An Overview of Markov Random Field and Application to Texture Segmentation
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1 An Ovrvw o Markov Random Fld and Applcaton to Txtur Sgmntaton Song-Wook Joo Octobr 003. What s MRF? MRF s an xtnson o Markov Procss MP (D squnc o r.v. s unlatral (causal: p(x t x, <t = p(x t x j, t m j t blatral (non-causal: p(x t x, t = p(x t x j, t m j t+m, j MRF ((D ld o r.v. s p(x,j x k,l, (k,l (,j = p(x,j x k,l, (k,l som nghborhood o (,j. Notatons ( Lablng problm: mappng sts to labls a sts: S = {,,m} b labls: L = {l,,l M } c lablng: ach st s assgnd a labl,, rgardd as a uncton : S L Sts can b rgular/rrgular, labls can b contnuous/dscrt. In th cas o rgular sts, th ordr o ndcs can b th rastr scan ordr o n n lattc and m=n. (Rgular assumd hr on Th conguraton {,, m } s an nstanc o lablng o th whol sts. F = L m s th st o all possbl conguratons. ( Nghborhood systm & clqus N I = { sts nghborng I } = { j S dst(,j r } : r th ordr C k = { {,, k } all sts wthn ar nghbors to vry othr } C = k C k * s g. o []
2 3. Modl Formulaton ( MRF Dnton: MRF on S wrt N P( > 0, F (postvty P( S-{} = P( N (Markovanty Spcyng an MRF can b don thr by (local condtonal probablty P( N or by (global jont probablty P( ( Gbbs Random Fld (GRF U ( T p( =, partton uncton Z U( = c ( : nrgy uncton c C Z T = = F U ( ( C : potntal uncton dpnds on s n th clqus Otn usd xprsson or U( s U( = ( + (, + (,, + j 3 j k {} C= S {, j} C {, j, k} C3 I w consdr only th clqus o sz on and two Not (, j ( j, s allowd. U( = ( + (, j S S j N (3 Markov-Gbbs Equvalnc F s an MRF wrt N F s a GRF wrt N ( Gvn GRF p(, thn c ( ' p(, c A S-{} p( p(, N S-{} = = = = = p c ( N p( S-{} p(, S-{} ' p( c A N L L p( ( whr ' = {, }. Ths s how w gt condtonal probablts rom jont probablty. S-{} ( proos xst (complcatd
3 (4 Som xampls o MRF modls a auto-logstc modl L = {0,}, β j : ntracton cocnts U( = α + βj j {} C {, j} C I st ordr nghborhood modl (4-nghbors s usd ths s calld th Isng modl. b auto-normal modl (Gaussan MRF, [] Assum zro-man r.v. y µ = j NI β I y y j +, whr N(0,σ. So assumd contnuous. MRF rprsntaton: Gbbs rprsntaton: P( N = πσ y σ ( ( y βjyj σ j N β yy = σ =, ( j j c mult-lvl logstc modl (MLL L = {0,,M}. Consdrng clqu sz up to two ( par-ws MLL, ( ( = α (dpnds on labl βc, = j = β, c j Not dpnds on th clqu typ c {,,3,4}. I β =β =β 3 =β 4.., sotropc, MLL producs blob-lk rgons and s usd or modlng rgons wth sam (dscrt labl. 3
4 4. Problm Formulaton: MAP-MRF lablng Problm: Gvn an MRF and obsrvd data d, nd an optmal lablng soluton. Paramtrs θ or th MRF may not b gvn. For th optmalty crtron, MAP (maxmum a postror s otn usd: * ( d = arg max P F = arg max P F ( d P( { U( d U( } = arg mn + F ( MRF Paramtr stmaton a Maxmum Lklhood arg max p ( θ = Z ( θ U ( θ Intractabl n gnral snc Z(θ s hard to valuat and dpnds on θ. b Psudo Lklhood: Assum som ndpndnc. Us PL( θ = P( N, θ c Codng Mthod: Partton S nto dsjont sts S (k. Wthn ach S (k PL s th tru lklhood. d Last Squars Mthod Usng potntal uncton rprsntaton and stmatd probablty, solv or paramtrs. ( P (, N P( N U(, N, θ P = = (, N, θ N U P( N can b stmatd usng hstogram mthod. Tak log, bcoms lnar n θ, us all 's to orm ovr dtrmnd lnar systm, solv LS soluton. L ( Optmzaton Local optmzaton: Itratd Condtonal Mods, Rlaxaton Lablng, Dynamc Programmng, Global optmzaton: Smulatd Annalng, Man Fld Annalng, Graduatd Non-Convxty, 4
5 5. Applcaton: Txtur Sgmntaton [3,4] ( Problm ( Modl Two drnt MRF wr usd. A 4th ordr GMRF s usd as th ndvdual txtur ld and a smpl nd ordr par-ws MRF s usd or th txtur labl ld. Txtur ld For ach txtur typ, an mag block (64 64 pxls s ttd to a 4th ordr GMRF modl. Th zroman (all mags ar assumd to b zro-man txtur mag rgon, {y(s s Ω}, Ω={ s=(,j,j M} s obtand by subtractng rom ach pxl th avrag o th local (7 7 nghborhood. Th 4th ordr nghborhood s dnd by th ollowng s s dnots th st whr th nghborhood s dnd. Th shadd rgon dnots th asymmtrcal nghbor st N s. Th stmats o th paramtrs ar obtand by th last squars mthod θ * = [ Σ s Ω q(s q (s T ] - ( s Ω q (s y(s, q (s = col [y(s+r + y(s-r, r N s ] ν * = /M s Ω ( y(s θ *T q (s 5
6 Ω s assumd to b wrappd around n a torodal mannr. Txtur labl ld Th labl valu can hav {,,3,4,5}. Th ollowng modl (nrgy uncton s usd U( L(s L(s+r, r N s = -β r Ns δ(l(s L(s+r whr N s s th nd ordr nghborhood (8-nghbor and δ(. s th Kronckr dlta. β s th paramtr ndcatng th dgr o clustrng. (3 Sgmntaton algorthm Th MRF lablng problm posd as a MAP (maxmum a postror problm: Fnd L that maxmzs P(L y P(y LP(L. Th ICM mthod s usd or th optmzaton. Th man stp o ICM s to maxmz th (local condtonal postror probablty whch satss P(L(s L(s+r, y(s, r N Ls P(y(s L(s P(L(s L(s+r, r N Ls or ach s. Hr, y(s s assumd to b ndpndnt o L(s+r, r N L. Not that or ach updat, th only unknown s L(s vrythng ls s gvn. P(y(s L(s s actually P(y(s L(s, y(s+r, r Ny s. Th probablts on th rght hand sd ar gvn by P (y(s L(s = (/(π ν l / xp[ {y s - r Nys -Nys θ l (r y(s+r } /ν l ] P(L(s L(s+r, r N Ls = xp(β r NLs δ(l(s L(s+r / Z whr Z s a normalzng constant, and subscrpt l dnots a paramtr dpndnt o th labl l. Th algorthm s smpl: ( Intalz y wth som valus. ( For ach s n th mag, updat L(s accordng to th abov stp (. Rpat stp ( or max traton. 6
7 Nots on ICM: ICM can b usd whn th unknown lablng s dscrt. Intal lablng s usually assgnd by th maxmum lklhood valus. Convrgnc s guarantd to a local mnmum. Th nal rsult tnds to b hghly dpndnt on th ntal lablng. Rrncs [] S. Z. L, Markov Random Fld Modlng n Imag Analyss, Sprngr, 00. [] R. Chllappa, Two Dmnsonal Dscrt Gaussan Markov Random Fld Modls or Imag Procssng, Progrss n Pattrn Rcognton, volum, pags 79-. Elsvr Scnc Publshrs B.., 985. [3] B. S. Manjunath, T. Smchony, and R. Chllappa. Stochastc and Dtrmnstc Ntworks or Txtur Sgmntaton, IEEE Transactons on Acoustcs, Spch, and Sgnal Procssng, ASSP- 38: , Jun 990. [4] R. C. Dubs and A. K. Jan. Random Fld Modls n Imag Analyss, Journal o Appld Statstcs, 6:3-64,
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