COMPUTATIONALLY EFFICIENT WAVELET AFFINE INVARIANT FUNCTIONS FOR SHAPE RECOGNITION. Erdem Bala, Dept. of Electrical and Computer Engineering,

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1 COMPUTATIONALLY EFFICIENT WAVELET AFFINE INVARIANT FUNCTIONS FOR SHAPE RECOGNITION Erdem Bala, Dept. of Electrcal and Computer Engneerng, Unversty of Delaware, 40 Evans Hall, Newar, DE, 976 A. Ens Cetn, Dept. of Electrcal and Electroncs Engneerng, Blent Unversty, Anara, 06533, Turey ABSTRACT An affne nvarant functon for obect recognton s constructed from wavelet coeffcents of the obect boundary. In prevous wors, undecmated wavelet transform was used for affne nvarant functons. In ths paper, an algorthm based on decmated wavelet transform s developed to compute the same affne nvarant functons. As a result computatonal complexty s reduced wthout decreasng recognton performance. Expermental results are presented. INTRODUCTION Obect recognton s an mportant problem n computer vson and pattern analyss [-6]. In ths paper, recognton of obects from ther boundares that are subect to affne transformatons s consdered. The affne transformaton ncludes rotaton, scalng, sewng and translaton. It preserves parallel lnes and equspaced ponts along a lne. In some cases, the affne transformaton can also be used to approxmate the perspectve transformaton [].

2 Several features that are lnear under an affne transformaton were developed n the lterature. The most commonly used ones are affne arc length [7], affne nvarant Fourer descrptors [], and moment nvarants [3]. Recently, dyadc wavelet transform was also used to develop several affne nvarant functons [5,0]. These functons are constructed from undecmated wavelet coeffcents whch are produced after computng the wavelet transform of a curve correspondng to the boundary of the obect. In ths paper, an algorthm based on decmated wavelet transform s developed to compute the affne nvarant functons proposed n [5]. Ths leads to a computatonally effcent obect recognton scheme. The paper s organzed as follows: In Secton, some bacground nformaton on affne nvarant functons s presented. In Secton 3, the computatonally effcent algorthm s presented. In Secton 4, expermental results are presented. In addton, a new obect recognton scheme based on lnear combnaton of affne nvarant functons constructed from multple resoluton wavelet coeffcents s presented. It s observed that recognton performance s comparable to other wavelet based schemes. BACKGROUND Consder a parametrc curve { x(, y( } wth parametert on a plane. A pont on the curve under an affne transformaton becomes and ~ x ( a + a x( + a y( ) () 0 t ~ y ( b + b x( + b y( ) () 0 t

3 Equatons () and () can be rewrtten n matrx form as follows: ~ x ( a ~ y t ( ) b a x( a0 x( A + B b + y t b y t ( ) 0 ( ) (3) where the nonsngular matrx A represents the scalng, rotatng, and sewng transformaton and the vector B corresponds to the translaton. Jacobean, J, of the transformaton s J a b a b det( ). A Let ~ I ( be an affne nvarant functon and I ( t ) be the same nvarant functon calculated usng the ponts that are subect to the affne transformaton. The relaton between the two nvarant functons can be formulated as: ~ ω I IJ (4) The exponent ω s called the weght of the nvarance. Ifω 0, then I s called an absolute nvarant, else t s called a relatve nvarant. 3 AFFINE INVARIANT FUNCTIONS USING DECIMATED WAVELET COEFFICIENTS Wavelet transform was used to recognze planar obects under the smlarty transformaton n [8, 9]. Affne nvarant functons usng the dyadc wavelet transform was derved by Teng and Boles [0] and Khall and Bayoum [5]. The man dfference between [0] and [5] s that, n [0] two dyadc levels were used, whereas n [5], a 3

4 wavelet-based conc equaton was ntroduced. Ths leads to an affne nvarant functon of sx or more dyadc levels. Dscrete dyadc wavelet transform (DWT) of a sgnal s mplemented usng halfband lowpass and hghpass flters formng a flterban together wth downsamplers []. The flterban produces two sets of coeffcents: orthogonal detal (or wavele coeffcents whch are the even outputs of the hghpass flter, and the approxmaton coeffcents whch are the even outputs of the lowpass flter. Samples wth odd ndces are dropped by the downsamplers n decmated mplementaton. Due to downsamplng computatonal cost of mplementng DWT drops to O(NlogN) (even to O(N) for some wavelets). Let us denote the wavelet transform of the sgnal x( at the resoluton level (or scale) asw x(, then the wavelet transform of () and () wll be W ~ x ( a W x( a W y( ) (5) + t W ~ y ( b W x( b W y( ) (6) + t Note that W a W b 0. because of the hghpass flter. 0 0 Let the sgnal par x ( and y( represent the boundary of an obect. An affne nvarant functon for an obect usng the wavelet coeffcents of sgnals x ( and y( for two, ( ) can be defned as f ( W x( W y( W y( W x( (7) 4

5 It can be easly shown that ~ f ( W ~ x ( W ~ y ( W ~ y ( W ~ x ( det( A) f ( (8) Ths nvarant functon ( f defned n [5] uses only the detal coeffcents calculated at two dfferent levels. In [0] another affne nvarant functon usng both the detal and approxmaton coeffcents of the same dyadc level s defned. In [5] Equaton (7) s also used to construct a wavelet-based conc equaton leadng to an affne nvarant functon based on sx dyadc levels. All of the nvarant functons defned n [5, 0] are computed usng the undecmated mplementaton of the wavelet transform (WT) whch does not use downsamplng operaton after flterng. Ths dramatcally ncreases the computatonal cost of the wavelet transform. If the length of the orgnal sgnal s N, then for the undecmated wavelet transform, length-n sgnals are fltered at each level. However, n the decmated mplementaton of the wavelet transform, the sgnal length s halved due to downsamplng operaton performed after each flterng step. In ths paper, we develop an algorthm to compute the affne nvarant functon defned n (7) usng the orthogonal decmated wavelet transform scheme. The wavelet sgnal W can be expressed as x(, at resoluton scale W x( d w( t ), (9) 5

6 where d are the wavelet coeffcents computed usng a decmated flterban at resoluton scale and w( s the so-called mother wavelet. If the length of the data s N (N5 s chosen n ths paper) then the lmts of summaton n (9) go from 0 to N assumng a crcular computaton of the WT. Smlarly, W y( can be expressed for as follows W y( el w( t / l) (0) where e l are the wavelet coeffcents at resoluton scale. In ths case the lmts of the summaton go from l 0 to l N / due to downsamplng. Let us assume that w ( s the Haar wavelet,.e., w ( for 0 < t < 0.5, w( for 0.5 < t <, w( 0, otherwse () The frst term of (7) can be expressed as d el w( t ) w( t / l) for, W x( W y( () Drect computaton of () and the affne nvarant functon defned n (7) requres N N / and N N multplcatons, respectvely. However, notce that w( w( t / ) w(, w ( w( t / ) 0, for >, snce the Haar wavelet has a compact support wth length. Smlarly, w ( t ) w( t / ) w( t ) 6

7 w( t 3) w( t / ) w( t 3), etc. By tang advantage of these relatons the double sum n () can be reduced to a sngle summaton as follows: W x( W y( N N d e / w t ) d e( ) / w( t ), for, 0, even, odd ( (3) Computaton of the rght hand sde of (3) requres only N multplcatons. The affne nvarant functon, ( for +, can be expressed as f f (4) ( d e / w ( t ) d e + ( ) / w ( t ) e d w t e / ( ) d ( ) / w ( t ), even, odd, even, odd where w ( w( t / ) s the wavelet of the resoluton scale,, and e are the wavelet coeffcents of the sgnals x and y at resoluton level, respectvely. An mportant feature of ths equaton s that t can be computed usng the computatonally effcent orthogonal wavelet transform as the wavelet coeffcents, and can be computed usng a flterban havng downsamplers. Equatons (3) and (4) are developed for the d d e specfc case of, +. However smlar equatons wth O(N) complexty can be easly developed to any, values because w( w( t / ) w(,..., w( t ) w( t / ) w( t ), and 0, otherwse etc due to the fact that w( has a compact support. Snce all the affne nvarant functons developed n [5] are based on ( f they can be computed usng decmated wavelet transform. As a result sgnfcant amount of computatonal savngs can be acheved. In the undecmated WT mplementaton, length-n sgnals are fltered at each level whereas n decmated 7

8 mplementaton length - N / sgnals are fltered at resoluton level and the fnal stage of constructng ( f requres only O(N) arthmetc. Although the decmated wavelet coeffcents are translaton varant Equaton (4) s translaton nvarant as the contnuous-tme functon ( f can be computed for all t values usng the rght hand sde of (4). In practce ( f s computed for unformly spaced N 5 ponts t 0,,..., 5 n [0] and n ths paper. Equaton (4) s obtaned by tang advantage of the fact Haar wavelet has compact support. Some computatonally effcent sgnal reconstructon algorthms from WT also tae advantage of ths fact []. In fact, all wavelets constructed from FIR flters have compact support. Therefore the double summaton n (7) can be reduced to a set of sngle summatons as n (3) for all compactly supported wavelets and equatons smlar to (4) can be obtaned as well. For example, wdely used Daubeches-4 wavelet has a compact support of length 6,.e., w( 0, for t > 6, and t < 0. In the case of Daubeches-4 wavelet w( w( t / ) 0, for > 3. Ths leads to a slghtly hgher computatonal cost than Haar wavelet but longer wavelets are more robust to nose compared to Haar wavelet. In general the length of data N (e.g., N5) s much hgher than the support length of most wavelets. Therefore computatonal savngs are sgnfcant. 4 EXPERIMENTAL RESULTS Snce a computatonally effcent algorthm s developed n the prevous secton for the affne nvarant functons developed n [5] t s natural that we get the same smulaton results. In [5] smulaton results are obtaned by usng a conc equaton based affne nvarant functon usng sx dyadc resoluton levels. In addton, we also present a new 8

9 practcal obect recognton scheme usng multple resoluton wavelet coeffcents n ths secton. In ths scheme, nvarant functons ( for a gven test obect are calculated by f usng consequtve pars of resoluton levels ( ), (, ),..., (, )., Correspondng nvarant functons for each model obect are ept n a database. The correlatons between the nvarant functons of the test obect and each model obect are calculated to get correlaton values R, R,..., R, whch are defned as R( I (, I I( I ( dt ( ) (5) I I where I ( ) and I ( represent the nvarant functons. The fnal decson functon t between the test obect and any model obect s found by lnearly combnng the correlaton values as follows: R fnal ν... + R + ν R + ν R (6) where ν ν ν. As a rule of thumb more weght should be gven to + resoluton levels contanng more sgnal energy to obtan robustness aganst nose. Ths approach gves us also the flexblty of samplng ( f n a nonunform manner,.e., at the resoluton level par (, + ), f ( ) f(,+) ( can be computed at N5 ponts (, + ) t but at the next resoluton level par,, f ( t can be computed at N56 ( ) + (, + ) ) 9

10 ponts etc. to acheve computatonal savngs n computng the correlaton functons defned n (5). The experments to test the effectveness of the proposed obect recognton method are carred out wth arplane mages that were also used n [5]. The same type of wavelet used n [5] are used n the experments. There are 0 model mages n the database. 0 test mages are constructed by applyng random affne transformatons to randomly chosen 0 of the model mages. The boundary sgnals of all the obects are normalzed to length 5. The correlaton values between the test mage and the model mages are calculated and the result s determned accordng to the model producng the hghest correlaton value. The experments are carred out wth two dfferent levels of unformly dstrbuted random nose whch s added to the boundares of the test mages. The sgnal to nose rato (SNR) s defned as n [5]. In the frst set of experments the SNR s about 50 db, and n the second set of experments the SNR s about 0 db. Table gves the hghest fve correlaton values for each test mage wth SNR 50 db, and Table gves the hghest fve correlaton values for each test mage wth SNR 0 db. In all experments, correct recognton results are obtaned as n [5]. In both cases of hgh and low nose power, the hghest correlaton value s produced wth the model mage from whch the test mage s constructed by applyng a random affne transformaton. In all experments summarzed n Tables and, resoluton level pars (4,5), (5,6) and (6,7) are used to calculate the nvarant functons ( and the correspondng weghts are chosen as ν.4, ν 0.3, ν 0 3 f 0.3, respectvely. In these experments, low and hgh nose levels are used, and the recognton success rate s 00%. In another set of experments the correlatons of affne nvarant functons of resoluton level pars (,3), 0

11 (3,4), (4,5), (5,6) and (6,7) are lnearly combned wth weghts 0., 0.3, 0.3, 0., and 0., respectvely. Perfect recognton results are acheved at low nose case. However at hgh nose power, t turns out that the wavelet coeffcents of levels and 3 are very nosy and obects are not recognzed correctly. 5 CONCLUSION The problem of D obect recognton usng affne nvarant functons s consdered. In prevous wors, undecmated wavelet transform was used for constructng affne nvarant functons. In ths paper, an algorthm based on decmated wavelet transform s developed to compute the same affne nvarant functons. As a result computatonal complexty s reduced wthout decreasng recognton performance. It s expermentally shown that the nvarant functon detects the affne transformed obects wth hgh accuracy. ACKNOWLEDGMENTS We would le to than Dr. Mahmoud I. Khall who provded the set of plane mages used n ths paper. REFERENCES [] J.L. Mundy and A. Zsserman, Geometrc Invarance n Computer Vson. Cambrdge, Mass.: MIT Press, 99. [] K. Arbter, W.E. Synder, H. Burhardt, and G. Hrznger, Applcaton of Affne- Invarant Fourer Descrptors to Recognton 3-D Obects, IEEE Trans. Pattern Analyss and Machne Intellgence, vol., no. 7, pp , July 990. [3] T.H. Ress,``The Revsed Fundamental Theorem of Moment Invarants, IEEE Trans. Pattern Analyss and Machne Intellgence, vol. 3, no. 8, pp , Aug. 99.

12 [4] H. Freeman, Shape Descrpton va the Use of Crtcal Ponts, Pattern Recognton, vol. 0, no. 3, pp , 978. [5] M. I. Khall and M. M. Bayoum, A Dyadc Wavelet Affne Invarant Functon for D Shape Recognton, IEEE Trans. Pattern Analyss and Machne Intellgence, vol. 3, no. 0, pp. 5-64, Oct. 00. [6] I. Wess, Geometrc Invarants and Obect Recognton, Int'l J. Computer Vson, vol. 0, no. 3, pp. 07-3, 993. [7] H.W. Guggenhemer, Dfferental Geometry. New Yor: McGraw Hll, 963. [8] Q.M. Teng and W.W. Boles, Recognton of D Obect Contours Usng the Wavelet Transform Zero-Crossng Representaton, IEEE Trans. Pattern Analyss and Machne Intellgence, vol. 9, no. 8, pp , Aug [9] M. Khall and M. Bayoum, Invarant D Obect Recognton Usng the Wavelet Modulus Maxma, Pattern Recognton Letters, vol., no. 9, pp , 000. [0] Q.M. Teng and W.W. Boles, Wavelet-Based Affne Invarant Representaton: A Tool for Recognzng Planar Obects n 3D Space, IEEE Trans. Pattern Analyss and Machne Intellgence, vol. 9, no. 8, pp , Aug [] S. Mallat, A Theory for Multresoluton Sgnal Decomposton: The Wavelet Representaton, IEEE Trans. Pattern Analyss and Machne Intellgence, vol., no. 7, pp , July 989. [] A. E. Cetn, R. Ansar, `Sgnal recovery from wavelet transform maxma,' IEEE Trans. Sgnal Processng, vol. 4, No., pp , Jan. 994.

13 Table The Best Fve Matches Between the Test Images and the Model Images for Small Nose Level Test Image Table The Best Fve Matches Between the Test Images and the Model Images for Hgh Nose Level Test Image