1992 - a method for registration of 3-d shapes - icp - ok

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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



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.



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.



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



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,



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].



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



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



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



BESL AND MCKAY: ME T H OD FO R REGISTRATION OF 3- D SHAPES

I

/1 I/1

4

f

\1 _,_1 ‘___» 1

1-"\ 1 ,

1 ‘ I ;I I, \ I

1' \ f r

-

P \ 1 I

T ~I 1‘tr 1'

,1 I1 H , 1

1 I ,

J ,r 6‘T 1

I’ II/I 4

I" J I

/ » 4 »1

I

I J ]’ 1

1 1/ 1 1 1

, 1 1 I

/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

_, I I 1F , ,

I, II , 1 11 I

' 11 i /I-IfI If f I

I '

If

1 11 r ,/I

I I 1

1 I f ’1 I

If1 ’ I 1 I

11- I I 1. 1

1 ’ 1'.1 1 'r J 1

____ I I

I’

~.

-.

\

\

I

Ir

11/

J1I

I

1

1-’ 1

I

1I

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,



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 ' \

F\\

\ \. \ ,‘ \| ,-. \ ~ \_ I

i _ _

I

I

v'

\‘ .

\

I

I

I, |\

\

_-P J

r.I'_-)1

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.

.._, ‘ ~ .-r _ . , -

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'-F; — . ' ; ¢2'~r ; , . . » ; . ‘ ~ < ’ - # * ¢ Z = --1 r‘.. ' / . ¢ - - — * - ;71

'1I“

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I-~$ 1 . 1 - 5 :1 -/.

1n_ _ / , ' \ ='. 1 - A » _»_ { _ ' 1 , ». / _ ' - , , / , /

“ 5 1 ~ “ ' = 5 § b ' 5 ~_ ' / ré i~’ ; ~ . - = ' - ' “= Y , - / 1. / » , ¢ -:f~ 3 5. ' . p l > ~ ' ' “ L - . 4» ,/ ',_ ‘ , - 4 - 5 : - 1 : ’t“1, ’ 1_- - _ . , ' - ’ ,1 3 ;-0;__. P ' , ’ , ’'‘-r _ _ \ ; 1 - Q ’)‘:‘-r _ 5 ' » ’ ¢v

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- 1 - _ , ’ < ’ ; ; /“|_ _ _ :J-\

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- . — - _ , 2 _ “ ' t ‘‘ r _ / - - - - - ~ _

. ., _ _ . ' |_ _ _ |_ _ _ _ -

\*

\\ m‘? 1..\\.",' 1 _.,

I ¢’\\

I \ I, \

u ' \\/‘v ‘

Q. \ 1I1 u \ IQ 4 Q

I) ‘II \\ I

K. IV“t. “ \ ‘ \' .. ‘ n“ Nu‘ K

_ i“.' -t-r“‘l‘| -\ v

‘iii’

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iv

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or

. 3 ‘ ,

m 9‘\\ \ 1'0

\ \ ‘~ \\

--—=_ _ . -

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§ ‘..

4

-_ u-..

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--_

'\

,-

\

"0— -.

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-



BESL AND MCKAY: ME T H OD FO R REGISTRATION O F 3 -D S HA PE S

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//J

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F . _ ., _ _

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.._

II'1: _ _ _ . _

-‘Y

__-

Fig. 10 . Noisy po int subset and surface patch prior to registration: 13 8 pointsand 45 0 triangles.

_¢

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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I‘ r / I 0 / 3 ' 5 5 ? 1 ' o 9 ? ' 5 $ * $ ' 3 ' 3 ' 3 ' 3 ' $ : $ $ $ $ i / 0 1 5 5 i $ $ ‘ o § 5 $ ' ¢ ' ¢ § ' $ ' 3 : ' i \ \ “ § § i i i \

\i“\

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r ; - - ' » - 4 ’'; I _ . ' a - - /.p i - - k i4 | p I I ' 5 '

II/;:'I)'’I)’: ””fIIIzII‘§‘§.’I”Iéglgi f f3Z;4; :41;z0:z; : I . : I ’ :¢.O.§‘§‘\“\\“\

\ 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-.

\ \\\.

.

|' -. . . .t. .',,. .

t.1 -‘ t

. 1 . ' L|‘ t __ - . ‘ . I

\ , r ' . . \ '. ’ ~ \0 r“\ - . ; , ‘ '\-

\

- ; : : - —

- - ‘ L ’ : - - _ -

-— - ; 4 : ‘ J - i f / _ Z_ _ _ ‘

: _ - - - * 1

- — — * ‘ _ ' _. ,-_ f ? T E § r . ‘ _ ' : ‘ -< = _ _ _ _

_ _ _ _ _ _ _ _ _ . ‘ : . ' - ' - + 5 . .“ - — _ ;_ ‘ § _ % 7 T - :‘ - _ ~ “ 7 ‘ \'

.. Z ’ . - 1 - ‘'-. . _ : - - 1- _ _ - _ \ . ' ; - _ -

- 4 ..

1,.

r

_.-»

1 ,o.\t.: ' : ‘

0 ‘.:4 :00

0 ‘ , - 0

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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“ t n - * > . - ~ ' @ = . . r . ~ ‘i-f;i '. /A T

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/>4’

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\m i n i

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 ,

i '1 “ fit W i

I

- \ i s\ \ - ' , ' \ \ i . \ \

\§\\\\\\’

= -

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

In

}Y

\;1Fig. 15 . Surface patch boundaries of a s e t of parametric surfaces: 97 cubic

Bezier patches.

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o % n . L ' * = r * = ' - 5 . - , = t ~ v ~ 4 r - t ~ ; < ~ . * : 1 * . % = . - . = t t ‘ . : . . . . , - 1t\ ,. + ’@ § > \ .\ I | , @ * a ” t ; ia i § ' ? ¢ i § ~ § , § ; § § ; & > ' { i § a , $ § \ _

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i 1' __ ,_.._.'-an-|_|4l ' <'_;_1I'{‘; ‘l__‘i- Q15 - ~..i'

_ ' - H - ' 7 ' _. .' . .151 - 1 '.0 . ' ; f £ u l 1 1 I I 3 , ' , . . F 5 ' t - I - , H , \ \ \ \ ‘ l _ @ § ' _ ‘ , . \ & n , § | _ \ \ \H , 'l‘lWH' ”.lIJ:'@------""..wv"::.:-.-w1 ,\.u. u1I| 'mmlW\l ' .' .' .' f--'-5----~--' .W| l F “ ‘ ''1 I """""""-i|_.ltll-lI"7"" I‘-ll*‘Hnw‘\I, , .w"*":;u,nug_1IF,f,"§yru:lIlW_U'\ m f '

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



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



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.

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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-

1992 - a method for registration of 3-d shapes - icp - ok

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