Sequence Mirroring Properties of Orthogonal Transforms Having Even and Odd Symmetric Vectors

Transcription

1 ECT TANSACTONS ON COMPUTE AND NFOMATON TECNOLOGY VOL., NO. 2 NOVEMBE 2 Squnc Mirroring Proprti of Orthogonal Tranform aving Evn and Odd Symmtric Vctor Do Nyon Kim and K.. ao Dpartmnt of Elctrical Enginring, Univrity of Ta at Arlington, USA cooldnk@yahoo.com, rao@uta.du ABSTACT Thi papr prnt qunc mirroring proprti for dicrt orthogonal tranform coniting of th vn and odd ymmtric row vctor. A an application, mirroring an imag (horizontally or vrtically and th rotation of th imag by angl 9, and 2 in th patial domain ar implmntd by changing th ign of th 2D orthogonal tranform cofficint of th original imag appropriatly. Kyword: Comprd domain, mag mirroring, mag procing, mag rotation, Vido coding, Tranform. NTODUCTON Thi papr prnt qunc mirroring proprti for dicrt orthogonal tranform coniting of th vn and odd ymmtric row vctor. Th proprti ar valid for DCT [, DST (alo all thir intgr rviion intgr DCT (ntdct [2, ntdst [, intgr coin tranform (CT [, ST [, [5, tc., Slant [ and adamard tranform [. Variant of CT ar dvlopd for.2 [[9, WMV9 [ and AVS China [, and thy alo hold th inhrnt proprti. Thi proprty for th DCT ha bn prntd by Shn and Sthi [2, but it i gnralizd for othr tranform in thi papr. W how pcific application uing om of th tranform. Orthogonal tranform that ar not ymmtric includ MDCT, aar tranform and artly tranform. 2. POPOSED ALGOTM W can prov qunc mirroring proprti for dicrt orthogonal tranform coniting of vn and odd ymmtric row vctor a follow. Lt b on of th orthogonal matric of iz ( and b dfind a a c g S b d f h b d f h a c g vn odd vn odd Sinc [ i orthogonal, it follow that T [ (, th uprcript T rprnt tranpo, and column vctor of T b [ i th idntity matri. For latr u, lt [ T and column vctor of [ b C n S (,,,,,,2 (,, (2, o ( o rprnt vn and odd ymmtric S T [ (, C,. Alo, lt column vctor of S,, C, C,2 ( C, S., for n,, ar column vctor of [ Lt an input qunc or column vctor b. W can gt th mirrord imag tranform cofficint M if w t th input qunc in rvr ordr and thn tranform it. th prmutation matri X M [ J (5 [ J i th oppoit diagonal unit matri. Th matri multiplication [ J arrang th column vctor of in rvr ordr. Thu th ign of lmnt of odd ymmtric row vctor of ar changd a th ign of lmnt of vn ymmtric row vctor ar not changd whil abolut valu of lmnt of ar not changd, inc i a matri having ithr vn or odd ymmtric row. Thu X (

2 ECT TANSACTONS ON COMPUTE AND NFOMATON TECNOLOGY VOL., NO. 2 NOVEMBE 2 and C, C, [ (, C, C, 2 ( K diag,,. ( t i notd that a diagonal lmnt of [ i th poitiv numbr on if th corrponding column vctor of i of vn ymmtry, and th lmnt i th ngativ numbr on if th corrponding column vctor i of odd ymmtry. From (, [ (, J,, C, C,2 C, C, K [ ( C,, C,, C,2, C, ( C,, C,, C,2, C, S. (9 Lt th mirrord qunc of an input qunc b M. From (9 w can comput X M tranform cofficint of or M a X M [ J [ K [ K M S [ T T M X ( X. ( X M W can gnraliz tranform rprntd by ( for iz N. T [ (, S N,,,,, N 2,, N (,, for N vn (,,, for N odd. (2 W rfr to orthogonal tranform rprntd by ( a typ orthogonal tranform, and th typ includ DCT, DST and Slant tranform. Th adamard tranform ha a imilar proprty. Lt b a matri of iz ( rprnting th adamard tranform. Thn T [ (, S o,,,. ( To chang an input vctor into rvr ordr, th ign of lmnt of vn ymmtric row vctor of ar not changd; howvr, th ign of lmnt of odd ymmtric row vctor ar changd. Similarly with (, w can comput X M tranform cofficint of th mirrord qunc of an input qunc a M J X [ X ( [ J i a prmutation matri and th oppoit diagonal unit matri of iz (, and [ (, K diag,,,,,,. (5 W can gt th ign of (5 in th am mannr a plaind in (. W rfr to orthogonal tranform rprntd by ( a typ 2 orthogonal tranform, and th typ 2 includ th adamard tranform. Thu w can mirror a qunc in th tranformd domain jut by changing ign of tranform cofficint. Application of th vry uful proprti for comprd-domain imag diting by uing th DCT hav bn prntd [2.. TWO-DMENSONAL EXTENSON Mirroring or th rotation of an imag i illutratd uing an ( block. Givn [ th horizontally mirrord block of [ i [ h, (. (, Similarly, w can gt th vrtically mirrord block of [ [ v. Lt [ rotatd clockwi by 9 o b [ O 9. (

3 ECT TANSACTONS ON COMPUTE AND NFOMATON TECNOLOGY VOL., NO. 2 NOVEMBE 2 Lt [ rotatd clockwi by o b [ O, and lt [ rotatd clockwi by 2 o b [ 2 O. mag can b dividd into nonovrlapping block of iz N N (for ampl N. Each block can b rprntd a a matri [ ( n, n } a hown in { 2 n, n2 (. Lt [ b an orthogonal tranform of iz (. S Thn w can comput th two-dimnional tranform cofficint [ X of an input imag block [ a [ X [ T. (9 Lt th 2D-tranform of horizontally mirrord qunc [ b and th 2D-tranform of h [ X vrtically mirrord qunc [ v b [ X. Thn V [ X and [ can b computd a X V [ X [ X [ [ X V [ [ X K K K [ (2 (2 K [ th matri [ can b or dpnding on th aociatd orthogonal tranform. r i dfind a [ (, K diag,,,,,, K and [ i dfind in (5. K (22 Apply th ( 2D-tranform to th nonovrlapping block of iz ( of th imag. 2 St th iz of a rctangular block to b horizontally flippd. orizontal and vrtical iz hould b intgr multipl of ight according to th tranform iz. Comput tranform-domain imag flipping for ach 2D-tranform block by uing (2. otat tranform block horizontally within th rctangular block. Th mot lft tranform block go to th mot right, and vic vra. n gnral, thi procdur i applicabl to ( N N [ X 2 o 2D-tranform having vn and odd ymmtric vctor by chooing appropriatly. W can alo rotat an imag block by 9 in th 2D-tranform domain by uing th propod chm. Lt u dnot th 2D-tranform of th 9, and 2 rotatd block in th patial domain [ 9, and o [ o [ 2 o a [ X 9, o [ X o and rpctivly. Thn w can comput 2Dtranform of rotatd block in th patial domain uing (9 a T T T T [ [ [ J [ X o 9 T [ X [ X T. (2 Lt th ymbol rprnt lmnt-by-lmnt multiplication. Thn [ X o [ X [ W [ X th matri [ V (2 W can b [ W or [ W dpnding on th aociatd orthogonal tranform. r th matric W ar dfind a [ W and [ [ W [ W Similarly, uing (9 [ [ J [ T T [ T T X o 2 T [ X [ X T (25. (2. (2 W can alo how (2 and (2 by uing [ T [ X T, th ymbol indicat a forward and invr 2D-tranform pair.. APPLCATONS. ntgr DCT for.2 W u th intgr DCT dignd for th Fidlity ang Etnion (FEt [ of th.2/mpeg- Advancd Vido Coding (AVC tandard. Th intgr DCT i dfind a follow:

4 ECT TANSACTONS ON COMPUTE AND NFOMATON TECNOLOGY VOL., NO. 2 NOVEMBE 2 [ diag{ ( 2, and FE FE FE F [ (2 SF FE ( 2, ( 5, ( 2, ( 2, ( 2, ( 5, ( 2} FE [ FE Matri [ ha bn hown in Eq. ( of [9..2 DCT and DST 2 Th typ 2 DCT for N and th typ DST for N 9 ar dfind in matri form a follow: C 2 2 N [ k k co m( n + mn m S in( mnπ N mn n [ π N 2 m, n,,, N- (29 m, n, 2,, N ( k 2 for p and othrwi. p. Slant Tranform k p Th Slant tranform [ i dfind a follow: SL SL SL F [ ( SL [ SF diag (,, 5, 5,,, 5, 5 and SL [ 5 9. adamard Tranform 9 5. Th adamard tranform [ i dfind by Eq. (5. in [..5 Simulation Uing Eq. (-(2, th 2D-tranform domain manipulation i utilizd to obtain imag mirroring and rotation by 9, and 2 in patial domain. Not that imag mirroring and rotation by 9, and 2 in patial domain i accomplihd by changing th ign of th 2D-tranform cofficint of th original imag appropriatly. Th opration ar implmntd on th Lna imag ( Fig... Concluion By appropriatly changing th ign of th cofficint in th 2D-tranform domain, mirroring and th rotation by 9, and 2 of an imag can b accomplihd in th 2D-patial domain. 5. ACKNOWLEDGEMENT Do Nyon Kim acknowldg th upport by th ntitut for nformation Tchnology Advancmnt and th Minitry of nformation and Communication, public of Kora, undr th T Scholarhip Program.

5 ECT TANSACTONS ON COMPUTE AND NFOMATON TECNOLOGY VOL., NO. 2 NOVEMBE 2 Proc., Commu., Spch & Viion, Vol., pp. 2-22, 99. [5 W. K. Cham and P. P. Yip, ntgr inuoid tranform for imag procing, nt. J. Elctronic, Vol., No., pp. 5-, 99. [ W. K. Pratt, W.-. Chn and L.. Wlch, Slant tranform imag coding, EEE Tran. Commun., Vol. 22, pp. 5-9, 9. [ A. K. Jain, Fundamntal of digital imag procing. Englwood Cliff, NJ: Prntic all, 99. [ G. J. Sullivan, P. Topiwala and A. Luthra, Th.2/AVC advancd vido coding tandard: ovrviw and introduction to th fidlity rang tnion, SPE Conf. on Application of Digital mag Procing XXV, pp. 5-, 2. [9 S.-K. Kwon, A. Tamhankar and K.. ao, Ovrviw of.2/mpeg- part, J. Vi. Commun. mag., Vol., pp. -2, 2.. Srinivaan t al., Window mdia vido 9: ovrviw and application, Signal Procing: mag Communication, Vol. 9, pp. 5-5, 2. [ W. Gao t al., AVS Th Chin nt-gnration vido coding tandard, in Proc. of NAB, La Vga, Nvada, 2. [2 B. Shn and. K. Sthi, nnr-block opration on comprd imag, in Proc. of th Third ACM ntrnational Confrnc on Multimdia, San Francico, CA, pp. 9-9, 995. (a (c (i (b (d (iii (iv (ii (v ( (f Fig.: Mirroring or rotation of portion of Lna imag in th patial domain by manipulating th 2D-tranform cofficint of th original block: (i horizontal mirroring, (ii vrtical mirroring, (iii rotation by 9, (iv rotation by, and (v rotation by 2 for (a ntdct of.2, (b Slant tranform, (c adamard tranform (d DST, and ( DCT. r th numbr of ach imag block i hown in (f.. EFEENCES.. ao and P. Yip, Th tranform and data comprion handbook. Boca aton, FL: CC Pr, Boca aton, FL, 2. [2 Y.-J. Chn, S. Oraintara and T. Nguyn, Vido comprion uing intgr DCT, in 2 Proc. EEE CP, Vol. 2, Vancouvr, BC, pp. -5. [ V. Britanak, P. Yip and K.. ao, Dicrt coin and in tranform. Orlando, FL: Acadmic Pr (Elvir, 2. [ W. K. Cham, Dvlopmnt of intgr coin tranform by th principl of dyadic ymmtry, EE