Labeling Problem & Graph-Based Solution

Transcription

1 Lling Prolm & Grph-B Soluion Am M. Ali

2 Lling Prolm In lling Prolm w hv o i P n o ll L : rprn img ur {.g. pixl, g, img gmn,.}. Fur my hv om nurl ruur pixl r rrng in 2D rry. : rprn innii, iprii,. P L Lling prolm i mpping. W no h lling y P = 1;2;:::;ng L = l 1 ;l 2 ;:::;l k g = 1 ; 2 ;:::; n g S o ll lling i no y Simpl Exmpl: L n P L F P = 1;2;3;4g L = 5; ; 15g = ; 5; 5; 15g = 5; 5; 5; 15g = ; 5; ; 15g Th o ll lling oni o F=L =81 lling 2

3 Lling prolm onp giv ommon noion or ivr viion prolm, uh : P = 1;2;:::;R Cg Img Sgmnion L=;255g inpu oupu Img Rorion P = 1;2;:::;R Cg L = (;;);:::;(255;255;255)g inpu oupu 3

4 Lling prolm onp giv ommon noion or ivr viion prolm, uh : Sro Mhing 4

5 Lling prolm onp giv ommon noion or ivr viion prolm, uh : Sro Mhing 5

6 Lling prolm onp giv ommon noion or ivr viion prolm, uh : Sro Mhing x (x, y) (x +, y) y L img Righ img P = 1;2;:::;R Cg L= min : mx g Dipriy rng Dph/ipriy mp 6

7 Lling prolm onp giv ommon noion or ivr viion prolm, uh : Img Mhing Shkhovov, l CVPR 7 (x, y) (x+δx, y + δy) P = 1;2;:::;R Cg inpu L=(±x min ;±y min ):(±x mx ;±y mx )g Diplmn rng Oupu Digil Tpry (Rohr l CVPR 5) inpu oupu P = 1;2;:::;n Blok g L=I S 7

8 Prolm Formulion Img h nurl ruur in whih pixl r rrng in 2D rry. P = 1;2;:::;ng o img pixl Nighorhoo ym N in P i h o ll nighoring pirp;qg p; q 2 P Exmpl up o h 5 h orr p Th nighorhoo ym ii ) p62n p ) i p2 N q hn q2 N p 1 orr nighorhoo ym N p =;;;qg N q =z;p;y;xg N r =;g z p q x y r 3 x 5 img 8

9 Prolm Formulion Orv Img I Ll Img L = 1;2;:::;Kg o ll. K # ll/l (.g., K=2) I=I 1 ;I 2 ;:::;I n g orv img. = 1 ; 2 ;:::; n g ll img. :P! L F o ll lling L n (.g., in hi 2 36 irn lling ) F=F 1 ;F 2 ;:::;F n g o rnom vril in on P, n i onigurion o h il F 9

10 Prolm Formulion F i Mrkov Rnom Fil (MRF) w.r. N i i proiliy m union P(F = ) rvi y P() ii 1. P (F = ) > or ll 2 F, 2. P (F p = p jf P pg = P pg )= P(F p = p jf N p = N p ), 3. P (F p = p jf N p = N p )ihmorllip, Poiiviy Mrkov Propry Homogniy Mrkov propry lih h lol mol? q GRF ri h propri o n img in rm o h join iriuion o ll or ll pixl, n provi glol mol or h img.

11 Prolm Formulion GRF provi glol mol or n img y piying h (join) proiliy iriuion P() = Z 1 xp( X V ( )); 2 C Z V C normlizing onn ll h priion union, ponil union, liqu union, ummion ovr liqu Gi nrgy o ll liqu. A liqu i o pixl in whih ll pir o pixl r muul nighor. ingl-i ponil β wo-i ponil γ 1 wo-i ponil γ 2 wo-i ponil γ 3 wo-i ponil γ 4 hr-i ponil η 1 hr-i ponil η 2 hr-i ponil η 3 hr-i ponil η 4 our-i ponil ξ 11

12 Prolm Formulion In h pirwi inrion mol, Gi nrgy i in in rm o liqu o iz 2 Th img i rprn y MRF wih join iriuion: X P( ) = Z 1 xp( V( p ; q )) (A) Mximum-A-Poriori (MAP) Eimion p;qg2n Th inpu img I n h ll img r ri y join Mrkov- Gi rnom il (MGRF) MGRF mol i i wihin h Byin rmwork o MAP imion o im =rgmx P(Ij)P(): 2F Th porior iriuion P(I j ) i MRF y uming h noi h pixl i inpnn (Du n Jin 89) P(I j ) = Y p2 P P(I p j p ) (B) 12

13 Prolm Formulion Mximum-A-Poriori (MAP) Eimion From (A) & (B) h MAP imor =rgmx 2F xp(x p2 P log(p(i p j p )) X p;qg2n V( p ; q )): Equivln o minimiz h nrgy X E() = V( p ; q ) p;qg2n X p2 P log(p(i p j p )): Fir rm xpr moohing onrin on lling. Ll vri moohly vrywhr xp h oj ounri ioninuiy. Son rm mur how muh igning ll o pixl p igr wih h orvion I p p 13

14 Prolm Soluion Morn nrgy minimizion mho uh : Grph u (Zih PAMI 1) Loopy Bli Propgion (LBP) (Flznzwl CVPR 4) Tr-RWigh mg ping (TRW) (Winwrigh Ino Thory 5) Quri Puo-Booln Opimizion (QPBOP) (Kolmogorov CVPR 7) Clil mho uh : Ir Coniionl Mo (ICM) (Bg 74) Simul Annling (Gmn & Gmn 84) 14

15 Prolm Soluion F Approxim Enrgy Minimizion vi Grph Cu Boykov, Olg, n Zih, PAMI 1 GC SA A Compriv Suy o Enrgy Minimizion Mho or Mrkov Rnom Fil Szliki, Zih, Shrin, Olg, Kolmogorov, Agrwl, Tppn, n Rohr, ECCV 6 Sro Sgmnion 15

16 Prolm Soluion Comprion o Enrgy Minimizion Algorihm or Highly Conn Grph, Kolmogorov n Rohr, ECCV 6 Opimizing Binry MRF vi Exn Roo Duliy Rohr, Kolmogorov, Lmpiky, Szummr, CVPR 7 16

17 Grph Cu 17

18 Grph Cu Grph Cu Bi Diniion & Noion Th wigh grph G= hv;ei V i h o vri in grph orrpon o pixl, or ohr ur. ;g (ink & our) r wo iinguih vri ll rminl. E u o pir (p;q) o lmn rom V Eg A ph i qun o g. N-link: onn pir o nighoring vri. Co/wigh: pnly or ioninuii wn vri T-link: onn vrx wih rminl. Co/wigh : pnly or igning h orrponing ll o h vrx 18

19 19 CVIP L Min-Cu & Mx-Flow A u i o g uh h rminl r pr in h inu grph No propr u o pr h rminl in Co o h u, no, h um o i g wigh Min-u i o in h u wih minimum o mong ll u. Min-Cu n olv y ompuing Mx-Flow wn rminl For& Fulkron 62 C½E G(C) = hv; E Ci C G(C) C j Cj g h i Cu No Cu g h i g h i g h i g h i Grph Cu g h i

20 Grph Cu Min-Cu =Min Cpiy= Mx-Flow Cu Co = Cu Co = Cu Co =3 2

21 21 CVIP L Grph Cu Min-Cu & Mx-Flow Exmpl Mx low = Th Grph Nw Grph Mx low = Nw Grph Mx Flow= Nw Grph Mx Flow= Nw Grph Min Cu= - Min-Cu/Mx-Flow lgorihm o Boykov & Kolmogorov 4

22 Grph Cu Grph u minimizion hniqu E() = X V( p ; q ) X log(p(i p j p )): p;qg2n p2 P Evry pixl rprn vrx in h grph. N-link (p;q) wigh V( p j q ) T-link (;p);(;p) wigh logp(i p j p ) Compu - MinCu 22

23 Grph Cu Grph u minimizion hniqu (Exmpl) On row img Orv img Thrhol img Piwi Conn Prior Po mol ½ =5 i V( p ; q )= p 6= q ; i p = q D pnly rm P(I p j p )/ xp( ji p p j) N-link wigh T-link wigh ji p p j 23

24 Grph Cu Grph u minimizion hniqu (Exmpl) 2 E() = X p;qg2n 5 ±( p 6= q )+ X p2 P ji p p j: B lling Thrhol lling E = = 52 E = = 55 24

25 Grph Cu Grph u minimizion hniqu (Exmpl) Mx-Flow =

26 Grph Cu Grph u minimizion hniqu (Exmpl) Mx-Flow =

27 Grph Cu Muli wy Grph-u lgorihm α-xpnion lgorihm (Boykov l. 1): Minimiz n nrgy union wih non inry vril y rply minimizing n nrgy union wih inry vril uing Mx- low/ min- u mho n Whih union n minimiz y α-xpnion Algorihm? α-xpnion lgorihm n ppli o pir-wi inrion h r mri on h p o ll V pq (l;l) = V pq (l 1 ;l 2 )> i l 1 6=l 2 V pq (l 1 ;l 2 )= V pq (l 2 ;l 1 ) V pq (l 1 ;l 2 )+ V pq (l 2 ;l 3 ) V pq (l 1 ;l 3 ) 27

28 α-xpnion Algorihm Grph Cu Algorihm 1. Sr wih ny rirry lling µ 2. S u = 3. For h ll 2 L (rnom orr) Irion () in ^ = rg min E( ) mong wihin on α-xpnion 1 o µ yl () i E( )> µ E(^ ) µ = ^ & u = 1 4. I u = 1 goo 2 5. Rurn 2 µ Evry pixl ihr kp i ol ll or wih o α 2- Thr i no α-xpnion mov, or ny ll α, wih lowr nrgy. α 28

29 Som Rul 29 Rorion Sgmnion Sro Mhing Diplmn

30 Thnk You

31 Mximum-A-Poriori (MAP) Eimion Prolm Formulion & Img Lling Iriv rrh o MAP im ohi (.g., imul nnling) or rminii (.g., ir oniionl mo) Simul Annling (Gmn & Gmn 84) Simul pro in mllurgy whih rmin h low nrgy o mril y grully lowring h nrgy Fin MAP imor or ll pixl imulnouly Fin h glol oluion wih rin mprur hul Compuionlly xpniv; h hul h l o h glol r vry low in pri. Ir Coniionl Mo (ICM) (Bg 74) Pixl r pro qunilly, n or h pixl h lgorihm l h ll h mximiz P(I P j p )P( p j ^ N p ) Fr hn imul Annling Vry niiv o h iniil lling Lol nrgy opimizion hniqu 31