No. 6 Cell image ecognition with adial hamonic Fouie moments 611 ecognition expeiment is pefomed. Results show that as good image featues, RHFMs ae su

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1 Vol 12 No 6, June 23 cfl 23 Chin. Phys. Soc /23/12(6)/61-5 Chinese Physics and IOP Publishing Ltd Cell image ecognition with adial hamonic Fouie moments * Ren Hai-Ping(Φ ±) a)y, Ping Zi-Liang( fflξ) b), Bo Wu-Ri-Gen( flψ ) c) Sheng Yun-Long(ΩffiΠ) d), Chen Sheng-Zu( Ω ) a), and Wu Wen-Kai(fiffΛ) a) a) Depatment of Nuclea Medicine, Cance Hospital, Chinese Academy of Medical Sciences, Peking Union Medical College, Beijing 121, China b) Depatment of Physics, Inne Mongolia Nomal Univesity, Huhhot 122, China c) Mathematics Institute, Peking Univesity, Beijing 1871, China d) Depatment of Physics, Laval Univesity, Quebec, GIK 7P4, Canada (Received 18 Novembe 22; evised manuscipt eceived 18 Febuay 23) Vaious types of moments have been utilized to ecognize image pattens in a numbe of applications. Multidistotion invaiant adial hamonic Fouie moments wee investigated in the ecognition of cell smea images. Image ecognition expeiments showed that adial hamonic Fouie moments ae not only highly concentated image featues but also have a good popety of antidistotion and antinoise. They may be used in fast and accuate automatic cell ecognition. Keywods: cell ecognition, moment invaiant PACC: 423, 423K, 423S 1. Intoduction Moment is a highly concentated image featue that has the invaiant popeties of moment function in multi-distotion of an image such as tanslation, scaling, otation, and changing intensity. Theefoe, it has been extensively employed as the invaiant global featues of an image in patten ecognition, image classification, taget identification, and scene analysis. [1] Hu [2] fist intoduced geometic moments invaiant in 1961, fom which image was had to be ecoveed. Teague [3] intoduced Zenike moments based on the theoy of othogonal polynomials. Othe othogonal moments ae otational moments, [4] complex moments, [5] and Legende moments. [6] Teh and Chin [6] evaluated vaious types of moments and found that Zenike moments outpefomed the othes in tems of oveall pefomances fo image desciption and obustness to noise. Sheng and Shen [7] poposed othogonal Fouie Mellin moments (OFMMs) in 1994, which have bette popeties than Zenike moments fo descibing an image. In 22, Ping et al [8] intoduced Chebyshev Fouie moments (CHFMs) which have nealy the same pefomance as OFMMs. We also suggested to choose a tiangula function as the adial function and poposed new moments named adial hamonic Fouie moments (RHFMs). [9] Compaed with CHFMs, RHFMs have a supeio pefomance nea the oigin and bette desciption ability fo small images in tems of image econstuction eos and noise sensitivity. Cytological examination is indispensable fo clinical diagnosis, theapy and eseach. It is extensively used fo diagnosis of haemal system diseases, contagious diseases and tumous by samples of blood, ceebospinal fluid and pleual effusion. Howeve, the examination is nealy completely done by expeienced expets, which is time consuming and had sledding, thus not suitable fo disease sceening. Automatic, fast and accuate cell image ecognition is needed to be ealized ugently. So, we use some cell slice images to investigate the application popeties of RHFMs. In this pape, vaious cell images ae descibed with RHFMs as the featue and then the multi-distotion invaiant patten Λ Poject suppoted by the National Natual Science Foundation (Gant No 62621) and by the Inne Mongolia Natual Science Foundation, China (Gant No 2131). y Coesponding autho. haipingen@163.com, Tel: ext 8599(O), Fax:

2 No. 6 Cell image ecognition with adial hamonic Fouie moments 611 ecognition expeiment is pefomed. Results show that as good image featues, RHFMs ae suitable fo fast and accuate cell ecognition, and thus have geat potential fo futhe applications. 2.Radial hamonic Fouie moments 2.1. Definition A function set P nm (; ) defined in a pola coodinate system (; ) contains a adial function T n () and a Fouie facto in the angle diection exp(jm ): whee T n () = 8 >< >: P nm (; ) = T n ()exp(j m ); (1) 1 p ; if n = ; 2 sin[(n +1)ß]; if n = odd; 2 cos(nß); if n = even: (2) The set of P nm (; ) is othogonal ove the ange»» 1, Z 2ß Z 1 P nm (; )P kl (; ) d d = ffi nmkl ; (3) whee ffi nmkl is the Konecke symbol, and =1 is the maximum size of the objects that can be encounteed in a paticula application. The image f(; ) can be decomposed with the set of P nm (; ) as whee f(; ) = Z 2ß Z 1 Φ nm = 1X +1X n= m= 1 We define Φ nm as RHFMs. Φ nm T n ()exp(j m ); (4) f(; )T n ()exp( j m ) d d : (5) 2.2. Nomalization and invaiance RHFMs ae not invaiant themselves, howeve, they can be nomalized into invaiant fo shifting, scaling, otation and intensity distotion of an image. [5] Fistly, we use the fist-ode geometical moments to detemine the cente of the image, which is taken as the oigin of the coodinate system. All the moments calculated in this coodinate system ae shift invaiant. Secondly, since the angle function is a Fouie facto, a otation of the image by an angle ' will esult in a phase facto e j m' fo all odes of Φnm. The modulus of RHFMs, jφnm j, ae otational invaiant. Finally, lowe-ode Fouie Mellin tansfe M i 1 M i [4] is computed fo evey distoted image in the taining set. We use the following fomulae (6) and (7) to calculate k i and g i espectively fo each image with the atio M 1 being a constant and slightly smalle than the M minimum M i 1 M i of all the images in the taining set to ensue that the nomalized images emain inside the unit cicle. g i = ffi M i k i = 1 M1 M i ; (6) M» M1 M i 2 1 M ffi M M i : (7) M Φ nm is then calculated by fomulae (8) and (9) fo all images of the taining set, which is scaling and intensity distotion invaiant. ffi i nm = Z 2ß Z ki g i f(=k i ; )T n (=k i )e j m d d ; (8) Φ i nm = ffi i nm =g ik 2 i : (9) Hee, Φ i nm is the invaiant moment of the image i. 3. Multi-distotion invaiant patten ecognition expeiment Twenty-six types of blood cell images, odeed as A to Z, wee chosen to be the taining set with intensities anging fom to 255 in pixel matices afte nomalization (Fig.1). In nomalization, the aveaged M value of all images was taken as a constant, which was The efeence set was obtained though otating each efeence object by 15 ffi, 3 ffi and 45 ffi and scaling it by 1.5 and 2 times, so as to detemine the inclass vaiance of RHFMs. The testing set images wee the otated- (6 ffi and 15 ffi ), scaled- (.4 and 3 times), and intensity-changed (.5 and 3 times) vesions of the taining images, without o with zeo-mean additive noises ff=4 and ff=25. Figue 2 shows the testing set of othochomatic nomoblast (image R).

3 612 Ren Hai-Ping et al Vol. 12 Fig.1. The taining set of cell images A Z afte nomalization. Fig.2. Testing image fo the image R. Fom top left to bottom ight: the image R has been intensity changed by.5 and 3 times, scaled by 3 and.4 times, otated by 6 ffi and 15 ffi, with zeo-mean additive noise of ff=4 and ff=25. We used 25 RHFMs with m, n=, 1, 2, 3, 4 to descibe the images, and then obtained a 25-dimensional featue space (x 1 ;x 2 ; ;x 25 ) T. Testing objects wee classified in the featue space using the weighted minimum-mean-distance ule. The weighted distance was calculated as ρ N;M X ff [jφ nm j (jφ nm j) i ] 2 1=2 d i (N;M) = ; (1) (ff nm ) 2 i n;m= whee jφ nm j is the modulus of the RHFMs of the testing object, (jφ nm j) i ae the RHFMs of the efeence object of class i, and (ff nm ) 2 i is the in-class vaiance of the (jφ nm j) i. M and N ae the maximum cicula hamonic ode and highest degee of the adial polynomials espectively. A testing image was classified to the class i fo which the distance d i was minimum. d i (M;N) ae shown in Figs.3 1 as a function of M and N with vaious distotions. Take the ecognition of intensitychanged (3 times) image R as an example (Fig.3), the distance between it and the efeence image R is the shotest, afte which ae efeence images of images N, X, O, Z and C when M=N=4. The diffeences between the minimal distance and othes ae athe lage, so it is easy to classify coectly. We can also find that in some cases, the minimal distance is not that of the test image to its efeence image when lowe odes ae chosen. In this situation, a highe and pope ode should be used. All the expeimental esults show that ecognitions ae well pefomed fo vaious distoted image R with RHFMs being the featue. The misclassification ates ae zeo fo all testing images, except fo image B with a high-level noise ff=25. Figue 1 shows that images B and V ae vey simila, which causes the misclassification when high-level noise exists. Moe RHFMs ae needed in this situation. When we used M =N =6, the above misclassification could be avoided. Images B and H with a high-level noise ff=25 ae misclassified when OFMMs ae used as image featue.

4 No. 6 Cell image ecognition with adial hamonic Fouie moments 613 When weighted minimum-mean-distance ule is used fo classification, the bigge the diffeence between minimal distance d min and sub-minimal distance d sub-min, the stonge the distinguishing ability of the featue is. So, we define the absolute diffeence as the subtaction of d sub-min and d min, and the elative diffeence as = d sub-min d min : (11) Table 1 shows the absolute and elative diffeences of the image R unde vaious distoting situations. The values in the table ae athe lage, indicating that the distinguishing and descibing ability of RHFMs is stong. At the same time, it is highly concentated since the coect cell image ecognition is pefomed when only 25 featues ae extacted (with M =N =4). Using moe RHFMs should be consideed in moe complicated cases. Fig.5. d i (M; N) as a function of M and N fo scaled image R (.4 times). Fig.6. d i (M; N) as a function of M and N fo scaled image R (3 times). Fig.3. d i (M; N) as a function of M and N fo intensity-changed image R (.5 times). Fig.4. d i (M; N) as a function of M and N fo intensity-changed image R (3 times). Fig.7. d i (M; N) asa function of M and N fo otated image R (6 ffi ).

5 614 Ren Hai-Ping et al Vol. 12 Table 1. The absolute and elative diffeences of the testing image R. Distotion Distoting facto dmin d sub-min Intensity Scale Rotation Noise Fig.8. d i (M; N) asa function of M and N fo otated image R (15 ffi ). Fig.1. d i (M; N) as a function of M and N fo image R added with noise (ff=25). Fig.9. d i (M; N) as a function of M and N fo image R added with noise (ff=4). Refeences [1] Pokop R J and Reeves A P 1992 CVGIP-Gaph. Model. Image Poc [2] Hu M K 1962 IEEE Tans. Inf. Theoy IT [3] Teague M R 198 J. Opt. Soc. Am [4] Boyce J F and Hossack W J 1983 Patten Recogn. Lett [5] Abu-Mostafa Y S and Psaltis D 1984 Patten Anal. Mach. Intell. PAMI Conclusion We have investigated the application of RHFMs in vaious cell images ecognition in this pape. Without any pepocessing of images, RHFMs can extact enough mophology infomation fo accuate cell classification, which makes it easy fo clinical cytological examination. Expeiments show that classification of all the 28 distoted images of 26 kinds of cells can be finished within 2 min. So, as good image featues, RHFMs ae suitable fo quick and coect cell image ecognition. It will be ou eseach focus to optimize the pogam, make it compatible with the clinical hadwae and do some necessay case validation. [6] Teh C H and Chin R T 1988 Patten Anal. Mach. Intell [7] Sheng Y L and Shen L X 1994 J. Opt. Soc. Am. A [8] Ping Z L, Wu R G and Sheng Y L 22 J. Opt. Soc. Am. A [9] Ren H P et al 23 J. Opt. Soc. Am. A 2 631