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IEEE TRANSACFI ON S O N P A TT ER N ANALYSIS AND MACHINE INTELLIGENCE, VOL. 14 , NO . 2, FEBRUARY 1 992 239
A Method for Registration of 3 -D ShapesPaul J . Besl, Member, IEEE, and Neil D. McKay
Abstract—This pape r describes a general-purpose, representa-tion-independent method for th e a cc ura te a nd computationallyef fic ient regist ration of 3 -D shapes including free-form curvesan d surfaces. The method handles the full six degrees of freedoman d is based on the iterative closest point (ICP) algorithm,which requires only a procedure to n d the closest point on ageometric entity to a given point. The ICP algorithm alwaysconverges monotonical ly to the nearest local minimum of a mean-square distance metric, an d experience shows that the rate ofconvergence is rapid during the rst few i terations. Therefore,given an adequate set of initial rotations an d translations fora particular class of objects with a certain level of “shapecomplexity,” one can globally minimize the mean-square distancemetric over all six degrees of fre ed om b y te stin g each initialregistration. For example, a given “model” shape an d a sensed“data” shape that represents a major portion of the model shapecan be registered in minutes by testing one initial translationan d a relatively small set of rotations to allow for t he g iv en
level of model complexity. On e important application of thismethod is to register sensed data f ro m u n x tu re d rigid objectswith an i dea l g eometri c model prior to shape i nspect ion . T hedescr ibed method is also useful for deciding fundamental issuessuch as the congruence (shape equivalence) of different geometricrepresentations as well as for estimating the motion between pointsets where the correspondences are n ot k no wn . Experimentalresults show the capabilities of the registration algorithm on pointsets, curves, an d surfaces.
Index Terms— Free-form curve matching, f ree- form surfacematching , mo tion estimation, pose estimation, quaternions, 3- Dregistration.
I. I N T R O D U C T I O N
GLOBAL AND local shape m a tc h in g m e tr ic s for free-
form curves an d surfaces as well as point sets weredescribed in [3 ] in an at tempt to formalize an d unify the
description of a ke y problem in computer vision: Given 3-
D data in a sensor coordinate system, which describes a data
shape th at m ay correspond to a model shape, and given a
model shape in a model coordinate system in a different
geometric shape representation, estimate the optimal rotationand translation t ha t a l igns , or registers, the model s ha pe a nd
the data shape minimizing the distance between the shapes
an d thereby allowing determination of the equivalence of the
shapes via a mean-square distance metric Of ke y interest to
many applications is the following question: Does a segmented
region from a range image match a subset of B-spline surfaces
on a computer-aided-design (CAD) model? This paper pro-
vides a solution to this free-form surface m a tc hi ng p r ob le m
a s dened in [3 ] and [5 ] a s a special c a s e of a simple,
Manuscript received October 30 , 1990; revised Ma y 6 , 1 99 1.The authors are with the Computer Science Department, General Motors
Research Laboratories, Warren, MI 48090 -9055.IE EE L og N u mb er 9 1 0 2 6 86 .
general, unied approach, which generalizes to n dimensions
and provides solut ions to 1 ) the point-set matching problemwithout correspondence an d 2 ) the free-form curve matching
problem. Th e algorithm requires no extracted f ea tu re s, n o
curve or surface derivatives, and no preprocessing of 3-D data,
except for the removal of statis tica l out liers.
The main application of the proposed method a s described
h ere is to register digitized data from unxtured rigid objects
with a n i dea li zed geometr ic model prior to shape inspec-
tion. When inspecting shapes using high-accuracy noncontact
measurement devices [4 ] over a shallow depth of eld, the
uncerta inty in di fferent sensed points does not vary b y m uch.Therefore, fo r purposes of simplicity and relevance to inspec-
tion applications based on thousands of digitized points, the
c a s e of unequal uncertainty among points is not considered.
Similarly, the removal of statistical outliers is considered apreprocessing step, is probably best implemented as such,
and will also not be addressed. In the context of inspection
appl ications, the assumption that a high-accuracy noncontact
measurement device does not generate bad data is reasonable
since some sensors have the ability to reject highly uncertainmeasurements.
The proposed shape registration algorithm can be used withthe following representations of geometric data:
1) Point s e t s
2 ) line segment s e t s (polylines)3) implicit curves: §'(a:,y,z) = 0
4‘ : parametric curves:
triangle s e t s (faceted surfaces)
implicit surfaces: g($,y,z) = 0parametric surfaces: (a:(u,v),y(u, v),z(u,'v)).Eax
This c ov ers m os t appl ications that would utilize a method
to register 3-D shapes. Other representations a r e handled byproviding a procedure for evaluating the closest point on the
given shape to a given digitized point.
This paper is structured a s follows: Several relevant papers
from the literature are rst reviewed. Next, the mathematical
preliminaries of computing the closest point on a shape to a
given point are covered for the geometric representat ions men-
tioned above. Then, the iterative closest po in t ( ICP) a lgori thmis s ta te d, a nd a theorem is proven concer ning i ts monotonic
convergence property. Th e issue of the initial registration
states is addressed next. Finally, experimental results for point
sets, curves, an d surfaces are presented to demonstrate the
capabilities of the ICP registration algorithm.
II. L I T E R A T U R E REVIEW
Relatively little work has been published in the area of regis-
tration (pose estimation, alignment, motion estimation) of 3-D
0162-8828/92$03.00 © 1 9 9 2 IEEE
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24 0 IEEE T R A NS A C TI ON S ON PATTERN ANALYSIS AND MACHINE I N TE L L IG E N C E , V OL . 14 , NO . 2, FEBRUARY 1992
free-form shapes. Most of the existing literature addressing
global shape matching or registration have addressed limitedclasses of shapes, namely, 1 ) polyhedral m o de ls , 2 ) p i ec ew is e -
(super)quadric models [2], [27], and 3) point s e t s with knowncorrespondence. Th e reader m ay c o ns u lt [ 6 ] an d [14 ] for pre-
1985 work in these a r e a s . For a sampling of other more recent
related work not discussed below, s e e [8 ], [1 0 ], [1 2 ], [1 3],
[ 1 9 ] , [ 2 0 ], [ 2 4 ] , [ 2 6 ] , [ 3 4 ], [ 3 5 ], [ 3 7 ]. [ 3 9 ], [ 4 4 ] . [ 4 6 ] , [ 4 8 ] ,[5 3 ], [5 8 ], [59].Historically, free-form shape matching using 3 -D data wa s
done earliest by Faugeras and his group a t INRIA [18], where
they demonstrated effective matching with a Renault auto part
(steering knuckle) in the early 1980’s. This work popularizedthe u s e of quaternions for least squares registration of cor-
responding 3-D point sets in the computer vis ion community.The alternative u s e of the singular value decomposition (SVD)algorithm [23], [1], [49 ] wa s no t as widely known in this time
frame. The primary limitation of this work was that it reliedon the probable existence of reasonably large planar regions
within a free-form shape.
Schwartz and Shari r [50] developed a solution to the free-form space curve matching pr ob lem without feature extraction
in late 1985. They used a nonquaternion approach to comput-ing t he l ea st squares rotation matrix. Th e method works well
with reasonable quality curve data bu t has difculty with verynoisy curves because the method u s e s arclength sampling of
the c ur ve s to obtain corresponding point sets.
Haralick er al. [28] addressed the 3-D point-set pose e s -
timation problem u sin g r ob u st m eth od s combined with the
least squares SVD registration approach, which provided a
robust statistical alternative to t he l ea st squares quaternion or
SVD point set matching. This algorithm is able to h an dle
statistical outliers and could theoretically be substituted fo ro ur quate mio n-based algorithm as long as the determinant of
the orthonormal matrix is strictly a positive one. A recent
conf erence pro ce e dings [47] c on ta in s n ew contributions on
this subject.
Horn [31] derived an alternative formulation of Faugeras’s
method [18 ] of least squares quaternion matching that uses
the maximum eigenvalue of a 4 > < 4 matrix instead of the
minimum eigenvalue. Horn [30] an d Brou [11 ] also developed
the extended Gaussian image (EGI) methods allowing the
matching of convex an d restricted sets of nonconvex shapes
based on sur face normal histograms.
Taubin [55 ] h a s done some interesting work in the area ofimplicit a lgebra ic no np lanar 3-D curve and surface estimation
with applications to position estimation without f ea tu re e x-
traction. H e describes a method of approximating data points
with implicit algebraic forms up to the tenth degree using
a n approximate distance metr ic. Global shapes (not occluded
shapes) can be identied based on generalized eigenvalues, an d
the registration transformation can be recovered. The methodis s how n to be useful for c om p le te p l an ar c ur ve an d space
curve shapes, bu t it is unclear that the effectiveness generalizeswell to more complicated su rfac es, su ch as terrain data or a
human face. Taubin h a s stated that the numerical methods ofthe approximate distance t tend to break down above the tenthd eg re e. H e later [56 ] extended his work in shape description
by investigating shape matching based on generalized shape
polynomials. This demonstrated some interesting theoreticalresults bu t rema ins to b e demonstrated fo r practical use on
complex surfaces.
Szelisk i [54] also describes a method for estimating motionfrom sparse range data without correspondence between the
points and without feature extraction. H is p ri ma ry goal was
to create a method for estimating the motion of the observer
between tw o range image frames of the sa me terrain. Given
the s e t of points from one frame, h e applies a smoothness
assumption to create a smoothing spl ine approximation of the
points. Then, a conventional steepest descent algorithm is used
to rotate and translate the second data s e t s o that it minimizesthe sum of the covariance-weighted z differences between the
points and the surface. H is approach is ba sed on a regular
my-grid structure, and true 3-D point-to-surface distances are
n ot c om p u te d. Th e s te ep e st -d es ce nt a p p ro ac h is a slower
alternati ve to reaching the l oc al m in ima than ou r proposed
IC P algorithm described below. Szeliski uses optimal Bayesian
mathematics to allow him to downweight nois ier values a t
longer ranges from a simulated range nder . For navigationrange imaging sensors, the uncertainty in data p o in ts v ary
signicantly from the foreground to the background. For high-accuracy sensors with shallow depths of eld, the uncertainty
variation between points is orders of magnitude less and is
of much less concern. Szeliski provides experimental resultsfor synthetic terrain d at a a nd a block. Th e terrain data motion
test was a simple translat ion along one axis: a 1 -D correlationproblem. His block test did involve si x degrees of freedom,
bu t the block is a very simple shape. Overall, this workpresents some interesting ideas, bu t the experimental results
are unconvincing for applications.
Horn and H a rr is [33 ] a ls o a ddres sed the problem of e s -
timating th e e xa ct rigid-body motion of the observer given
sequentially digitized range image f rame s of the same terrain.
They describe a range rate constraint equation an d an elevation
rate constraint equat ion. The result is a noniterative least
squares method that provides a six-degree-of-freedom motionestimate as long as the motion between frames of data is
relatively small. This method is much quicker than the one
p ro pos ed b y Szeliski, bu t it is no t clear tha t th is m eth od
generalizes to arbitrary rotations an d translations of a shape.
Kamgar-Parsi er al. [36 ] also describe a method for the
registration of multiple overlapping range images without
distinctive feature extraction. This method works very well
using the level sets of 2.5-D ra ng e d ata bu t is essentially
restricted t o t he t hr ee degrees of freedom in the p la ne s in c e the
work was addressed toward piecing together terrain map data.
Li [38] addressed free-form surface matching with arbitrary
rotations and translations. Hi s method forms a n attributed
relational graph of fundamental surface regions for d ata a nd
model shapes an d then performs graph matching using aninexact approach that allows for var iabil ity in attributes a s
well as in g ra p h a dj ac en cy r el at io n sh ip s . This seems to be
a reasonable approach but rel ies on extraction of derivative-based quantit ies. Experimental results are shown for a coffee
cu p a nd the Renault auto part; see also W o n g er al . [ 6 0 ] for
other re lated work using attributed graphs for 3-D matching.
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24 2 IEEE TRANSACTIONS O N PA TT E RN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 14 , NO . 2, FEBRUARY 1992
is
W e e l i i : i i i ] ewhere the partial derivatives of the objective function are given
b y
Mil’) = 2 ' F f . ( 1 1 ' ) ( ' F ( 1 1 ' ) ~ 2 5 )f . ) ( i - 1 ’ ) = 2 t ’ i ( t 1 ' ) ( ' F ' ( ' J ) — 1 5 ' )
f m . (t 7 ) = 2 t i . . . ( t » 1 ) ( t * ( t 1 ' ) - 1 5 ' ) + 2 F i , ( 1 1 ' ) F u ( ' J )
f . . . . ( i i ) = 2 1 * ’ § . » . ( 1 1 ' ) ( t ’ ( — t i) + 2 F i ( ' @ 1 ) t % ( iT T )f U ’ U ( , J ) = 2 1 * i » . . . ( t 1 ' ) ( F ( — I 5 ) + 2 r ’ Z ( 1 ? ) F t ( t 1 ' ) -
The curve c a s e requires only computation of f,, and
f,,,,. Th e Newton’s update formula for either entity
is
( 1 0 )
( 1 1 )
( 1 2 )
( 1 3 )
( 1 4 )if i
tnI=ar-Wv%naar*vnnJ a nwhere rig = 11,. W h e n using the starting point selection
method described above based on a simplex approximation
with a reasonably smal l 6 , Newton’s method for computing
the closest point generally converges in one to v e iterationsan d typically in three. Th e computat ional cost of Newton’s
method is very low in contrast with nding good starting
points.
B. Point to Implicit Entity Distance
An implicit geometric entity is dened as th e ze ro set of
a possibly vector-valued mult ivar iate function §(F') :0 . The
distance from a given point 1 5 ' to an implicit entity I is
( K 2 5 2 1 ) = 3 1 1 1 1 1 d ( 1 5 ' » F ) = P 1 1 1 1 | l I — r i l l » ( 1 6 )s(1 )=0 9( )=0
Th e calculations to c om p u te t hi s distance are also no t closed
form an d are relatively involved. On e method for computing
point-to-curve an d point-to-surface distances is outlined below.Sets of implicit entities a r e straightforward once the distance
metric for an individual entity is implemented. Let J be the
set of NI parametric entit ies denoted I), and J = {Ik} fo ris = l,N1 . The d is tance between a point 1 5 ' a nd the implicit
entity set J is
aao:%g@Mgaa> an
Th e closest point 3 7 , - on the implicit entity I, satises the
equality d(15',§',-) = d(ji',.]).
Ou r rst step towards computing the distance from a pointto a n implicit entity is creating a simplex-based approximation(line segments or triangles) as w a s d on e for parametric entities
[7]. Computing the point-to-line set or point-to-triangle set
distance yields an approx imate closest point Fa, which ca n beused to c om p u te the exa ct d is tan c e.
Th e implicit entity distance problem is quite different from
the parametric entity c a s e where unconstrained optimizationsufces. To nd the closest point on a n implicit entitydened by § ' ( ' F ' ) = 0 to a given point p ’, on e must
solve a constrained optimization problem to minimize
a quadratic objective function su bject to a nonlinear
constraint
Minimize f('F') = | | '. F ' — p ' ] | 2 where § ' ( ' F ' ) : . (18)
One approach to this problem is to form the augmented
Lagrange multiplier system of equations [40]:
vne+Yvao-0 anan = 0
where V — 6/6f]* and solve this system of nonlinearequations via numer ica l methods. T he n um be r of equations
and unknowns fo r the nonlinear system is three for planar
curves, four for surfaces, an d ve for implicitly dened space
curves. Continuation methods [41] can b e used to solve thisproblem for a lgebra ic e nti ti es even without a good starting
point, bu t a good starting point will allow the u s e of faster
methods, such as the multidimensional Newton’s root nding
method. From a numer ical po in t of view, the parametric
methods are much easier to deal with. From a n applied
point of view, no industrial CAD s ys te m s s to re free-form
curves or surfaces in implicit form. For t his reason, implicit
surfaces of interest are dealt with in our imp lemented system
either via special c a s e mathematics (e.g., spheres) or vi aa parametric form. Of course, if there were an application
where it was necessary t o h an dl e free-form implicit entitiesin their implicit form [51], the above algorithm could be
implemented.
Taubin [55] uses an approximate distance algorithm that
implies a simple update formula for surfaces an d planar curves
when g(F0) is nearly zero:
_ . _ . V9 ( (Ft)
I i t 'm < 2 ‘ )it
’ 5 { ‘ 3 L = 5 - 1 1 : ;
This method is only exact if the innite line with the directionVg(F) at the starting point F 0 intersects the implicit entity at a
point where the normal vector h a s that same direction. This isno t true in general, an d the approximation is general ly worse
the further the point is from the implicit entity. Therefore,
th is res ul t cannot be used if precise d is tance results are
required.
C. Corresponding Point Set Registration
All closest point (minimum distance) algorithms have been
mentioned in forms t ha t ge nera li ze to n dimensions. On e more
necessary procedure fo r yielding the least squares rotation and
translation is reviewed. F or o ur purposes, the quaternion-based
algorithm is preferred over the singular value decomposit ion(SVD) method in two a nd t hr ee dimensions since reections
are not desired. The SV D approach, based on the cross-
covariance matrix of two point distributions, does, however,generalize easily to n dimensions and would be our method ofchoice for rt > 3 in an y rt-dimensional appl icat ions. T he b a si c
solution of Horn [31 ] is described below, although the method
of Faugeras [18] is equivalent. Ou r summary stresses the roleof the SVD cross-covariance matrix, which is an important
relationship no t discussed in other work.
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BESL AND MCKAY: ME T H OD FOR REGISTRATION OF 3- D SHAPES
The unit quaternion is a four vector ( IR = [q0q1q2q3]*, where(1 0 2 0 , and qg + qf + qg + qg = 1. Th e 3 x 3 rotation matrix
generated by a unit rotation quaternion is found a t the bottomof this page. Let 5 1 - = [q4q5q6]’ be a translation vector. The
complete registration state vector rj ‘ is denoted rj ’ — [rj'R|rj’T]‘.
Let P = be a measured data point set to be aligned witha model point se t X = {:i:',}, where N, , = Np an d where each
point 5 , corresponds to the point 5, with the same i nd ex . T he
mean square objective function to be minimized is
NP
fa) = Z I l a - R e n a - a - | | * \ < 2 2 )1 1
The “center of mass” / 1 } , of the measured point set P and the
center of mass x for the X point s e t are given by
“B2_ I I M 3‘ $ 5 1 1 5 -3
82-1 . ; M FS“p = — an d /I = — S ’ - . ( 23 )
The cross-covariance matrix Em of the s e t s P and X is given
W
‘U2_‘ . § M 5 .3—§ —§ —§ — 1 ‘"0 —§ -5 —§
3 1 0 $ = — [ ( 1 % — / 1 - p ) ( @ 1 1 : — / 1 - a l t ] = 3; ZZ;lPt$il — Hatti-
(24)The cyclic components of the anti-symmetr ic matrix A , _ , - =
(E m — Z I g : , , ) , , - a r e used to form the column vector A =
[A23 A3 1 A1g]T. This vector is then used to form the
symmetric 4 X 4 matrix Q(E,,,,)
where I is the 3 > < 3 identity matrix. The unit eigenvector
q}} = [qg q 1 Q 2 q3]* corresponding to the maximumeigenvalue of the matrix Q(E,,,) is selected a s the optimalrotation. Th e optimal t ranslation vector is given by
(I T = t ie — R ( T R ) / I n : ( 2 6 )
This least squares q u a te rn io n o p e ra ti on is O(N,,) and is
denoted as
(til d m s ) = Q(P, X) (27)
where rim, is t he m e an square point matching e rr or . T he n ot a-
tion rj'(P) is used to denote the point set P after transformationby the registration vector Q ’.
24 3
IV . THE I T E R A T I V E C L O S E S T P O I N T ALGORITHM
Now that the methods for computing the closest point on
a geometric shape to a given point and fo r computing a
least squares registration vector h av e b ee n outlined the IC P
algorithm can be described in terms of a n abstract geometricshape X whose internal repre se nta tion must be known to
execute the algorithm bu t is not of concern fo r this discussion.
Thus, all that follows appl ies equally well to 1 ) s ets of points,2 ) sets of line segments, 3) sets of parametric curves, 4 ) sets
of implicit curves, 5) s e t s of triangles, 6 ) s e t s of parametric
su rfac es, a nd 7) sets of implicit surfaces.
In the description of the algorithm, a “data” shape P is
mo ve d (re g is te red , positioned) to be in best alignment with
a “model” shape X. The data and the model shape may be
represented in an y of the allowable fo rm s . F or ou r purposes,
the data shape must be decomposed into a point set if it is
no t already in point set form. Fortuna tely , this is easy; the
points to be used fo r triangle and line s e t s are the verticesan d the endpoints, and if the d ata s hap e comes in a surface or
curve form, then the vertices and endpoints of the triangle/line
approximation ( a s described above) a r e used. The numberof points in th e d ata shape will be d en ot ed N P . Let N, , be
the number of points, line segments, or triangles involved inthe model shape. As described above, the curve an d surface
closest-point evaluators i m p le m e nt ed i n ou r system require a
framework of lines or triangles to yield the initial parameter
values for the Newton’s iteration; therefore, the number N, is
still relevant fo r these smooth ent it ies but var ies according to
the accuracy of the approx imat ion.
Th e distance metric d b et we en an individual data point p
an d a model shape X will be denoted
-0
d ( i 5 Z X ) = - £ 1 1 i 1 1 | | i — 1 5 ' | | - ( 2 3 ):rEX
The closest p oin t in X that yields the minimum distance is
denoted 3 ] ’ such that d ( j 5 ' , g , T ) :d(p',X), where '5 7 E X. Note
that computing the closest point is O(N,,) worst c a s e withexpected costloghen the closest point computation
(from 1 3 ’ to X) is performed for each point in P, that process is
worst c a s e O(NpN,,) . Let Y denote the resulting s e t of closest
points, and let C be the closest point operator:
Y=qRxy am
Given the resultant corresponding point se t Y, the least squares
registration is computed as described above:
(ii < 1 ) = Q ( P = Y) ( 3 9 )
The positions of the data shape point set a r e then updated viaP = rj'(P).
2 2 _< 1 0 '1 ' Q 1 " ( 1 % ( I i i 2 2 ( q 1 q 2 -qoqal 2 (q1q3 + ( I 0 9 2 )
R= 2@ m+ mm) %+——£ 2@m~mm) (3 )2(q1q3 — qoq2) 2(q2qa + qoqil < 1 3 + q g — qj — q g
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24 4 IEEE TRANSACTIONS O N PA TT E RN ANALYSIS AND MACHINE I N T E LL IG E N C E , V OL . 14 , NO . 2, FEBRUARY 1992
A . ICP Algorithm S t a t e m e n t
The ICP algorithm can now be stated:
' Th e point set P with Np points from the d ata s hap e
and the model shape X (with N, supporting geometric
primitives: points, l ines, or triangles) a r e given.' Th e iteration is initialized b y s ettin g P0 = P, (I 0 =
[ 1 ,0 ,0 ,0 ,0 ,0 ,0 ] ‘ * and k = 0 . The registration vectors
are dened relative to the initial data se t P0 so that thenal registration represents the complete transformation.
Steps 1, 2 , 3, an d 4 are appl ied until convergence within
a t ol er an ce ' r. T he computat ional cost of each operation
is given in brackets.
a . Compute the closest points: Yk :C (Pk, X) (cost:
0(NpN, , ) worst c a s e , 0(Np log N,,,) average).
b. Compute the registration: (fIl==d;,) = Q(P0,Y;,)
(cost: O(Np)).c . Apply the registration: P ; , , + 1 : q ; ' , ( P 0 ) (cost:
0 ( N p ) ) -d. Terminate the iteration when the change in mean-
square error falls below a preset threshold * r > 0
specifying the desired precision of the registration:
dk — dk+1 < '7'.
If a dimensionless threshold is desired, one ca n replace r
with rt/tr(Z,,), where the square root of th e tra ce of the
covariance of the model shape indicates the rough size of the
model shape.
B. Convergence Theorem
A convergence theorem for the IC P algorithm is no w stated
and proved. The key ideas a r e that 1 ) least squares registrationgenerically reduces the average distance between correspond-ing points during each iteration, w he re as 2 ) th e c lo se st point
determination generically reduces the distance for each pointindividually. Of course, this individual distance reduction also
reduces the average distance because the average of a s e t ofsmaller positive numbers is smaller. W e offer a more elaborate
explanation in the proof below.
Theorem: Th e iterative closest point algorithm a lw a ys c on -
verges monotonically to a local minimum with respect to the
mean-square distance objective function.Proof.‘ Given Pk = { 1 3 ' , ; , , } = rj},(P0) an d X, compute the
s e t of closest points Y k : a s prescribed above given
the internal geometric representation of X. Th e mean squa red
error e;, of that correspondence is given by
N1 ” _ _
B k I — Z“ /tr *Ptk | |2- (31)Np 1 1
Th e Q operator is appl ied to get Q }, and dk from that corre-
spondence:.N
1 P % "F —§ —D
d k = - A 7Zlytt — R ( q k R ) P v I 0 — q k T | | 2 ~ (32)p 1 1
It is always the case that dk § ek. Suppose that dk > ek . If
this were so, then the identity transformation on the point se t
would y ie ld a smaller mean square error than the least squares
registration, which cannot possibly be the c a s e . Nex t, l et the
least squares registration r j } , be appl ied to the point se t P0 ,
yielding the point se t P; , +1 . If the previous correspondence to
the set of points Yk were maintained, then the m ea n s qu are
error is still dk , th at is
‘G2_i $ 3 . 2k = — We -i5¥,k+1||2- ( 3 3 )
However, during the application of the subsequent closest
point operator, a ne w point set Y;,,+1 is obtained: Y;,+1 I
C(P;,+1,X). It is c le ar th at
| l ? I i . i = + 1 -Z 7 t , k + 1 | | S lltjtk -}7t,k+1l | fol‘ @ 3 6 1 1 i = 11NP (34)
because the point 3 1 1 ; , was the closest point prior to transforma-tion by r j } , and resides a t some new distance relative to 1 3 ' , ; , ; , + 1 .
If g ] ' , ; ,; , + 1 were further from g ' 5 ' , , ; , + 1 than Qk, then this would
directly contradict the basic operation of the C closest pointoperator. Therefore, the mean square errors e k and dk must
o be y th e following inequality:
0 5 d , . + 1 g e ; , + 1 g a t g 6 , , f o r a ll k . ( 3 5 )
The lower bound occurs, of course, since mean-square errors
cannot be negative. Beca u se the mea n-squ a re error sequence
is nonincreasing an d bounded below, the algorithm as stated
above must converge monotonically to a minimum value.
Q.E.D.Experimentally, we n d fast convergence during the rst
few iterations that slows down a s it approaches the local
minimum. Even a t this slow pace, somewhere between 30
an d S0 iterations yields excellent results: dk 2 : 0.1% of model
shape size. The convergence can be accelerated using a simpleadditional operat ion descr ibed in the ne xt se ct io n .
C. An Accelerated ICP Algorithm
The accelerated ICP algorithm u s e s a minor variation onthe basic line search methods of multivariate unconstrainedminimization [45]. As the iterative closest point algorithm
proceeds, a sequence of registration vectors is generated: ( I 1 ,
(jg, ( Y 3 , 6 ' 4 , (I5, (I6, . . ., which traces ou t a path in the registration
state space from the identity transformation toward a locallyoptimal shape match . Co nside r the difference vector sequence
de n ed b y
Alli IIi i - § i ¢ ~ 1 (36)
which denes a direction in the registration state space. Let
the a n gle in 7 space between the two last directions be denoted
_ Afli<tA(Il¢-1 )91 ¢ ICO5 1Em37)
||Aqt||llAqt_1l|
an d let 6 6 be a sufciently s m a ll a n gu l ar tolerance (e.g., 10° ) .
If
Elk < and l9;, ,_1 < (38)
th en th ere is good direction alignment for the la st th re e
registration state vectors: Q ’ , - 1 , , §}.,_1, a nd r j} ,_ 2 . Let dk , dk_1,
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BESL AND MCKAY: ME T H OD FOR REGISTRATION OF 3- D SHAPES
Linear Approximation The Accelerated
I PAl 'hParabola C gem m
ivlI
d(i-2)\.
Mean d-1 Plane ‘EmswSquare ~._\ Represents . _ . _ ,_,,,_A_,, ___r ’_ _ ._
Em“ 7-D SpaceAxis Q
q(i-2in “ :
Q
q t i - 3 )8e(1_I) Linear Update
5 8 ( 1 )
Parabola Update
Fig. 1 . Consistent direction allows acceleration of the ICP algorithm.
and d;,_2 be t he a ss oc ia te d m ea n square errors, an d let ' 0 1 , ,
v;,_1, and v ; . , _ ; _ > b e associated approximate arc length argumentvalues:
U h = Uvt-1 = -|lA¢li<||, ’va-2 = l|A(lit-1|] + ' v x t ~ 1 - (39)
S e e Fig. 1 fo r a picture of the situation. Next, a linearapproximation an d a parabolic interpolant to the last three
data points are computed:
d1(o) = alt; + bl d2(o) = 0 . 2 2 1 2 + b 2 ’ U + C 2 ( 40 )
which gives u s a possible linear update, based on the zero
crossing of the line, and a possible parabola update, based on
the extremum point of the parabola:
’U 1 = -01/(11 > 0 U2I02/20,2.
To be on the safe side, we adopt a maximum allowable value
omax. Th e following logic is used to perform an attempted
update:
1)If0<’Ug <'u1 <vm,,Xor0 <'0 2
< vmax < 0 1 , use the pa rab o la -ba sed u pda ted registration vector:
(fl, = (Ir + 'ugAcj}, /|[Atj ' ; , | | instead of th e u su al vector
if}, when performing the u pda te on the point set, i.e.,
Pa+1 = <I'i,(P0)-
2 ) If0<v1 <v2<vm.-ix 0 r0< v 1 < v I 1 1 & X <v20r‘U 2 < 0 and 0 < o 1 < o m a x , u s e the line-based updated
registrat ion vector c = (T r + o1Aq'}, /HArj' ; , | | instead of
the usual vector if}.3 ) If both v1 > omax an d '0 2 > vmax, use the maximum
allowable update ( T 2 , = iii + t1m,,,,,Atj’;,/||Arj';,|| instead
of the usual vector c j } , , .
W e have found experimentally that setting vmax :25|]Atj 'k|]
adaptively has provided a g oo d s an ity c he ck on the updates
allowing the iterative closest point algorithm to mo ve to the
local minimum with a given degree of precision in many fewersteps. A nominal ru n of more th an 5 0 basic ICP iterations for a
given value of 1 ' is typically accelerated to 1 5 or 20 i te ra tions.
If the updated registration vector were s om e ho w to over-
shoot the minimum enough to yield a worse mean square
error, it would be a dva n ta g eou s to construct a n e w p a ra b ol a
q ( 1 ) ‘ ' A , , _ , ,_
:x
24 5
__ _ *_* ' __H_z;LT
J - < s > § m = . : : r . . . . * . . = : . . . .\____1~§mmumBmr
Z 0 ’ - 1 ' - I > l " 0 l Z > W " |
zA-_ ._ _
m_ 1__~1g>4_
'5
e i d o
AL
l‘ ; L . 5 ,::, r 1’—q —---_ f
z o -Jm__ ___ _ 1 ll 1_ l Ill)B» — _- "C
__ _ i_if__I -— C Clm T ’ 7 *
4-4-—_-i
--7 CumuIltivcArcL€tglh1
— 1 -%== Eua Fig. 2 . V ar ious quant it ies p lot ted against iteration count fo r t he b as ic
nonaccelerated ICP algorithm.
using the new registrat ion with the last two steps and moveto the appropriate minimum. This h a s not been necessary in
ou r experience. To b e rigorous, one can simply ignore the
suggested update if it causes a worse mean square error.
To give a quantitative example comparison, the registrationvalues, RM S error, maximum e rr or , a n gu l ar change, an d cu -
mulative arc length values we re re co rd e d during 50 iterations
of both the basic and accelerated ICP algori thms dur ing the
same free-form surface matching test. The results fo r the basic
ICP algor ithm are shown in F ig . 2. Note the smooth character
of all the c u rves. T he most important feature is that the cos(6t9)
plot indicates a consistent direction of updates fo r all bu t the
rst few iterations. In contrast, the accelerated IC P algorithms ho ws th e desirable jumpy behavior as seen in Fig. 3. In
addition, note how most quant it ies get close to th ei r na lvalues after the rst acceleration step and very close after two.The acceleration steps occur whenever a V-shaped dip occurs
in the plot of cos(6l9) versus the iteration count.
D. Alternative Minimization Approaches
T he I CP algorithm allows us to move from a g iven s ta r ting
point to a local minima in 7 space relatively quickly incomparison with o th er p o s si bl e a lt er na ti ve s. Each iteration
requires only one evaluation of the closest point operator:the m o st e xp e n siv e computation. Any optimization method
that does not u s e explicit vector gradient estimates, such
a s Powell’s direction set method, the Nelder-Mead downhil lsimplex method, o r s im u la te d anneall ing, requires literally
hundreds to tens of thousands of closest point evaluations.
These numbers are based on tests done t o s im u l at e the action
of the least squares registration step involved in on e IC P
iteration b u t u s in g instead Powell’s direction set method an d
the Nelder-Mead method from [45].
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24 6 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE I N TE L L IG E N C E , V OL . 14 , NO . 2, FEBRUARY 1992
/\r v ‘ v V 1 :Horizontal Axis:L V ytos6 9) Number of Iterations
XIUmrror
ESrrorl. _. .
\%2&9’)Exi-1
>< *<
in >
2 Q ‘ - “ l > ‘ % C ) ' F U) - - — ] > [ — < g ; Z j > 7 J '
J
Z
Cumulat ive Arc Length
Fig. 3. Various quanti t ies p lotted against iteration count fo r the accelerated
ICP algorithm.
An y optimization method that u s e s explicit vector gradients,
such a s steepest descent, conjugate gradient, and variable
metric schemes, will require at least seven closest point
evaluations fo r each numerical gradient evaluation. Therefore,
such a method would have to converge in three or four
iterations to be competitive with the accelerated ICP method.Such generic methods general ly r eq u ir e m a n y more tha n three
iterations with the numbe r of required closest point evaluations
running well over 1 0 0 even in ideal c i rcumstances. If a pure
numerical Hessian-based Newton’s m ethod were used, the
numerical gradient and Hessian computations would require
a t least 1 3 closest point evaluations per iteration, implying
that the iteration would have t o c on v er ge in two iterationsto b e competitive with the accelerated ICP algorithm. A pure
Newton’s method might require only three iterations if the
initial point were a lready well into the region of attraction
surrounding a local minimum, bu t the initial iterations wouldno t be handled well by Newton’s method.
Whenever an accelerated parabolic u p da te t ak es p l ac e after
three basic IC P steps, we can ge t ne arly quad ra t ic conve rge nce
fo r less than steepest descent cost. This is an interesting
accomplishment for a function where derivatives cannot be
evaluated. Note that the steepest descent gradient directionis no t deliberately computed; we merely observe when a
consistent direction is being followed.Other prob lems involved with us ing general -purpose op-
timizationmethods
are the following: 1 ) If a ny a ng le s are
used a s in [54], angular cycles across 3 60 ° must be handled
correctly, and 2 ) if a unit quaternion becomes a nonunit quater-
nion, as would be expected taking arbitrary direction steps
in 4 space, the quaternion must be renormalized somewhere.
Unfortunately, if the objective function evaluator c ha ng es the
values in the state vector during the optimization iteration, this
has a ba d effect on most nonlinear optimization algorithms.
To summarize, any method that allows one to move froman initial state to its corresponding local minimum could
theoretically be used in place of the ICP algorithm. For
e xample , co ns ide r Szeliski’s [54 ] work with steepest descent
an d three rotation angles. However , the arguments above
indicate that on e would have a hard t ime trying to nd an
algorithm that was only ten times slower on average. The key
benet of the IC P algorithm is that the convergence is fast an d
m o n ot on ic . N o expensive closest point evaluations are spent
on registration vectors that have worse mean square errors than
the current state. Because of the ICP convergence theorem, on e
does not hav e to “feel around” in the multidimensional space
t o d e te rm ine the direction in which to m ov e.
V. THE SET OF INITIAL REGISTRATIONS
Even though the IC P algorithm must converge monotoni-
cally to a local minimum from an y given rotation an d trans-
lation of the data point set, it may or may no t converge on
t he d es ir ed global minimum. Th e only way to be sure is to
n d the minimum of al l the local minima. The problem withreaching the desired global minimum with certainty is that it
is difcult to precisely characterize in general the partitioningof the registration state space into local minima wells (regions
of attraction) because this partitioning is potentially differentfo r every possible different shape encountered.
To be precise, consider a 6 -D state sp ace Q , where the
quaternion component q o is determined from the other quater-nion components: q o = (/1 — (qf + q§ + ( 1 1 % Th e actual
state space Q is a subset of the space Q ’ = [-1, 1 ] ? ’ > < R3 ,where ‘R :(—oe,+oo) is the real line. The subset Q is
s p ec i ed b y the “inside or on the unit 3 sph e re ” co nst ra int
that qf + q g + q g 5 1 . Therefore, Q may be viewed a s a type
of hyper-cylinder in 6 space.
For any given nonpathological shape X that represents
a real-world surface o r ob je ct (e.g., pathological shape de -
scriptions based on sin (1 /: 12 ) n ea r zero not allowed) and foran y given point set Preg already correctly registered with
X, consider that any initial state Q ’ G Q of the point set
P = q ’ ( P ; @ g ) will converge to a local minimum a s it is
m a tc he d to X. There are a nite number of local minima
Nm(X, P) after one h a s xed X and P. (The shape X is
considered pathological if this is n ot t ru e .) Let \I1(X, P) be
the s e t of all local minima:
\ I 1 t X . P ) = { ¢ . . } . t ’ : 1 » < 4 2 )This induces a natural partitioning of Q into equivalence
classes, labeled \ I 1 , , , where every value of if that converges
via the IC P algorithm to ip n is a member of t he c la ss \I1n.
This a llows us to state that
Nm
ti: U q r , a n d \ I 1 . , , r “ | \ I 1 , . , , =¢irn¢m. ( 4 3 )'n.=1
Let 1 1 1 be the equivalence class that maps to the correct global
minimum 1 1 1 1 .
T o g u aran tee that the global minimum is found for a given
shape X and a given s e t P not already registered with X, one
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BESL AND MCKAY: ME T H OD FO R REGISTRATION OF 3 -D S H AP E S
must u se an appropr iate se t of initial states so that transforming
P by a t l east one initial registration will p lace the point set
into the correct equivalence class \ I 11 of registrations. Thisallows it to conve rge to the correct global minimum 1 / ) 1 . Two
fundamental questions are 1 ) how to construct an initial s e t
of states for an y g iv e n o b je c t that guarantees a correct global
minimum and 2 ) how to construct a n initial s e t of states that
guarantees all shapes in a given class of shapes when those
shapes converge to the correct respective g loba l minimum.By using a sufciently dense uniform sampling of quater-
nions on the unit sphere combined with a sufciently dense
sampl ing of translation vectors occupying the total volume
about the shape X, it is possible to determine the complete
nite se t of local minima with a sufciently smal l probability
of error for one given object. O ne could construct a set of
initial states to include all these local minima solut ions along
with the halfway states between th e h n ea re st neighbors (e.g.,
h = 12) to a ttemp t to a vo id having a s e t of initial states that
lie on or near the boundaries between the equivalence classes.
A method that could be useful for computing guaranteed
initial states for a se t of model shapes is to us e a 6-D
occupancy array to compute hypervoxel-based descriptions
of the equivalence classes for ea ch s ha pe of interest an dthen computing a n overall partitioning by rening the rst
shape’s partition by intersecting the equivalence classes of
each subsequent shape. Such methods can b e very memory
intensive; a 20 > < 20 > < 2 0 > < 20 > < 20 > < 20 hypercubic-hypervoxelgrid of the smallest hypercylinder containing all relevant
registrations of all shapes of interest requires a n 8-Mbyte(6 4 Mb) array for a s ing le o b je c t. Clearly, this is a test of
patience fo r anyone wishing to construct a n initial set of states
customized to a given se t of objects to yield guaranteed resu lts ,
bu t indeed, it is possible and relatively straightforward, except
fo r the high dimension of the problem.
A. Initial States for Global Matching
There a r e simpler methods of dealing with the initial stateproblem that are very effective on most shapes on e com es
across. Let us adopt the following denition of the rst two
moments of the distribution of geomet ry in P and X: / 1 } , =
Elle P ] . /It = Elflfé X l» E n = E [ ( r 5 ' — / It - )( 1 5 ' — / I p ) ‘| r 5 ‘ €P], an d Z3 , 2 E[(F— ;1’,,,)(F— )ti',_,)t[F E X], where
represents the sample expectation (averaging) operator. If the
point data set P covers a signicant portion of the model
shape X such that the condition
a1\/tr(E,,) 5 , / i t ~ ( E p ) 5 \/tr(E,,) ( 4 4 )
holds for a sufciently large factor, say 0 : 1 = 1/\/2m 0.71 ,
then we have found that it is general ly no t necessary to use
multiple initial translation states, a s long a s enough rotation
states are used. This factor ( 1 3 1 is the allowable occlusionpercentage for global matching. Th e e xac t value of O 5 1 could be
computed for an y given class of object shapes via exhaustive
testing if that is desired.
There are two reasonable options for the initial translation
s ta te: 1 ) Apply the ICP algorithm directly to the point s e t P
using multiple rotation states a bo u t i ts center of mass / 1 1 , , or
24 7
2 ) transform it rst so that the c enters of mass of P an d X
coincide, and then apply ICP. W e have found no differences
in the nal registration results between 1) not translating and
2 ) pretranslating the point se t w he n u sin g an adequately large
set of initial rotations (e.g., 24). In fact, any translation state
sufces because the IC P algorithm is very insensitive to the
initial translation state when used fo r global shape matching.
W e h av e o bs er ve d that pretranslating the data point set’s center
of mass to the model shape’s center of mass generally saves
a few iterations (e.g., 2 to 4) ou t of the u su al 2 0 or so total.A further simplication in the global shape matching al-
gorithm can be accomplished for most generic shapes, whereprincipal moments demonstrate some level of distinctness. Letpr Z pg Z pz be the square roots of the eigenvalues of Zlp,
an d le t 1 1 , , Z ry Z rz be the square roots of the eigenvalues
of E m . If the following s e t s of conditions hold:
P y S 0 1 2 % P i . » S 0 4 2 % (45)
ry § agrm T 2 § O 2 ' l y ( 4 6 )
for a s p e ci ed v a lu e of 0:2, e.g., O 5 2 = 1/\/2 z 0.71 , on e
can reliably match the basic shape structure of data and model
using only the eigenvectors of the matrices E 1 , and E m . Again,the exact value of 0 :2 could be computed for an y given set of
objects an d an y given level of sensor noise via exhaustive
testing if needed. In this case of eigenvalue distinctness,
the identity transformation and the 180° rotations about the
eigenvector axes c or re s p on d in g t o 1 ,, ry, an d T 2 provide a
very good set of only four initial ro ta t io ns that will yield the
correct global minimum for a wide class of model shapes.
If two of the three eigenvalues are approximately equal bu t
signicantly different from the third for both d ata a nd model
shapes, the number of initial states need only be expanded fo rrotations about the n o na m b ig u ou s a x is , t he re by reducing the
t ot al n u m b er of initial rotational states.
If neither of the above c a s e s fo r global matching hold true
(i.e., p a , z p g m pz and 1 ,, m ry re rz), then one must use a
ne sampl ing of quaternion states that cover the entire surface
of the northern hemisphere of the unit 4 sphere uniformly.
Th e rotation groups of the regular polyhedra, which have
been well known to crystallographers since the 1800 ’s [29],provide a convenient se t of uniformly sampled initial rotations:
(a) 12 tetrahedral group states, (b) 24 octahedral/hexahedralgroup states , and (c ) 6 0 icosahedral/dodecahedral group states.
The tetrahedral states a r e a proper subset (subgroup) of the
octahedral/hexahedral s ta te s and the icosahedral/dodecahedral
states. The octahedral/hexahedral states are no t properly con-
tained in the icosahedral/dodecahedral states. Fo r a convenient
listing of these rotations in quaternion form, see Appendix A
of [32 ].
From an implementation point of view, one has the option
of us ing precomputed l is ts or nested loops. For the nestedloop case, all normal ized combinations of q o = [1, 0}, ql =
{+1,0, -1}, Q 2 = {+1, 0 , -1], an d Q 3 ={-l-1,0, -1} provide
an easy-to-compute se t of 4 0 rotation states that includes the
tetrahedral and the octahedral/hexahedral groups . (One mustensure that the rst no nze ro quate rnion co mpo nent is positive
to avoid dupl ication of states.) For a real ly complicated s e t
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24 8 IEEE TRANSACTIONS O N PA TT E RN ANALYSIS AND MACHINE I N TE L L IG E N C E , V OL . 14 , NO. 2, FEBRUARY 1992
of shapes, all normalized combinations of Q 9 = - {1,0.5,0},
q , = {+ 1 ,0 .5 ,0 , -0 .5 , -1}, q g : { +1 ,0 .5 ,0 , -0.5, -1},and ( 1 3 = {+1,0.5, 0, -0.5 l provides another easy-to-
compute se t of 3 1 2 initial rotation states. Another scheme for
a very dense sampling of states is to rene the 6 0 states of
the icosahedral group by subdividing triangles using a 1 to 4
split an d using an increased number of rotations about each
axis s p e ci e d b y each vertex of the rened icosahedron.
The general rule of thumb is the more compl icated theobject, the more initial states required. Any method of getting
a sufciently dense, uniform distribution of quate rnions o ve r
the no r th e rn h emisph e re of the unit 4 sphere (o r over the
full interior and surface of the unit 3 sphere) is adequate. Ingeneral, every application ma y want to u s e a customized set
of initial quaternions that will maximize the probability of
choosing a good starting point early fo r the shapes of interest.
B. A Counter Example
Although the above methods for global shape matching will
work very well for many shapes with a small probability of
error, we can a lso state categorically that fo r any given xed
set of initial rotation states, on e ca n construct a shape X thatcannot be correctly reg is tere d by the algorithm. Begin with
a sphere of radius R. Ad d a thin spike of length S to the
surface of the sphere for each specied axis and fo r each
rotation a bo ut tha t a xis a s indicated by the fixed s e t of initialrotation states. Next, a dd o ne or more spikes of length S + e
anywhere that the extra spikes will t. If the extras will no t
t, make the original spikes thinner. Then, sample points on
the surface of this shape with any desired scheme so that there
is at least on e point p er spike, including one at the tip of each
spike. The IC P algorithm combined with the given xed set
of initial rotation states will not be a ble to register a generic
repositioning of this point se t with the original object in such a
way that the longer (or shorter) spikes are correctly registered
with each other. It can safely be predicted that the proposed
registration algorithm will have difculty correctly registering“sea urchins” and “planets.” These shapes are characterizedas having almost exactly equal eigenvalues of the covariance
matrix E, an d as having smal l shape features at a n e scale
relative to the overall shape. Of course, for an y given xed se t
of object shapes, the set of initial rotations can be increased
to guarantee correct registration.
C. Local Shape Matching
The proposed registration algorithm is denitely not useful
if only a subset of the da ta point shape P corresponds to
the model shape X or a subset of the model shape X, that
is, the data point set includes a signicant number of data
points that do not correspond to any points on the modelshape. Unfortunately, this is a trait of the majority of the
shape matching algorithms that have ever been implemented.
Moreover , this is a common problem in computer vision since
data segmentation algorithms of ten misg roup one se t of data
points with another distinct group of points t ha t sh ou ld no t be
grouped together.
The ICP algorithm is still useful for local matching problems
where the entire se t of data points P matches a subset of
the model shape X. The dra wb ac k is th at m ore than one
initial translation must be used, which in cr ea se s t he c os t of
computing the correct registration. If N, is the number ofinitial translation states, Nq is the numbe r of initial rotation
states, a nd each closest point evaluation of P relative to Xcosts O(N,,N$), in the worst case, the total cost of local
matching is O(N,NqN,,N,,) as opposed to the cost of global
matching O(NqN,,N,,). T he n u mb er of initial translations is
a lso d ep en de nt on the relative size of the data point se t P
compared with the model shape X. By d enin g a quantity n
as the ratio of the “sizes” of the d ata a nd model shapes
4 = 2 1 ( 4 1 )
where m(-) is a general measure that measures approximate1) arc length ifX is a c ur ve , 2 ) area ifX is a surface, an d 3 )
volume ifX is a volumetric point conguration, t he n o ne can
es ti mate the basic qualitative behavior of the required number
of translation states as N, m c(X)r7 for most shapes, where
c(X) is a n approximate proportionality factor that depends on
the complexity of the shape X being matched.Computing such a n estimate m(X) is straightforward fo r
the shape X, bu t eva lua ting m(P) is more difcult because P
ma y simply be a point set, an d the corresponding length , area,
and volume on the shape model X is unknown. One can u s e
a convex hull, surface Delaunay, or closest point connection
algorithm to get accurate measures for volumes, areas, and
arc lengths, bu t it is more likely that on e would design an
algorithm to tolerate up to a particular percentage of occlusion,
an d a xed se t of translations would be computed t o h an dl e
all objects in a given class of objects with up to that level of
o cclus io n. Example s of local shape matching are demonstrated
in the next section.
VI. EXPERIMENTAL RESULTS
This s ec tio n is d iv id ed in to three sections: 1 ) point set
m a tc hin g, 2 ) cu rv e m atc hin g, an d 3 ) s ur fa ce matching. All
programs were written in C. An y quoted approximate times are
given for execution on a single-processor computer rated a t 1.6
Mops on the 10 0 > < 10 0 double-precision Linpack benchmark.
A. Point Se t Matching
In this s ec tio n, w e d em on str ate the ability of the IC P
algorithm to p e rf or m loc al p o in t set matching without cor-
respondence. Table I lists a point set with eight points thatis to b e matched against a s e t of 1 1 points. Fig. 4 shows the
two point s e t s prior to registration. Fig. 5 shows the two po in ts e t s after registration by the ICP algorithm after six i terat ions,which took less than 1 s . The computed registration is
Translation: -48.078 6.65685 119.479Rotation Axis: (0.0321865 0.998188 -0.0508331)
Rotation Angle: 55.7188 degs
RMS Error: 0.437608 m m
Th e algorithm does no t p ay attention to the extra points or
to the ordering of the points because it always pairs a given
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BESL AND MCKAY: ME T H OD FO R REGISTRATION OF 3- D SHAPES
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/I I
/I 1 ‘I
_____L___
1,1 1 1 /' ’ 1’ I I
PT 1i 1
1 11
1 1j 1
11
1
I
1fl
(1I
f
I
,’ I rI j j 1
Fig. 4 . Set 1 and set 2 prior to registration.
\\
1 \_1-"\
1 \
I
r 1'1 I ,
1 \ I\1 1
1 \ ,. II .._ \ /
-_- I ,1
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\
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Ir
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Fig. 5 . Set 1 and set 2 af ter ICP regist rat ion.
TWO 3-D P OI N T S ET S: S E T 1 IS A SUBSET OF SE T 2
TABLE I
1 ' 1u_ ..
yl 2 1 I2 y2 2 2
43.89 -5.88 1 0 6 . 9 9 72.78
- it
7.12 146.10
42.02 20.52
T _
112.52 70.19 24.80 1 4 8 . 67
4 2 . 0 1 25.39 113.25 76.21 1 8 . 2 8 147.20
4 4 . 9 5 4.69 112.60 72.71 17.69 148.09
44.12 17.96 115.15 70.67 4.62 145.95
48.26 -1.37 113.59 64.38 -5 .4 0 1 4 3 .5 9
46.28 7.03 1 1 4 . 5 8 72.47 -1.16 1 4 3 . 8 5
4 7 . 0 6 18.52 1 1 7 . 65 69.82 1 9 .8 1 148.32
77.00 25.00 150.00l
80.00 -10.0 140.00
83.00 30.00 145.00
point on the rst s e t to the closest point on the other s e t . Onlyon e initial rotation sta te a nd o ne initial translation state were
used in t his e x am p l e of local matching. In general, one must
use an initial se t of rotations combined with an initial set of
24 9
translations to ach ieve local matching. Compared with basic
point se t matching, which requires the same number of points
lis ted in d irec t correspondence, we a r e essentially trading off
additional CP U time for l oc a l m a tc h in g capability an d point
order insensitivity.1 ) Computational Issues: If on e were to use a brute-force
tree search (testing every possible correspondence and choos-
ing the best one) in order to nd the best m atch , th is typ e
of registration would require operations of the basic
least squares quate rn io n match . Fo r the s im p le e xa m pl e of
local matching above with Np :8 an d N,,, :1 1 , a brute
force test would require 6 6 52 80 0 quaternion registration
operations. Th e above registration required only s i x o pe ra tions
since the initial state wa s already a member of the equivalence
class of the global minimum. Even with 2 4 initial translation
s ta te s and 6 0 initial rotation states allowing ten iterations for
each combined initial state, we would require only 1 4 4 00
operations to prov ide an exhaustive an d very capable matching
ability. Moreover , it only takes a few minutes for these types of
small point sets. Th e compu ta tion r eduction r a tio of the IC P
algorithm compared with brute-force testing for t his s im p le
case is 4 6 2 : 1 . Of course, other alternatives are possible, bu t
we see that considering 1 4 4 0 initial states is no t unreasonablewhen the IC P algorithm is u sed to move from initial state to
local minimum.
In the African mask examp le in the surface section below,
we accura te ly reg is te red a point set with N,,=2500 points
to a point set with N,,=4200 points using 6 0 initial rotation
states in less than 8 min. The amount of brute-force enumer-
at io n test ing re qu i re d for this case is ridiculously large; more
than 17002°°°(> 107500) operations are r eq u ir ed . E v en at 1
TeraOp/s (1012 ) for the age of the universe 1018 s, this exact
brute-force enumeration of all possible combinations wouldrequire well over 10250 universe lifetimes
B. Curve Matching
I n th is section, the ability of the ICP algorithm to do localfree-form curve matching is demonstrated. A 3-D parametric
space c ur ve s p lin e wa s dened as a linear combination of
cubic B-splines and control points. A copy o f it was translatedan d rotated to be relatively difcult to m atch. The rotated
an d t rans la te d curve wa s converted to a polyline description
with 6 4 points. Each a:,y, z component of each point of the
polyline was then corrupted by zero-mean Gaussian noise witha standard deviation of o :0.1 (compared with a curve size
of 2 .3 > < 2 > < 1 units). Th e i3o range of 0 .6 units is clearly
visible compared with the s ize of the object. W e th en c ut off
over half of the noisy polyline leaving a partial noisy curve
shape. Fig. 6 shows the two space curves prior to registration.
Fig. 7 s ho w s t he two space curve s af te r ICP registration using
12 initial rotation states and six initial translation states for atotal of 72 initial registration states. If the registration to mo ve
the curve away from the original curve is post-multiplied by
the registration recovered using the IC P algorithm, w e s ho ul d
obtain a matrix close to the identity matrix except fo r the
effects of noise. T he m a tc h of the partial noisy cu rv e to the
original spl ine curve yielded the following registration matrix,
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25O IEEE TRANSACTIONS O N P AT TE RN ANALYSIS AND MACHINE I N TE L L IG E N C E , V OL . 14 , NO . 2, FEBRUARY 1992
I * " "7 " T F
i _ _K
‘iy ' \
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i _ _
I
I
v'
\‘ .
\
I
I
I, |\
\
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I
Fig. 6 . Ideal and a partial noisy space curve before regist rat ion.
T“ .\ “
i\ .\ .
\
I I \
.r\ 1
1 , /’I |
,I,’
l/
F
. ‘ 0 .4
Fig. 7. Ideal and a part ia l noisy space curve af ter regist rat ion.
which is very c lose to the identity matrix:
Translation: 0.023540 0.006925 -0.015471
Rotation: 0.999925 —0.012227 -0.001007
Matrix: 0.012262 0.998647 0.050550
0.000387 —0.050557 0.998721
C. Surface Matching
In this section, the ability of the IC P algorithm to register
free-form surface shapes is demonstrated.
1 ) A Bezier Surface Patch: A simple parametr ic Bezier sur-
face pa tch was constructed fo r quick testing of the free-formsurface matching capability of the ICP algorithm. A s e t of2 5 0 randomly positioned points wa s evaluated in the interior
of the domain of the surface patch an d translated an d rotated
in a random manner. Th e points of this point s et a re connected
by l ines indicating the point list sequence; t he y d o not indicate
line geometry to be matched. The su rface p atc h is drawn
-./1.
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. ., _ _ . ' |_ _ _ |_ _ _ _ -
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l l \ ‘ n \ \ \ .I I I I l ; \ \ \ \: I l , L . l L \ \ \
I l . . 1 . sI l i l f - 4 . . l . I A A l
r I l l l 1 ? : 7 . £ i i l l , n : ' . : z Illn g e rf l l l l l l g . / ’, \ I I % % /M
-H __
W/._ §Q§ ‘..
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Fig. 8 . N oi sy p o in t s e t and surface patch prior to regist rat ion: 25 0 pointsm a tc hed to 450 t riang les.
H I . - { i i i . = * f . ~ l : ; _ ' f ‘ : a § % f i " i ; i ’ ” ” % j a a t i ’;/%
-1 -v--_
'\
,-
\
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w
0S"31-
‘v
Fig. 9 . Noisy po int set and surface patch after registration.
via isoparametric lines indicating 4 50 triangles an d ts in a
3 > < 3 > < 1 units box. Following the space c u rv e e x am p l e, 3-
D v ecto r noise with a standard deviation of 0 .1 units in each
direction was added to the point set data to create a noisy
point set. Th e surface patch a nd th e noisy point set are shown
prior to registration in Fig. 8 . After running the ICP algorithmwith same 24 initial rotation states for a total of about 3 min,
we obtained the res ul t s hown in F ig . 9 . The initial positioningtransformation multiplied by the recovered transformation for
the noisy point set yields the following approximate identitytransformation:
Translation: -0.057329 0.013923 0.018430Rotation: 0.999357 0.034041 0.011264
Matrix: -0.033883 0.999328 -0.013935-0.011731 0.013545 0.999840
This demonstrates that global matching under noisy condi-
tions works quite well.A subset of 138 n ois y p o in ts wa s used to te st th e local
matching ability. The sur face pa tch and the noisy point subset
are shown prior to r eg is tr at io n i n F ig . 10 . After running the
ICP algorithm with 24 initial rotation states and six initialtranslation states for a total of about 6 min, w e o bta in ed
the result shown in Fig. 11 . Th e initial positioning trans-
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BESL AND MCKAY: ME T H OD FO R REGISTRATION O F 3 -D S HA PE S
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Fig. 10 . Noisy po int subset and surface patch prior to registration: 13 8 pointsand 45 0 triangles.
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rt - & 5 § E ‘ . 8 ‘ . % i ; 2 : \/”|l- i \ \ \ \ ” ” ’
I
U _
a
2 / i t s * 1Fig. 11 . Noisy point subset and surface patch after registration.
formation multiplied by the recovered transformation for thenoisy point subset yields the following approximate identity
transformation:
Translation: -0.166476 0.159480 0.128289Rotation: 0.990806 0.113548 0.073548
Matrix: -0.113595 0.993521 —0.003545
-0.073474 -0.004842 0.997285
This matr ix approx imation to the identity is much less
precise than the global matching c a s e , bu t the data is s o noisy
and the shap e is s o featureless that we found it surprising that
the registration came ou t a s well a s it did.
2 ) The NRCC Aican Mask: In t hi s e xpe r ime nt , range data
from the National Research Council of Canada’sAfricanmask example was obtained using the commercially available
Hyscan laser triangulation sensor from Hymarc, Ltd. A low-resolution 6 4 > < 6 8 gridded image wa s computed from the
original data set for use in o u r e x p er im e nt s an d is shown in
Fig. 1 2 . This data will serve a s the model surface description
with 8 4 4 2 triangles ( 4 2 2 1 quadr ila tera l polygons). A thinned
\ I’ ' u »5$ 'o o ' \\\\\~ ~ . I l l l l i i i f i i i i i i i i i i i i i i i z i i z g t s t i t ; 7 : ; ; : ; § : ; = : ; : § § 5 5 5 : ? § : ; $ : § $ : § : ; $ ; Z v § t ; § i . t § i \ ; § t tr ,0 ;¢,$;;r'i/ )
x i. . . . ' / f i l m r m » - a t 7 1 / 1 i t I a t ' : ~ : ~ : ~ : - : ' : . ~ ~ w . ~ ‘ t \
I.--
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r ; - - ' » - 4 ’'; I _ . ' a - - /.p i - - k i4 | p I I ' 5 '
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\ l\ \/ l I I I 1 ' ; - ' / / I / I I 1 , ' I { l ;0 ; ' o ' 0 ' o ' o titgtgit t \““ \\\\-‘
‘l i $ § 7 ' £ i r / ' I : I t I o t 0 W t ‘ f t i t ii t y’ t , t } t l ~ 2 . 2 = =3 ,W3:
Fig. 1 2 . Model surface: Range image of mask: 84 4 2 triangles.
r-.
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. 1 . ' L|‘ t __ - . ‘ . I
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: _ - - - * 1
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_ _ _ _ _ _ _ _ _ . ‘ : . ' - ' - + 5 . .“ - — _ ;_ ‘ § _ % 7 T - :‘ - _ ~ “ 7 ‘ \'
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- 4 ..
1,.
r
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Fig. 13 . Data: Rotated and translated point set of m as k: 2 5 4 6 points.
version of the measured data point set containing 2 5 4 6 points
is u sed as the data point set an d is shown as scan lines in a
test registration view in Fig. 13 .
All trial positionings of the 9 0 - m m object, including the
on e shown above, converged to the correct solution in about
10 min, an d all cases had a 0 .59-mm RMS error. This includes
six i terations worth of testing o n e ac h of 2 4 initial state vectors
an d full iteration on the best state of the six initial iterations. A
side view of both the digital surface model an d the mea su red
point set as registered is shown in Fig. 14 . Th e registrationis quite accurate.
A Bezier surface p a tc h m o de l of the mask w as c re at ed to
te st th e parametric surface matching capability on the given
shape. This model is shown in in Fig. 15 . Th e point set
in various rotat ions and translations was then registered to
the parametric surface model. W e had expected about a 1.2-
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- = ; ‘ f _ - r e / J ’
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Fig. 1 4 . Side view of range image model and registered point set.
mm rms distance, bu t the na l solution had a 3 .4 -m m rm s
distance. After examining the results closely, w e d is co v er ed
that surfaces had no t been created in reg io ns wh ere there were
measured poin ts . Note the extra data points for which there
was no possible surface correspondence and the evidence ofmisregistration in Fig. 1 6 . Overall, this match was not bad
considering the circumstances. After a post-processing step
t o ig n or e a n y measu red points whose point-to-surface vectors
were n ot within a few degrees of the surface normal at the
corresponding points, the misregistration disappeared, an d the
data locked in on the surface with the expected rms distance.
Although the IC P algorithm is no t designed to handle data
points that do not correspond to the model shape, one canconclude that a minor misgrouping of nearby points will
usually have a minor effect. Another important point to k ee p
in mind is that the matching algorithm does not care about
the partitioning of the composite surface model into separate
surface patches.
As a nal test on the m ask, abou t 3 0 % of the points from
t he m e as ur ed data point se t were deleted as shown in Fig.
1 7 . The registration algorithm locked in on the solution and
gave a slightly improved nn s distance in less time than the
full data s e t .
3) Terrain Data: For the nal set of exper imenta l results,
some terrain data for an a re a near Tucson wa s obtained from
the University of Arizona. Fig. 1 8 shows a shaded image
of the r ug ge d te rr ain . T he d im e ns io ns of the model surfacea r e 6700 > < 6840 > < 1400 units. A point s e t was extracted
by performing 56 planar s ec tio n c uts at regular intervals
along one axis and then thinning ou t the data using a chord
length deviation check. An interior section of this data se t
was extracted such that about 6 0 % of the surface area of
the original data set w as c ov er ed . Th e resulting data se t
6 , 1 ,
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25 2 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE I N TE L L IG E N C E , V OL . 14 , NO. 2, FEBRUARY 1992
””1"* L
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\;1Fig. 15 . Surface patch boundaries of a s e t of parametric surfaces: 97 cubic
Bezier patches.
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7. ’ : ' ' “ ii i ii 5 r # ‘ “ _ ” ” t i § i i‘ ¢ $ ' $ ? “ 3 5 5 § ‘ ? . ’ I. d f t \ ‘ I f 4 lVH '9 i@@ t . - . ~ . w~ ;5 ’ ; f , . , , - , M, , r~ , ,* , ' ; , '§9 ;_ m. . § _ f i j ’E 1 @ £ ' ¥ , 4 tW Y ¢ ? , j g _ : ; ; 5 ? , 1 % ‘ i 5 ‘ A 2 ' a % ‘ 5 # , g ? § § * ' ~ *' i ~ '~ 7~ o ' e r » / 2 7 % ~ ’ 1 é “ ‘ *./7 '
Fig. 1 6 . Side view of parametric surface model and registered point set.
it s a s a . § % s ~ . = . = ; ,~ ~ = ; % ? = = § * I s i a ' -a s ;"I ' . Z . ~_ = ; l =t - 2 : ? 47 ' " - ‘ L II
~ . ~ i t t\
Y i\
? - ‘ . 12 - 1 3 0 -
contained 13 6 55 points an d is shown in Fig . 19 . Th e data
point set was then lifted above the model surface and rotated
to be approximately orthogonal to the model s e t a s shown
in Fig. 2 0 . Th e IC P algorithm per formed local matching tothe model surface using 2 4 initial rotations and one initial
translation. Th e registration process for these larger data sets
took about 1 hr. Th e r es ult s are shown in Fig. 21. The
initial positioning transformation multipl ied by the recovered
transformation yields the following approximate identity trans-
formation, which demons tra tes tha t su rfac e matching for very
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BESL AND MCKAY: ME T H OD FO R REGISTRATION OF 3 - D S H AP E S
.
\§_:,‘ 1/ ,1: - _ ii I-* ~»
.1 "T‘ ‘ — 1 T ~ ' € ' ¢ = _ . .
i 7 } ' { } 1 \ 1 ‘ - , | ] r » . _ ~\\\ ‘
. “ ‘ . \ l i l \\\. f '5 J“;-\| M “ [Ir ‘ C ‘
1 '| |i ll‘
\ ‘ \‘\\ \.
""' 4'
Fig. 17. Subset of the data point set: 1781 points.
tr t
Fig. 18 . Model surface: Rugged terrain near Tucson, AZ : 45 90 0 triangles.
complex surface shapes works quite well:
Translation: -11.330336 1.404177 0.934043
Rotation: 0.999994 0.003308 —0.000231
Matrix: -0.003309 0.999994 —0.000505
0.000230 0.000505 1.000000
VII. C O N C L U S I O N S
The iterative closest point (ICP) algorithm is able to register
a data shape with Np points to a model shape with NI
primitives. Th e model shape can be a point set, a se t of
polylines, a set of parametric curves, a set of implicit curves,
a se t of triangles, a set of parametric surfaces, or a set of
implicit surfaces. Any other type of shape representation ca n
25 3
)
"‘ -\\-
iiiiW J R -,-
4“
5
.-._
Fig. 19 . Data: Rugged terrain near Tucson, AZ: 13 6 55 points.
Fig. 20 . Data and model of rugged terrain prior to registration.
Fig. 21 . Data and model of rugged terrain af ter regist rat ion.
be incorporated if a procedure fo r computing the closest point
o n th e model shape to a given point is available. Ifa data shape
were to c om e in a form other than point se t form, a dense set
of points on the data s ha pe c an serve as th e d ata point set.Th e accelerated IC P algorithm converges to a local mini-
m um quickly in comparison with generic nonlinear optimiza-
tion methods. It is f as t e no u gh that global shape matching
can be achieve d us ing a sufciently dense sampl ing of unit
quaternions used a s initial rotation states, an d local shape
matching can be achieved by combining a sufciently dense
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BESL AND MCKAY: ME T H OD FOR REGISTRATION OF 3 -D S HA PE S
performance in the closest point operation. Actual testing onparallel architectures also needs to be done.
Given a large bu t nite amount of off-line processing, it
seems reasonable to make the following statements: For a xedse t of objects, a given level of allowable occlusion, a given
maximum possible (Gaussian) noise level, and a set of in it ia lregistration states, it is possible to estimate the probability of
registration failure by carrying ou t exhaustive tests. Fo r a 1%
(of the smallest shape s iz e) n o ise level an d a 4 0 % maximum
allowable occlusion, ou r tests indicate that very low probabil-
ities of failure should be achievable with minimal extra work.T he m e th od needs to be characterized in such detail.
A single complicated object in a s e t of simpler objects may
require a large se t of initial registrations t o h an dl e the entire
set of objects with certainty. Unfortunately, much t ime i s then
spent testing initial registrations with simpler objects such that
on e is continually homing in on the sam e local minimum
over and over again. With some extra bookkeeping, it may
be possible to quickly recognize a familiar local minimum
well th at y ou have already fallen into a few t imes an d abort
that iteration or to us e penalty function methods to penalize
walking down a path that has already been explored. Th e
appropr iate shape of the penalty functions would d ep en d o nth e s ha pe of the region of a t tr act ion in 6-D, which is difcult
to quantify an d analyze.
It may be possible to extend the basic least squares registra-tion solution to allow deformations ( ind epe nd ent ax is sca ling
and bending) of the model shapes when matching to the
data shapes. S he ar s a nd s ep a ra te axis scaling ca n be easily
accomodated by allowing a general afne transformation;allowing even quadratic bending about the center of mass
signicantly complicates matters.
Finally, these free-form shape matching methods a r e likelyto be useful a s part of a 3-D object recognition system.
ACKNOWLEDGMENT
The authors wish to thank A. Morgan, R. Khetan, R. Tilove,W. Wiitanen, R. Bartels, D. Field W . Meyer, C. Wample r ,
D . B ak er, R. Smith, N. Sapidis, an d the reviewers for their
comments.
R E F E R E N C E S
ili K . S . Arun, T. S . Huang, and S . D. Blostein, “Least square tting of tw o3-D point sets,” IEEE Trans. Part. Anal. Machine Intell. vol. PAMI-9,
no. 5, pp . 698-700 , 1987._ 2 i R. Bajcsy and F . Solina, “Three-dimensional object representation
revisited,” Proc. 1s t I nt . C on C om pu t. V is ion (London), June 8-11 ,1989, pp . 2 3 1 - 2 4 0 .
I3 : P . J . Besl, “Geometric modeling and computer vision,” Proc. IEEE, vol.7 6, no. 8, pp . 936-958, Aug. 1 98 8 .
4 1 P . J . Besl, “Active optical range imaging sen sors , ” in Advances in
Machine Vision (J. Sanz, Ed.). New York : Sp ringer-Verlag, 1989; s e e
also Machine Vision an d App l icat ions , vo l. 1 , pp . 12 7-152 , 1989.
[5]i,Th e free-form surface matching problem,” Machine Vision forThree-Dimensional Scenes (H . Freeman, Ed.). New York: Academic,1990 .
[6 ] P . J . Besl a nd R. C. Jain, “Three dimensional object recogni t ion,”ACMComput. Surveys, vol. 17, no. 1 , 75-145 , March 1985,
[7] J . Blumenthal, “Polygonizing implicit surfaces,” Xerox Parc Tech. Rep.
EDL-88-4 , 1988.[8 ] B. Bhanu and C. C. Ho , “CAD-based 3-D object representation fo r robot
v i s ion ,” IEEE Comput., vol. 2 0, no. 8 , pp . 1 9 - 36 , A ug . 1987.
I
[ 1 9 ]
[ 1 1 ]
I 1 2 ]
[ 1 3 ]
I 1 4 ]
[ 1 5 ]
[ 1 6 ]
[ 1 7 ]
[ 1 8 ]
[ 1 9 ]
[ 3 9 ]
[ 3 1 ]
[ 3 2 ]
[2
[ 2 4 ]
[ Z 5 ]
[ 2 6 ]
I 2 7 ]
[ 2 8 ]
2 9
30
31
3 2
3 3
3 4
I 3 5 ]
[ 3 6 ]
[37]
25 5
W . B oe hm , G. Farin, and J . Kahmann, “A survey of curve and surface
methods in CAGD,” Comput. Aided Geometric Des., vol. 1, no. 1, pp .
1 - 6 0 , July 1984.R. C. Bolles an d P. Horaud, “3DPO: A three-dimensional part ori-
entation s ys tem, ” In t. J. Robotic Res., vol. 5, no. 3, pp . 3-26 , Fall198 6.P . Brou, “Using the Gaussian image to n d the orientation of an object,”Int. J. Robotics Res., vol. 3, no . 4, pp . 89-125, 1 9 83 .J . Callahan and R. Weiss, “A model fo r describing surface shape,” inProc. Conf Comput. Vision. Part. Recogn. (San Francisco, CA), June
1 98 5, p p . 240-247.C. H. Chen and A. C. Kak, “3DPOLY: A robo t v is ion system forrecognizing 3-D objects in low-order polynomial time,” T ec h. R ep .88-48 , Elect. Eng. Dept., Purdue Univ., West Lafayette, IN, 1988.R. T. Chin and C. R. Dyer, “Model-based recognition i n r ob ot vision,”ACM Comput. Surveys, vol. 1 8 , no. l, pp . 6 7 - 1 0 8 , Mar. 1986.C. DeBoor, A Practical Guide t o Splines. New York: Spr inger-Ver lag,1978.C. DeBoor and K . Holl ig, “ B- sp l in e s w it ho u t d iv id ed differences,”Geometric Modeling:Algorithms andNew Trends (G. Farin, Ed.), SIAM,
pp . 2 1 - 2 8 , 1987.G. Far in, Curves an d Surfaces i n Computer Aided Geometric Design: APractical Guide. New York: Academic, 1990 .O. D. Faugeras and M. Hebert, “The representation, recognition, andlocating of 3- D objects,” Int. J. Robotic R e s . vol. 5, no. 3, pp . 27-52 ,Fall 1986.T. J . Fan, “Describing and recognizing 3- D ob ject s us ing surfaceproperties,” Ph.D. dissertation, IRIS-237, Inst. Robotics Intell. Syst.,Univ. of Southern California, Los Angeles, 1988.P . J . Flynn and A. K . Jain, “CAD-based computer vision: From CA D
models to relational graphs,” IEEE Trans. Patt. Anal. Machine Intell.,vol. 1 3 , no. 2 , pp . 114-132 , 1991; s e e also Ph.D. Thesis, Comput. Sci.Dept., Michigan State Univ., E. Lansing, M I.E. G. Gilbert and C. P . Foo, “Comput ing the distance between smootho bje cts in 3 D space,” RSD-TR-13-88 , Univ . of M ic hi ga n , A n n A rb o r,1988.E. G. Gilbert, D. W . Johnson, S . S . Keerthi, “A fas t p roc edu re forcomputing the distance between complex objects in 3 D space,” IEEE J.Robotics Automat, vol. 4, pp . 193-203, 1 9 88 .G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore,MD : Johns Hopkins Univ. P ress , 198 3.W . E. L. Grimson, “The combinatorics of l oca l cons traint s in model -based recognition and localization from sparse data,” J. ACM, vol. 33 ,no. 4, pp . 658-686 , 198 6.W. E. L. Grimson an d T. Lozano-Pérez, “Model-based recognition an d
localization from sparse range or tactile data,” Int. J . Robotics Res., vol.3, no. 3, pp . 3-35 , Fall 1984.K . T. Gunnarsson and F . B. Prinz, “CAD model-based localization of
parts in manufacturing,” IEEE Comput., vol. 2 0, no. 8 , pp . 6 6 - 74 , A ug .1987.
E. Hal l , J . Tio, C. McPherson, and F. Sadjadi, “Measuring curvedsurfaces for robot vision,” Comput. vol. 15 , no. 12 , pp . 4 2 - 5 4 , Dec.1982.R. M . H a r al ic k et al ., “Pose estimation from corresponding point data,”inMachine Vision for Inspection and Measurement (H . Freeman, Ed.).N ew Y or k: A ca de m ic , 1989.H Hilton, Mathematical Crystallography and th e Theory of Groups of
Movements. Oxford: Clarendon, 1963; London: Dov er , 1963.B. K . P . Horn, “Extended Gaussian images,” Proc. I EE E , v o l. 7 2, no .12 , pp . 1656-1678, D ec. 1 98 4__.._., “Closed-form solution of absolute orientation using unit quater-nions,” J . O pt. Soc. Amer. A v ol. 4, no. 4, pp . 6 2 9 - 6 4 2 , Apr. 1987_ _ _ _ _ _ _ , “Relat ive orientat ion,” A.I. Memo 994, AI Lab, Mass. Ins t.
Technol. , Cambridge, Se pt . 1 9 87 .B. K . P . Horn and J . G. Harris, “Rigid body motion f rom range imagesequences,” Comput. Vision Graphics Image Processing, 1989.K. Ikeuchi, “Generating a n interpretation tree from a CA D model for3-D object recognition in bin-picking tasks,” Int. J. Comput. Vision, vol.
1, no . 2 , pp . 145-165, 1 9 87 .A. K . Jain and R. Hoffman, “Evidence-based recognition of 3- D
objects,” IEEE Trans. Part. Anal. Machine Intell., vol. 1 0 , no. 6, pp .793-802 , 1988.B. Kamgar-Parsi, J . L. Jones, and A. Rosenfeld, “Registration of
mult ip le overlapping range images: Scenes without distinct ive features,”Proc. IEEE Comput. Vision Patt. Recogn. Conf (San Diego, CA), June
1989.Y. Lamdan and H. J . Wolfson, “Geometric hashing: A general andefcient model-based recognition scheme,” Proc. 2ndInt. Conf Comput.
Vision (Tarpon Springs, FL), Dec. 5-8, 1 9 8 8 , pp . 238-251.
8/13/2019 1992 - A Method for Registration of 3-D Shapes - ICP - OK
http://slidepdf.com/reader/full/1992-a-method-for-registration-of-3-d-shapes-icp-ok 18/18
25 6
[ 3 8 ]
I 3 9 ]
[40]
[41]
I 4 2 :4 3 -
- 4 4 :. 4 5 -
- 4 6 :
i 4 7 1I 4 8 1 ‘
[ 4 9 1
[ 5 0 1
[ 5 1 I
1 5 2 *[ 5 3 8
I 5 4 1
[ 5 5 1
[56]
I57]
IEEE TRANSACTIONS O N P A TT ER N ANALYSIS AND MACHINE I N TE L L IG E N C E , V OL . 14 , NO . 2, FEBRUARY 1992
S . Z. Li, “Inexact matching of 3D s urf aces , ” VSSP-TR-3-90 , Univ . of
Surrey, England, 1990 .P . Liang, “Measurement, orientation determination, and recognitionof surface shapes in range images,” Cent. Robotics Syst., Univ. of
California, Santa B arba ra , 1 9 87.D. Luenberger, Linear and Nonlinear Programming. Reading, MA:
Addison-Wesley, 1984.A. P . Morgan, Solving Polynomial Systems Using Continuation forEngineering and Scient ic Problems. Englewood Cliffs, NJ: Prentice-Hal l , 1987.M. E. Mortenson, Geometr ic Modeling. Ne w York : Wi ley, 1985.J . Mundy et al., “The PACE system,” in Proc. C VPR ’88 Workshop ; s e e
also DARPA IUW.
D. W . Mu rray , “Mode l-based recogni tion using 3D shape alone.,”C o m p u t . Vision GraphicsImage Processing, vol. 4 0 , p p. 250-266, 1 9 87 .
Press, Flannery, Teukolsky, and Vetterling, Numerical Recipes in C.
Cambridge, UK : Cambridge University Press, 1988.J . Rieger, “O n the classication of views of piecewise smooth objects,”I m ag e a nd Vision Computing, vol. 5 , no. 2 , p p. 91-97 , 1987.Proc. IEEE Robust Methods Workshop. Univ. of Washington, Seattle.P . Sander, “ O n r el ia b ly inferring differential structure from 3D im -ages,” Ph.D. dissertation, Dept. of Elect. Eng., McGill Univ., Montreal,C an ad a, 1 9 88 .P . H. Schonemann, “A generalized solution to the orthogonal procrustesprobIem,” Psychometrika, vol. 31 , no. 1 , 1966J . T. Schwartz and M. Sharir, “Identication of partially obscured objectsin tw o and three dimens ions by match ing nois y charact er is tic curves,”Int. J . Robotics Res., vol. 6 , no . 2 , pp . 29-44, 1 9 87 .
T. W . Sederberg, “Piecewise algebraic surface patches,” Comput. AidedGeometric Des., vol. 2 , no. 1 , pp . 53-59 , 1985.B. M. Smith, “IGES: A key to CAD/CAM systems integration,”
IEEEComput. Graphics Applications, vol. 3, no. 8, p p. 78-83, . 1 9 8 3 .G. Stockman, “Object recognition and localization via pose clustering,”Comput. Vision Graphics Image Processing, vol. 4 0 , pp . 361-387, 1 9 87 .R. Szeliski, “Estimating motion from sparse range data without corre-spondence,” 2 nd Int. C on C om pu t. V is ion ( Ta rpon Spring s, FL), Dec.5-8, 1 9 88 , p p. 2 0 7 - 2 1 6 .G. Taubin, “Algebraic nonplanar curve and surface es timation in 3-
space with applications to position estimation,” Tech. Rep . LEMS-43,Div. Eng., Brown Univ., Providence, RI, 1988.____, “About shape descriptors a nd s ha pe matching,” T ec h. R ep .
LEMS-57, Div. Eng., Brown Univ . , Providence, RI, 1989.D. Terzopoulos, J . Platt, A. Barr, and K . Fleischer, “Elastically de-formable models.” Comput. Graphics, vol. 2 1 , no . 4, pp . 2 0 5 - 2 1 4 , July1987.
B. C. Vemuri, A . M iti ch e, and J . K . A gg ar wa l, “Curvature-basedrepresentation of objects from range data , ” Image Vision Comput., vol.4 , no. 2 , pp . 1 0 7 -1 1 4 , M ay 1986.B. C. Vemuri and J . K . A g ga rw a l, “Representation and recognitionof objects from dense range maps,” IEEE Trans. Circuits Syst., vol.CAS-34, no. 11 , pp . 1 3 5 1 -1 3 6 3 , N ov . 1987.A. K . C. Wong, S . W . Lu , and M . R io u x, “ Re co g ni ti on and shape
synthesis of 3- D objects based on attributed hypergraphs,” IEEE Trans.Patt. Anal. Machine Intell., vol. 11 , no. 3, pp . 279-290 , Mar. 1 . 9 8 9 .
[ 5 3 ]
I 5 9 ]
[ 6 9 ]
Paul J . B es l (M’86) received the B.S. degree summacum laude in physics from Princeton University,Princeton, NJ , in 1978 a nd the M. S . and Ph. D.degrees in electrical engineering and computer sci-ence from the University of M ic hi ga n , A n n A rb o r,in 1 98 1 and 1 98 6, respectively.
Prior to 1984, he worked for Bendix AerospaceSystems in An n Arbor and Structural DynamicsResearch Corporation, Cincinnati, OH . Since 198 6 ,he has been a research scientist at General MotorsResearch Laboratories, Warren, MI, where h is pri-
mary research interest is computer vision, especially the practical applicationsof range imaging sensors.
Dr . B es l is a member of the Association fo r Computing Machinery. Hereceived a Rackham Distinguished Dissertat ion Award fo r his Ph. D. worko n r an ge i ma ge understanding, which was l at er p u bl is he d i n b oo k form by
Springer Verlag.
Neil D. McKay received the B. S . degree in elec-trical engineering from the University of Rochester,Rochester, NY , in 1980, the M.S. degree in co m-
technic Institute in 1982, and the Ph. D. degreein electr ical engineering from the University of
Michigan in 1985.
He has been with the Computer Science Depart-ment of General Motors Research Laboratories since1985. Hi s interests include robot path planning andopt imal control.
puter and systems engineering from Rensselaer Poly-