I HC BCH KHOA H NI VIN CNG NGH THNG TIN V TRUYN THNG ---------- BI TP LN MN X L NH TI: Tm hiu cc c trng sinh trc nh khun mt, nghin cu ng dng ca php bin i KL v phn tch thnh cc thnh phn chnh PCA trong trch chn c trng khun mt Ging vin hng dn: PGS.TS Nguyn Th Hong Lan Sinh vin thc hin: Nguyn Vn Thnh SHSV: 20072604 Lp: H thng thng tin v truyn thng KSCLC-K52 H Ni, 12/2011 BI TP LN MN X L NH2011 2 Nguyn Vn Thnh HTTT&TT KSCLC-K52 MC LC LI NI U .......................................................................................................................................... 3 I.C TRNG SINH TRC NH KHUN MT ...................................................................... 4 II.NGHIN CU NG DNG CA PHP BIN I KL ...................................................... 7 1.Php bin i KL ........................................................................................................................ 7 2.ng dng ca php bin i KL ............................................................................................... 9 III.PHN TCH THNH CC THNH PHN CHNH PCA TRONG TRCH CHN C TRNG KHUN MT ....................................................................................................................... 14 1.Php bin i PCA.................................................................................................................... 14 2.Phn tch thnh chnh PCA trong trch chn c trng khun mt ................................... 14 2.1.Tnh ton cc vector ring ............................................................................................... 14 2.2.Biu din khun mt theo c s tm c ...................................................................... 17 IV.KT LUN ............................................................................................................................... 18 TI LIU THAM KHO .......................................................................................................... 19 BI TP LN MN X L NH2011 3 Nguyn Vn Thnh HTTT&TT KSCLC-K52 LI NI U X l nh l mn hc quan trng i vi sinh vin ngnh cng ngh thng tin. y l mn hc kh i vi hu ht nhiu sinh vin do yu cu kin thc v ton v xc sut. Vi mc tiu c thm kin thc c bn trong lnh vc v x l nh, em chn tiTmhiuccctrngsinhtrcnhkhunmt,nghincung dngcaphpbiniKLvphntchthnhccthnhphnchnhPCA trong trch chn c trng khun mt. Qua bi tp ln, em c thm c ci nhn chung v h thng nhn dng khun mt, cc c trng sinh trc khun mtgipphthinvtrchrtcccctrngchovicnhndng.ng thi em c cng c thm v kin thc ton v xc sut thng k, c c hiu bit tt hn v ng dng ca kin thc c bn trong cc bi ton thc t. D rt c gng trong vic tm hiu ti liu, nhng do thiu st v kin thc c bn,hnchtmhiubitmbibococnrtnhiuthiukhuyt.Knh mong nhn c nhng kin gp em hon thin hn. Nhndpny,emxingilicmnchnthnhtiPGS.TSNguynTh Hong Lan nhit tnh hng dn gp gip em hon thnh bo co mn hc ny. Em xin chn thnh cm n! H Ni, ngy 12 thng 12 nm 2011 BI TP LN MN X L NH2011 4 Nguyn Vn Thnh HTTT&TT KSCLC-K52 I.C TRNG SINH TRC NH KHUN MT Cc c trng khun mt bao gm: -ctrnghnhhc:cutrc,hnhdngvccthnhphntrnkhunmt: ming, mt, mi, lng my. Khong cch gia mt, mi, ming v hm; ng bao cc hc mt; cc cnh ca ming; v tr ca mi, hai mt v cc vng xung quanh.Cc thnh phn khun mt c trch rt hnh thnh vector c trng biu din hnh hc khun mt. BI TP LN MN X L NH2011 5 Nguyn Vn Thnh HTTT&TT KSCLC-K52 Hnh 1.1. Minh ha c trng hnh hc ca khun mt -c trng v din mo biu din s thay i v b ngoi: kt cu da nh cc np nhn trn khun mt; biu nhit ca khun mt: cc mu nhit khun mtlduynhtvimingivctrngvnci.Ccctrngv dinmocthctrchrttrnckhunmthocphnnotrn khun mt. - BI TP LN MN X L NH2011 6 Nguyn Vn Thnh HTTT&TT KSCLC-K52 Hnh 1.2. Minh ha c trng v din mo khun mt BI TP LN MN X L NH2011 7 Nguyn Vn Thnh HTTT&TT KSCLC-K52 II.NGHIN CU NG DNG CA PHP BIN I KL 1.Php bin i KL Xt khng gian mu S = {x} gm n vector d liu mu. Trong x l vector cc bin ngu nhin N chiu:x = [ x1 x2 xN ]T Php bin i KL i vi cc vector x c nh ngha nh sau: y = WTx (1.1) Trong x = [ x1 x2 xN ]T, y = [ y1 y2 yM ]T v ma trn W l ma trn php bin i vi kch thc NxM (M N) c dng: | |((((
= =MN MNTiTw ww ww W...... ... ......11 11 Ta nh ngha ma trn hip tng quan ca cc vector x:[C]x = E[xxT](1.2) Ma trn tng quan c c lng t n mu trong khng gian d liu quan st c xc nh bi biu thc: ==niTi i xx xnC11 Php bin i KL l bi ton tm mt ma trn bin i W tha mn (1.1) Mi ct ((((
=iNiiwww ...1ca W l vector c s trc giao ca khng gian mi hay: 1 . =Tk j w w nu j = k(1.3) BI TP LN MN X L NH2011 8 Nguyn Vn Thnh HTTT&TT KSCLC-K52 V vy mi phn t yi ca y s c tnh: N iN i iTi ix w x w x w x w y + + + = = ...2 2 1 1(1.4) Do wi l cc vector c s trc giao theo (1.3) nn W l ma trn trc chun tha mn: WTW = I = WWT(I l ma trn n v) (1.5) T : WT = W-1 v ta c dng bin i ngc ca (1.1): x = Wy(1.6) t Cy l ma trn ng cho mong mun ca vector bin ngu nhin y: Cy = (((((
N...... .......... 0.....1 Trong cc phn t ng cho l cc phng sai ca d liu c bin i. Ma trn ng cho ny c th tnh ton t ma trn hip tng quan gc nh sau: Cy =| |Tyy E=( )( ) | |TT Tx W x W E=( ) | | W xx W ET T =W C WxT Hay yWC =W Cx (1.7) Cy l ma trn ng cho nn biu thc (1.7) c a v sng Cxwi = i wi (1.8) T (1.8) ta thy i v wi l cp gi tr ring v vector ring ca ma trn tng quan Cx trong biu thc (1.2). Hay cc ct wi ca ma trn W l cc vector ring ca ma trn Cx. mboWltrcchuncnpdngtrcgiaovchunhaGram-Schmidtvi cc vector ring tm c. Nh vy, php bin i KL chnh l tm cc vector ring wi ng vi M gi tr ring ln nht ca ma trn hip tng quan ca cc vector ngu nhin quan st c. Php bin iKLlcchtiunhmgimthnguyntkhnggiandliucschiuln BI TP LN MN X L NH2011 9 Nguyn Vn Thnh HTTT&TT KSCLC-K52 thnhkhnggianmicschiubhnrtnhiuvisaislbnhtnmcc vector ring ng vi cc gi tr ring nh nht. 2.ng dng ca php bin i KL Hnh 2.1 biu din mt nh a mc xm kch thc 512 x 512 vi gi tr mc xm ca mi im nh c biu din bng 8 bit (gi tr trong [0 255]). Gi tr mc xm ca cc im nh k nhau c xu hng tng t nhau. Hnh 2.2 minh ha gi tr mc xm ca cc cp im nh lin k. Hnh 2.1. nh a mc xm kch thc 512 x 512 Tronghnh2.2,michmbiudinmtimnhtrongbcnhhnh2.1vi honh x l gi tr ca im nh v tung y l gi tr ca im nh lin k bn phi n.T th, ta thy quan h x =y thhin tng quanmnh gia cc im nh lin k. Chia bc nh thnh cc khi 1x2 ri nhau nh hnh 2.3, chng ta biu din BI TP LN MN X L NH2011 10 Nguyn Vn Thnh HTTT&TT KSCLC-K52 mt bc nhthnh tp cc vector2 chiu xi.Phn bgi trmc xmcami thnh phn c v nh hnh 2.4. Chng ta thy rng phn b mc xm ca mi thnh phn tngirngvphhuhtdi0255.Hnna,haiphnbnykhgingvi phn b chung ca mi im nh trong bc nh hnh 2.1. Hnh 2.2. th biu din cc cp gi tr mc xm ca cc im nh BI TP LN MN X L NH2011 11 Nguyn Vn Thnh HTTT&TT KSCLC-K52 Hnh 2.3. Chia bc nh ban u thnh cc khi im nh 1 x 2 BI TP LN MN X L NH2011 12 Nguyn Vn Thnh HTTT&TT KSCLC-K52 Hnh 2.4. Phn b gi tr mc xm ca mi thnh phn trong cc khi By gi, chng ta quay phn b trong hnh 2.2 i mt gc 45o. Kt qu trong hnh 2.5 cho thy, hai thnh phn mi khng tng quan ngha l bit gi tr ca thnh phn th nht s khng gip ta xc nh c gi tr ca thnh phn th hai. Phn b ca hai thnh phn mi c v nh hnh 2.6. Thnh phn th nht vn kh ging vi phn b trc, tc l phn b rng, tri hu ht khong gi tr. Tuy nhin thnh phn th hai th khc,nhphnrtnhiuvgitrnhti0.Donckhongbininhhn nhiu nn chng ta cn t s bit m ha gi tr ca n.V vy, chng ta c th gim s bit cn thit m ha mt mt nh khi gii tng quan. Hnh 2.5. th cc cp mc xm khi quay 45o BI TP LN MN X L NH2011 13 Nguyn Vn Thnh HTTT&TT KSCLC-K52 Hnh 2.6. Phn b mc xm ca hai thnh phn khi xoay tng ng vi hnh 2.5 Hu ht cc bc nh u cha s tng quan ngu nhin v d liu do dn n sdthadliu.PhpbiniKLloibcsdthad liunhvicgii tng quan d liu v gim s chiu v vy nh c th c lu tr hiu qu hn. l ng dng ca php bin i KL. BI TP LN MN X L NH2011 14 Nguyn Vn Thnh HTTT&TT KSCLC-K52 III.PHN TCH THNH CC THNH PHN CHNH PCA TRONG TRCH CHN C TRNG KHUN MT 1.Php bin i PCA Tng t nh php bin i KL, ta c khng gian d liu quan st S = {x} gm n vectordliumuNchiu.Dliuthtntistngquanngunhingiacc thnh phn do c s d tha d liu. tng ca php bin i PCA l phn tch d liu thnh cc thnh phn khng tng quan (gi l cc thnh phn chnh) gim d tha d liu. Php bin i PCA c nh ngha nh sau: x uT+ = (2.1) Trong x = [ x1 x2 xN ]T , u = [u1 u2 uN ]T cc thnh phn th i v j khng tng quan trong khng gian mi. V ma trn l ma trn php bin i vi kch thc N2 c dng: ||((((
= = +NN NNTiTvv vvv...... ... ......11 11(2.2) vi l cc vector ring tng ng vi ma trn hip phng sai ca cc x quan st c. ( )( ) | |Tx x E C = (2.3) Trong ((((
=N ...1v ==nkik ixn11 2.Phn tch thnh chnh PCA trong trch chn c trng khun mt 2.1.Tnh ton cc vector ring Gi s chng ta mt tp luyn gm M bc nh khun mt I1, I2IM. Cc bc nh ny c cng kch thc v c chnh tm.BI TP LN MN X L NH2011 15 Nguyn Vn Thnh HTTT&TT KSCLC-K52 ChngtabiudinmibcnhIikchthcNxNbngmtvectorI ckch thc N2 chiu. Bc 1: Tnh vector trung bnhBI TP LN MN X L NH2011 16 Nguyn Vn Thnh HTTT&TT KSCLC-K52 Bc 2: Tnh+ I = ui ivi i = 1, 2N2 Bc 3: Tnh ma trn hip phng sai C ca cc vector quan st: TMiTi iAAMC = u u ==11(ma trn C c kch thc N2xN2) (2.4) Trong | |MA u u u = ...2 1 (kch thc N2xM) Bc 4: Tm cc vector ring ui ca ma trn hip phng sai C hay ca AAT. Tuy nhin kch thc ca ma trn ny l N2xN2 qu ln nn vic tm vector ring ca ma trn ny l khng kh thi. Chng ta xem xt ma trn ATA c kch thc MxM Tm cc vector ring vi ca ma trn ATA, ta c: i i iTv Av A =(2.5) Quan h gia ui v vi: i i i i i iTi i iTAv CAv Av Av AA v Av A = = = (2.6) T suy ra ui = Avi
V vy, AAT v ATA c cng cc gi tr ring v cc vector ring ca chng quan h vi nhau theo ui = Avi Ch : - ATA c th c M gi tr ring v vector ring. -AAT c th c N2 gi tr ring v vector ring. -M gi tr ring ca ATA cng vi cc vector ring tng ng vi M gi tr ring ln nht ca AAT. BI TP LN MN X L NH2011 17 Nguyn Vn Thnh HTTT&TT KSCLC-K52 Sau tnh M vector ring ng vi M gi tr ring ln nht ca AAT: ui = Avi ng thi chun ha vector ui sao cho1 =iuBc 5: Chn K vector ring ng vi K gi tr ring ln nht. 2.2.Biu din khun mt theo c s tm c Mi vector biu din khun mt (tr i vector trung bnh) iutrong tp luyn c th c biu din bng mt t hp tuyn tnh ca K gi tr ring tnh trn. == uKjj j iu w mean1i=1,2M. Trong iTj ju w u =chnh l thnh phn chnh th j trong khng gian mi. V uj gi l cc nh ring. Minhcchunhatrongtpluynscbiudintrongcsnybi vector: ((((((
= OiKiiiwww...21 trong i = 1, 2 M BI TP LN MN X L NH2011 18 Nguyn Vn Thnh HTTT&TT KSCLC-K52 IV.KT LUN Nh vy, em trnhby v cc c trng sinh trc nh khunmt, gm cc c trnghnhhcvccctrngvdinmokhunmt,cngvillthuytv ngdngcaphpbiniKLvphntchthnhccthnhphnchnhPCAtrong trch chn c trng khun mt. Php bin i KL ng dng trong bi ton gim s chiu ca khng gian d liu s lnthnhkhnggiancschiunhhnnhmgimgitnhtonvtnghiuqu cacckthuttrongxlnh.TrongkhiPCAgiitngtngquandliuv phn tch thnh cc thnh phn chnh ng dng trong trch chn c trng khun mt. BI TP LN MN X L NH2011 19 Nguyn Vn Thnh HTTT&TT KSCLC-K52 TI LIU THAM KHO [1]. Bi ging mn X l nh PGS.TS Nguyn Hong Lan i hc Bch Khoa H Ni, 2010. [2]. Digital Image Processing William K. Pratt- Wiley, 2007. [3]. Fundamental of Image Processing Ian T. Young et al., 1998.[4]. Principal Component Analysis I.T.Jolliffe Springer, 2002. [5].R.D.DonyKarhunen-LeveTransform-TheTransformandData Compression Handbook, 2001.
