4.1 Proteinìmenoc Algìrijmoc
Sth sunèqeia ja perigrafeÐ h aploÔsterh èkdosh tou algorÐjmou kajìdou kat thn klÐsh pou apaiteÐ h arqikopoÐhsh na brÐsketai sqetik kont sthn telik lÔsh. H arqikopoÐhsh mporeÐ na eÐnai mÐa lÔsh tou proseggistikoÔ algìrijmou pou perigrfthke parapnw. Oi parmetroi thc arqikopoÐhshc eÐnai to arqikì m koc r0 twn tmhmtwn, ta shmeÐa twn tmhmtwn uk, k ∈ {0, 1, · · · , N }, u0= 0, uN = 1, h mègisth apìstash thc pragmatik c lÔshc apì ta shmeÐa twn tmhmtwn δ kai to sflma sÔgklishc T pou eÐnai h apìstash tou teleutaÐou shmeÐou thc diamèrishc apì to B. EpÐshc dÐdetai o rujmìc prosarmog c (learning rate) 0 < λ < 1. O algìrijmoc kajìdou kat thn klÐsh ja perioristeÐ sthn perioq kont sthn arqik prosèggish, me mègisth apìstash an shmeÐo mikrìterh apì to δ. Epomènwc ja brÐsketai sthn perioq twn diamerÐsewn Sj = {tji, i ∈ {0, · · · , N }, tj0 = 0, |C(tji) − C(ui)|2 < δ}. H sunj kh ¸ste na ektelesteÐ me epituqÐa o algìrijmoc eÐnai sthn perioq twn diamerÐsewn Sj na mhn uprqoun yeÔtikeς lÔseic, dhlad topik elqista. Paraktw perigrfetai o algìrijmoc. H telik diamèrish dÐdetai apì ta shmeÐa Pi.
Algìrijmoc Kajìdou kat thn KlÐsh
s = r0
t0= 0 P0= A Repeat
for i=1:N
Pi= C(ti): (ti> ti−1) ∧ (|Pi−1Pi| = s) ∧(|Pi− C(ui)| < δ)aþ end
s =
s + λ|B−PNN|, PN brÐsketai entìc thc arqik c kampÔlhc C(t) s − λ|B−PNN|, PN brÐsketai ektìc thc arqik c kampÔlhc C(t) Until |PN − B| < T
aþEn den uprqei lÔsh, upologÐzetai to Pi qrhsimopoi¸ntac thn teqnik mirroring proekteÐnontac thn kampÔlh, sthn perÐptwsh aut to Piìpwc kai ta epìmena apì autì ja brÐskontai ektìc arqik c kampÔlhc.
H taqÔthta ektèleshc tou parapnw algorÐjmou exarttai apì thn kampÔlh, ton arijmì tmhmtwn N, to pìso kont briskìmaste sthn telik lÔsh kai apì to λ pou kajorÐzei thn taqÔthta sÔgklishc thc mejìdou. Gia na exasfalisteÐ omal all kai tautìqrona gr gorh sÔgklish protimtai to λ na xekin sei meglo λ ≈ 0.5 kai stadiak na mei¸netai met apì lÐgec epanal yeic na stajeropoihjeÐ se λ ≈ 0.05.
4.2 Proteinìmenoc Genetikìc Algìrijmoc
Sto tm ma autì exetzoume mia epèktash tou algìrijmou kajìdou kat thn klÐsh pou de qreizetai arqikopoÐhsh. To prìblhma èqei jewrhtikì endiafèron afoÔ ja prèpei o algìrijmoc na persei apì pijan
topik elqista kai telik na katal xei se kpoio sunolikì. Arqik xekinoÔn dÔo algìrijmoi kajìdou
kat thn klÐsh me m kh tm matoc r0, r1 thn elqisth kai th mègisth dunat tim tou m kouc tm matoc.
Sth sunèqeia ekteleÐtai to epanalhptikì mèroc thc mejìdou. UpologÐzontai ìlec oi dunatèc diamerÐseic gia ta antÐstoiqa m kh tm matoc r pou uprqoun, an kpoia diamèrish apoklÐnei met apì pollèc epanal yeic diagrfetai kai antikajÐstatai apì mia nèa me tuqaÐo m koc tm matoc r. EpÐshc nèec diamerÐseic me tuqaÐo m koc tm matoc r dhmiourgoÔntai an kpoio arijmì epanal yewn. Ta m kh tm matoc r kje diamèrishc anane¸nontai katllhla me antÐstoiqo trìpo ìpwc sthn apl èkdosh tou algorÐjmou kajìdou kat thn klÐsh. O algìrijmoc termatÐzei ìtan to kro kpoiac apì tic diamerÐseic brÐsketai arket kont sto B4.
H taqÔthta ektèleshc tou parapnw algorÐjmou exarttai kurÐwc apì thn kampÔlh kai ton arijmì tmhmtwn N. Se peript¸seic pou sthn kampÔlh uprqoun poll topik elqista o algìrijmoc argeÐ polÔ na sugklÐnei, en¸ stic peript¸seic pou uprqei mÐa lÔsh (l.q. gia meglo arijmì tmhmtwn) o algìrijmoc sugklÐnei pnta. Jewrhtik, lìgw thc qr shc genetikoÔ algorÐjmou, met apì kpoio arijmì epanal yewn ja uprqei pntote sÔgklish, ìmwc suqn o arijmìc autìc gÐnetai polÔ megloc, kai sthn prxh den ufÐ-statai sÔgklish. Autì sun jwc sumbaÐnei stic peript¸seic pou o arijmìc twn diamerÐsewn pou exetzontai auxnetai polÔ. Mia parallag tou algorÐjmou pou diorj¸nei to parapnw prìblhma, epitaqÔnontac th sÔgklish, eÐnai na mhn af nei na auxnontai aperiìrista oi diamerÐseic pou exetzontai all na uprqei èna
nw ìrio eÐte to krit rio diagraf c all kai eisagwg c miac diamèrishc na gÐnei austhrìtero.
4.3 Apotelèsmata
Sto tm ma autì parousizontai ta apotelèsmata tou algorÐjmou kajìdou kat thn klÐsh, o opoÐoc sugklÐnei akrib¸c se mÐa lÔsh. (Sq ma 4.1). Ta upologizìmena eujÔgramma tm mata èqoun akrib¸c to Ðdio m koc kai mìno to kro thc diamèrishc apèqei apì to B èna elqisto m koc pou teÐnei sto mhdèn kaj¸c auxnetai o arijmìc twn epanal yewn thc mejìdou.
4Se apìstash mikrìterh apì T (sflma sÔgklishc thc mejìdou)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
Sq ma 4.1: Sta Sq mata eikonÐzontai apotelèsmata thc mejìdou kajìdou kat thn klÐsh se diforec kampÔlec kai gia diaforetikì arijmì tmhmtwn.
Keflaio 5
Efarmog KataskeuastikoÔ AlgorÐjmou Apìdeixhc
Sto tm ma autì ja perigrafoÔn dÔo algìrijmoi, ènac proseggistikìc kai ènac akrib c, pou basÐzontai sth mèjodo thc apìdeixhc (Tm ma 2.5). To pleonèkthma pou èqoun oi mèjodoi autoÐ eÐnai pwc upologÐzoun lÔseic mìno efìson uprqoun, se antÐjesh me ton proseggistikì algìrijmo (Tm ma 3) ston opoÐo den uprqei h parapnw eggÔhsh. En¸ wc meionèkthma mporeÐ na jewrhjeÐ to gegonìc pwc uprqei pijanìthta na mhn upologisteÐ kpoia lÔsh, dhlad na d¸sei ligìterec apì tic pragmatikèc. Bèbaia uprqei eggÔhsh (ìpwc fnhke kai sthn apìdeixh) pwc ja upologÐzetai pntote toulqiston mÐa lÔsh.
O proseggistikìc algìrijmoc upologÐzei me uyhl akrÐbeia th lÔsh kai sth sunèqeia mporeÐ na ako-louj sei kpoioc algìrijmoc tÔpou kajìdou kat thn klÐsh gia thn akrib eÔresh twn lÔsewn. O akrib c algìrijmoc upologÐzei thn akrib lÔsh kai èqei exetasteÐ sthn perÐptwsh polugwnik¸n kampÔlwn. Gia th mèjodo aut , uprqei eggÔhsh, pwc ja termatisteÐ se peperasmèno arijmì bhmat¸n (se antÐjesh me th mèjodo kajìdou kat thn klÐsh pou suneq¸c proseggÐzei tic lÔseic).
5.1 Proseggistikìc Algìrijmoc
O proseggistikìc algìrijmoc efarmìzei th mèjodou apìdeixhc tou tm matoc 2.5 me thn prosèggish pwc h sunrthsh d(x, y), eÐnai tmhmatik grammik kai suneq c, uprqoun stajerèc wij, vij, cij kai ta tm mata Dij = [xi, xi+1] × [yj, yj+1] ⊂ [0, 1]2, me x1= y1= 0 ≤ x2= y2≤ · · · ≤ xM −1= yM −1≤ xM = yM = 1.
d(x, y) = wˆ ij· x + vij· y + cij, (x, y) ∈ Dij (5.1.1)
An Ðsque kti tètoio tìte kai h fN(x, y)ja tan tmhmatik grammik , sta Ðdia tm mata Dij. Epomènwc kai h mhdenik isostjmh thc fN(x, y), ja èdine polugwnik kampÔlh, kai tèloc h qN(s)ja tan tmhmatik
grammik sunrthsh. Gia ton prosdiorismì twn lÔsewn arkeÐ na brejoÔn oi rÐzec thc qN(s). Lìgw tou ìti ta tm mata paramènoun, all kai ìlec oi ekfrseic eÐnai grammikèc, h ulopoÐhsh thc ìpwc ja doÔme paraktw eÐnai arket apl kai h ektèlesh thc arket gr gorh.
Gia thn ulopoÐhsh thc parapnw mejìdou jewroÔme deigmatoleiyÐa C(tk). Sth sunèqeia mporeÐ na efarmosteÐ h mèjodoc tou tm matoc 2.5 kai ìpou qreizetai na gÐnetai grammik parembol afoÔ ìlec oi ekfrseic èqoun jewrhjeÐ grammikèc. Epeid h grammik prosèggish ˆd(x, y)thc d(x, y) apì ta M shmeÐa thc C(t)mporeÐ na plhsisei osod pote kont thn d(x, y), ja isqÔei kai gia tic isostajmikèc to Ðdio, ìpwc kai gia tic telikèc lÔseic. Oi isostajmikèc upologÐzontai apì thn allag pros mou thc fN(xk, yk), l.q. isqÔei fN(xk, yk) · fN(xk, yk+1) < 0, ra to mègisto sflma thc diaforc thc jèsh thc pragmatik c isostajmik c apì thn upologizìmenh eÐnai |yk− yk+1|, ìpwc kai to sflma twn telik¸n lÔsewn. Sth prxh h parapnw prosèggish dÐnei sflma thc mèshc apìstashc twn upologizìmenwn tmhmtwn apì thn akrib lÔsh perÐpou 50 forèc mikrìterec apì to maxk(C(tk) − C(tk+1))gia tic kampÔlec pou èqoume dokimsei (Sq ma 4.1) me M = 200.
Sth sunèqeia ja perigrafeÐ analutik o proseggistikìc algìrijmoc.
• Arqik gÐnetai deigmatoleiyÐa thc kampÔlhc se M shmeÐa, C(tk), k ∈ UM me t1 = 0 ≥ t2 ≥ · · · ≥ tM −1≥ tM = 1}.
• Sth sunèqeia upologÐzetai h d(ti, tj), f2(ti, tj) = d(ti, tj) − d(ti, 0), i, j ∈ UM, j ≥ i.
• O upologismìc thc mhdenik c isostjmhc thc f2(i, j) pou pernei apì to f2(1, 1) gÐnetai proodeu-tik. Apì upìjesh gnwrÐzoume pwc h (a2(s), b2(s)) ja eÐnai tmhmatik grammik , epomènwc arkeÐ mia deigmatoleiyÐa pnw stic korufèc (a2(s) ∈ {t1, t2, · · · , tM}). O upologismìc thc (a2(k), b2(k)) prag-matopoieÐtai me ènan algìrijmo disqishc sunìrou. Arqik, upologÐzetai o pÐnakac V (ti, tj)pou eÐnai Ðsoc me 1 stic perioqèc pou brÐskontai oi kampÔlec mhdenik c isostjmhc kai 0 alloÔ. Dhlad , an isqÔei f2(ti, tj) · f2(ti, tj+1) < 0 f2(ti, tj) = 0tìte V (ti, tj) = 1kai V (ti, tj+1) = 1. Pardeigma tou pÐnaka V eikonÐzetai sto Sq ma 5.1(aþ). Sth sunèqeia upologÐzetai h zhtoÔmenh mhdenik isostjmh (a2(k), b2(k))apì ton paraktw algìrijmo disqishc sunìrou.
20 40 60 80 100 120 140 160 180 200
Segments = 3 Number of solutions = 3
(dþ)
Sq ma 5.1: Endimesa kai telik apotelèsmata proseggistikoÔ algorÐjmou gia N = 3. (a') Sthn f2(x, y) probllontai me mple qr¸ma ta shmeÐa tou pÐnaka V pou èqoun tim 1 sthn perioq twn opoÐwn brÐskontai sta shmeÐa twn kampul¸n mhdenik c isostjmhc thc f2(x, y). (b') Apotèlesma tou algorÐjmou Upologismìc Mhdeni-k c Isostjmhc proballìmeno sthn d(x, y), me mple tetrgwna emfanÐzontai oi telikèc lÔseic. (b') H kampÔlh [a2(s), b2(s)]kai h q(s), me kìkkinouc kÔklouc èqoun shmeiwjeÐ oi treic lÔseic pnw se autèc. (d') Probol twn tri¸n lÔsewn pnw sthn kampÔlh.
Upologismìc Mhdenik c Isostjmhc
a(1) = 0, b(1) = 0, w = 0, k = 0 V (t1, t1) = 2aþ
Repeat w = w + 1 for i1=-1:1 for i2=-w:w
x = a(k) + i1
y = round(b(k)) + i2
if V (tx, ty) = 1 ∧ V (tx, ty+1) = 1 V (tx, ty) = 2
V (tx, ty+1) = 2 w = 0
a(k + 1) = tx,
b(k + 1) = ty−f f2(tx,ty)
2(tx,ty+1)−f2(tx,ty) bþ
k = k + 1 end
end end
Until b(k) >= 1
aþMe tim 2 ston V (ti, tj)shmei¸nontai ta shmeÐa pou èqoun episkeuteÐ.
bþO upologismìc tou b(k + 1) prokÔptei me thn paradoq pwc h f2(tx, t), t ∈ [ty, ty+1]eÐnai grammik kai kai zhtme b(k + 1)t.¸. f2(tx, b(k + 1)) = 0
• Dedomènhc thc (a2(s), b2(s)) mporoÔme na upologÐsoume me qr sh grammik c parembol c thn q(s) = d(a2(s), b2(s))−d(1, b2(s))h opoÐa eÐnai tmhmatik grammik . Oi rÐzec thc (ρ) upologÐzontai brÐskontac tic diadoqikèc jèseic pou allzei prìshmo (q(s) · q(s + 1) < 0) kai efarmìzontac th paraktw sqèsh.
ρ = s − q(s)
q(s + 1) − q(s) (5.1.2)
• O parapnw algìrijmoc mporeÐ na epektajeÐ eÔkola kai stic peript¸seic N > 3 kai na upologistoÔn diadoqik oi f3(x, y), · · · , fN(x, y). O algìrijmoc gia thn eÔresh thc mhdenik c isostjmhc dedomènhc gia kje N eÐnai akrib¸c autìc pou perigryame parapnw (Upologismìc Mhdenik c Isostjmhc).
Sto Sq ma 5.1 eikonÐzontai endimesa tou algorÐjmou apotelèsmata gia N = 3 tm mata.
H akrÐbeia upologismoÔ twn lÔsewn exarttai apì to M, megalÔtera M dÐnoun kalÔterec proseggÐseic pou sugklÐnoun sthn pragmatik lÔsh. To meionèkthma eÐnai pwc h mn mh pou apaiteÐtai eÐnai O(M2)en¸
to kìstoc upologismoÔ twn lÔsewn eÐnai O(NM2).
Sq ma 5.2: O diaqwrismìc thc kampÔlhc mhdenik c isostjmhc se tmhmatik monìtonec kampÔlec thc morf c (x, Y (x)).
5.2 Akrib c Algìrijmoc
Sth sunèqeia ja perigrafeÐ ènac akrib c algìrijmoc pou sthrÐzetai sth mèjodo apìdeixhc tou tm matoc 2.5. O algìrijmoc èqei ulopoihjeÐ gia polugwnikèc kampÔlec, en¸ ja mporoÔse me antÐstoiqo trìpo na ulopoihjeÐ gia algebrikèc kai genik gia kampÔlec me gnwstì tÔpo1. Ousiastik perièqei ta Ðdia b mata me ton proseggistikì me th diafor pwc den knei kpoia upìjesh gia thn d(x, y) all ìpou qreiasteÐ thn upologÐzei analutik apì thn exÐswsh thc kampÔlhc kai lÔnei tic exis¸seic pou prokÔptoun.
Me thn upìjesh ìti h kampÔlh eÐnai polugwnik , oi exis¸seic pou prokÔptoun eÐnai deuterobjmiec, opìte lÔnontai akrib¸c se O(1). Pio analutik, h d(x, y) = |c(x) − c(y)| ja eÐnai thc morf c |A + Bx + Cy|, ta A, B, C eÐnai gnwstèc stajerèc pou prokÔptoun apì thn exÐswsh thc kampÔlhc. To tetrgwno thc d(x, y) eÐnai tmhmatik algebrik sunrthsh deutèrou bajmoÔ2. 'Idiac morf c ja eÐnai kai h f2(x, y), orismènh sta Ðdia tm mata me thn d(x, y). Gia na upologisteÐ h mhdenik isostjmh thc f2(x, y)arkeÐ o algìrijmoc pou anaptÔxame parapnw (Upologismìc Mhdenik c Isostjmhc), me allag thc sqèshc upologismoÔ tou b(k+1)to opoÐo pleìn ja prokÔyei apì th rÐza gnwstoÔ poluwnÔmou thc morf c |A+Ba(s+1)+Cb(s+1)| =
|D|, mlista, sthn perÐptwsh dÔo riz¸n, h mÐa pijanìn ja aporrÐptetai apì touc periorismoÔc gia to b(s+1).
Tèloc o upologismìc tou q(s) eÐnai epÐshc efiktìc ìpwc kai o upologismìc twn riz¸n tou, prokÔptoun antÐstoiqa deuterobjmiec exis¸seic.
Gia N > 3 to prìblhma gÐnetai pio polÔploko epeid ta tm mata (Dij gia N = 3) sta opoÐa orÐzo-ntai oi isostajmikèc ja prèpei na epanaupologÐzetai. To gegonìc autì ofeÐletai sto ìti de diathreÐtai h tmhmatik monotonÐa entìc tou kje tm matoc, epeid èqoume deuterobjmiec ekfrseic, all uprqei perÐptwsh diamerismoÔ tou kje tm matoc to polÔ se dÔo (Sq ma 5.2). Epomènwc gÐnetai diaqwrismìc thc mhdenik c isostjmhc thc fN(x, y)se tmhmatik monìtona tm mata, kai sth sunèqeia upologÐzetai h fN +1(x, y) = d(x, y) − d(x, Y (x))sta tm mata aut. Ta upìloipa b mata eÐnai akrib¸c ìmoia me ta b mata tou proseggistikoÔ algorÐjmou.
1H kampÔlh mhdenik c isostjmhc pou pernei apì to (0, 0) ja prèpei na dÐdetai apì kleistì tÔpo.
2Sth prxh mporoÔme na lÔsoume to prìblhma qrhsimopoi¸ntac thn d2(x, y), ¸ste na èqoume pntote algebrikèc ekfrseic, kai oi lÔseic pou ja upologistoÔn ja ikanopoioÔn kai thn d(x, y).
Keflaio 6
Efarmogèc - Sumpersmata
6.1 Efarmogèc
Sto tm ma autì ja anafèroume tic efarmogèc pou èqei to prìblhma. Katarq n to prìblhma proèkuye apì mia grafik efarmog , thn 3D anaparstash tou fidioÔ. Se èna tm ma, apì th mèjodo pou protjhke sto [8], up rxe angkh gia thn tmhmatopoÐhsh 2D kampÔlhc se Ðsa tm mata. Skopìc tan na upologisteÐ sugkekrimènoc arijmìc shmeÐwn tou skeletoÔ tou fidioÔ pou na isapèqoun metaxÔ touc.
'Amesh efarmog thc epÐlushc tou probl matoc eÐnai h montelopoÐhsh kampÔlhc. O upologismìc N − 1 shmeÐwn pou isapèqoun metaxÔ touc pnw sthn kampÔlh, kai h grammik eÐte kubik touc parem-bol , dÐnei mia prosèggish thc kampÔlhc h opoÐa teÐnei proc thn kampÔlh kaj¸c auxnei to N. 'Estw Pi, i ∈ {0, 1, · · · , N }, P0 = A, PN = B ta upologizìmena shmeÐa apì ton proteinìmeno algìrijmo (Curve Segmentation into Equal Segments (CSES)). Ta shmeÐa (2 · (N + 1) sto pl joc arijmoÐ) aut mporoÔn na antikatastajoÔn apì to s ma thc gwnÐac Ai, i ∈ {1, · · · , N }metaxÔ dÔo diadoqik¸n shmeÐwn, thn jèsh enìc mìno shmeÐou P0, kai thn apìstash r = |Pi+1− Pi|2metaxÔ dÔo diadoqik¸n. 'Etsi to pl joc twn arijm¸n pou apaitoÔntai gia thn perigraf mei¸netai se N + 2 dhlad sqedìn sto misì. Mlista an efarmosteÐ me-tasqhmatismìc Fourier sto s ma thc gwnÐac, sth perÐptwsh pou h arqik kampÔlh eÐnai paragwgÐsimh, tìte h enèrgeia tou s matoc sugkentr¸netai stouc pr¸touc suntelestèc1. Epomènwc apaiteÐtai mikrìc arijmìc suntelest¸n se sqèsh me to N gia thn perigraf tou s matoc. O arijmìc pou exarttai mìno apì thn arqik kampÔlh gÐnetai mikrìteroc ìso pio omal eÐnai h kampÔlh C(t). Oi suntelestèc Fourier metafèroun plhroforÐa isodÔnamh me to s ma thc gwnÐac miac kai me ton antÐstrofo metasqhmatismì Fourier mporeÐ na anakataskeuasteÐ to s ma thc gwnÐac.
O proteinìmenoc metasqhmatismìc eÐnai antistrèyimoc, epomènwc apì touc perigrafeÐc mporoÔme na upologÐsoume xan thn kampÔlh. Sto Sq ma 6.1 emfanÐzetai to digramma tou eujÔ kai antÐstrofou metasqhmatismoÔ ìpwc kai oi upologizìmenoi perigrafeÐc. Oi proteinìmenoi perigrafeÐc diaqwrÐzontai se
1Gia na sumbeÐ uyhl sugkèntrwsh thc enèrgeiac stouc pr¸touc suntelestèc, pèra thc omal c kampÔlhc apaiteÐtai kai periodikìthta, h opoÐa mporeÐ na epiteuqjeÐ wc ex c. Prin thn efarmog tou metasqhmatismoÔ Fourier prostÐjentai epiplèon deÐgmata sto s ma thc gwnÐac ¸ste to s ma na gÐnei periodikì. O upologismìc touc mporeÐ na gÐnei me parembol tou tèlouc kai thc arq c tou s matoc. Ta deÐgmata aut ja afairejoÔn kat thn anakataskeu .
Sq ma 6.1: To digramma eujÔ kai antÐstrofou metasqhmatismoÔ apì thn arqik kampÔlh, stouc perigrafeÐc thc.
ekeÐnouc pou perigrfoun to sq ma thc kampÔlhc, thn kateÔjunsh thc, thn klimkwsh thc kai th jèsh thc sto epÐpedo. Oi suntelestèc Fourier (sk, k > 1) perigrfoun to sq ma thc kampÔlhc kai paramènoun analloÐwtoi se peristrof , metafor kai klimkwsh thc kampÔlhc. O s1 (pr¸toc suntelest c Fourier) perièqei thn plhroforÐa thc kateÔjunshc thc kampÔlhc. O suntelest c r perièqei thn plhroforÐa gia to m koc thc kampÔlhc2(thn klimkwsh thc) kai o P0 gia th jèsh thc kampÔlhc.
H perigraf aut epeid perièqei mikrì arijmì suntelest¸n mporeÐ na d¸sei polÔ uyhlì bajmì sumpÐeshc gia omalèc kampÔlec. Sto Sq ma 6.2 emfanÐzontai apotelèsmata anakataskeu c omal¸n kampul¸n me qr sh mikroÔ arijmì suntelest¸n Fourier. Sto Sq ma 6.3 emfanÐzontai apotelèsmata anakataskeu c kampÔlhc pou perièqei gwnÐec, opìte apaiteÐtai megloc arijmìc suntelest¸n gia thn anakataskeu thc.
6.2 EpÐlogoc
Sthn ergasÐa aut exetsthke to prìblhma thc tmhmatopoÐhshc 2D suneqoÔc kampÔlhc se isom kh eujÔgramma tm mata. Arqik melet jhke to prìblhma kai apodeÐkthke pwc èqei toulqiston mÐa lÔsh gia kje arijmì tmhmtwn kai se opoiad pote kampÔlh. O algìrijmoc pou basÐzetai sth mèjodo thc apìdeixhc upologÐzei toulqiston mÐa lÔsh tou probl matoc kai mporeÐ na ulopoihjeÐ se proseggistik kai se akrib morf . Epiplèon, protjhkan kai melet jhkan dÔo mèjodoi epÐlushc tou probl matoc, me stìqouc ton proseggistikì upologismì ìlwn twn lÔsewn kai ton akrib upologismì miac lÔshc, antÐstoiqa. Exetsthke h sumperifor twn dÔo proteinìmenwn mejìdwn se diforec kampÔlec kai gia diaforetikì arijmì tmhmtwn.
O proseggistikìc algìrijmoc ekteleÐtai se sÔntomo qronikì disthma gia mikrì arijmì tmhmtwn kai eÐnai anexrthtoc tou pl jouc twn lÔsewn kai thc morf c thc kampÔlhc se antÐjesh me ton algìrijmo kajìdou kat thn klÐsh pou o qrìnoc ektèles c tou ephrezetai apì th morf (duskolÐa) tou probl matoc gia to dedomèno N all kai thn arqikopoÐhsh. Oi dÔo algìrijmoi emfanÐzoun mia sumplhrwmatik sumperifor, me ton proseggistikì na eÐnai sun jwc upologistik efiktìc gia mikr N en¸ o genetikìc algìrijmoc tÔpou kajìdou kat thn klÐsh na eÐnai sÐgoura efiktìc gia megla N, ìpou kai to prìblhma emfanÐzei mÐa lÔsh. O algìrijmoc kajìdou kat thn klÐsh mporeÐ na ektelesteÐ me arqikopoÐhsh ta apotelèsmata
2Gia meglo N isqÔei ìti to m koc thc kampÔlhc proseggÐzetai apì Nr pou eÐnai pnta mikrìtero apì to sunolikì m koc thc kampÔlhc.
−3 −2 −1 0 1 2
Sq ma 6.2: Sta sq mata eikonÐzontai oi arqikèc kampÔlec (me mple qr¸ma) kai h anakataskeu touc (me kìkkino qr¸ma). H arqik tmhmatopoÐhsh ègine se 200 Ðsa tm mata en¸ qrhsimopoi jhkan sto (a') 7, sto (b') 7, sto (g') 8, sto (d') 10, sto (e') 14 kai sto (st') 16 suntelestèc Fourier.
0 2 4 6 8 10 12 14
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
(aþ)
0 2 4 6 8 10 12 14
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
(bþ)
Sq ma 6.3: Sta sq mata eikonÐzontai oi arqikèc kampÔlec (me mple qr¸ma) kai h anakataskeu touc (me kìkkino qr¸ma). H arqik tmhmatopoÐhsh ègine se 200 Ðsa tm mata en¸ (a') 25 kai sto (b') 50 suntelestèc Fourier.
tou proseggistikoÔ algorÐjmou. Epomènwc, me touc proteinìmenouc algorÐjmouc epilÔetai to prìblhma gia kje N.
Megloc arijmìc problhmtwn kai efarmog¸n faÐnetai pwc angetai sto upì melèth prìblhma, ìpwc eÐnai h sumpÐesh kai montelopoÐhsh s matoc. Wc mellontik ergasÐa, ja eÐqe endiafèron na entopistoÔn tètoiec efarmogèc all kai na anaptuqjoÔn nèec mèjodoi pou ja epilÔoun to prìblhma.
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