Optimal Mapping of Deep Gray Scale Images to a Coarser Scale of Gray

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1 DIMACS Techcl Reot 4-53 Novembe 4 Otml M of Dee Gy Scle Imes to Cose Scle of Gy by Solomo Bood CACI Itetol, Ic. 483 Wlde Le, Lhm, MD 76 sbood@cc.com Alesey Bood BAE Systems, Msschusetts Ave, Wshto, DC Alesey.Bood@besystems.com Ily Much DIMACS Rutes Uvesty P.O. Box 88, Psctwy, NJ 8855 much@dmcs.utes.edu DIMACS s collbotve oject of Rutes Uvesty, Pceto Uvesty, AT&T Lbs-Resech, Bell Lbs, NEC Lbotoes Amec d Telcod Techoloes, s well s fflte membes Avy Lbs, HP Lbs, IBM Resech d Mcosoft Resech. DIMACS ws fouded s NSF Scece d Techoloy Cete.

2 Abstct The oblem of otml m of mult-level y scle mes to cose y scle s fomulted d exloed. A dstce betwee mes hv dffeet testy es s toduced. Mmzto of such dstce c be vewed s lest sques oxmto of souce hh e me by the best tet me wth ve umbe of levels of y. Follow S.Lloyd [], we oved tht the ltte oblem s equvlet to otml tto of the souce me testy e to ve umbe of tevls, ovded tht the sum of t-tevl vtos eches mmum. A effcet lothm fo otml tto bsed o dymc omm s used, whch s the sme comlexty s the oes ow fom ltetue, but bette tems of equed memoy. The oosed och s led to vsulzto of dee y scle medcl mes. Advtes of the method ove le m d hstom equlzto e demostted o the smle mes. The othe lcto felds my clude me otmzto fo t/fx/coy.

3 3 Itoducto Medcl m equmet (CT- d MRI-sces, X-y, etc. oduces mes of testy e 9-3 bts e xel, whle covetol me esetto devces d med ovde lowe e. Pesol comutes usully llow fo oly 8 bts, o 56 levels of y. I ths e, we toduce oblem sttemet fo otml m of dee y scle mes to cose y scle otml y scle equtzto (OGSR. It s bsed o the best oxmto of fe souce me by ts cose veso fom clss of mes wth the lmted umbe of shdes of y. We ove tht such oxmto s equvlet to otml tto of the souce testy e to ve umbe of tevls, ovded tht the vee t-tevl sctte eches mmum. Thus, fo ou secfc oblem, we e-ceted esult deved ele fo dffeet dt model the sl qutzto theoy []. Sub-otml heustc tto ocedues wee oosed umbe of wos (e.., []-[6]. Whle these ocedues e usully smle, fst, d ccetble most of cses, they do't ovde lobl otmum,.e. do t utee esoble outut fo y ut me (see [7]. O the othe hd, loblly otml d effcet lothm of scl semetto, bsed o dymc omm, ws oosed [8]-[]. I ou e, we eset lothm equvlet to [] tems of otetl comlexty d efomce, yet bette tems of equed memoy. Accod to ou method, testy of ech xel the tet me s equl to the odl umbe of the tevl cot testy vlue of the coesod xel the souce me. As esult, the vbly sced souce testy tevls e med to eqully sced destto testy vlues. It s ot cle o whethe such o-le tsfomto would yeld stll ecozble ctues; smle mes Secto V c be coet umet o. The otmlly med mes e sfctly moe elstc th those obted by the hstom equlzto (HE techque.

4 4 Atteto s lso dw to ctcl sects of medcl me esetto. I ode to te to ccout cotst dstotos due to o-lety of testy, we oose double-vew dsly: lely med (LM me toethe wth the oe obted by the OGSR method; tht mes t ossble to vew mxmum of me detls whle efe the tue shdes of y. The method lso c be used fo e-ocess mes med t hh qulty t o fx. Poblem sttemet Cosde souce me A:{,..., ˆ ˆ } s odeed set of M xels; ech xel ˆ :{ x, y ; } s defed by ts coodtes d testy vlue M. The ode of xels s ssumed fxed, so me A c be detfed by ts testy vecto {,...,M } the M-dmesol sce. Comoets e ssumed, fo smlcty, el o-etve umbes, whch defe souce testy scle. Let B:{ b,...,bm } deote othe me vecto wth comoets evluted the tet scle, whch s set of tees,. Gve souce me A, ode to fd ts best esetto the tet scle, we eed to defe dstce betwee the two mes wth the testy vlues belo to dffeet scles. Cosde set of mootoclly ces metes β - vlues the souce scle, such tht: β β... β mx ( { M } (mx ( Let D defe dstce betwee A d B s follows: D(A, B M b m ( β { β,..., β } ( The mjo oblem of ths wo s fd the best oxmto of A the tet scle: B m D( A, B A { B} (3

5 5 Let Ψ deote clss of ll mes tht hve equl o less th dstct levels of y. Obvously, A belos to cet N Ψ, whee N M, whle B Ψ ; t s ssumed < N. Let lso B :{ b,...,b } be me the souce scle, defed by me B d mete set { β }, M so tht b β b. Note tht lthouh the bsolute vlues of y e dffeet B d B, both belo to Ψ, d the equ-testy eos (EIRs e couet. Fom ( d (3, obt smle fom fo D(A, B A cteo vlue: m M { B Ψ } ( b (4 Thus, otml me m to lowe y scle c be slt to the two stes A B B : Fo ve souce me A fd the best (lest sques oxmto B Ψ ccod to (4; Sot the obted dstct testy vlues of B sced ode, stt fom β, d defe B A (otml B fo ve A by sett b, f b β. Itesty vlues ovde two-fold dffeetto of the ts of the whole me: fst, they defe EIRs s dsjot eos the ctue (stl detls; secod, they defe secfc scl ttbute to ech EIR bsolute vlue of y (cotst detls. Du A B ste, we eseve s much stl d cotst detls s ossble. Du B B ste, we eseve ll the stl detls but toduce ddtol cotst dstoto due to m the vbly sced vlues of B to eqully sced vlues of B (oly ode of y shdes s coseved. I the follow two sectos, we exloe the A B ste. 3 Ime oxmto d tto of testy e Exesso (4 defes lest sques me oxmto oblem, close to qutzto of dom sl. If e-fomulted tems of ou e, t ws show tht ude cet codtos, oblem (4 could be educed to the follow oe:

6 6 fd ( vlues {,..., } d vlues { β,..., β ovded: } - totlly ( vlues the souce scle, β < β <... < β < mx(,..., + (5 M m ( β +, β ;, } < { (6 I othe wods, slt the whole testy e [,, wth clusve, o-clusve, by (- ots { ;, } to sem-oe tevls, d fd cet vlue β wth ech tevl, - to ovde mmum of t-tevl testy vtos. I [], the (4 (6 educto ws obted ude the dffeet codtos: cotuous sce, obblstc model wth ddtve ose. Eve moe mott s tht heustc lothm used fo qutzto dd ot eque ccute tetmet of wht ws vewed s stutos of obblstc mesue zeo (whee xel testes fell to edots of the tevls. So some of the stutos, commo to ou oblem, wee oed []. I fct, educto (4 (6 tus out to be tue, s well, eve f the metoed stutos hve ozeo obblty, le t s the cse the me oxmto; - moeove, equvlece: (4 (6. Po to fomult the equvlece theoem, we toduce othe defto. Gve mes A d B, defe set of dsjuctve eos { A,..., A } of A, whee A s set of ll xels â such tht â A f d oly f b β : A :{ˆ b β ;, M}, < (7 The follow Theoem (oved Aedx c be vewed s moe ccute e-ceto of the S.Lloyd s esult [], f led to me oxmto. Theoem If me B Ψ s the best oxmto of souce me A Ψ N tems of lest sques (4, the: Reos A defed by (7 e stctly odeed by testes,.e.

7 7 < j fo y xels ˆ A, q ˆ j A ( <q <. Fo y ( <, vlue β s vee of testy vlues of xels eo A, d thee exsts exctly dffeet testy vlues set { β ;, }: β < β <... < β mx ( (8 { M } Theoem sttes tht thee s oe-to-oe coesodece betwee eos A + d tevls [, of the souce scle - ths s why exessos (4 d (6 e equl (whch oves (4 (6 : m M { B Ψ } ( b m ( β { A, β ;, } ˆ A ( β + m {, β ;, } < (9 Bsed o the Theoem d (9, t s esy to ove the covese sseto, (6 (4 : f tevls [, + d vlues β (, ovde mmum (6, the the coesod me B wth comoets b β whee s defed fom + [,, - ovdes mmum (4. Swtch fom (4 to (6 s sfct smlfcto, becuse the umbe of uow vbles s educed fom M (4 to - (6, whee usully <<M. Note tht oblem (6 my be teeted s othe lest sques oxmto oblem: fd the best oxmto of the souce me testy hstom by the -ste-wse fucto []. It s dffeet fom the ol oxmto oblem (4; howeve, ccod to the Theoem, both e equvlet the follow sese: ve soluto fo oe of them, we c esly et soluto fo the othe oe. 4 Otml tto by dymc omm Let m be multlcty of testy the souce me,.e. umbe of xels wth testy. I ode to smlfy otto, fom ow, we wll ssume tht souce vlues of y e tees, N.

8 8 We lso ssume tht lthouh some of souce testy slots my be emty ( m, the umbe of dstct levels of y (wth m > s ete th. A sequece {,...,, m N m m } defes the souce me hstom; so the sum to be mmzed (6 c be educed to: m m m m m m +, ( f, ( whee, ( f m m m, ( N < ( By us thee oe-dmesol ys Q, S, H ( N, whch c be clculted dvce: Q m, S m, H m, ( N < ; Q S H, ( we c vod eettve summto ( whe clcult vlues of, ( f, o vod us le two -dmesol y of (N*(N-/ double-ecso umbes clculted dvce (le, e.., []:, ( f (Q - Q - (S - S / (H -H, ( N <, (3 As esult, cteo (6 ets the follow cocl fom: D + },...,, {, ( m f, ( < < < < <... N (4 Vecto mmzto (4 c be slt to sequece of smle scl otmzto ocedues s show below. Isted of desto D fo the cteo vlue, we wll use moe eel ( ~ D, - to

9 9 ~ dcte tht the fl cteo vlue s just secl cse of whole fmly of sml cte D (, ~ whch should be clculted ode to obt the fl vlue of D D ( : ~ D D ( { m,,..., } f (, + + m + f (, m f (, 3 { } {,,..., } ~ m [ f (, + D ( ] ~ m f (, + m [ f (, + D ( ] { } { } { }... (5 To esolve (5, we c use the follow system of ecuet fuctol equtos of dymc omm (stt fom the lst oe: ~ D ( f (, N, ( < N ; (6 ~ + ~ + + ( (, + ( + ~ + D f D, whee m [ f (, + D ( ] (7 { } By tbult (7 fo -,,;,,N--, d oult y, we et D D (. The, ~ stt fom,, d us y, - ll the ed ots of otml tevls c be estoed. The bsc tme-cosum t of ths lothm s tbulto (7, equ CN oetos, - the sme s [8]-[], whee C s costt, whch deeds o mlemetto. M mes fom N 48 to 56 levels of y too sevel secods o the GHz PC. 5 Exmles The OGSR method eseves mxmum of stl me detls t the exese of cotst dstoto. I medcl lctos, the bsolute levels (o, t lest, the ootos e usully mott d cot be ched o oed. To ecocle both equemets (tue cotst d hh me detl esoluto, we oose the follow double-vew esetto (.e. show s of ctues: ( the vew obted by smle le m (LM, (b the vew obted by OGSR s descbed ths e.

10 The fst oe s efeetl vew: t ovdes tue shdes of y but lcs stl detls; the secod oe mes t vsble stl detls by us tfcl shd. Below, we eset two dee y scle medcl mes dowloded fom the Web ste of Mllcodt Isttute of Rdoloy of Wshto Uvesty, St. Lous, Mssou. We do ot dscuss medcl teetto of mes, tht mht be toc of the secl esech. Isted, we demostte eel cbltes of the method, dw tteto to ddtol me detls tht become vsble. The esults e eseted the tet scle..55. I ddto to the ecommeded "Double-vew" (- (b, we lso fo comso, demostte oe moe vew (c, obted by otml HE []. F. shows smle medcl me MR-ANGIO. Vew (b ovdes sfctly hhe qulty of me detls, th vew (. Not oly s the vscul et bette vsble (b, but lso moe detls the mjo vessels wlls e esolved. Vew (c hdes me detls the bhtest (whte e, whch e clely vsble vew (b, d eve vew (. Tht s becuse HE ms the sse bhtest testy vlues to to sle tet testy vlue to ovde eqully oulted testy tevls the equlzed me. F. ovdes moe sht by show hstoms fo ths me. I F., the ol testy e of me MR-ANGIO s slt to 56 otml tevls ( tems of (6. The tevls e show s ltet whte d lht y vetcl stes; ech tevl stts fom the fst y vlue fll the tevl, d ss utl the ext tevl stts. The to hlf of the dm esets the ol me hstom; the bottom hlf (wth the vetcl xs dected dowwds esets the 56 es-wse hstom of the coesod best oxmto me B. The e-wse tue of the bottom hstom s clely vsble the ht t, whee the esoluto s suffcet to detfy sle e ech tevl. I F. b, the to t s the sme souce me hstom, whle the bottom t s e-wse hstom coesod to the HE vew (ech e s ostoed t the tevl s vee testy. The ltet vetcl whte d lht y stes show the coesod testy tevls bult by HE.

11 Com the bottom hstoms F. d F. b oe c see tht the HE es e vtully equl sze, whle the OGSR es closely follow the she of the souce hstom. Ths s why the OGSR me F. b loos moe elstc th the HE me F. c. Also ote tht the bhtest, ooly oulted testy levels e esolved F by bout tevls, whle F. b, ll the vlues of the ht hlf of the testy e fll to oly 3 tevls. Ths s why the bhtest me detls the equlzed me F. c e lost. These obsevtos e tycl. F. 3 esets othe exmle, MR-SHOULDER. The souce me testy vlues fll to e (Sce me sze s 4 4, oly t of t s show the fme wdow 56 56; howeve, the whole e of y s eseted ths vsble t. The souce testy e s oly twce s wde s the tet e; tht s why smle le m vew ( ovdes suffcet dsly qulty. Vew (b ovdes futhe vsulzto movemet, e.., some detls suoud the boe hed e bette vsble th vew (. It s moe elstc comed to vew (c whee detls the bhtest e e lost. 6 Cocluso A method of m dee y scle me to lowe esoluto scle s oosed. The method mes t ossble to dsly mes wth thousds levels of y o the eul motos. It s fully utomtc d me cotet deedet techque, fee of heustc elemets, ovd the best me esetto tems of the exlctly fomulted d tutvely justfed cteo. Comuttol comlexty of the m lothm s O(N, whee N s the souce me testy e, d s the tet testy e. The eseted me exmles demostte hh qulty of the obted vews. Deste the essetl o-lety of the method, t cuses less o-le dstotos th, e.., HE. It ovdes obust d vtully elstc vew, whch es the close to the le m whle sfctly che tems of eel ece d detl esoluto.

12 Aedx Poof of Theoem. Let A std fo ou of xels (7, coesod to testy vlue β B.. Fst, show tht otml me B hs exctly dffeet levels of y. If mmum (4 s eched wth oly m levels (m<, the t lest oe of A cots xels wth dffeet testes. By sltt ths ou, we c futhe lesse the cteo- cotdcto wth (4. Secod, β must be vee testy A. Let D A B ( β... + ( β + R, whee R s deedet of + β, hece fom D / β, obt β ( /.. Now, ove tht bsed o (4, xel ous A defe dsjot sem-oe tevls of the souce testy e, whch hve emty tesectos wth ech othe. Cosde y two testy levels of me B, - fo cetty, β d β, ( β < β. Wthout lmtto of eelty, we c ssume tht xels ech of ous A, A e odeed by testy vlues:... ;... q, (to smlfy otto, we use sted of, d q sted of. We eed to ove tht <. We wll show tht y ltetve would esult cotdctos.. The fst ooste cse > c be excluded sml wy s [], s follows. Let D A B ( β + + ( β + ( β + + ( β + S, q whee S stds fo ll the tems the sum tht e deedet of both β d β. If we just teche xels d, whch mes elc B by B so tht b, β b β, - the ew cteo vlue s: D A B ( β + + ( β + ( β +

13 3 ( β + ( β + + ( q β + S, d D -D ( β β (. The lst exesso s etve, whch mes tht B s the bette oxmto th B, cotdcto wth ou bsc ssumto. b. Cosde the secod ltetve:. Ths s the stuto, whch ws ssumed [] to be of obblstc mesue, hece t ws oed. I ou oblem - obblstc tems - t s eul cse, whch c t be oed. The follow Lemm s used to ove tht stll Lemm. s ot ossble, - howeve, fo qute dffeet eso. Add oe xel of testy x to ou of xels A (fo cetty, A cuses oetve che of ths ou s cotbuto D to the dstce A B by vlue : (S - x / [(+], whee S Ideed, let Q (... + ( +, the D ( β + + Q ( S / ( β. Afte dd xel wth testy x, obt D Q + x - ( S + x / ( + ; by subtct D fom D et. The follow two coolles esult fom the Lemm: C. Add xel to ou eve educes the ou s cotbuto to the dstce; the cese s oly f the dded testy s equl to the me: x S. C. Remov xel xˆ A fom ou A (f t s ot sle membe A eve ceses the ou s cotbuto; the decese s ( S - x / [(-], d t s oly f x S. Us these coolles, we c me sue tht by mov oe xel fom A to A (o fom A to A, the esumbly mml dstce c be futhe decesed, whch s bsud. Sevel cses should be

14 4 cosdeed; fo the se of bevty, we omt tvl cses, e.. whe oe of the ous A, A does ot cot xels wth testy dffeet fom, - these cses c be excluded esly. Cosde moe tycl stuto < q < (both ous hve membes wth testy vlues dffeet fom, d ty to move â fom A to A. Icese of dstce due to dd â to A s D ( S - / [(+] ; decese of dstce due to emov â fom A s D ( S - q / [q(q-]. If D < D the we obted cotdcto; so suose D D. We wll show tht ths cse, mov ˆ fom A to A would educe the dstce, whch s bsud. Cosde the follow ch of estmtos: ( S - / [(-] ( S - / [(-] > ( S - / [(+] ( S - q / [q(q-] ( S - q / [q(q-] > ( S - q / [q(q+], - t esults ( S - /[(-] > ( S -q /[q(q+], whch mes tht mov ˆ fom A to A would decese the vlue of cteo, - cotdcto wth the tl ssumto, Q.E.D. LITERATURE []. Lloyd, S.P. Lest Sques Qutzto PCM, IEEE Ts. O Ifomto Theoy, IT- 8(,. 9-37, Mch 98. []. Wesz, J. d Rosefeld A. Hstom modfcto fo theshold selecto, IEEE Ts. Syst. M Cybeet. 9, 38-5, 979. [3]. Redd, S.S., Rud, S.F., d Keshv, H.R. A Otml Multle Theshold Scheme fo Ime Semetto, IEEE Ts. Systems, M d Cybeetcs, 4(4, , July 984. [4]. Bhttchy P., Y Y.-K. Itetve hstom modfcto of y mes, IEEE Ts. Systems, M d Cybeetcs, 5(3,. 5-53, Mch 995.

15 5 [5]. Khotzd A., Bouf A. Ime semetto by llel, o-metc hstom bsed cluste lothm. Ptte Recoto, 3(9, , 99. [6]. Ts D.-M. d Che Y.-H. A fst hstom-cluste och fo mult-level theshold, Ptte Recoto Lettes, 3,. 45-5, Al 99. [7]. Lee, H., P R.-H., Commets o A Otml Multle Theshold Scheme fo Ime Semetto, IEEE Ts. Systems, M d Cybeetcs, (3,. 74, My/Jue 99. [8]. Fshe, W.D. O ou fo mxmum homoeety, J.Am.Stt.Assoc. 53, (958. [9]. Bood, S.M. Otml Gou of Iteelted Odeed Objects, Automto d Remote Cotol, Pleum Publsh Cooto, 98, []. Kudu, S. A soluto to hstom-equlzto d othe elted oblems by shotest th methods, Ptte Recoto, 3(3,. 3-34, Mch 998. []. Bood, S., Bood, A. Otml Hstom Equlzto Bsed o Dymc Pomm, Poceeds of the Itetol Cofeece o Im Scece, Systems, d Techoloy (CISST, vol.,.46-49, CSREA Pess,.

16 6 F.. Ime MR-ANGIO (Souce: ft://ft.el.wustl.edu/ub/dcom/mes/veso3/othe/hls/m-o.dcm.z. Le m b. Otml y scle educto c. Hstom equlzto

17 7 F.. Itesty hstoms fo me MR-ANGIO fom F.. OGSR me vesus souce me; b. HE me vesus souce me

18 8 F. 3. Ime MR-SHOULDER (Souce: ft://ft.el.wustl.edu/ub/dcom/mes/veso3/othe/hls/m-shoulde.dcm.z. Le m b. Otml y scle educto c. Hstom equlzto