Invertible Linear Transforms Implemented by Integer Mapping. (Published in Chinese, Science in China, Series E, April 2000, V30, N2, pp )

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1 Invertble Lnear Transfors Ipleented by Integer Mappng Pblshed n Chnese, Scence n Chna, Seres E, prl, V,, pp- HO Pengwe SHI Qngyn Center for Inforaton Scence, atonal Lab of Machne Percepton Pekng Unversty, Bejng, 87 bstract The general nteger appng theory of nvertble lnear transfors s consdered n ths paper Two ters of nteger factor and eleentary tranglar atr are ntrodced We prove that there ests an nteger pleentaton f a lnear transfor s nvertble and n fnte densonal space The nteger pleentaton of an nvertble lnear transfor has several advantages sch as appng ntegers to ntegers, perfectly nvertble, and n-place calclaton constrctve approach of nteger pleentaton and a splfed approach are also presented, whch can be converted to soe atoatc steps The error of nteger pleentaton s estated, and an eaple s gven Keywords: Lnear transfors, appng ntegers to ntegers, tranglar factorzaton, perfect nverson Introdcton It s necessary for a fnte word-length copter that a transfor aps ntegers to ntegers and s perfectly nvertble for hardware pleentaton and lossless copresson of dgtal sgnals by eans of transforaton nfed lossy/lossless copresson syste also enables regon-of-nterest ROI based age codng and progressve qalty fro the coarsest levels to the fner levels or even the lossless level So nteger transfors are sgnfcant for sorce codng Orgnally, people devsed soe nteger transfors or soe nteger approaches of one-to-one appng for soe sple transfors, sch as TS transfor by Zand et al 995 [] and ther referred S transfor by Ble & Fand989 [], color space transfor by Gorsh et al 997 [] In 99, Brekers & van den Enden [] proposed a ladder strctre of flterng networks n flter bank desgn for perfect nverson and perfect reconstrcton, whch broke a new path for nteger transfor research, bt t was not pad enogh attenton to To stdy nteger transfors systeatcally wth slar networks s for dscrete wavelet transfors, n whch the networks are tlzed to constrct wavelet and to lft the vanshng oent orgnally Sweldens, 996 [5], and the ethod s called the lftng schee paper by Dewtte & Cornels997 [6] sees to apply the lftng schee to pleent wavelet transfors by nteger appng frstly Later, Dabeches & Sweldens et al 998 [7,8] and Sh 998 [9] etended and appled the lftng schee to general wavelet transfors, and presented the schees to convert dscrete wavelet transfor nto nteger transfor steps, whch were wdely accepted and adopted The nteger pleentaton of general lnear transfors s addressed n ths paper The eleentary strctres of nteger lnear transfors are stded, and t s ponted ot that ladder strctres are the proper strctres for nteger pleentaton Then, an approach to factorze an nvertble lnear transfor nto a seres of eleentary strctres s presented, and an optzed factorzaton s proposed However, to copare wth the nfnte precson sybolc calclaton, an nteger transfor has errors reslted fro arthetc rondng operatons, of whch the pper bonds s also dscssed To lessen the errors, soe prncples are sggested t the end of ths paper, a typcal eaple of -by- dscrete Forer transfor s gven to show the avalablty and practcablty of the pleentaton

2 Eleentary strctres of nteger lnear transfors In followng dscsson, a lnear transfor s of cople nber doan n nteger referred below n cople nber feld s consdered as a cople nber that both the real and agnary coponents are ntegers n arthetc rondng operaton for a cople nber ples an nteger converson for both of ts real and agnary parts Perfect nverson of a transfor s sggested that both of the forward and the reverse transfor are nvertble lnear transfor that aps nber to nber y can be yab If varable can be assgned to an arbtrary nteger, so the condton of nteger otpt, y, s that both nber a and nber b are ntegers There ght be soe processng procedres n the transfor doan, so the range of y s consdered as the set of ntegers wthot restrctons, and a condton for the nverse lnear transfor that aps ntegers to ntegers s that both /a and /b are ntegers Therefore, the nteger perfectly nvertble condton of the lnear transfor s that a ± or ± and b s an nteger, or the defnton range of a or /a s {, -,, -}, where s the agnary nt For rectatve convenence, any eleent of the defnton set s called an nteger factor, and denoted as j and j /j Obvosly, j s also an nteger factor If the varables are defned n Z transfor doan, an nteger factor ay also be jz k, where k s an nteger If b s not an nteger, the coptatons ay be stll done wth floatng pont nbers, bt an nteger representaton, [b], can be a sbsttton, and the reslt s garanteed to be nteger and nvertblty s preserved The nteger converson ethods can be rondng, choppng, carry-n, or soe other styles It follows that a generc for of an nvertble lnear transfor of a nber pleented by nteger appng s: yj[b], and ts nverson: j y-[b] The flow chart s as follows: b b b [ ] [ ] j y - j' Fg Forward and reverse lnear transfor of a nber pleented by nteger appng s a atter of fact, the chart s also an eleentary strctre for the nteger pleentaton For nvertble transfors, nber b s necessary to satsfy two followng condtons: For forward transfor, b does OT depend pon drectly or ndrectly; For reverse transfor, b does OT depend pon y drectly or ndrectly Whether n real or cople nber feld, the above strctre s a knd of ladder strctre In the case of a, the strctre s the sae as the ladder strctre proposed by Brekers & van den Enden 99 [] and the lftng schee strctre by Sweldens 996 [5] So t can be sarzed n bref that ladder strctres are approprate eleentary strctres for nteger pleentaton of nvertble lnear transfors For a general densonal nvertble lnear transfor y, f t allows perfectly nvertble nteger pleentaton, t st eet two necessary condtons of nber b It shows that f the calclatonal orderng s properly arranged there wll be an eleentary strctre for perfectly nvertble nteger pleentaton, e the transfor atr s an pper or lower tranglar atr whose dagonal eleents are nteger factors, whch s called an eleentary tranglar atr If all the dagonal eleents of an

3 eleentary atr are s, the atr wll be a nt tranglar atr n eleentary tranglar atr has two portant propertes as follows: Prodct of two eleentary pper tranglar atrces s stll an eleentary pper tranglar atr, and prodct of two eleentary lower tranglar atrces s stll an eleentary lower tranglar atr Deternant of an eleentary tranglar atr s an nteger factor If s an eleentary pper tranglar atr, the calclatonal orderng of lnear transfor y can be arranged as top-down: y j ann j n Its nverse orderng s also nverted: [ b],,,, j' y a n n n j' y [ b],,,, If s an eleentary lower tranglar atr, the calclatonal orderng of lnear transfor y can be arranged as botto-p, and ts nverson top-down: y j an n j n [ b],,,, j' y n a n n j' y [ b],,,, Factorzaton of nvertble lnear transfors For arbtrary nvertble lnear transfors n fnte denson, whether there ests soe nteger pleentaton depends pon whether there ests a factorzaton consstng of a seres eleentary strctres of nteger pleentaton, or a seres of eleentary tranglar atrces For above lnear transfor, y, the proble s whether there ests a factorzaton: V V V PV V V where P s a pertaton atr, V are eleentary tranglar atrces M or M, For arbtrary -by- nvertble atr, f det α, the atr can be odfed to satsfy det Wthot regard to the physcal sense, the odfcaton ethod can be a dvson by constants, α, α,, α, fro rows of The constants satsfy: k α det α k n advantage of above odfcaton s that the dynac ranges of the transfor coponents can be controlled, whch s very helpfl for the applcatons gnorng the physcal eanng of a transfor, sch as lossless copresson Take the physcal plcaton nto accont, the relatve agntde of the transfor egenvales cannot 5 be odfed, so a proportonal odfcaton s the alternatve, or let α α α α Therefore, the case of det s consdered herenafter For det, followng theore can be

4 derved: Theore: Matr has an eleentary tranglar factorzaton of PV V V M DR, f and only f det, where M s fnte, V k,,, M are eleentary tranglar atrces, P s a pertaton k atr, and D R s a rotaton atr for one cople nber Proof: Its necessty s obvos, and a constrctve proof for ts sffcency s gven as follows s dedced n atr theory, for a nonsnglar atr, there ests a tranglar factorzaton of PLDU, where P, L, D, U are pertaton atr, nt lower tranglar atr, dagonal atr and nt pper tranglar atr, respectvely det D det PD det PLDU det Sppose D dag d, d,, d, λ n dd dn n,, L,, D R dag, L,, λ, and denote D O dag λ, λ dag λ, λ, λ, λ, λ, λ,, λ,, λ, λ, λ, f seven f s odd D E dag, λ, λ, λ, λ,, λ dag, λ, λ, λ, λ,, λ, λ, λ, f seven f sodd then we obtan D DO DE D or R D DE DO DR θ and det D λ det D e, where D R s a R dagonal atr of rotaton transfor for the last cople coponent For a second-order atr wth two recprocal dagonal eleents, there est any fors of nt tranglar factorzaton besdes those gven n the lteratre [8], sch as α α α α α α α α VV V V Therefore, wth the -by- parttonng sklls, D and O D E can be decoposed as a prodct of nt tranglar atrces: D O VO VO VOV, O D E VEV EVEV Sch that a coplete factorzaton s: E PLDU PLV V V V V V V V D U 7 O O O O E E E E R 6 or PLV V V V V V V V D U E E E E O O O O R 8 The rotaton atr D R can be relocated as the last ter, D RU U RDR, then two farthest left nt lower tranglar atrces can be nted as one nt lower tranglar atr, V, and two farthest rght nt pper tranglar atrces can be nted as one nt pper tranglar atr V S ddle atrces can be 8 renaed as V, V, V, V5, V6, V So, the coplete factorzaton of the transfor has a for of 7 8 PVV VVV5V6V7V D R 9

5 whch has the sae strctre as epected to prove The proof s copleted The proven factorzaton ehbts at ost 8 eleentary nteger transfors ecept a rotaton and a θ pertaton transfor for an nvertble lnear transfor If e s an nteger factor, the rotaton atr can be ltpled by the last nt pper tranglar atr to ake an eleentary pper tranglar atr n order to left ot the rotaton transfor Wthot the rotaton atr, two corollares can be dedced fro the theore: Corollary : Matr has an eleentary tranglar factorzaton of PV V VM f and only f det s an nteger factor Corollary : Matr has a nt tranglar factorzaton of PV V VM f and only f det θ If e s not an nteger factor, a cople rotaton transfor can be pleented wth the real and the agnary coponents rotaton transfor can be decoposed nto steps of eleentary nteger transfors, and a rotaton atr also has any nt tranglar factorzatons [8] : cosθ snθ snθ snθ cosθ cosθ snθ cosθ snθ cosθ snθ cosθ snθ snθ V V V R R R Ths we proved that arbtrary fnte densonal nvertble lnear transfor can be pleented by nteger appng or ts atr can be decoposed nto a seres of nteger-nvertble eleentary tranglar atrces f and only f the absolte vale of the atr deternant s Ths, an arbtrary fnte densonal nvertble lnear transfor has nteger pleentatons, and every transfor blt wth eleentary tranglar atrces s edately nvertble where the nverse transfor has eactly the sae coptatonal coplety as the forward transfor Ipleentaton strctres and optzaton In forer secton, we proved the theore and presented an approach of nteger pleentaton of an nvertble lnear transfor The nber of eleentary atrces s p to 8, the frst and the last eleentary tranglar atrces are pleented n-place seqentally, and the ddle 6 can be pleented n-place and parallel Frtherore, t s easy to fnd that the poston of D s fleble It can be oved forward to any poston n front of a nt tranglar atr, bt then all the passed-by tranglar atrces are changed whle the nt tranglarty s preserved In order to speed p the pleentaton, t s necessary to optze the eleentary tranglar factorzaton Optzaton schee can be to nze the nber of factorzed atrces, to nze the coptatonal coplety of each step If the physcal sense of a lnear transfor s abandoned, the transfor can be odfed to be a prodct of two eleentary nt tranglar atrces If PLDU, factorzaton of the odfed transfor atr wll T be PD P PD LDU PLDU, whch can be pleented by nteger appng wth two eleentary nteger transfors, where LD D LD s a nt lower tranglar atr If the physcal sgnfcance of a lnear transfor s consdered, the relatve agntde of the transfor egenvales can not be odfed, so the odfcaton can only be wth a scalng ethod and then to optze the factorzaton In fact, the factorzaton stated above can be cobned The prodct of V V5 R 5

6 wth followng eleents can be: / λ V V5 / λ / λ / λ / λ O / λ O O / λ / λ O O O The cobned atr can be calclated n-place and parallel So, the whole transfor can be pleented by nteger appng n 7 steps wth a strctre lke: V8 V7 V6 VV5 V V V O y P DR Fg flowchart strctre of lnear transfor pleented by nteger appng The seqental calclatng order of the frst eleentary tranglar atr s fro top to botto, and that of the last eleentary tranglar atr s fro botto to top, whle the ddle 5 steps can be pleented wth the nherent parallels The nverson can be pleented wth slar coptng network bt transferrng data backwards, sbtractng nstead of addng at the coptng nodes and reversng the seqental calclatng order The strctre allows the transfor to be calclated n-place and wthot allocatng alary eory Eleentary tranglar factorzaton of a atr can be obtaned atoatcally by Gassan elnaton ethod If a Gassan elnaton ethod s sed wth partal pvotng or coplete pvotng, all the absolte vales of the off-dagonal eleents are not greater than n the lower tranglar atr of LDU factorzaton, whch s very helpfl for the error control Bt ndobtedly, the fors of optzed factorzaton are not nqe, and the optal factorzaton stll erts frther nvestgaton 5 Error Estaton De to arthetc rondng operatons are appled n each eleentary transfor to ake whole lnear transfor nvertble by nteger appng, coparng to the orgnal lnear transfor of nfnte precson 6

7 sybolc calclaton, there st be errors If a rondng-off arthetc s adopted for a cople nber, the rondng errors of the real and the agnary part are n the nterval of [-5,5, and they wll be n that of [, f a choppng ethod s sed The ntervals are half-open rectanglar regon n cople plane For a fnte eleentary factorzaton of PV V VM, wth a denotaton of for the rondng error vector reslted fro the transfor of the -th eleentary atr V, and V P, then the total error vector of the whole transfor s: We se M V V V L M VM M L Vk k for nfnty nor of vectors or atrces and U for the pper bond of a nt rondng error Snce Re U, U I and U, an error bond can be estated as: M k Vk M k V k M U V k k Splttng the cople atr nto an addton of a real and an agnary atr, the error bonds of the real and the agnary part of whole transfor can also be gven analogosly pparently, f all the eleents n a row of V are ntegers, the correspondng eleent of wll be If all the eleents of V are ntegers,, or the atr does not brng n fresh errors bt t stll transfers and acclates estent errors Referrng to the forlae 7,8,9 for eleentary tranglar factorzaton, there are at least tranglar atr aong V, L, V n whch all the eleents are ntegers, 8 and they brng n no prtve rondng errors Ths, the acclatve error wll not reach the pper bond of forla anyway The pper bond estaton shows: The frther left a factor atr s, the ore ts nflence s on the fnal error Therefore, optzed factorzaton shold confor to followng prncples: Close to large nbers: In order not to aplfy the progressve error, f there s an eleent absoltely larger than n a factor atr, the larger the nber s, the nearer the atr shold be The absoltely largest nber had better appear n the last tranglar atr way fro large nbers: If there s an absoltely large nber n a row of a atr, all of the atrces on ts left shold avod pttng absoltely large nbers n the correspondng coln better choce s the absolte vale of all off-dagonal nbers n the correspondng coln to be saller than Pror for ntegers: n nteger shold be chosen f possble, so as not to rond off error v Factorzng wth pvotng: transfor atr shold be factorzed nto eleentary tranglar atrces wth partal or coplete pvotng ethods n order that all the off-dagonal eleents of frthest left atr are not larger than 6 Eaples In order to show the avalablty and the practcablty of the ethod presented n ths paper, the -by- 7

8 8 dscrete Forer transfor n cople nber doan s consdered to constrct and to analyze the nteger pleentatons The -by- dscrete Forer transfor atr and ts nverse are: The atr can ap ntegers to ntegers wthot dobt, bt can not ap ntegers to ntegers edately and perfectly For det6, the nteger pleentaton needs a odfcaton for the transfor atr If the physcal sense s gnored, the transfor can be odfed to be pleented wth eleentary steps De to PLDU, Ddag, --, -, -, PI, the odfcaton and the decoposton can be: / / / / / / P T PD For the factorzaton,, and sch odfcaton akes, n whch the frst coponent s If the dynac ranges of all the eleents of the orgnal sgnal s [, L], then the dynac ranges of the transfored data wll be [, L], [-L, L] a rectanglar regon n cople plane, [-L, L], and [-L, L] If we epect the dynac ranges of the frst and the thrd transfored coponents to be [, L] and [-L/, L/] respectvely, then the odfcaton can be a ltplcaton wth a dagonal atr Ddag/,, /,, whch akes detddetd det beng an nteger factor The factorzaton s a lttle bt ore coplcated: / /8 / /8 / / / /8 / /8 / D The n-between dagonal atr can be factorzed nto nt tranglar atrces sng the standard factorzaton forla for order dagonal atr wth two recprocal eleents: / / / Then cobne the frthest left lower atrces and the frthest rght pper atrces, we get: / / /8 / /8 / D Here, and they contrbte to the fnal error Ecept that the second coponent of

9 9 ght be 5 ±, all the others are, whch are added p to as or [ ] T 8, / /8,, ± ± ± Two forer eleents of are, and two latter possbly non-zero eleents are perted to sbscrbe for the frst and the thrd eleent of the fnal error s a reslt, the pper bonds of the total error vector can be clated accordng to forla, as [ ] T U U,,,/8 /8 for absolte real part and [ ] T U U, /8,, /8 for absolte agnary part Consderng the physcal sgnfcance, we have to odfy the transfor by scalng: / / / / / / 6 / / The n-between dagonal atr can also be factorzed nto nt tranglar atrces sng parttonng sklls: / / / / The frst lower tranglar atr can be cobned wth the lower atr n forer forla, and the last pper tranglar atr can be cobned wth the pper atr n forer forla, so the entre factorzaton can be of eleentary tranglar atrces: / / The factorzaton akes, whch contrbte nothng to Two forer eleents of are, and the real and the agnary part of the thrd possbly non-zero eleent and the real part of the forth eleent ght be 5 ± ccordngly, the spre of the absolte vale of entre error vector s [ ] T 5,,5, for real part and [ ] T,5,,5 for agnary part respectvely 7 Conclsons lnear transfor has nteger pleentatons as long as t s nvertble and fnte densonal The nteger pleentaton of a lnear transfor s perfectly nvertble and n-place coptable The steps of eleentary nteger transfors ecept a possble pertaton operaton s not ore than 7 f the atr deternant of the lnear transfor s an nteger factor v The optzaton of the lnear transfor factorzaton erts frther nvestgaton

10 References [] Zand, J llen, E Schwartz, and M Bolek, CREW: Copresson wth reversble ebedded wavelets, IEEE Data Copresson Conference, Snowbrd, UT, March 995, pp - [] H Ble and Fand, Reversble and rreversble age data copresson sng the S-transfor and Lepel-Zv codng, Proceedngs of SPIE, 989, Vol 9Medcal agng III: Iage Captre and Dsplay: -8 [] M J Gorsh, E L Schwartz, F Keth, M P Bolek, and Zand, Lossless and nearly lossless copresson for hgh qalty ages, Proceedngs of SPIE, 997, Vol 5Very Hgh Resolton and Qalty Iagng II, March, 997: 6-7 [] F M L Brekers, W M van den Enden, ew etworks for Perfect Inverson and Perfect Reconstrcton, IEEE J on Selected reas n Concatons, 99, : -7 [5] W Sweldens, The Lftng Schee: Csto-Desgn Constrcton of Borthogonal Wavelets, J of ppled and Coptatonal Haronc nalyss, 996, : 86- [6] S Dewtte, J Cornels, Lossless Integer Wavelet Transfor, IEEE Sgnal Processng Letters, 997, 6: 58-6 [7] R Calderbank, I Dabeches, W Sweldens, B-L Yeo, Wavelet Transfors That Map Integers to Integers, J of ppled and Coptatonal Haronc nalyss, 5: -69 [8] I Dabeches, W Sweldens, Factorng Wavelet Transfors nto Lftng Steps, J of Forer nalyss and pplcaton, 998, : 7-69 [9] Sh Qngyn, Borthogonal Wavelet Theory and Technqes for Iage Codng, Proceedngs of SPIE, 998, Vol 55ISMIP'98, Oct 998: -