Graph Cuts Approach to the Problems of Image Segmentation. CS7640 Ross Whitaker (Adapted from Antong Chen)

Transcription

1 Graph C Approach o he Problem o Image Segmenaion CS7640 Ro Whiaker Adaped rom Anong Chen

2 Oline Image Segmenaion and Graph C Inrodcion Glance a Graph Theory Graph C and Max-Flow/Min-C Algorihm Solving Image Segmenaion Problem Experimenal Example

3 Image Segmenaion and Graph C Inrodcion An image egmenaion problem can be inerpreed a pariioning he image elemen pixel/voxel ino dieren caegorie A C o a graph i a pariion o he verice in he graph ino wo dijoin be Conrcing a graph wih an image we can olve he egmenaion problem ing echniqe or graph c in graph heory

4 Conrcing a Graph rom an Image Two kind o verice Two kind o edge C - Segmenaion

5 Glance a Graph Theory e E Undireced Graph An ndireced graph i deined a a e o G = V E node verice V and a e o ndireced edge E ha connec he node. Aigning each edge a weigh he graph become an ndireced weighed graph. we 3

6 Glance a Graph Theory Direced Graph o A direced graph i deined a a e o node verice V and a e o ordered e o verice or direced edge E ha connec he node For an edge e=v i called he ail o e v i called he head o e. Thi edge i dieren rom he edge e =v G = V E e ' = v e = v 3 3

7 Glance a Graph Theory A c i a e o edge C E ch ha he wo erminal become eparaed on he indced graph G ' = V E \ C Denoing a orce erminal a and a ink erminal a a c S T o G = V E i a pariion o V ino S and T = V \S ch ha T and S

8 Graph C and Max-Flow/Min-C Algorihm A low nework i deined a a direced graph where an edge ha a nonnegaive capaciy A low in G i a real-valed oen ineger ncion ha aiie he ollowing hree properie: o Capaciy Conrain: o For all v V v c v o Skew Symmery o For all v V v = v o Flow Conervaion o For all V \{ } v = 0 v V

9 How o Find he Minimm C? Theorem: o In graph G he maximm orce-o-ink low poible i eqal o he capaciy o he minimm c in G. o L. R. Fold Graph Theory Applicaion 99 Springer-Verlag New York Inc

10 Maximm Flow and Minimm C Some Concep Problem o I i a low hen he ne low acro he c S T i deined o be S T which i he m o all edge capaciie rom S o T braced by he m o all edge capaciie rom T o S o The capaciy o he c S T i c S T which i he m o he capaciie o all edge rom S o T o A minimm c i a c whoe capaciy i he minimm over all c o G.

11 Algorihm o Solve Max-Flow Problem Ford-Flkeron Algorihm Ph-Relabel Algorihm New Algorihm by Boykov ec.

12 Ford-Flkeron Algorihm Main Operaion Saring rom zero low increae he low gradally by inding a pah rom o along which more low can be en nil a max-low i achieved The pah or low o be phed hrogh i called an agmening pah

13 Ford-Flkeron Algorihm The Ford-Flkeron algorihm e a reidal nework o low in order o ind he olion The reidal nework i deined a he nework o edge conaining low ha ha already been en In he graph hown below here i an iniial pah rom he orce o he ink and he middle edge ha a oal capaciy o 3 and a reidal capaciy o 3-= 0/ 0/ / /3 / 0/ 0/

14 Ford-Flkeron Algorihm Aming here are wo verice and v le v denoe he low beween hem c v be he oal capaciy c v be he reidal capaciy and here hold be o c v = c v v Given a low nework and a low he reidal nework o G i G = V E where E = { v V V : c v > 0} Given a low nework and a low an agmening pah P i a imple pah rom o in he reidal nework We call he maximm amon by which we can increae he low on each edge in an agmening pah P he reidal capaciy o P given by o c P = min{c v : v i on P}

15 Algorihm Execion Example 0/ = 0 = 0/ 0/ 0/ 0/3 0/ 0/ a d = 0 = 3 3 b e = / = 3 3 / /3 / / / c /

16 Finding he Min-C Aer he max-low i ond he minimm c i deermined by S = {All verice reachable rom } T = G \ S

17 Special Cae A in ome applicaion only ndireced graph i conrced when we wan o ind he min-c we aign wo edge wih he ame capaciy o ake he place o he original ndireced edge a b

18 Algorihm Framework The baic Ford- Flkeron algorihm or each edge do v 0 v 0 v E while here exi a pah P rom o in he reidal nework G do c P min{c v : v i on P} or each edge v in P do v v + c P v v

19 Ford-Flkeron Algorihm Analyi Rnning ime o he algorihm depend on how he agmening pah i deermined o I he earching or agmening pah i realized by a breadh-ir earch he algorihm rn in polynomial ime o O E max In exreme cae eiciency can be redced draically. o Below applying Ford-Flkeron algorihm need 400 ieraion o ge he max low o 400 v v

20 Ph-Relabel Algorihm Thi algorihm doe no mainain a low conervaion rle here i V \{ } V 0 We call he oal ne low a a verex he exce low ino given by e = V. We ay ha a verex V \{ } i overlowing i e = V > 0 Heigh Fncion: o Le G = V E be a low nework wih orce and ink le be a prelow he low aiying he kew ymmery capaciy conrain and he condiion above in G. A ncion h :V Ν i a heigh ncion i h = V h = 0 and or every reidal edge v E

21 Ph-Relabel Algorihm Sage Ini$alize- Prelow G or each verex V do 0 ] [ h 0 ] [ e or each edge E v do 0 ] [ v 0 ] [ v V h or each verex Adj do ] [ c c ] [ c e ] [ ] [ c e e

22 Ph-Relabel Algorihm Sage Ph v Applied when: i overlowing 0 > v c and + = v h h AcDon: Ph ] [ min v c e v d = ni o low rom o v: ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ min v d v e v e v d e e v v v d v v v c e v d + +

23 Ph-Relabel Algorihm Sage Relable Applied when: i overlowing and or all v we have h h v E v V ch ha AcDon: Increae he heigh o : h + min h v : v E

24 Ph-Relabel Algorihm Framework Generic-Ph-Relabel G o Iniialize-Prelow G while here exi an applicable ph or relabel operaion do elec an applicable ph or relabel operaion and perorm i

25 Ph-Relabel Algorihm Example 0/ 0/ 0/ 0/ 0/3 0/ 0/ 3 h=6 e=-3 =0 3 h=6 e=-3 e= e= =0 Iniial Ph -= 3-= h=6 e=-3 h= e=-=0 h= e=-=0 ++=3 =0 Ph h=6 e=-3 h= h= h= e=3 =0 Relabel 3 h=6 e=-3 h= e= h= e= =0 Relabel

26 Ph-Relabel Algorihm Example h=6 e=-3 h= h= h= e=3-= =0+= Ph h=6 e=-3 h= h= h=+= e= = Relabel += h=6 e=-3 h= += h= h= e=-=0 = Ph h=6 e=-3 h=+=3 e= h= h= = Relabel -= h=6 e=-3 h=3 e=-=0 h= h= += = Ph -= h=6 e=-3 h=3 h= += h= e=-=0 = Ph +=

27 Ph-Relabel Algorihm Example -= h=6 e=-3 h=3 h= e=-=0 h= += = Ph h=6 e=-3 h=3 h= h= h=+0= e= Relabel = -= h=6 e=-3 h=3 h= h= h= e=-=0 Ph =+=3 h=6 e=-3 h=3 h= h= h= Finalize =3

28 Ph-Relabel Algorihm Analyi The algorihm complexiy i bonded o OVE Improved algorihm ing highe-level elecion rle and FIFO-rle qee baed elecion rle can redce he complexiy o O V E Opimizing he program by ing bcke daa rcre baed on any o he improved algorihm can yield OVE

29 Solving Image Segmenaion Problem Daa elemen e P repreening he image pixel/ voxel Neighborhood yem a a e N repreening all pair {pq} o neighboring elemen in P ordered or nordered A be a vecor peciying he aignmen o pixel p in P each Ap can be eiher in he backgrond or he objec A = A A... A p... A P A deine a egmenaion o P

30 Co o Segmenaion RA deine he penalie or aigning Ap o objec or backgrond which are Rp obj and Rp bkg BA decribe he bondary properie o he egmenaion B{pq}i large when p and q are imilar i i cloe o 0 when p and q are very dieren = = P p N q p q p q p P p p p A A B A B A R A R } { } { δ A B A R A E + = λ = = q p q p q p A A A A A A 0 δ

31 Graph Conrcion Edge Weigh Condiion n-link {p q} B {p q} {p q} N -link {p } λ R p bkg p P p OUB K p O 0 p B -link {p } λ R p obj p P p OUB 0 p O K p B

32 n-link Conrcion B { p q} exp I p σ I q di p q Ip and Iq are he ineniie o pixel p and q σe he penaly o diconiniie beween pixel o imilar ineniie when Ip Iq <σ he penaly i large when Ip Iq >σ he penaly i mall dip q penalize he diance beween p and q and generally when hey are in he neighborhood yem hi erm i one

33 -link Conrcion Rp obj = ln PrIp O Rp bkg = ln PrIp B K = + max B p P q:{ p q} N { p q} EDmaed Backgrond Ineniy DiribDon Baed on Backgrond Seed EDmaed Objec Ineniy DiribDon Baed on Backgrond Seed

34 Experimenal Daa

35 Reerence Yri. Boykov and Marie-Pierre Jolly Ineracive Graph C or Opimal Bondary & Regiion Segmenaion o Objec in N-D Image In Proceeding o Inernaional Conerence on Comper Viion Volme I 05- Jly 00 Yri. Boykov and Vladimir Kolmogorov An Experimen Comparion o Min-C / Max-Flow Algorihm or Energy Minimizaion in Viion IEEE Tranacion on PAMI 6 9: 4-37 Sepember 004 Yri. Boykov and Vladimir Kolmogorov Comping Geodeic and Minimal Srace via Graph C In Proceeding o Inernaional Conerence on Comper Viion Volme II 6-33 Ocober 003 Vladimir Kolmogorov and Ramin Zabih Wha Energy Fncion can be Minimized via Graph C? IEEE Tranacion on PAMI 6 : Febrary 004 Y. Boykov O. Vekler and R. Zabih Fa Approximae Energy Minimizaion via Graph C IEEE Tranacion on PAMI 3 : -39 November 004 Sdipa Sinha Graph C Algorihm in Viion Graphic and Machine learning An Inegraed Paper UNC Chapel Hill November 004 Yri Boykov and Olga Vekler Graph C in Viion and Graphic: Theorie and Applicaion Chaper 5 o The Handbook o Mahemaical Model in Comper Viion Springer 005 Thoma Cormen Charle Leieron Ronald Rive and Cliord Sein Maximm Flow Chaper 6 o Inrodcion o algorihm econd ediion McGraw-Hill 005 L. R. Flold An Inrodcion o Tranporaion Nework Secion.5.3 o Graph Theory Applicaion Springer 99 L. Ford and D. Flkeron Flow in Nework Princeon Univeriy Pre 96. Andrew V. Goldberg and Rober E. Tarjan A New Approach o he Maximm-Flow Problem Jornal o he Aociaion or Comping Machinery 354:9 940 Ocober 988 E. A. Dinic Algorihm or Solion o a Problem o Maximm Flow in Nework wih Power Eimaion Sovie Mah. Dokl. :