Sensitivity analysis for matching and pose computation using dihedral junctions

0031-3203/91 $3.00 + .0~ Pergamon Press plc 1991 Pattern Recognition Society

Pattern Recognition, Vol. 24, No. 6, pp. 505-513, 1991 Printed in Great Britain

SENSITIVITY ANALYSIS FOR MATCHING AND POSE COMPUTATION USING D I H E D R A L JUNCTIONS S. M. BHANDARKAR*t and MINSOO SUK:~ *Department of Computer Science, 415 Graduate Studies Research Center, University of Georgia, Athens, GA 30602, U.S.A.; :~Department of Electrical and Computer Engineering, 111 Link Hall, Syracuse University, Syracuse, NY 13244--1240, U.S.A.

(Received 27 October 1989; in revised form 3 May 1990; received for publication 29 May 1990) Abstract--Recognition-via-localization is a popular approach in 3-D object recognition. This approach relies on the propagation of constraints that arise from the matches of local geometric features and could therefore be treated as a constraint satisfaction problem. Hough clustering, which verifies the consistency of local geometric constraints by determining the pose of the object in parameter space, is a popular technique owing to its conceptual simplicity and potential ease of parallelization. Our previous work has shown the usefulness of dihedral junctions for the recognition and localization of polyhedral objects and dihedral feature junctions for the recognition and localization of curved obiects made up of piecewise combinations of conical, cylindrical and spherical surfaces. Experimental results from our previous work showed that the computed pose parameters are sensitive to the difference in the included angle between the scene and model dihedral junctions or the scene and model dihedral feature junctions. A formal analysis of the sensitivity of the computed pose to the difference in the included angle between the scene and model dihedral junctions or the scene and model dihedral feature junctions is presented in this paper. The results of the formal sensitivity analysis were found to be in conformity with the experimental results from our previous work and so the work presented in this paper could be treated as a sequel to our previous work. Based on the results of the sensitivity analysis, the rotation parameters were found to be more sensitive than the translation parameters which, in comparison, were far more robust. It is also shown how the introduction of redundancy in parameter space results in greater robustness in the computed pose. Although the analysis in this paper is based on the matching of dihedral junctions or dihedral feature junctions, the approach taken in the sensitivity analysis is general and can be applied to the matching based on other feature types. Model-based vision

Pose clustering

Pose computation

1. I N T R O D U C T I O N

Recognition-via-localization is a popular paradigm in 3-D object recognition. This approach relies on the propagation of constraints that arise from the matches of local geometric features and could therefore be treated as a constraint satisfaction problem. Hough clustering and searching the interpretation tree (6) are commonly used techniques for constraint propagation/constraint satisfaction in the recognition-via-localization approach to 3-D object recognition. Hough clustering, which verifies the consistency of local geometric constraints by determining the pose of the object in parameter space, is a popular technique owing to its conceptual simplicity and potential ease of parallelization. Each match of a scene feature with a model feature is used to compute a geometric transform which would place the geometric feature in registration with the scene feature. In 3-D space with six degrees of freedom this transform can be uniquely specified by six parameters---three translations one along each of the

t Author to whom correspondence should be addressed.

Sensitivity analysis

coordinate axes and three rotations one about each of the coordinate axes. Thus each geometric transform that is computed by matching a scene feature with a model feature can be represented by a point in six-dimensional Hough space or parameter space. Clustering or histogramming of points in Hough space is used to detect maxima or peaks which represent pose hypotheses. In the application of Hough clustering to 3-D object recognition,(~-3) there exists a wide variety in the choice of features for matching and pose computation. The features chosen for recognition and localization should satisfy the following criteria: (i) two non-parallel edges (which may be coplanar); (ii) one edge and one face normal such that they are neither parallel nor perpendicular; (iii) three face normals no two of which are parallel; (iv) two parallel edges and one face normal which is not perpendicular to either of them; (v) two face normals and one edge which is not perpendicular to either of the face normals. These feature types are illustrated in Fig. 1.

505

506

S.M. BHANDARKARand MINSOOSUK

B

A C

Feature type (ii)

Feature type (i)

B D

Feature

type (iii)

Feature type (iv)

Feature type

(v)

Fig. 1. Feature types of polyhedral object recognition.

In our previous work (4) we had developed a Hough clustering technique for the recognition and localization of polyhedral solids based oll matching and pose computation using dihedral junctions. As shown in Fig. 2, dihedral junctions are junctions with a single vertex and two incident edges. Dihedral junctions are a subset of feature type (i). The results for the recognition and localization of polyhedral objects are extended to the recognition and localization of objects made up of piecewise combinations of conical, cylindrical and spherical surfaces/5) Dihed-

ral feature junctions are chosen as features for matching and localization. Examples of dihedral feature junctions are shown in Figs 3 and 4. Dihedral feature junctions are junctions formed from localization features for each of the surface types, i.e. conical, cylindrical, spherical and planar surface types. (7) The mathematical analysis underlying the matching and pose computation for both dihedral junctions and

bk,81

82 /

c F

S2

C1

I" I S I ~"~D In I

Fig. 2. Matching candidate model and scene dihedral junctions.

d

C2

*1

C1 C2 : apex of conical surface or centroid of cylindrical surface or centroid of planar surface, a 1

a2 : axis of conical or cylindrical surface or normal to the planar surface

Fig. 3. Type I: dihedral feature junction pair.

Sensitivity analysis for matching and pose computation

507

[U0, V0, Wo, 1] T. The goal is to find a transformation T such that

T[xO,Yo, Zo,1]T =[uO, Vo, Wo,1] T.

(1)

T is determined in a stepwise manner as follows: c

d

i~1

C2 I=,I

C1 :

apex of the conical surface or centroid of the cylindrical surface or centroid of the planar surface

al:

axis of the conical or cylindrical surface or normal to the planar surface

C2 :

centroid of the spherical surface

Fig. 4. Type II: dihedral feature junction.

dihedral feature junctions is identical, and hence only the matching and pose computation using dihedral junctions will be considered in this paper. Owing to the errors introduced during the process of segmentation and feature extraction, the matching of scene features with model features is not exact. A certain amount of tolerance has to be incorporated during the process of matching a scene feature with a model feature. The tolerance introduced in the matching process is reflected as an error in the computed pose parameters. During the course of our experiments based on the matching and pose computations using dihedral junctions, the computed pose parameters were found to be sensitive to the difference in the included angles of the model dihedral junction (Ore) and the scene dihedral junction (0,). The rotation parameters were found to be much more sensitive than the translation parameters which were more robust to differences in the included angle between the model and scene dihedral junctions. The objective of this paper is to present a formal analysis of sensitivity of the computed pose to the difference in the included angles of the model and scene dihedral junctions. The analysis substantiates experimental results cited in our previous work (45) and hence can be treated as a sequel to this work. Although the analysis is based on the matching of dihedral or dihedral feature junctions, the approach taken in the sensitivity analysis is general and can be applied to matching based on other feature types such as the ones shown in Fig. 1.

(1) Points B and E are translated to their respective origins. Let T R A N S ( - B ) and T R A N S ( - E ) denote the respective homogeneous transformation. This ensures that both junctions have their vertices translated to the origin. (2) The vectors ml and m2 are rotated about k through an angle 0 so as to end up aligned with sl and s:, respectively, k is determined by the requirement that it be perpendicular to both ml - s~ and m2 - s2, or equivalently that the projections of ml and s~ along k be equal and the projections of m2 and s2 along k be equal, i.e. k.ml

=k-sl~k'(mt-sl)=0

k ' m2 = k .

S2

(2)

~ k- (m2

-

s2)

O.

=

Thus ( m l - s l ) ® (m2 - s2) k = I(ml _ s l ) ® ( m 2 - s2)[

(4)

where ® denotes the vector cross product. From Fig. 5, we have = ( m l • k)k = (st " k ) k

(5)

---9

AC = m~ - (m I • k)k

(6)

AC" k = [ml - (ml • k)k]. k = 0

(7)

(since k is a unit vector as in (4)). Hence OAC is a right angle triangle with OC as the hypotenuse. Also, IA-~I = IA--BI = ~ / 1 - ( m l • k ) ( s t • k)

(8)

(since m i is a unit vector). Therefore, 0 is determined by

AB.AC c o s 0 = ~

=

[ m ] - ( m l . k ) k ] - [ s l - ( s l .k)k] [ 1 - ( m ] "k)(sl-k)] (9)

=1

[1 - ( m l • s t ) ]

(10)

[1-(k.mt)(k-sl)]"

2. JUNCTION MATCHING AND POSE DETERMINATION

Figure 2 shows a model dihedral junction which is to be matched to a scene dihedral junction. With reference to Fig. 2, let mt be the unit vector in the direction BA and let m2 be the unit vector in the direction BC. Similarly, let sl be the unit vector in the direction ED and s 2 be the unit vector in the direction EF. Let the homogeneous coordinates of B in the model coordinate system be Ix0, Y0, z0, 1] T and let those of E in the scene coordinate system be

(3)

k

O

A

B

Fig. 5. Geometry for k and O.

508

S.M. BHANDARKARand MINSOOSuK

(3) The final transformation can thus be written as:

~,.

R O T ( k , 0) T R A N S ( - B ) Ix0, Y0, z0, 1] r = T R A N S ( - E ) [u0, v0, w0, 1] r.

%

I

,

(11)

/! II

.

O.

I

From (1) and (11) T = T R A N S - ~( - E) R O T ( k , 0 ) T R A N S ( - B). Fig. 6. Error in matching dihedral junctions.

(12) The transformation T from the model coordinate system to the scene coordinate system could be thus written as: (8,9)

11 r12

r13

I r 21 r22

r23

0o0d

[~3 1 r32

r33

0

0

1

T = R O T ( k , O ) T R A N S ( t x , ty, tz)

0

0 0 l O t y 0

0

1

0

0

(13)

where 0m =~ Os let 0s = 0m + dO,, then s~ = Se + ds2 where 1s21 = dora. We explore three possible cases which arise in the matching of dihedral junctions. Case 1.

ml=Sl~k=sl---m

where

(17)

1

thus rll = kx2(1 - cos O) + cos 0

d k = 0,

r12 = k x k y ( 1 - cos O) - k~ sin 0

m 1 . m 2 = k.1112 = cos 0 m and sl • s2 = k - 82 = cos Om

r13 = k , kz(1 - cos O) + ky sin 0

r21 = kxk>,(1 - cos O) + k~ sin 0 r22

=

ky2(1 - cos 0) + cos 0

(18) (19)

since (14)

[1 - (m2 • s2)] [1 - ( k ' m2)(k • s2)]

cos 0 = 1

r23 = kyk~(1 - cos 0) - kx sin 0

[ 1-_(m~:~)_]

r31 = k , k ~ ( 1 - cos 0) - ky sin 0 d(cos 0) -- - d

r32 = k y k z ( 1 - cos 0) + k~ sin 0

Using the identities d ( a / b ) = ( b d a - a d b ) / b d ( a . b) = d a - b + a . db,

tx = u0

d(cos 0) =

-

rllX

0 -

rl2Yo

-

-

rl3Z 0

(21)

"1 -- (k-" m2)(k • s2)J"

r33 = k~(1 - cos 0) + cos 0 and

(20)

2 and

m 2 • ds 2

[1 - (m2" k)(s2, k)]

ty = 0 0 -- r21x 0 - r22Y0 -- r23z 0 tz = Wo -- ralxo -- ra2Yo - r33zo.

(15)

(1 --

m 2 • s2)(m 2 • k)(ds2, k) (m2" k)(s2" k)] 2

The axis of rotation k could be alternatively expressed by the pair (~, r/) where [d(cos 0)1 =

k , = cos ~sin r/, ky = sin ~sin r/, k~ = cos rl

(16) where - : r < r / < :r and 0 < ~ < 2:r. The transformation T is thus uniquely specified by the 6-tuple (tx, ty, t~, ~, ~1, 0). 3. S E N S I T I V I T Y

ANALYSIS

In this section, sensitivity of the pose (i.e. rotation) parameters (~, r/, 0) and translation parameters (t,, tv, tz) to the difference in the included angles of ihe scene dihedral junction and the model dihedral junction is formally analysed. As shown in Fig. 6, two dihedral junctions are said to match if 0s = Om. In order to explore the case

(22)

[1 -

m2" ds2 (1 - cos O)dO m . (23) sin 2 On ~ tan 0 m

Since d(cos 0) = sin 0d0 Id0{ =

m2 "ds2 (1-c°sO)dOm sin 0 s i n 20m + sin 0 t a n 0m

I . (24)

Using Schwartz's inequality we get an upper bound on the value of the error as Id(cos 0)1 -< rn2 "ds2 sin 2 0m m2 • d s ]dO[--- s i n ~ - ~

2

+

(1 - cos O)dOm tan 0,.

(25)

cos 0)d0,, sin~an'-~

(1 -

mI +

(26)

Sensitivity analysis for matching and pose computation Since Im2" ds2i = Im2l tds21 cos/~ ~ dO., where fl is the angle between ds 2 and m2, dO,. 1 Id(cos 0)1-< sin2 0---~+ I(1 - cos 0)1 ~ d O m

509

Using Schwartz's inequality td(cos 0) t ICOS Om I

-<

I(1 - cos 0)1 o------S (Imlsi.2

• ds, I + Is, • ds21)

(27) d0m (1 - cos 0)] IdOl-< Isin 01 sin 2 0 m "t" S'~n-O I ~

1 doM" (28)

The asymptotic behavior of Id01 and Id(cos 0)l with respect to 0," and 0 is as follows:

(36) idol_< ( l - c o s O ) I c o s O m J ~(ml"ds21+ls,'ds21). sinO

(37) Since

Im," ds21 = Im,I Ids21 cos ~,-< dora

d0," As 0--~ 0, !d01 ~ ~, Id(cos 0)1 ~ sinZ 0m

where y is the angle between m~ and ds 2 and ISl "ds21 = ISll Ids2l cos o : - dO,,,

As 0--~ -~, Id0[--, dora

+

Id(cos 0)1 ~ dora ~

,

where o: is the angle between sl and ds2 and

Icos 0,. I Id(cos 0)l-< 2/(1 - cos 0)l ~ d 0 m

+

Id01---, ~,

As 0 - + ~,

1

2

(39)

The asymptotic behaviour of Id01 and Id(cos 0)l with respect to 0m and 0 is as follows:

As O~ ---~0, IdOl ~ 0% Id(cos Om)l---' ~

As 0--> O, Id01 ~ 0, Id(cos 0)I--* 0

Z dO M As Om --->~, IdO[ ~ ~ , Id(cos 0)[ ~ dO,"

:r As 0--~ ~,

As Om ~ ~, Id01---' ~, Id(cos 0)1 ~

2(1 - cos 0) Icos 0m I si-n--ff ~ dora.

Id0/-<

(38)

21cos0ml

Id01 ~

sin2 0---~ dO,n,

~.

21cos o,"1 For a given value of dO,, we have an error surface E = E(0, 0m).

]d(cos 0 ) 1 ~

41cos 0," I As 0---, z, Id01--, ~, Id(cos 0)l-~ sin2 dO,,,

0----~

Case 2. m 2 = s2 ~

(29)

k = S2 = m 2

thus

As 0 m - ~ 0, td01---> ~, Id(cos .71

As 0," ~ ~ Id 01 - , dk = ds2,

0,

o,")1--,

Id (cos 0l ~ 0

(30) As Om ~ :r, IdOl---->~, td(cos O)l--~ oc.

m 1 • m 2 = m~ • k = cos 0 m andsl "82 = sl" k = cos 0m

(31)

Case 3.

ml 4: k :/: sl andm2 :/: k ¢ s~.

since [1 - (ml • sl)1 cos 0 = 1 [1 ( k . m~)(k" Sl) ]

From (4), (32) k=

-

d(cos 0) = - d

[

i-(ml'sl) I" 1 - -(k" mll ;(-if:s, )

(mj - Sl) ® (m2 -- s2)

M

cos 0) ~[COS Om I

11111

• ds 2 -4" S 1

• ds2l

(34)

M & I(mt - sl) ® (m2 - 82) I = [((ml - sl) @ (m2 - s2))" ((ml - sl) @ (m2 - 82))] ~

IdOl

(40)

where

Id(cos 0) 1 -

(m, - s l ) ® (m2 - s2) [(ml - sl) ® (m2 - s2)l

(33)

Using the results in (30) and (31)

= I(1

sin20-----~dOm

II(1 -sinoC°SO)

0mlI ~Icos l m

"ds2 + s l " ds21.

(35)

[ (___m.a-s,)® (m 2 _--82)] dk = d [tml - s l ) ® ( m 2 - s2)t3

(41)

(42)

510

S.M. BHANDARKARand MINSOOSUK

dk = ( M d [ ( m l - st) ® (m2 - s2)] - [(ml - ss) ® (m2 - s2)]

dM)/M 2.

Iml'dkl=lds211mt®si[c°sfl--
(43)

where fl is the angle between ds 2 and ml ® ss

On simplification this yields

[cos ~.J

1 dk = ~ [ d s z

® (mr -

(59)

[d(cos 0)I = 2[(1 - cos 0)[ ~

sz)]

Ires" dk[.

(60)

1

- ~ k[k" [ds2 ® (ml - Sl)]].

(44)

From (59) and (60)

Since k • k = 1, k . dk = O.

[d(cos 0)l --<

(45)

2](1 - cos 0)I IcosXl Msin 2 ~,

dO,".

(61)

Since d(cos 0) = - s i n 0dO, (60) yields

From (44) and (45), 1 dk = ~ [ds2 ® (ms - s 0 ]

(46)

1 ddk[ = ~ [ds2 ]Jml - s1 ] sin fl

(47)

(62)

From (59) and (62) we get 2(1 - cos 0) [cos;t I . . Id01<:-

1 Idk[ = ~ d O m [ m s - sl[sinfl

(48)

where fl is the angle between ds2 and mt - Sl

-?ffs~nb ] ~ a v , " "

The asymptotic analysis for follows:

[d0[ and [d(cos 0)[

(63) is as

As 0 ~ 0, IdOl--, 0, Id(cos o)l---, 0

1 Idk[ ~ ~ d O,". Thus dk is directly proportional to proportional to M.

id0l = 12(1 - cos 0) [cosA I sin 0 [~ Ira1" dk[.

(49)

doraand

~: 2 ]cosA[ As O--->~, IdOl--~ -~si-~-~dv,",

inversely 2 ]cos ~.I [d(cos 0)1 --' ~ sin2------~d 0m

Case 3.1. [1 - ( m l "

cos O = 1

st)]

[1 - ( k . m t ) ( k "

Sl)]'

(50)

AS 0--'~ :r, [d0[---, oo, Id(cos 0 ) 1 ~ 4 IcosZl M M sin z ~. dO,,

A s ; t ~ 0, Id01--, oo, [d(cos 0)l---->oo

Thus, d(c°s 0) = - d [l/

.1-.(m,-s,) - (k" ml)(k"

(1 - m t • s l ) d [ 1 =

[1 - ( m l

yf

]

(51)

sl)_[

- (mt • k)(ss

• k)(st

As ;t---, :r, Id01---, 0% Id(cos 0)I--, oo.

• k)]

• k)] 2

As ~---, ~, Id01--, 0, Id(cos 0)I--, 0

(52)

Case 3.2.

Since mt • k = sl • k (1 - m l

d(cos 0) = =

cos 0 = 1 • sl)d[1

- (ml

" k ) 2]

[1 - (ml • k)2] 2

(53)

- 2 ( 1 - ms • s t ) ( m t • k ) ( m l " dk) [1 - (ml • k)2] 2

(54)

[1 + ( m x • k ) 1 2 [ 1 - ( m s

• k)] 2

(k" m2)(k" s2)]"

(64)

Thus, d(cos O) = - d -[1 1 - ( m 2 . s Q ] - (k" m 2 ) ( k , s2)J"

(65)

Since mz • k = s2 • k,

- 2 ( 1 - ms ' sl)(ms • k ) ( m l • dk) =

[1 - ( m 2 • s 2 ) ] [1 -

(55)

d(cos O) = - d 11 - ( k ' mz)ZJ

From (46),

ml ml'dk=~-'[dsz®(ml-sl)].

m2 • dsz

(56)

= [1 - ( m 2 • k ) 2]

Using

the

trigonometric

identity

A. (B®C)=

( A ® B ) • C = ( C ® A ) • B a n d w i t h cos;. -~- - m I . k , ds2 m j . dk = " ~ - " [(ml - s l ) ® m l ] ds2 m i " dk = --~-. [m i @ s t]

(57) (58)

(1 -

-

m2 • sz)2(m2 • k)(m2 • dk) [1 - (m2" k)2] 2

(66)

m2 • ds2

Id(cos 0)[ -- [1 -(-m2:iO 21 -~ (1 - m 2 • ['~ "--k-)T~s2)(-2)(m2 --- (--~2 • k)(m2 • dk) .

(67)

Sensitivity analysis for matching and pose computation Using Schwartz's inequality [d(cos 0)1 --+

(1

m2 - s 2 ) ( - 2 ) ( m 2 " ~ ( m 2 • dk)

-

k = [kx, ky,

k~] =

dk x = -sin ~sin r/d~ + cos ~cos r/dr/

(75)

dkv = cos ~sin t/d_~ + sin ~ cos 0 d 0

(76)

dkz = -sin r/dr/.

(77)

Since the axis of rotation [cos ~ sin r/, sin ~ sin r/, cos r/]

[1 m2 ' ds2 :~n~2:iQ2 l "

(68)

From (46),

511

(76) and (77) yield m2

m 2 - dk = ~ - . [ds 2 @ (ml - sO].

(69)

d~ = ~

Using the trigonometric identity A • (B ~ C) = (A ® B ) . C = (C ® A) . B and with cos ~ = mE . k, ds2 m2 .dk = - - ~ - . [(ml - s l ) ® m 2 ] = Ids2l { [ ( m l - sl) ®m2]l cosfl M Ids21 m

dora

M

M

(70)

(71)

dr/= ~

1

1

(sin ~dkx - cos ~dk~,)

(78)

(cos ~dk~ + sin ~dky).

(79)

Thus, as t/--->0 or ~r, d ~ - ~ and as r/--~.(yr/2), dr/--* ~. Assuming that the errors dky and dk~ are of the same order as ]dk], the errors in ~ and 7/ are greater than ldkl. From (15) and assuming that there is no error in the model coordinates x0, Y0 and z0,

dtx = duo - druxo - drl2Y 0 - drl3z 0 dty = dvo - dr21x 0 - dr22Yo - dr23zo

m2 • ds2 = Ids21 Ira21 cos o:--
(72)

where tr is the angle between ds2 and m2, (68)-(73) yield dO,, 2(1 - cos 0) Icos 3.1 Id(cos O)l -< sin2--~ + ~ ~ dora.

Using the fact that td(cos 0)1 = Isin 0d0 I

as

--.

IdOl--,

dO m Id(cos 0)1-,

As ~. --~ ~r, [d01 ~ ~, td(cos 0)1--~ dGn

As 0--, 0, Id01--, ~, ld(cos 0)1--, sin2 ~. :r d0m 2 Icos 3.I As 0--* ~, Id 01---' sin2---~ + ~/~;t d0,,

[d(cos 0)l~ d0m 2 [cosX[ sin2"--~ +~ d0m dO,. 4 ]cosAI As 0--" [d 01--, ~, Id(cos 0) t ~ si--~-f~+ ~ si--n-T~-zd 0,,.

In all the above three cases it can be seen that the error in 101 is higher than the error in Icos 0 I, of course, since PR 2 4 : 6 - D

-~ max[duo, d(cos 0), d(sin 0), dkx, dky. dk~]

(81) max[dvo, d(cos 0), d(sin 0), dk x, dk~,, dk~]

(82) (dtz)ma× max[d w0, d(cos 0), d(sin 0), dkx, dky, dkz].

As X--* 0, ]dO]--* 0% ]d(cos 0)1 ~ do

As

From (81) and the expressions for rll , rt2 , r13, r21, r22, r23, r31, r32 and r33 in (14),

(dty)max

d0~ 2(1 - cos 0) ] ]cos ~.1 Id01 <-Isin 0l sin-' 2 + ~'/sm~9 I 2-----~d0'~" sin (74)

Id01 and Id(cos 0)I is

(80)

(dtx)max

(73)

The asymptotic analysis for follows:

dG = dwo - dr31xo - dr32yo - dr33zo.

Id cos 0{ IdOl = Isin 0-------~and 0 < tsin 0[ < 1.

(83) d(sin 0) is of the same order of magnitude as d(cos 0) if d(sin 0) ~ 10.0d(cos 0), which implies that 5.71 -< 0 - 174.29 which is very often the case. From Equations (76)-(78) and (82)-(84) it can be seen that the estimates of the rotation parameters ~, r/and 0 are more sensitive than the translation parameters t~, ty, and tz (assuming that the error in (u0, v0, w0) is small, which seems reasonable). A straightforward way of making the computed pose more robust to the difference in the angle between the model and scene dihedral junctions or dihedral feature junctions is by introducing redundancy in the parameter space. For example the axis of rotation k could be presented by the triple (kx, ky, k~) rather than the double (~, r/) since parameters kx, ky and k~ are more robust than the parameters and r/. Similarly, the angle of rotation 0 can be replaced by the double (cos 0, sin 0) which uniquely specifies the angle 0 in the range -¢r -< 0 -< :t. Since the values of sin 0 and cos 0 are less sensitive than the value of 0, replacing 0 by the double (cos 0,

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S.M. BHANDARKARand MINSOOSUK

sin 0) would ensure greater robustness in the computed pose. 4. CONCLUSIONS In this paper we have formally analysed the sensitivity of the computed pose parameters to the difference in the angles of the scene and the model dihedral junctions. We have shown how the rotation parameters ~, ~7 and 0 are more sensitive to the difference in the included angles of the scene and model dihedral junctions or dihedral feature junctions as compared to the translation parameters ix, ty and tz. We have also shown how the pose compu.tation could be made more robust at the cost of adding redundancy in the parameter space. The formal analysis presented in this paper is supported by the experiments in our previous work. (4,5) SUMMARY Recognition-via-localization is a popular paradigm in model-based vision. Primitive geometric features are extracted from the scene. These features are matched against similar features in the model. The matches are constrained by length and angle measurements. The constraints that arise from the matching of these features are propagated using a constraint propagation/constraint satisfaction technique. A consistent set of constraints constitutes a valid scene interpretation. In this sense, a scene interpretation problem could be looked upon as a constraint propagation/constraint satisfaction problem. Hough clustering is a popular constraint propagation/constraint satisfaction technique owing to its conceptual simplicity and potential ease of parallelization. Each match of a scene feature with a model feature is used to compute a geometric transformation that places the model feature in registration with the scene feature. The geometric transformation is characterized by a feature vector. The dimensionality of the feature vector is determined by the number of degrees of freedom that the objects in the scene possess. For three-dimensional object recognition with six degrees of freedom, the feature vector is a six-tuple. Each match of a scene feature to a model feature can thus be represented as a point in six-dimensional parameter (Hough) space. Clustering of points in the Hough space is used to determine the global pose of the object. In our previous work we had shown the advantages of using dihedral junctions as features for recognition

and localization of polyhedral objects. These were further generalized to dihedral feature junctions for the recognition and localization of complex objects made up of piecewise combinations of conical, cylindrical, spherical and planar surfaces. In our experimental results we observed that the rotational parameters of the computed pose were more sensitive than the translational parameters to the difference in the included angle of the scene and the model dihedral junction. In this paper we have carried out a formal sensitivity analysis where the sensitivity of the computed pose to the difference in the included angle of the scene and the model dihedral junction is analytically computed. The work in this paper could be thus treated as a sequel to our previous work. The results of the sensitivity analysis were found to be in conformity with the experimental results from our previous work. It has been shown how the introduction of redundancy in the Hough pose results in greater robustness in the computed pose. Although the sensitivity analysis in this paper is based on matching and pose computation using dihedral junctions and dihedral feature junctions, the analysis technique is general enough to be extended to matching and pose computation using other feature types. REFERENCES

1. G. Stockman, Object recognition and localization via post clustering, Comput. Vision Graphics Image Process, 40, 361-387 (1987). 2. B. A, Boyter and J. K. Aggarwal, Recognition of polyhedra from range data, IEEE Expert, pp. 47-59 (1986). 3. M. Dhome and T. Kasavand, Polyhedra recognition by hypothesis accumulation IEEE Trans. Pattern Anal. Mach. lntell. 9, 429-438 (1987). 4. S. M. Bhandarkar and Minsoo Suk, Hough clustering technique for surface matching, Proc. IAPR Wkshop Comput. Vision, Tokyo, Japan, pp. 82-85. 5. S. M. Bhandarkar and Minsoo Suk, Recognition and localization of objects with curved surfaces, Mach. Vision Appl. (in press). 6. W. E. L. Grimson and T. Lozano-Perez, Localizing overlapping parts by searching the interpolation tree, IEEE Trans. Pattern Anal. Mach. Intell. 9, 469-482 (1987), 7. T. Nagata and H. B. Zha, Determining orientation, location and size of primitive surfaces by a modified Hough transformation technique, Pattern Recognition 21,481-491 (1988). 8. R. P. Paul, Robot Manipulators: Mathematics, Programmmg and Control. MIT Press, Cambridge, MA (1981). 9. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers. McGraw-Hill, New York (1972).

About the Author--S. M. BHANDARKARreceived his B.Tech. in Electrical Engineering from the Indian

Institute of Technology, Bombay, India, in 1983, and his M.S. and Ph.D. in Computer Engineering from Syracuse University, Syracuse, New York, in 1985 and 1989, respectively. He is currently an Assistant Professor in the Department of Computer Science at the University of Georgia in Athens. He was nominated University Fellow for the academic years 1986-87 and 1987-88. He is a member of IEEE, AAAI and Phi Kappa Phi. His research interests include computer vision, pattern recognition, image processing and artificial intelligence.

Sensitivity analysis for matching and pose computation

About the Author--M1NSOO SUK received his B.S., M.S. and Ph.D. degrees, all in Electrical Engineering in 1970, 1972 and 1974, respectively, from the University of California at Davis. He is currently an Associate Professor in the Department of Electrical and Computer Engineering at Syracuse University~ Syracuse, New York. Before joining Syracuse University, he was a member of Technical Staff at Rockwell International, and an Associate Professor at the Korea Advanced Institute of Science and Technology. His research interests include computer vision, pattern recognition, image processing, computational geometry, artificial intelligence, knowledge-based systems and parallel architectures and algorithms.

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