Fast Algorithm for Walsh Hadamard Transform on Sliding Windows

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1 Fs Algohm fo Wlsh Hdmd Tnsfom on Sldng Wndows Wnl Oung W.K. Chm Asc Ths ppe poposes fs lgohm fo Wlsh Hdmd Tnsfom on sldng wndows whch cn e used o mplemen pen mchng mos effcenl. The compuonl equemen of he poposed lgohm s ou 3 ddons pe poecon veco pe smple whch s he lowes mong esng fs lgohms fo Wlsh Hdmd Tnsfom on sldng wndows. Inde Tems Fs lgohm Compuon of nsfoms Wlsh Hdmd Tnsfom Pen chng I. ITROUCTIO Pen mchng lso nmed s emple mchng s wdel used n sgnl pocessng compue vson mge nd vdeo pocessng. Pen mchng hs found pplcon n mnufcung fo qul conol ] mge sed endeng ] mge compesson 3] oec deecon ] nd vdeo compesson. The lock mchng lgohm used fo vdeo compesson cn e consdeed s pen mchng polem 5]-]. To eleve he uden of hgh comple nd hgh equemen of compuonl me fo pen mchng lo of fs lgohms hve een poposed 9]-3]. I hs een found h pen mchng cn e pefomed effcenl n Wlsh Hdmd Tnsfom WHT domn 9]. In pen mchng sgnl vecos oned sldng wndow need o e comped o sough pen. Hel-O nd Hel-O s lgohm 9] eques - ddons fo onng ll WHT poecon vlues n ech wndow of sze. oe h one sucon s consdeed o e one ddon egdng he compuonl comple n hs ppe. The lgohm cheves effcenc ulzng pevousl compued vlues n he nenl veces of he ee sucue n Fg.. Recenl he G Code Kenel GCK lgohm ] whch ulzes pevousl compued vlues n he leves of he ee sucue n Fg. ws poposed. The GCK lgohm eques sml compuon s 9] when ll poecon vlues e compued nd eques less compuon when onl smll nume of poecon vecos e compued. The uhos e wh he epmen of Eleconc Engneeng The Chnese Unves of Hong Kong e-ml: wloung@ee.cuhk.edu.hk; wkchm@ee.cuhk.edu.hk

2 ] -] ] - -] - - ] ] ] ] ] - -] Fg. Tee sucue fo Wlsh-Hdmd Tnsfom n sequenc ode Ths ppe poposes fs lgohm fo WHT on sldng wndows. Insed of pefomng ode- WHT mens of ode- WHT nd ddons n he ee sucue whch s he echnque doped n 9] he poposed lgohm compues ode- WHT mens of ode- WHT nd ddons. In hs w he poposed lgohm cn on ll WHT poecon vlues usng ou 3 ddons pe wndow. In he compuon of pl poecon vlues fo sldng wndows he poposed lgohm eques onl.5 ddons pe poecon veco fo ech wndow. As shown epemenl esuls n Secon VII he compuonl me equed he poposed lgohm fo compung en o moe poecon vlues s ou 75% of h of he GCK lgohm. The es of he ppe s ognzed s follows. Secon II defnes ems nd smols used n hs ppe. Then he WHT lgohm n 9] s efl noduced. In Secon III we noduce wo emples of he poposed lgohm. Secon IV lluses he poposed lgohm fo - ode- WHT. The lgohm compues ode- WHT usng ode- nd ode- WHT. In Secon V he nume of ddons equed he poposed lgohm s deved. Secon VI gves he epemenl esul of moon esmon n vdeo codng pplcon whch ulzes he poposed lgohm fo compung - WHT on sldng wndows. Fnll Secon VII pesens conclusons. II. WALSH HAAAR TRASFOR O SLIIG WIOWS A. efnons Consde K npu sgnl elemens n whee n K- whch wll e dvded no ovelppng wndows of sze K>. Le he h npu wndow e: X - ] T fo K-. A - ode- WHT nsfoms numes no poecon vlues. Le e n ode- WHT m nd -] T whee s he h WHT ss veco.

3 3 Le e he h WHT poecon vlue fo he h wndow nd In ] T X fo -; K-. 3 nd e clled he h poecon kenel nd poecon esul especvel. Le Y e he poecon veco connng ll poecon vlues of he h wndow nd Y - ] T X. Fo emple when we hve: v Y 3 3 X. 5 The WHT n 5 s n sequenc ode. Is ss vecos e odeed ccodng o he nume of zeo cossngs. The elonshp eween WHTs n sequenc ode ddc ode nd nul ode cn e found n 5]. B. Pevous WHT compuon mehods In 9] Hel-O nd Hel-O poposed fs lgohm h compues Y fom Y nd Y usng 6. % o 3 6 % o whee % s he module opeon; nd s he floo funcon. As shown n Fg. he lgohm fs compues WHT poecon vlues fo wndow sze eng whch e hen used o compue WHT poecon vlues fo wndow sze eng nd so on. The compuon ss he oo nd moves down he ee unl he poecon vlues epesened he leves e compued. The lgohm n 9] eques ddon pe wndow long ech node of he ee n Fg.. The GCK lgohm ] ulzes pevousl compued ode- poecon vlues fo compung he cuen ode- poecon vlue. When smll nume of poecon vlues e compued he GCK lgohm eques ddons pe wndow fo ech poecon vlue whle he lgohm n 9] eques Olog ddons. When ll poecon vlues e compued oh he lgohms n 9] nd ] eques ou ddons. A new fs lgohm whch s moe effcen hn h epoed n 9]] s poposed n hs ppe. I cn effcenl compue ode- WHT on sldng wndows.e. fo P- P nd K-.

4 III. FAST ALGORITH FOR WHT O SLIIG WIOWS FOR WIOW SIES A Ths secon gves emples of compung ode- WHT on sldng wndows of szes nd usng he poposed lgohm. A. Fs Algohm fo Wndow Sze The poposed lgohm nd he GCK lgohm ] fo wndow sze eng e desced n Tle I. The poposed lgohm compues Y he WHT poecon vlues n wndow usng Y he compued poecon vlues n wndow s shown n Tle I. Ecep fo he h poecon vlue he GCK lgohm ulzes he pevous ode- poecon vlues o compue he cuen ode- poecon vlue. TABLE I FAST ALGORITH WHE WIOW SIE IS 3 Poposed lgohm GCK lgohm] efne d s: d -. 7 We cn see fom Tle I h he WHT poecon vlues n wndow cn e compued fom hose n wndow nd d. Theefoe we hve d d d 3 3 d nd Fg. shows he sgnl flow dgm. Thus fe onng Y he poposed lgohm on Y 5 ddons s shown n 7- whees he lgohm n 9] eques 6 ddons nd he GCK lgohm eques ddons.

5 5 Fg. Sgnl flow dgm of he Boom up lgohm fo wndow sze equl B. Fs Algohm fo Wndow Sze The poposed lgohm nd he GCK lgohm ] fo wndow sze eng e desced n Tle II. The poposed lgohm compues Y 7] T whch s he WHT poecon veco n wndow fo K- usng Y whch s he compued poecon veco n wndow. TABLE II FAST ALGORITH WHE WIOW SIE IS Poposed lgohm GCK efne s: - fo -; K-5. 9 Fo we hve:

6 6. Fom 3 nd hen 7 we hve: 9. 9 d d As gven s he h ode- WHT poecon vlue of d d ] T. Accodng o Tle II nd WHT poecon veco n wndow cn e compued fom hose n wndow s well s s follows: In summ cn e epesened : 56 } ] { 37 } { v v v v whee v. TABLE III COPUTATIO OF ALL ORER- WHT PROJECTIO VALUES I WIOW J Sep d One ddon s equed. Sep ] d d ] T nd -] d d ] T. - Two ddons e equed. oe h d ws oned dung compuon of Y n Sep. Sep c 56 } ] { 37 } { v v v v whee v. - Egh ddons e equed n hs sep fo 7.

7 7 Tle III gves he hee seps fo compung Y fom Y s well s Y nd he coespondng nume of opeons equed. Theefoe he poposed lgohm eques ddons whees he lgohm n 9] eques ddons fo onng he poecon vlues n Y. The GCK lgohm eques 6 ddons. IV. FAST ALGORITH FOR WHT O SLIIG WIOWS FOR WIOW SIE A. The lgohm Le e he ode- evese-den m.e. elemens he evese-dgonl posons e nd ohes. Fo emple s:. The equon elow s poved n he ppend: fo -. 3 Hence ode- WHT s poned no goups of ode- WHT. Ulzng he mehod n fo ech of he ode- WHT n 3 WHT poecon vlues n wndow cn e compued usng poecon vlues n wndow s well s nd s follows: n whee s defned n 9. Equon whch s fo ode- WHT ecomes when. Le us fs consde he compuon of n. Ulzng he d n 7 we defne:

8 v d d d -] T X - X ]. 5 Fom 3 9 nd hen 5 we hve: - T X - T X T X - X ] T v. 6 Equon 6 shows h s he h poecon vlue of he ode- WHT of v. When 6 ecomes. The sgnl flow dgm n Fg. 3 depcs he compuon of ode- WHT usng -6. Tle IV desces he compuon of Y fom Y nd he coespondng nume of opeons s well s memo equed. Snce mos sze K memo s equed fo he poposed lgohm ech sep he memo equed fo he poposed lgohm s K whch s he sme s he GCK lgohm. Ths ppe focuses on - WHT. Howeve s es o eend he poposed - WHT lgohm o hghe dmensons. Fo emple when he - WHT of sze s compued ou lgohm n Tle IV cn use GCK fo compung he WHT of sze n Sep nd usng he Sep c o on he poecon vlues of sze. In hs w we eque ddon n Sep ddons n Sep nd ddons n Sep c.e..5 ddons n ll. In hs w he poposed lgohm eques.5 ddons pe wndow pe poecon vlue ndependen of dmenson. In Secon VI we wll show he epemenl esul h uses he fs lgohm fo - WHT. Poecon Wndow Poecon Wndow d d d d - d W H T W H T Fg. 3 Sgnl flow dgm of he oom up lgohm fo ode- sequenc WHT

9 9 TABLE IV COPUTATIO OF ORER- WHT Ovell pocedue: Fo ech {Sep }; Fo ech { Fo ech { Sep ; } Fo ech { Sep c; } } Sep Compue d -. Ths sep povdes he v n 6. Anlss: One ddon pe wndow s equed. Sze K memo s equed fo song d nd he npu d fo K-. Sep Compue T v. Ths sep povdes he n. We cn use he GCK lgohm n ] fo compuon. Anlss: ddons pe wndow e equed fo he vlues of fo gven. As sed n ] sze K memo s equed GCK. Sep c On usng. oe h he n s compued pevousl. Anlss: ddons pe wndow e equed fo he vlues of. Sze K memo s equed fo song he fo K- whch e compued n Sep ; sze O memo s equed fo song mos wo poecon vlues fo < equed n he gh hnd sde of fo he gven. Snce we hve <<K fo mos cses he memo equemen s less hn K n hs sep. V. COPUTATIO REQUIREET OF THE PROPOSE FAST ALGORITHS FOR WIOW SIE A. When ll poecon vlues e compued Le he ol nume of ddons fo onng Y e B. Accodng o he nlss n Tle IV we hve: B 3.

10 The nume of ddons equed fo he GCK lgohm nd he poposed lgohm e summzed n Tle V whch shows h he poposed lgohm eques ou 3 ddons whle he GCK lgohm eques ddons. The nume of ddons equed ou lgohm fo ode- nd ode- WHT e 5 nd especvel ecuse we cn use dec compuon nsed of he GCK fo clculng n he Sep of Tle IV. Fo emple f hen d nd no compuon s equed n he Sep of Tle IV fo onng. TABLE V UBERS OF AITIOS REQUIRE BY THE GCK ALGORITH A THE PROPOSE ALGORITH FOR ALL PROJECTIO VALUES OF ORER- WHT Sze 6 3 GCK Poposed B. When no ll poecon vlues e compued In mn pplcons no ll poecon vlues e equed. In hs p we nlze he compuonl equemen when onl he fs P poecon vlues e compued fo wndow sze. Specfcll we shll deve he nume of ddons fo he compuon of nd P- fo K-. Hee we shll no consde he cses when < ecuse he compuonl comple s neglgle s << K n mos cses. A zeo pddng ppoch delng wh he cses when < s noduced n ]. TABLE VI COPUTATIO OF ORER- WHT WHE OT ALL PROJECTIO VALUES ARE REQUIRE

11 The ovell pocedue nd he hee seps e he sme s h n Tle IV. The onl dffeence s h he ol nume of s P now. Anlss: Sep : ddon pe wndow s equed n hs sep. Sep : P ddons e equed n hs sep usng he GCK fo he P vlues of. Sep c: If P % fo emple P s 6 o fo he compuon n he poposed lgohm need o compue P fo P. So P ddons e equed f he P % ; Ohewse P ddons e equed. Le he nume of ddons pe wndow equed o on fo P-; K-5 e B P. Tle VI lss he seps nd he coespondng nume of ddons equed. As shown n Tle VI we eque ddon n Sep P ddons n Sep nd mos P ddons n Sep c. The nume of ddons equed fo onng P poecon vlues n ode- WHT usng he poposed lgohm s gven n Tle VI hs he followng nequl: B P P P 3P 3. The compuon equed s ou.5 ddonspelkenel usng he poposed lgohm. VI. EXPERIETAL RESULTS To nvesge he compuonl effcenc of he poposed lgohm fo pen mchng n pccl pplcons lock mchng n moon esmon s ulzed. Block mchng n moon esmon usng fs WHT ws ced ou on he fs fmes of vdeo sequence empee whch hs esoluon of 35. The epemen consdes he eecuon me equed fo onng dffeen numes of WHT poecon vlues whch nges fom o. The poposed lgohm s comped wh he lgohm n ] whch ulzed he GCK lgohm. In sml epemen epoed n ] wo poecon vlue compuon odes wee used. The e he snke ode nd ncesng fequenc ode. Fg. shows he odeng of he fs poecon vlues of hese wo odes. The pecenge of he me equed he poposed lgohm wh espec o he GCK lgohm s gven n Fg. 6. The poposed lgohm oupefoms he GCK lgohm when he nume of poecons s gee hn 6. As he poposed lgohm compues 3 o poecon vlues ogehe o sve compuon whees he GCK lgohm does no so he pecenge of

12 compuonl me sved he poposed lgohm n compson wh he GCK lgohm depends on he nume of poecons. Genell he poposed lgohm cheves hghe svng when mos poecon vlues o e compued cn ke dvnge of hs pope. Ths s wh when he nume of poecon vlues ppoches 3 nd 6 fo snke ode he poposed lgohm eques he les pecenge of me comped wh he GCK lgohm. When he nume of poecon vlues s less hn 5 he poposed lgohm eques moe compuonl me ecuse poecon vlues cnno e gouped ogehe fo compuon. Theefoe we would sugges he use of he GCK lgohm when he nume of poecon vlues s less hn Snke ode Incesng fequenc ode Fg. Two dffeen poecon odes 5 5 Snke IF Snke 66 Snke IF 66 IF % 5 % Snke ode Incesng fequenc ode Fg. 5 The pecenge of me equed ou lgohm wh espec o GCK lgohm whee Snke snds fo he snke ode nd IF snds fo he ncesng fequenc ode. The epemen s mplemened on.3ghz PC usng C on wndows XP ssem wh complng envonmen VC 6.. VII. COCLUSIOS Ths ppe poposes fs compuonl lgohm fo Wlsh Hdmd Tnsfom on sldng wndows whch eques ou.5 ddons pe poecon veco pe wndow. The compuonl me of he poposed lgohm s ou 75% h of he GCK lgohm whch s he fses lgohm epoed so f. In cses whee no ll poecon vlues e needed he poposed lgohm cn

13 3 oupefom he GCK lgohm when he nume of poecon vlues s fve o ove. The poposed lgohm cheves s hgh effcenc n he compuon of ode- WHT usng ode- nd ode- WHT. Ths ppe povdes fs lgohm fo - WHT. In he fuue we e gong o seek even fse lgohm. We wll lso o see f hee ess he supese of GCK h cn e compued consn nume of ddons pe wndow pe poecon vlue ndependen of he sze nd dmenson of he nsfom. ACKOWLEGEET The wok desced n hs ppe ws pll suppoed gn fom he Resech Gns Councl of he Hong Kong Specl Admnsve Regon Chn CUHK37. The uhos e lso hnkful o Pof. Hel-Os fo povdng he code mplemenng he GCK lgohm. APPEIX A Ths ppend povdes he poof fo 3. Ecep fo hs ppend sequenc ode s used fo epesenng WHT. In hs ppend ddc-odeed WHT wll e ulzed fo povng 3. ul ode- WHT cn e epesened : whee s he Konecke poduc A B s mp nq m composed of he m n locks B nd. Boh sequenc nd ddc odes 5] e he eodeng fom of he nul ode fo WHT. Hee we denoe s he ode- sequenc-odeed WHT m; denoe ddc-odeed WHT m nd: -] T whee s he ode- s he h WHT ss veco. The n veco epesenon of n g ] T whee k e o fo k g nd: g- g- g- g. Fo ddc-odeed WHT fo - we hve:. Le nd e nde of sequenc-odeed nd ddc-odeed WHT especvel. As poned ou n 5] he elonshp eween he n veco epesenon of nd s: W ] g 3

14 whee W ] g g g. Accodng o 3 f whee < k hen s onl decded. So we hve: >. enoe f s: nume odd n s nume even n s f 5 The n s decded oh nd : fw ] whee he sze of W ] s log log. I s ovous h ff] so we hve: W ] f fw ] ]f. 6 cn e epesened s follows usng nd 6: ] ] f W. 7 Accodng o 7 we hve: ] ] f. Theefoe cn e epesened s follows: ] ] ] ]] ] ] ] ]} ] { z z z z z z z f X f X I f I X I I f X f. 9

15 5 The followng equon s vld usng 9:. Equ. 3 s vld when n. REFERECES ]. S. Akso O. Tokul nd I. H. Cedmoglu An ndusl vsul nspecon ssem h uses nducve lenng Jounl of Inellgen nufcung vol. 5 pp Aug.. ] A.W. Fzgon Y. Wele nd A. ssemn Imge-Bsed endeng usng mge-sed pos Poc. In l Conf. Compue Vson vol. pp Oc. 3. 3] T. Luczk nd W. Szpnkowsk A suopml loss d compesson sed on ppome pen mchng IEEE Tns. Infomon Theo vol. 3 pp Sep ] R.. ufou E. L. lle nd. P. Glsnos Temple mchng sed oec ecognon wh unknown geomec pmees IEEE Tns. Imge Pocess. vol. no. pp ec.. 5] C.. k. C.K. Fong nd W.K. Chm Fs moon esmon fo H.6AVC n Wlsh Hdmd domn IEEE Tns. Ccus Ss. Vdeo Technol. cceped. 6] ITU-T Rec. H.6 nd ISOIEC 96- AVC JVT-G5 ch 3. 7] C.. k. C.K. Fong W.K. Chm Fs moon esmon fo H.6AVC n Wlsh Hdmd domn IEEE Tns. Ccus Ss. Vdeo Technol. 6: Jun.. ] Y. oshe nd H. Hel-O A Fs Block oon Esmon Algohm Usng G Code Kenels n Poc. IEEE Smp. Sgnl Pocessng nd Infomon Technolog pp.5-9 Vncouve Cnd Aug. 6. 9] Y. Hel-O nd H. Hel-O Rel me pen mchng usng poecon kenels IEEE Tns. Pen Anlss nd chne Inellgence vol. 7 no. 9 pp. 3-5 Sep. 5. ] G. Ben-Az H. Hel-O nd Y. Hel-O The g-code fle kenels IEEE Tns. Pen Anlss nd chne Inellgence vol. 9 no. 3 pp ] S. Omchnd nd. Omch Fs emple mchng wh polnomls IEEE Tns. Imge Pocess. vol. 6 no. pp.39 9 Aug. 7.

16 6 ] G. J. VndeBug nd A. Rosenfeld Two-sge emple mchng IEEE Tns. Compu. vol. C-6 no. pp Ap ]. Ben-Yehud L. Cdn H. Hel-O nd Y. Hel-O Iegul Pen chng usng Poecon n Poc. IEEE Inenonl Confeence on Imge Pocessng Geno Il Sep. 5. ] J.L. Shnks Compuon of he Fs Wlsh-Foue Tnsfom IEEE Tns. Compu Vol. C- o. 5 pp ] W.K. Chm nd R.J. Clke dc Smme nd Wlsh ces IEE Poceedngs P.F. Vol.3 o. pp.-5 Ap. 97. Wnl Oung eceved he B.S. degee n compue scence fom Xngn Unves Hunn Chn n 3. He eceved he.s. degee wh he College of Compue Scence nd Technolog Beng Unves of Technolog Beng Chn. He s now pusung Ph. n he epmen of Eleconc Engneeng The Chnese Unves of Hong Kong. W-Kuen Chm gdued fom The Chnese Unves of Hong Kong n 979 n Eleconcs. He eceved hs.sc. nd Ph.. degees fom Loughoough Unves of Technolog U.K. n 9 nd 93 especvel. Snce 95 he hs een wh he epmen of Eleconc Engneeng The Chnese Unves of Hong Kong. Pof. Chm s Cheed Engnee nd seno meme of IEEE.

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