VNU Journal of Scicnce, N atural Sciences and Technology 24 (2008) 179-185
Moving parabolic approximation model o f point clouds and its application
Z houw ang Y ang', Tae-wan Kim^*
^Department o f N aval Architecture and Ocean Engineering Seoul National University, Seoul Ĩ5Ĩ-744, Korea 'D epartm ent o f N aval Architecture and Ocean Engineering
and Research Institute o f Marine Systems Engineering, Seoul National University, Seoul Ỉ 51-744, Korea
Received 31 O ctober 2007
A b s tra c t. W e propose the m oving parabolic approxim ation (M PA) m odel to reconstruct an im proved point-based surface im plied by an unorganized point cloud, while also estim ating the differential properties o f the underlying surface. W e present exam ples w hich show that our reconstructions o f the surface, and estim ates o f normal and curvature inform ation, are accurate for precise point clouds and robust in the presence o f noise. As an application, our proposed m odel is used to generate ưiangular m eshes approxim ating point clouds.
1. Introduction
A cquiring large am ounts o f point data from real objects has becom e m ore convenient because o f m odem sensing technologies and digital scanning devices. H ow ever, the data acquired is usually distorted by noise, arising out o f physical m easurem ent processes, and by the lim itations o f the acquisition technologies.
Even so, it is possible to obtain the smooth underlying shapes w hich are im plied by an unstructured p oint cloud. C onsequently, techniques o f reconstructing m odels from noisy data sets are receiving increasing attention.
Point-based surfaces [ 1 -3 ] have recently becom e an appealing shape representation in com puter graphics and can be used for
Corresponding author. Email: [email protected]
geom etric m odeling [4]. The point-based representation o f a surface should be as com pact as possible, m eaning that it is neither noisy nor redundant. It is therefore im portant to develop algorithm s w hich generate com pact point sets from nonuniform and noisy input, so as effectively to reconstruct the underlying surfaces. It should also be possible to recover the intrinsic geom etric properties o f the underlying surfaces as precisely as possible from point clouds.
D ifferential quantities such as norm als, principal curvatures, and principal directions o f curvature can be used for a variety o f tasks in com puter graphics, com puter vision, com puter- aided design, geom etric m odeling, com putational geom etry, and industrial and biom edical engineering. A num ber o f m ethods for curvature estim ation have been published by 179
180 Z h o u w a n g Yang, Tae-ĩvnn Kim / V N U journal of Science, Nnturnl Sciences and Technolo'^y 24 (200S) 179-185
various com m unities, but m ostly for m anifold representations o f the surface such as polyhedral m eshes, or oriented data sets such as points paired w ith normals. W e would like to recover the differential properties o f an underlying surface directly from an unstructured point cloud, even though it may be n o n u n ifo n n and noisy. O ur approach, m otivated by some recent work o f Levin [2], is based on local maps o f differential geometry [5] and practical algoriửims m optimization theory [6], The main conừibutìon of this work is a scheme to generate a pomt-based reconstruction o f an unorganized point cloud and simultaneously to estimate the differential properties o f the underlymg surface. As an application, we will used the proposed technique to reconstruct friangular meshes approximating given pomt clouds.
2. M oving parabolic approximation
R ecently, there has been increasing interest expressed in surface m odeling using unorganized data points. A pow erful approach is the use o f the moving least-squares (M LS) technique [2] for m odeling point-based surfaces [1]. O ne o f the m ain strengths o f M LS projection is its ability to handle noisy data. We extend the M LS technique to a moving parabolic approxim ation (M PA ), w hich is a m odel o f a second-order projection. The M PA m odel is naturally fram ed as an optim ization problem based on the following proposition:
P ro p o sitio n 1: A t every point p on a surface
s
, there exists an osculating paraboloid s ' such that the normal curvature o f s ‘ is identical to that o f 5 at p for any tangent vector.2.J. M PA m odel
Suppose ứiat a given set o f data points Ị"^
is noisy samplmg o f an underlying surface
s.
Generally, pj will not lie on ửie underlying shape
s
due to noise. We first define a neighborhood o f the given point cloud in the fonn:
i | x - p j | < r }
;=1
W ith an assum ption r > max min p , - p ,
(1)
(
2)
w e ensure that the neigh b orh ood ^ r ) contains the underlying surface as well as the approxim ation that we are going to construct. A num ber o f points in this neighborhood are chosen for reference, called reference points, w hich will be projected on to the underlying surface using M PA models.
Let X G jS(r) b e a re fe re n c e p o in t in the close neighborhood o f the given data points.
The foot-point o f X on the underlying surface
s
is denoted as
+ (3)
w here n is the unit norm al to s , and is the sig n e d d ista n c e fro m X to Ox a lo n g n. W e a im to com pute the foot-point Ox and the differential quantities at the foot-point. Let (ti(n ), t 2(n)}be the perpendicular unit basis vectors o f the tangent plane, so that {Ox; ti, Í2, n} forms a local orthogonal coordinate system. W riting = Py -
X, we fonnulate the moving parabolic approxim ation m odel as a constrained optim ization:
m in / (n, a, b,c) = ị ^ [q[n -
(4) where (n,(,,a,b,c) are decision variables and p IS
a scale param eter.
Once the optim um solution (n ’,^ ,a ,b ,c )of the M PA model o f Equation (4) has been obtained, we can recover the differential quantities o f the underlying surface
s
at theZhoimmng Ynnv,, Tne-wan Kim / V N U Journal o f Science, Natural Sciences and Technology 24 (2008) 179-185 181
foot-point 0.J = X + c,*n\ including the principal curvatures and the principal directions o f curvature. An osculating paraboloid o f the underlying surface at Ox can then be represented by the param etric expression
_ r 1 , ...
s ’ {u,v)= u,v, — [a'U' + 2 b ’uv + c ’v^) , (5)
V 2 y
in the local coordinate system {o^; t ’ , t ’ ,n*}.
The first fundamental form o f S (u ,v ) is given by /■ = Edu^ + I F d u d v + G á v \ (6) where E = \ , F =0 and Ơ =1 at the foot-point Ox- The second fonn o f S ‘(u,v) is given by
I I ' = Ldu^ + I M d v d v + N d v \ (7) where L = a , M = b ' and N = c . The mean curvature / / 'a n d the G aussian curvature AT'can now be calculated as follows:
_ L G - 2 M F + N E _ a + c
" = 2(e G ~ F - ]
L N - M * * 1^2
= a c - b . E G - F ^
From this calculation and P ro p o sitio n 1, we obtain the m inim um and maxim um principal curvatures o f the underlying surface
S a t 0^:
(9)
and the corresponding principal directions o f curvature in the tangent plane:
e L = = Ì2 ,iK a n = á < c = <
e... =liu ii
b tị + K . — a^min
(10) + h
S ' L - c ' ) ( +Ố'Í2
The principal directions and are always orthogonal to each other except at the um bilical points. At an um bilic, holds, and the surface is locally part o f sphere with a radius o f 1/H*. In the special case w here the identical principal curvatures vanish, the surface becom es locally Hat.
2.2. Im plem entation and examples
The M PA model o f Equation (4) is a constrained optim ization problem. W e solve this constrained optim ization by a practical algorithm based on Lagrange-N ew ton m ethod [6]. W e im plem ent our M PA approach and p erfonn it on a num ber o f point clouds.
The m oving parabolic approxim ation model was tested on several different shapes o f surface. Each shape is a graph o f a bivariate function defined over [-1,1] X [-1 ,1 ] and evaluated using a 41 X 41 grid.
[ x „ y , ) = { - \ + H 2 0 ,-1 + k /2 0 ) , k =Ồ^ . . .,40, to determ ine a set o f clean points that lie on the graph:
'Pcie.n = [ { ^ 1 {x,, y , ) f \l,k = 0, . . . , 40 In order to verify the stability o f the algorithm , we generated a point cloud noise by adding G aussian noise with a m agnitude o f 1%
o f the overall cloud dim ension to clean data.
The four test surfaces were a sphere { x , y , z Ỵ = [ x , y , ^ l 4 - x ^ - ) , a cylinder { x , y , z Ỵ = { x , y , ^ J 2 - , a paraboloid { x , y , z Ỵ =^ ( x , y , x ^ + y ^ Ỵ , and a hyperboloid { x , y , z Ỵ = { x , y , x ^ - y ' J . The estim ated curvature inform ation obtained from M PA model was com pared with the exact curvatures in each case. We m easured the
182 Zhoiiivang Yang, Tae-ivan Kim / VhJU journal o f Science, Natural Sciences and Technology 24 (2008) 179-185
difference in terms o f root-m ean-square (RM S) error, w hich we define as
E r r = - ẳ ( » r - v < r , (11) V /=1
w here v a ư " represents one o f the estimated
values k“ „. C . . I f " “ K " '. and v a /“ represents one o f the exact values k“ „, or IC"". Table I summarizes the RM S eưors that occurred in the estim ation o f principal, mean and Gaussian curvatures. From w hich, w e observe that our M PA algorithm can obtain robust and accurate estim ates in the presence o f noise as w ell as for clean data.
W e also applied the M PA algorithm to the scanning data set o f a mouse w hich contains 36036 points, and presented the point-based reconstruction and the estim ates o f curvature in Figure 1. The results show the confidence o f our M PA m ethod for reverse engineering applications.
Table 1. RMS errors in curvature estimation for the test surfaces
E x a m p l e K r r ( K „ „ J E r r ( K , _ ) E r r C / / ) E r r C K ^
S p h e r e
0 . 0 0 2 8 0 . 0 0 1 4 0 . 0 0 1 9 0 . 0 0 1 9
( c l e a n d a t a ) ( w i t h 1 % n o i s e )
0 . 0 4 1 2 0 . 0 2 6 4 0 . 0 2 3 3 0 . 0 2 3 8
C y l i n d e r
0 . 0 0 3 8 3 . 5 e - 0 7 0 . 0 0 1 9 2 . 5 e 0 7
( d e a n d a t a ) ( w i t h 1 % n o i s e )
0 . 0 7 4 7 0 . 0 2 8 1 0 . 0 4 4 6 0 . 0 2 1 5
P a r a b o l o i d
0 . 0 1 4 4 0 . 0 1 8 8 0 . 0 1 5 8 0 . 0 2 8 7
( c l e a n d a t a ) ( w i t h 1 % n o i s e )
0 . 0 9 5 7 0 . 1 0 7 5 0 . 0 8 8 5 0 . 1 8 2 8
H y p e r b o l o i d
0 . 0 1 1 7 0 . 0 0 1 7 0 . 0 0 2 8 0 . 0 1 3 8
( d e a n d a t a ) ( w i t h 1 % n o i s e )
0 . 1 2 7 8 0 . 1 2 9 7 0 . 0 6 8 4 0 . 1 5 0 5
(a) scanning ddtii ựKHi ì i s (b) pDÌiit-bascd rcvo»>ỉruc(ion
<c) IcM urc m a p a f csiiriw icd iiK-an c u n a U J fc s
(d ) tcx iu rc n ia p o f CNiimalcd (iau>M un c u r \a iu r c s
Fig. 1. Applying the MPA algorithm to the Mouse model.
3. Mesh reconstruction
As an application, our M PA m odel is used to generate a triangular m esh that approxim ates the underlying surface o f given point cloud.
Our m ethod o f m esh reconstruction from point clouds by m oving parabolic approxim ation can be outlined in the following scheme.
1. A rough initial m esh is
consừucted from given point cloud
V = Ị p c . Let be the
initial set o f new inserting vertices.
Zhouiuang Yang, Tae-wan Kim / V N U ]ournal o f Science, Natural Sciences and Technology 24 (200S) '179-185 183
2. R epeatedly apply the steps o f curvature- based refinem ent (a-b-c) until the approxim ation error is w ithin a predefined tolerance or the m axim al num ber o f times is reached:
a. For each e we project it on to the underlying surface o f the point cloud p using the M PA algorithm , and get the estim ate o f m ean curvature vector K;c(v) at the projection V = M PA(v^). A fter projection, the set o f potential vertices is denoted by
VP o te n tia l
V v '' G K ^ '^ a n d K r . w ) o
b. C alculate the m ean curvature norm al Km(v) via the differential geom etry operator [7], and define
K ^ ( v) _ K ^ ( v) ) £ K^(V) } as the collection o f active vertices,
c. Insert a new vertex at the m idpoint o f every edge adjacent to any V ị ^
and then renew
y N i - w _ | y ^ _ V G
and VV,- G £ }. T he approxim ating m esh is updated by adding the topological
connections for those new inserting vertices.
3. O utput the resulting m esh ytY = (V, E) as the final approxim ation to the input point cloud P.
Figures 2 to 4 show the m eshes reconstructed from given point clouds using our M PA algorithm .
u ) th e d a ta p tiin u fh ) tin: initial
Fig. 2. Mesh reconstruction for the Knot model
(a) the data points (b) the initial mesh
184 Z h o u w a n g Yang, Tae-wan Kim Ị V N U Journal o f Science, Natural Sciences and Technoỉo<Ịỵ 24 (2008) 179-185
(c) the mesh after one iteration (d) the mesh after two iterations Fig. 3. Mesh reconstruction for the Horse model.
(a) the data points (b) the initial mesh
(c) the mesh after one iteration (d) the mesh after two itenrations Fig. 4. Mesh reconstruction for the Sculptiưe model.
Z h o u w a n ^ Yang, Tae~wnn Kim / V N U Journal of Science, Natural Sciences and Technology 24 (200S) 179-185 185
4. Conclusion
the INUS Technology Inc for providing scanning data points o f the M ouse model.
W e have shown how to construct an improved point-based representation from a point cloud, at the same tim e as com puting the norm als and curvatures o f the underlying shape.
O ur algorithm is based on optim ization theory and works robustly in the presence o f noise, w hile yielding accurate estim ates for clean data.
The effectiveness o f the algorithm has been dem onstrated in the reconstruction o f point clouds obtained by sam pling several different surfaces, including a sphere, a cylinder, a paraboloid and a hyperboloid.
As an application, w e use the M PA algorithm to construct a triangular mesh approxim ating the underlying surface o f a given point cloud. W e expect that our M PA method will find further applications in many operations on point-based surfaces, such as sm oothing, sim plification, segmentation, feature extraction, global registration.
Acknowledgments. This w ork was supported by grant No. R O l-2005-000-11257-0 from the Basic Research Program o f the K orea Science and Engineering Foundation, and in part by Seoul R&BD Program* W e would like to thank
References
[1] A. A lexa, J. B chr, D. C o h cn -O r,
s.
F leishm an, D.L evin, c . Silva, “ P o in t set su rfa c e s’, In P ro c eed in g s o f IE E E V isualization (2 0 0 1 )2 1 ,.
[2] D. L evin, “ M esh -in d cp cn d cn t su rface in te rp o la tio n '’. In B ru n n ett, B. H am ann, and H.
M u eller, ed ito rs, G eo m etric M o d elin g f o r S cien tific Visualization^ S p rin g cr-V crlag , (2 0 0 3 ) 37.
[3] N. A m en ta, Y .J. K il, “ D efining p o in t-se t surfaces**, In P ro c eed in g s o f A C M S ỈG G R A P H (2 0 0 4 ) 264.
[4] M. P auly, R. K eiser, L .p. K obbclt, M. G ro ss,
“ S hape m o d elin g w ith p o in t-sam p led g e o m e try ” , In P ro c e e d in g s o f A C M S IG G R A P H (2 0 0 3 ) 6 [5] P.M . d o C arm o, '''D ifferential G eom etry o f C u rves
a n d Surfaces"', P rcn ticc-H all, 1987.
[6] R. F letcher, '"P ractical M e tiw d s o f O p tim iza tio n '', John W iley & Sons, 2nd ed itio n , 1987.
[7] M. M eyer, M. D csb ru n , p. S chrodcr, A .H . B arr,
“ D iscrete d iffcrcn tial-g co m ctry o p erato rs fo r trian g u lated 2 -m a n ifo ld s” , In H .c . H ege a n d K.
P o h h ier, editors. V isualization a n d M a th em a tics ///, S p rin g er-V crlag , (2 0 0 3 ) 35.
