1-4244-1355-9/07/$25.00 @2007 IEEE

International Conference on Intelligent and Advanced Systems 2007

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An Approach to 3-D Object Recognition UsingLegendre Moment Invariants

Lee-Yeng OngFaculty of

Information Science and TechnologyMultimedia UniversityJalan Air Keroh Lama,

75450 Melaka, Malaysia.

Chee-Way ChongFaculty of

Engineering and TechnologyMultimedia UniversityJalan Air Keroh Lama,

75450 Melaka, Malaysia.

Rosli BesarFaculty of

Engineering and TechnologyMultimedia UniversityJalan Air Keroh Lama,

75450 Melaka, Malaysia.

Abstract—Feature descriptors for 3-D images have recentlygained considerable attention in application for games, virtualreality environment and solid modeling. Numerous research hadbeen introduced for 3-D invariants of geometric moments, com-plex moments and Zernike moments. In this paper, we present atheoretical framework to derive translation and scale invariantsfor 3-D Legendre moments, by using indirect and direct methods.Indirect method generates 3-D Legendre invariants from theexisting 3-D geometric moment invariants. Direct method, onthe other hand, eliminates the displacement and scale factorsfrom Legendre polynomials to generate translation and scaleinvariants. Experiment using 3-D binary images are carried outto verify the proposed feature descriptors.

I. INTRODUCTION

The information archives of 3-D objects are becomingincreasingly popular with the recent advances of demand ingames, virtual reality environments and solid modeling ofindustrial parts [6], [12], [13]. Pattern recognition plays animportant role to observe, classify and process the raw data inarchive before it can be utilised for front-end applications.

Since 1962, moment functions are widely used in patternrecognition to obtain invariant features of 2-D or 3-D images,regardless of changes in image position and size [1]. Numeroustheoretical framework had been introduced for 2-D momentinvariants over the past few decades [2], [4], [8], [11]. In 1980,Sadjadi et al. introduced 3-D geometric moment invariants[3], which were further expanded by Yang et al. into higherdimensions [7]. Consequently, 3-D complex moment invariantswere presented by Lo et al. [5] and 3-D Zernike momentinvariants were derived by Canterakis [10]. However, to theauthors’ knowledge, no report has been published on 3-Dinvariants of Legendre moments.

According to Chong et al., translation (T) and scale (S)invariants of 2-D Legendre moments can be produced by twomethods, which are indirect method (IDM) and direct method(DM) [11]. IDM method generates 3-D Legendre TS invariantsfrom TS invariants of geometric moments, mpqr, whereas DMmethod applies solely with polynomials of Legendre moments,Lijk.

mpqr =N−1∑x=0

N−1∑y=0

N−1∑z=0

xpyqzrf(x, y, z) (1)

Lijk = λijk

N−1∑a=0

N−1∑b=0

N−1∑c=0

Pi(xa)Pj(yb)Pk(zc)f(a, b, c) (2)

This paper presents the theoretical framework of 3-DLegendre TS invariants by using IDM and DM methods.The organization of this paper is as follows: A briefexplanation on IDM method and its invariants derivationfrom geometric moments are briefly described in Section2. In Section 3, DM method is elaborated in details. Inorder to prove that these descriptors are invariant, Section4 simulates the comparison results between both methodswith 3-D binary images. Lastly, Section 6 concludes the study.

II. INDIRECT METHOD (IDM)

The computation of IDM method is illustrated by three stepsgiven below.

Step 1: Translation invariants are firstly defined by deriving3-D geometric moments with respect to centroid. The cen-troids, x, y and z, are then factored out to express centralmoments in terms of original moments [7].

μpqr =p∑

e=0

q∑f=0

r∑g=0

(pe

)(qf

)(rg

). (−x)e (−y)f (−z)g m(p−e)(q−f)(r−g) (3)

The centroid x, y and z can be replaced with (x−xs), (y−ys)and (z − zs) for symmetrical images. The shift terms, xs, ys

and zs are defined as [9]

xs =(

μ200

m000

) 12

, ys =(

μ020

m000

) 12

, zs =(

μ002

m000

) 12

(4)

Step 2: To obtain TS invariants, translation invariants, μpqr

are then applied into scale invariants as follows:

Υpqr =(μ000)

p+q+r+32 μpqr

(μ200)p+12 (μ020)

q+12 (μ002)

r+12

(5)

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Step 3: TS invariants of Legendre moments are finally ob-tained by extending the 2-D Legendre-geometric relationshipequation from [8] into 3-D .

μijk =(2i + 1)(2j + 1)(2k + 1)

N3

.

i∑p=0

j∑q=0

k∑r=0

AipAjqAkr Υpqr (6)

where Aip denotes the coefficient of xp in the expansion seriesof Legendre polynomials, Pi(x), as given below.

Pi(x) =i∑

p=0, i−p=even

{(−1)

i−p2

12i

(i + p)! xp

( i−p2 )! ( i+p

2 )! p!

}

=i∑

p=0, i−p=even

{Aip xp} (7)

III. DIRECT METHOD (DM)

Direct method is applied with the same procedures, however,the TS invariants are derived directly from Legendre polyno-mials as in Eq. (7).

Step 1: Translated Legendre moments are firstly expressedas central moments, which are noted as

Ψijk = λijk

N−1∑a=0

N−1∑b=0

N−1∑c=0

Pi(xa − x)Pj(yb − y)

. Pk(zc − z)f(a, b, c), (8)

It is followed by centroid factorisation to reform the originalLegendre polynomials, Pi(xa), Pj(yb), Pk(zc), as in Eq. (2).This can be done by grouping x, y and z from the translatedLegendre polynomials, Pp(xa− x), Pq(yb− y) and Pr(zc− z)respectively.

Pi(xa − x) = [Aiixia + Ai(i−2)x

i−2a + · · ·][x0]

− [Aii

(i1

)xi−1

a + Ai(i−2)

(i− 2

1

)xi−3

a

+ · · ·][−x 1] + [Aii

(i2

)xi−2

a

+ Ai(i−2)

(i− 2

2

)xi−4

a + · · ·][x 2] · · · . (9)

It can be observed that Eq. (9) is developed from de-creasing orders of Legendre polynomials, Pi(xa), Pi−1(xa),Pi−2(xa),· · ·. This condition is similar for the polynomi-als along y-axis, Pj(yb − y) and polynomials along z-axis,Pk(zc−z). Finally, the 3-D translation invariants are described

in terms of Legendre moments as

Ψijk = λijk

i∑u=0

j∑v=0

k∑w=0

ηi(i−u)ηj(j−v)ηk(k−w)

.L(i−u)(j−v)(k−w)

λ(i−u)(j−v)(k−w)(10)

with the coefficients listed below:

ηi(i−u) =1

A(i−u)(i−u)[

u∑s=1

(−x)(

i− u + ss

)Ai(i−u+s)

−u−1∑t=1

ηi(i−t)A(i−t)(i−u)], (11)

where

ηii = 1; i− s = even; u− t = even; u ≥ 1.

The shift term, x is defined by [11] to replace centroid incomputation for symmetrical images.

x ={

L200

L000− 5x

L100

L000+

15x2

2+

52

} 12

(12)

Step 2: Legendre polynomials are more complex than themonomials of geometric moments. Thus, it is difficult toextract scale factors directly from Legendre polynomials. Totackle this, scale factors d1, d2 and d3 have to be eliminatedrespectively by grouping all similar terms in original polyno-mials and scaled polynomials. The relation polynomials areconcluded as:

g∑i=0

δgiPi(d1x) = ag

g∑i=0

δgiPi(x); (13)

where the coefficients δgg = 1 and

δgi =t−2∑s=0

−B(g−s)i δg(g−s)

Bii;

g − i = even; t− s = even; t = (g − i) ≥ 2

Likewise, the same procedures can be implemented for poly-nomials, Pj(d2yb) and Pk(d3zc). The scaled polynomials inscaled moments are then replaced with relation polynomialsalong x-axis, y-axis and z-axis.

νgfe = dg+11 df+1

2 de+13

g∑i=0

f∑j=0

e∑k=0

.

{ϕgfe

ϕijkδgi δfj δek Lijk

}(14)

In order to develop TS invariants, Legendre moments, Lijk aresubstituted with the proposed translation invariants from Eq.

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(10). Using the first, g-th, f -th and e-th orders of Eq. (14), the3-D TS invariants of Legendre moments are finally deducedas:

Ψabc =νabc ν2

000

νa00 ν0b0 ν00c(15)

Selected orders (2 to 3) for 3-D Legendre TS invariants arelisted in Appendix.

IV. EXPERIMENTAL REMARKS

To verify TS invariants for IDM and DM methods, thefollowing experiment is carried out, with three non-uniformlyscaled and shifted 60x60x60-resolution binary image, whichare listed in Table I. They are shifted up, down, left andright as well as diagonally within an 200x200x200 imageframe. Scale factors namely, d1, d2 and d3, indicate theaspect ratio along x-axis, y-axis and z-axis respectively. Thepercentage moment spread from centroid, σ/μ%, whereσ and μ are standard deviation and mean of the selectedmoments, are used to measure the performance of theinvariants descriptors. From the result, it is concluded thatdirect method performs slightly better than indirect methodunder the same circumstances.

V. CONCLUSION

The theoretical framework of translation and scaleinvariants of Legendre moments have been presented with3-D binary images. It is shown that TS invariants of 3-DLegendre moments, can be either derived indirectly fromgeometric moments or directly from Legendre polynomials.The simulation results testify the validity of the proposedinvariants functions.

APPENDIX

Selected orders of 3-D Legendre TS invariants for DirectMethod

Order 1:

Ψ100 =ν100ν

2000

ν300ν020ν002; Ψ010 =

ν010ν2000

ν200ν030ν002

Ψ001 =ν001ν

2000

ν200ν020ν003

Order 2:

Ψ200 =ν200ν

2000

ν400ν020ν002; Ψ020 =

ν020ν2000

ν200ν040ν002

Ψ002 =ν002ν

2000

ν200ν020ν004; Ψ110 =

ν110ν2000

ν300ν030ν002

Ψ101 =ν101ν

2000

ν300ν020ν003; Ψ011 =

ν011ν2000

ν200ν030ν003

Order 3:

Ψ300 =ν300ν

2000

ν500ν020ν002; Ψ030 =

ν030ν2000

ν200ν050ν002

Ψ003 =ν003ν

2000

ν200ν020ν005; Ψ120 =

ν120ν2000

ν300ν040ν002

Ψ102 =ν102ν

2000

ν300ν020ν004; Ψ210 =

ν210ν2000

ν400ν030ν002

Ψ201 =ν201ν

2000

ν400ν020ν003; Ψ012 =

ν012ν2000

ν200ν030ν004

Ψ021 =ν021ν

2000

ν200ν040ν003; Ψ111 =

ν111ν2000

ν300ν030ν003

REFERENCES

[1] M.K. Hu, ”Visual pattern recognition by moment invariants,” IRE Trans-actions on Information Theory, 8(1) (1962) 179-187.

[2] S.A. Dudani, ”Aircraft identification by moment invariants,” IEEE Trans-actions on Computers, 26(1) (1977) 39-45.

[3] F.A. Sadjadi and E.L. Hall, ”Three-dimensional moment invariants,” IEEETransactions on Pattern Analysis and Machine Intelligence, 2 (1980) 127-136.

[4] C.H. Teh and R.T. Chin, ”On Image Analysis by the Method of Mo-ments,” IEEE Transactions on Pattern Analysis and Machine Intelligence,10(4) (1988) 485-513.

[5] C.H. Lo and H.S. Don, ”3D moment forms: Their construction andapplication to object identification and positioning,” IEEE Transactionson Pattern Analysis and Machine Intelligence, 11(10) (1989) 1053-1064.

[6] S. Panayiotou and A. Soper, ”Artificially Intelligent 3D Industrial In-spection System for Metal Inspection,” Proceedings of IEEE SouthwestSymposium on Image Analysis and Interpretation, (1994) 130-135.

[7] L. Yang, F. Albregsten and T. Taxt, ”Fast computation of three-dimensional geometric moments using a discrete divergence theorem anda generalization to higher dimensions,” Graphical Models and ImageProcessing – CVGIP, 59(2) (1997) 97-108.

[8] R. Mukundan and K.R. Ramakrishnan, Moment Functions in ImageAnalysis- Theory and Applications, World Scientific Publishing, 1998.

[9] R. Palaniappan, P. Raveendran and S. Omatu, ”Noise Tolerant Momentsfor Neural Network Classification,” International Joint Conference onNeural Networks, July 1999, (4) 2802-2807.

[10] N. Canterakis, ”3D Zernike moments and Zernike Affine Invariants for3D Image Analysis and Recognition,” 11th Scandinavian Conf. on ImageAnalysis, 1999.

[11] C.W. Chong, P. Raveendran and R. Mukundan, ”Translation and scaleinvariants of Legendre moments,” Pattern Recognition, 37 (2004) 119-129.

[12] Jeong-Dan Choi, Byung-Tae Jang and Chi-Jeong Hwang, ”Collab-orative Interactions on 3D Display for Multi-user Game Environments,”Lecture Notes in Computer Science, LNCS 3101, Springer-Verlag, (2004)81-90.

[13] J.F.Kok Arjan and Robert van Liere, ”A Multimodal Virtual RealityInterface for 3D Interaction with VTK,” Knowledge and InformationSystems, Springer-Verlag, (2007).

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⎯

Images

Scale Factorsd1=1.0, d2=2.0, d3 =3.0

d1=1.0, d2=3.0, d3 =1.5

d1=1.0, d2=2.0, d3 =3.0

d1=1.0, d2=3.0, d3 =1.5

d1=1.0, d2=2.0, d3 =3.0

d1=1.0, d2=3.0, d3 =1.5

Translation d4 = -2 d5 = -2 d6 = -2

d4 = +2 d5 = -2 d6 = -2

/μ%

d4 = -2 d5 = +2 d6 = -2

d4 = +2 d5 = -2 d6 = +2

/μ%

d4 = -2 d5 = -2 d6 = +2

d4 = -2 d5 = +2 d6 = +2

/μ%

002μ 16.60651 16.60724 0.003 18.83028 18.83096 0.003 10.86638 10.86663 0.002

101μ -3.89986 -3.90032 0.008 -4.42209 -4.42257 0.008 -5.85269 -5.85316 0.006

102μ 13.49626 13.49704 0.004 15.30400 15.30451 0.002 -18.7141 -18.7143 0.001

111μ -8.27329 -8.27401 0.006 -3.54580 -3.54605 0.005 11.47242 11.47294 0.003

210μ 18.38864 18.38891 0.001 7.88108 7.88107 0.000 -9.03058 -9.03048 0.001

Indi

rect

Met

hod

(ID

M)

300μ -43.8369 -43.8388 0.003 -49.7071 -49.7089 0.003 41.51045 41.51140 0.002

002ˆ 0.01275 0.01275 0.000 0.016395 0.016394 0.004 0.00507 0.00507 0.000

101ˆ -0.00992 -0.00992 0.000 -0.01275 -0.01275 0.000 -0.00465 -0.00465 0.000

021ˆ -0.00349 -0.00349 0.000 -0.00524 -0.00524 0.000 -0.00217 -0.00217 0.000

111ˆ 0.01020 0.01020 0.000 0.01120 0.01120 0.000 0.00442 0.00442 0.000

012ˆ -0.00456 -0.00456 0.000 -0.00501 -0.00501 0.000 -0.00136 -0.00136 0.000 D

irec

t M

etho

d (D

M)

300ˆ 0.00701 0.00701 0.000 0.00902 0.00902 0.000 0.00303 0.00303 0.000

TABLE I: TS Invariants for Scaled and Shifted 3-D Binary Image by using IDM and DM Methods