcted to dense low-nose datasets. Surface estmaton from nosy pont samples may be posed naturally as an nstance of the local regresson problem from classcal statstcs. A popular non-parametrc technque n ths category s locally weghted regresson, also known n ts more general form as Savtzky-Golay flterng. As explaned n [], t adapts well to non-unformly sampled data and exhbts less bas at boundares. The movng least squares (MLS) technque [] bulds on ths by frst computng a locally approxmatng hyperplane and then applyng a locally weghted regresson procedure to the data projected to the hyperplane. The tech-
3 nque works well wth nose but s unable to reproduce sharp features due to ts mplct assumpton of a sngle locally smooth surface. Fleshman et al. [8] ft quadratc polynomals locally to data and used standard technques from robust statstcs n the fttng process. The technque reled on an ntally fndng low-nose local regons to obtan a relable estmate of the quadratc ft, whch may not always be feasble. Wang et al. [3] proposed a more complcated procedure nvolvng a sequence of voxelzaton and gap-fllng, topologcal thnnng and mesh-generaton. Based on local connectvty, each voxel s classfed as beng at a juncton, boundary and surface nteror. The procedure has several ponts of falure, partcularly at regons that are not densely sampled wth respect to the voxel sze. The method presented n ths paper combnes the strengths of some of the prevous approaches. We modfy a locally weghted smoother to mplctly represent potentally multple lnear subspaces through a degenerate hgh-order polynomal. Ths allows us to explctly model edge ntersectons nstead of tryng to ft a hghly non-smooth surface. The use of a local smoother preserves the adaptablty to varyng sample densty. By posng the regresson as a constraned energy mnmzaton problem, we can easly ncorporate ansotropc error models n the data. We outlne the algorthm n Secton 2 and examne ts behavor through several experments n Secton 3. 2 Constraned Local Regresson In ths secton, we descrbe a modfed regresson algorthm that wll enable us to recover nose-free surfaces from nosy pont cloud data, whle preservng hgh-frequency features n the geometry. We wll frst consder the case of 2-D data to smplfy the explanaton of the man dea. 2. Problem defnton and approach We assume that we are gven a set of ponts {x } R d that are assumed to be nosy observatons of the postons of true ponts {ˆx } R d that le on a locally contnuous, but not necessarly smooth surface. The assocated nose covarances Λ S d + at each pont are assumed to be known, for nstance, through a nose model of the sensor used to acqure the ponts. The ponts are assumed to be rregular, n the sense that they do not follow a known regular samplng dstrbuton, and unstructured n the sense that the local connectvty of the ponts, such as n the form of a mesh, s not avalable. Our goal wll be to compute the true poston ˆx correspondng to each observed pont x. The operatng assumpton wll be that ponts n a local neghborhood, N (x ) of x may be modeled as belongng to one or more lnear subspaces. Ths naturally suggests a maxmum lkelhood (or equvalently defned mnmum energy) formulaton of the problem, subject to the constrant that the nose-free ponts le on one or more subspaces. Snce the parameters of the models, number of models, as well as the assocaton of the ponts to each subspaces are unknown, a popular strategy s to attempt a procedure of teratve model fttng and data assocaton, such as Expectaton-Maxmzaton (EM). However, such teratve procedures tend to be error prone when performed wth few and nosy data ponts, as may be expected for our problem. Instead, we propose to model the problem as one of maxmum lkelhood subject to two types of constrants. The frst type of constrant ensures that each nose-free pont n
4 the neghborhood of nterest les on a hgh-order polynomal whose degree s an upper bound on the number of subspaces n that neghborhood. The second type of constrant s a functon of the coeffcents of the polynomal, whch restrcts the famly of allowable polynomals to degenerate forms that can represent combnatons of lnear subspaces. In practce, we wll sometmes relax the constrant of degeneracy to make the optmzaton problem more tractable at the expense of admttng a sngle non-lnear manfold but restrct them to locally developable surfaces. 2.2 Constrants n the 2-D case In the case of 2-D data, each local neghborhood can be modeled as as consstng of a par of lnear subspaces. Thus locally the shape may be descrbed mplctly as a zero-level set of the equaton (γ T x + d )(γ T 2 x + d 2) = 0, where γ R 2, d R are the parameters for each of the two lnear subspaces (lnes n the case of 2-D data). Note that ths subsumes the case where the subspaces concde. Expandng out the terms yelds an nhomogeneous 2nd degree polynomal n 2 varables, whch we wll refer to as x and y correspondng to each spatal dmenson. Let us denote the coeffcents of each monomal n the polynomal as gven by the expresson θ x 2 + θ 2 y 2 + θ 3 xy + θ x + θ y + θ = 0. () Ths may be rewrtten n matrx form as [ ] x 2θ θ 3 θ [ θ 3 2θ 2 θ x = ] [ x ] [ x A = 0. (2) ] θ θ 2θ It s a known result n algebrac geometry that a quadratc n two varables reduces to a product of two lnear factors only f A s sngular []. In fact, the case where A has only rank one corresponds to the case where the subspaces (lnes) concde. The determnant n ths case may be wrtten explctly to yeld the equalty θ 2 θ 2 θ + θ 3 θ θ (θ 2 θ 2 + θ θ 2 + θ θ 2 3 ) = 0, (3) whch can be used to constran the soluton for the θ s. We wll denote such constrants on the coeffcents of the polynomal as φ(θ) = Constraned optmzaton Together wth the constrant on coeffcents we can pose the task as a constraned optmzaton problem defned at each pont of nterest x {x } gven by mn w (x)(x ˆx ) T Λ (x ˆx ), () subject to two sets of constrants. The frst set of constrants s θ T v(ˆx ) = 0 where θ R m s the vector of monomal coeffcents and v(x) : R d R m s the mappng from the d-dmensonal pont to the monomals formed by ts coordnates. For the 2-D case (d = 2), the number of monomal terms m =. The second constrant s that on the monomal coeffcents, whch s (3) n the case of 2-D data.
5 The weghtng term w (x) s used to gve more mportance to ponts closer to the pont of nterest x. We can defne w (x) usng a kernel loss functon, such as a truncated Gaussan functon centered at x, so as to sutably delneate the neghborhood of nterest N (x). Our mplementaton uses the Epanechnkov kernel w (x) = x x 2 /σ 2 for x x < σ and 0 elsewhere, chosen because of ts fnte support and asymptotcally optmal propertes n related tasks such as kernel regresson [2]. Here σ determnes the length scale, whch may be chosen dfferently for each x. We comment on ts selecton later n Secton 2.. In what follows, we wll sometmes drop the dependence on x n the notaton for clarty, wth the understandng that the optmzaton problem s beng solved for ponts n a local neghborhood of each x {x }. The standard approach to solvng such a constraned optmzaton problem s by frst formng the Lagrangan 2 w (x ˆx ) T Λ (x ˆx ) + λ θ T v(ˆx ) + α T φ(θ), () where {λ } and α are the Lagrange multplers. To proceed further, we lnearze the equatons around the current estmate of x s and θ. Let x = ˆx x and θ = θ 0 θ, where θ 0 s the current estmate of the true θ. To reduce notatonal clutter, we denote v(x ) by v and φ(θ 0 ) by φ 0. Ths yelds the equaton 2 w x T Λ x + λ (θ T 0v(x ) + v(x ) T θ + θ T 0 v x ) + α T (φ(θ 0 ) + φ 0 θ) = 0. () Takng dervatves wth respect to θ, x and the Lagrange multplers yelds the system of equatons: w Λ x + λ θ T 0 v = 0 λ v T (x ) + α T φ 0 = 0 (7) θ T 0v(x ) + v(x ) T θ + θ T 0 v x = 0 φ(θ 0 ) + φ 0 θ = 0. (8) The soluton to the above set of equatons can be wrtten as θ = φ(θ 0 )( φ T 0 φ 0 ) φ 0 (9) λ = w (θ T 0 v T Λ v θ 0 ) v(x ) T (θ 0 + θ) (0) x = w Λ v θ 0 λ = Λ v θ 0 (θ T 0 v T Λ v θ 0 ) v(x ) T (θ 0 + θ). () The above solutons to the lnearzed constraned optmzaton problem suggests an teratve technque n whch a canddate ntal value of θ 0 s computed and the values of θ and the ˆx s are progressvely modfed untl the constrants are satsfed. The ntal value of θ 0 may be chosen as the result of an unconstraned optmzaton usng the Fundamental Numercal Scheme (FNS) algorthm [2] or the related Heteroscedastc Errors n Varables (HEIV) method [] based on solvng a generalzed egenvalue problem. Related formulatons: At ths pont, we wsh to comment on some related work to clarfy some superfcal smlartes. The use of a hgh-order polynomal product to represent a combnaton of (low-order polynomal) subspaces s not new. Work by Taubn [9] ft complex 3-D curves to data, and used a hgh-order polynomal to represent the ntersecton of surfaces that formed the curve. It used an approxmaton to the dstance functon
6 (a) (b) (c) (d) Fgure 2: Illustraton of sequence of optmzaton steps n an example of global fttng of (a) nosy observatons of ponts lyng on two planes. Level-set surfaces are shown at values 0 (green), 0. (red) and 0. (blue), and are drawn for parameters estmated wth (b) TLS, whch are used to ntalze soluton to the (c) HEIV estmate, whch when subject to degeneracy constrants yelds the best ft at the ntersecton of the planes as shown n (d). that reduced the fttng problem to an easly solvable generalzed egenvector problem, but mplctly made the assumpton of unform nose covarance on the ponts. Vdal et al. [] proposed the Generalzed Prncpal Components Analyss (GPCA) algorthm to model combnatons of lnear subspaces. However, they dd not consder nose n the ponts, and have to resort to a separate estmaton procedure to compute the parameters of the ndvdual subspaces. In contrast, the formulaton n ths secton explctly ncorporates a heteroscedastc nose model on the ponts. We use a separate constrant to capture the desred degeneracy of the polynomal as part of the optmzaton procedure, nstead of resortng to postprocessng of the result. Lastly, our focus s on local rather than global fttng of the data, snce the data n our applcaton cannot necessary be descrbed globally by lnear subspaces. 2. Constrants n the 3-D case In the case of 3-D data, we consder the choce of model correspondng to an upper bound of 2 lnear subspaces (planes) n each local neghborhood under consderaton. Ths may be descrbed formally as a zero-level set of the equaton (γ T x + d )(γ T 2 x + d 2) = 0, where x R 3 and γ R 3, d R are the parameters for each of the two planes. Note that ths agan subsumes the case where the subspaces concde. Expandng out the terms yelds an nhomogeneous 2nd degree polynomal n 3 varables (denoted x, y and z). Let us denote the coeffcents of each monomal n the polynomal as gven by the expresson θ x 2 + θ 2 y 2 + θ 3 z 2 + θ xy + θ yz + θ xz + θ 7 x + θ 8 y + θ 9 z + θ 0 = 0. (3) Ths may be rewrtten n matrx form as 2θ θ θ θ 7 [ ] θ x 2θ 2 θ θ 8 θ θ 2θ 3 θ 9 θ 7 θ 8 θ 9 2θ 0 [ x = ] [ x ] [ x A = 0. () ]
7 Algorthm : DenoseByConstranedFttng({x }, {Λ }) 2 3 Data: Ponts = {x } R d wth nose covarance {Λ } begn for x do Compute weghts w = k(x x ) where k s a loss functon such a Gaussan Fnd the total least squares soluton θ TLS to the unconstraned fttng problem. The least square soluton s smply equal to the mnmal egenvector of the weghted covarance matrx formed by the v(x ) s,.e. the mnmal egenvector of w v(x )v(x ) T Use θ TLS to ntalze the teratve soluton to an unconstraned optmzaton procedure []. The soluton to the unconstraned problem θ HEIV can be obtaned through an fxed-pont teraton procedure gven by: S(θ(k))θ(k + ) = λ k C(θ(k))θ(k + ) (2) where λ k s the smallest generalzed egenvalue, and S and C are gven by: S(θ) = A θ T B θ C(θ) = B θ T A θ (θ T B θ) wth A = w v(x )v(x ) T and B = v T Λ v Iteratvely enforce the degeneracy constrant (3) usng equatons (9) and () (or () and () n the case of 3-D) along wth the unt norm constrant θ = and ntalzng wth θ HEIV end end Followng the argument n Secton 2.2, t s easy to see that matrx A must be of rank 2 for the assocated quadrc surface to represent a par of planes. Ths s equvalent to the constrants that the determnant of A as well as each of ts 3 3 mnors are zero. We have observed t suffcent to relax the constrants on the mnors and retan the constrants only on the prncpal mnor formed by the degree 2 coeffcents, as det(b) = det 2θ θ θ θ 2θ 2 θ = 0. () θ θ 2θ 3 Geometrcally, the use of ths partcular subset of constrants restrcts the famly of surfaces represented by the polynomal coeffcents to the famly of parallel or ntersectng planes, and cylnders. Usng the parameters estmated wth ths subset of constrants, we may then construct the matrx A, fnd ts rank-2 approxmaton usng ts SVD decomposton, and recover the parameters of the degenerate polynomal from the rank-2 matrx. Fgure 2 llustrates the sequence of steps nvolved n estmatng the polynomal coeffcents for a synthetc dataset conssts of nosy ponts lyng on two planes ntersectng at rght angles. Level-set surfaces are dsplayed for the polynomal coeffcents estmated at each step of fttng all the ponts. It can be seen that the TLS soluton msfts the geometry, the HEIV soluton tends to oversmooth the ntersecton (as n Fgure for 2-D data) and enforcng the degeneracy constrants recovers the true geometry n ths example.
8 Denosed ponts usng unconstraned estmate Denosed ponts from constraned estmate (a) (b) (c) Fgure 3: Example of denosng a toy dataset by local fttng of an mplct degenerate polynomal (a) Input data consstng of ponts from sx lne segments corrupted wth sphercal Gaussan nose of std. devaton σ = 0. (b) (b) Denosed data usng an mplct quadratc ft wth the HEIV estmator []. (c) Denosed output after mposng degeneracy constrants on coeffcents. 2. Algorthm and Implementaton From the soluton of the constraned optmzaton problem n the prevous secton, we may construct our denosng procedure as gven n Algorthm. We draw attenton to some detals that nfluence the performance of the proposed method. Support radus: The choce of support radus used to compute the weghts w n the kernel functon has a sgnfcant nfluence on the algorthm n two ways. Frst, the proposed method assumes an upper bound of 2 subspaces n the volume of nterest, whch need not be the case for any choce of support sze. The chosen support radus must be one for whch the modelng assumpton s vald, condtonal on there always exstng such a choce. Secondly, even when the assumpton of number of subspaces s vald, there s a tradeoff between choosng too small a radus, rskng poor estmates due to the fewer number of ponts, or too large a radus, rskng the unfavorable nfluence of ponts that do not belong to the local model. We currently use a heurstc strategy of choosng the support radus that gves the best ft, n a maxmum lkelhood sense, to the correspondng neghborhood of the nterest pont, excludng the pont tself to prevent the trval soluton of zero radus. In practce, we have observed that when the number of manfolds s under- or over-estmated, ths strategy tends to reduce the support radus and show bas toward a one-manfold soluton when enforcng the degeneracy constrant. However, ths s an area n need of further study. Robustness: The use of weghts w also suggests the use of robust statstcs to dentfy outlers to the model [8]. One strategy to dentfy ponts that have a large nfluence on the estmated model parameters, such as usng egenvector perturbaton bounds [0] for the generalzed egenvalue problem (2) or usng nfluence functons. In our experments, we use a smple greedy strategy of evaluatng leave-one-out fttng score and gnorng the pont as an outler f t s not a good ft wth ts neghbors. 3 Experments We performed a seres of controlled experments of synthetc data n known confguratons to evaluate the behavor of the denosng algorthm. Fgure 3 shows an example where
9 Nosy Input Fttng wth Radal Bass Functons (RBF) Denosng wth constraned estmator (a) (b) (c) Fgure : Example of denosng samples from a trangular wave functon (a) Input data corrupted wth sphercal Gaussan nose of std. devaton σ = 0. (b) Denosed data usng radal bass functon based smoother wth Gaussan kernel. (c) Denosed output after local degenerate polynomal fttng. Dstance error 0. Normal angle error (degrees) Z 0.0 Z (a) (b) (c) 0 Fgure : Example of denosng samples from 3 faces of a regular cube (a) Input data corrupted wth unform nose of std. devaton σ = Denosed ponts are shown wth patches color coded by (b) dstance error and (c) surface normal angle error. an Epanechnkov loss functon wth bandwdth 0.3 was used to denose a 2x2 square grd pattern of ponts. The use of a constrant enforcng degeneracy n the polynomal can be seen to preserve the ntersectons better than usng an HEIV smoother. Fgure compares the proposed fttng procedure wth a standard nterpolaton algorthm based on radal bass functons (RBF). The RBF algorthm has two parameters []. The frst controls the wdth of the Gaussan kernel whch nfluences the localty of the smoothng. The other controls the tolerance to fttng error,.e. a value of zero would lead to nterpolaton between the ponts, whle hgher values allow greater fttng error. The parameters were tuned so that the results best matched the ground-truth n the sense of least-square error. It can be seen that the proposed algorthm does a better job of preservng sharp changes n the functon and s more stable at the boundary, whle the RBF functon tends smooths over the hgh curvature regons. In Fgure, we test the proposed algorthm on 300 nosy 3-D samples (sphercal Gaussan wth std. dev. 0.0) from 3 faces of a unt cube, and compared t aganst usng the HEIV estmator from []. The use of the proposed estmator reduced the mnmum error n normal angle over the dataset from 0.7 to 0. and the medan dstance of the ponts to ther correspondng planes from 0.02 to unts.
10 Conclusons In ths document, we nvestgated the strategy of fttng local degenerate hgh-order polynomals to data to more fathfully represent and estmate hgh-frequency varatons n pont-sampled surfaces. The proposed strategy helps to address the nherent nablty to perform dfferental analyss at non-manfold regons, such as ntersectons of curves, wthout actually havng to estmate the parameters of component manfolds. A current open problem s the judcous selecton of the support regon of the loss functon. Too small a value results n a fragmented reconstructon, whle the use of too large a value degrades the soluton due to the nfluence of outlers to the mplct model. Work on an analytcal soluton to the optmal support radus to replace our current heurstc s n progress. References [] K. Barrett. Degenerate polynomal forms. Communcatons n numercal methods n engneerng, (), 999. [2] W. Chojnack, M. J. Brooks, A. van den Hengel, and D. Gawley. FNS, CFNS and HEIV: A unfyng approach. Journal of Mathematcal Imagng and Vson, 200. [3] T. K. Dey. Curve and Surface Reconstructon. Cambrdge Unversty Press, 200. [] T. Haste and C. Loader. Local regresson: Automatc kernel carpentry. Statstcal Scence, 8(2):20 29, 993. [] D. Levn. Mesh-ndependent surface nterpolaton. In Geometrc Modelng for Scentfc Vsualzaton, pages 37 39, [] B. Mate and P. Meer. Estmaton of nonlnear errors-n-varables models for computer vson applcatons. IEEE Trans. PAMI, 28(0):37 2, 200. [7] N. J. Mtra, A. Nguyen, and L. Gubas. Estmatng surface normals n nosy pont cloud data. Journal of Computatonal Geometry and Applcatons, (), 200. [8] S.Fleshman, D. Cohen-Or, and C. T.Slva. Robust movng least-squares fttng wth sharp features. In Proc. ACM SIGGRAPH, 200. [9] G. Taubn. An mproved algorthm for algebrac curve and surface fttng. In Intl. Conf. on Computer Vson, 993. [0] R. Unnkrshnan, J.-F. Lalonde, N. Vandapel, and M. Hebert. Scale selecton for the analyss of pont-sampled curves. In Proc. 3DPVT, 200. [] R. Vdal,. Ma, and S. Sastry. Generalzed prncpal component analyss (GPCA). IEEE Trans. Pattern Analyss and Machne Intellgence, 27(2):9 99, 200. [2] M. P. Wand and M. C. Jones. Kernel Smoothng. Chapman & Hall, 99. [3] J. Wang, M. M. Olvera, and A. E. Kaufman. Reconstructng manfold and nonmanfold surfaces from pont clouds. In Proc. IEEE Vsualzaton, 200. [] H. Wendland. Scattered Data Approxmaton. Cambrdge Unversty Press, 200.
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