A total variation approach

Transcription

1 Denosng n dgtal radograhy: A total varaton aroach I. Froso M. Lucchese. A. Borghese htt://as-lab.ds.unm.t / 46 I. Froso, M. Lucchese,. A. Borghese

2 Images are corruted by nose ) When measurement of some hyscal arameter s erformed, nose corruton cannot be avoded. ) Each xel ofa dgtal mage measures a number of hotons. Therefore, from ) and ) Images are corruted by nose! htt://as-lab.ds.unm.t / 46 I. Froso, M. Lucchese,. A. Borghese

3 Gaussan nose (not so useful for dgtal radograhs, but a good model for learnng ) Measurement nose s often modeled dld as Gaussan nose Let x be the measured hyscal arameter, let μ be the nose free arameter and let σ be the varance of the measured arameter (nose ower); the robablty densty functon for x s gven by: ( x μ, σ ) = ex σ π x μ σ htt://as-lab.ds.unm.t 3/ 46 I. Froso, M. Lucchese,. A. Borghese

4 Gaussan nose and lkelhood Images are comosedosed by a set of xels, x (x s a vector!) How can we quantfy the robablty to measure the mage x, gven the robablty densty functon for each xel? Let us assume that the varance s equal for each xel; Let x and μ be the measured and noseless values for the -th xel; Lkelhood functon, L(x μ): L x = = μ x μ = ex σ π σ ( x μ) = ( ) L(x μ) descrbes the robablty to measure the mage x, gven the nose free value for each xel, μ. htt://as-lab.ds.unm.t 4/ 46 I. Froso, M. Lucchese,. A. Borghese

5 What about denosng??? What s denosng then? Denosng = estmate μ from x. How can we estmate μ? Maxmze (μ x) => ths usually leads to an hard, nverse roblem. It s easer to maxmze (x μ), that s => maxmze the lkelhood functon (a smle, drect roblem). But Is maxmzaton of (μ x) dfferent from that of (x μ)? htt://as-lab.ds.unm.t 5/ 46 I. Froso, M. Lucchese,. A. Borghese

6 Bayes and lkelhood Bayes theorem: em ( μ x ) ( x ) = ( x μ ) ( μ ) Lkelhood ( μ x) = ( x μ) ( μ) ( x ) A ror hyothess h s on the estmated arameters μ. For the moment, let us suose (μ) = cost. Probablty densty functon for the data x Just a normalzaton factor!!! In ths case, maxmzng (μ x) or (x μ) s the same! htt://as-lab.ds.unm.t 6/ 46 I. Froso, M. Lucchese,. A. Borghese

7 So, let us maxmze the lkelhood L f Instead of maxmzng L(x μ), t s easer to mnze z log[l(x μ)]. When the nose s Gaussan, we get: x μ ex = = σ π σ ( x μ ) = ( x ) = μ μ = σ π σ = ( x μ) = ln L( x μ) [ ] = ln + ( x ) Maxmze L=> Least squares roblem! Least squares! Constant! htt://as-lab.ds.unm.t 7/ 46 I. Froso, M. Lucchese,. A. Borghese

8 However, what about nose n dgtal radograhy? ose n dgtal radograhy s Posson (hoton countng nose)! Let n, be the nosy (measured) number of hotons assocated to xel, and the unnosy number of hotons. o Then: ( ) n, = n, e! n, htt://as-lab.ds.unm.t 8/ 46 I. Froso, M. Lucchese,. A. Borghese

9 Gaussan nose: examle 000 Gaussan nose Constant varance htt://as-lab.ds.unm.t 9/ 46 I. Froso, M. Lucchese,. A. Borghese

10 Posson nose: examle 000 Posson nose Lower varance for low sgnal htt://as-lab.ds.unm.t 0 / 46 I. Froso, M. Lucchese,. A. Borghese

11 Lkelhood for Posson nose Let us wrte the negatve log lkelhood for the Posson case: L f = ( n ) = ( n, ) = = n, e! = n, ( ) = ln[ L( x μ) ] = ( ) n [ n, ln ( ) ] = [ n, ln ] + + ln( n,! ) = = = L( n ) s also known as Kullback-Lebler bl dvergence (aart from a constant term, whch does not affect the mnmzaton rocess), KL( n ). = htt://as-lab.ds.unm.t / 46 I. Froso, M. Lucchese,. A. Borghese

12 Maxmze L! L s maxmzed <=> f s mnmzed; Otmzaton (Gaussan nose) can be erformed osng: ( ) x f ( ) f ( ) j μ j x μ x μ j= = 0 = 0, = 0, μ μ μ ( x μ ) = 0, x =, μ The nosy mage gves the hghest lkelhood!!! Ths soluton s not so nterestng The lkelhood aroach suffers from a severe overfttng roblem. htt://as-lab.ds.unm.t / 46 I. Froso, M. Lucchese,. A. Borghese

13 Maxmze L! L s maxmzed <=> f s mnmzed; Otmzaton (Posson nose) can be erformed osng: f ( ) f ( ) = [ ln( )] n, n n = = 0 0, = 0, n, = 0, = n,, The nosy mage gves the hghest lkelhood!!! Ths soluton s not so nterestng The lkelhood aroach suffers from a severe overfttng roblem. htt://as-lab.ds.unm.t 3 / 46 I. Froso, M. Lucchese,. A. Borghese

14 Back to Bayes Bayes theorem: em Lkelhood ( ) = n ( ) ( ) n ( ) n A ror hyothess on the estmated arameters μ. Probablty densty functon for the data x Just a normalzaton factor!!! If we ntroduce a-ror knowledge about the soluton μ, we get a Maxmum A Posteror (MAP) soluton ( n ) s maxmzed! htt://as-lab.ds.unm.t 4 / 46 I. Froso, M. Lucchese,. A. Borghese

15 What do we have to mnmze now? We wantto maxmze ( n ) ~( n ) (), that s: ln = [ ( )] = ln ( ) ( ) ln = ln n [ ] = ln [ ( ) ( )] n n, = = [ ( n, ) ( )] = ln ( n, ) ln ( ) [ L( ) ] ln ( ) n = egatve log lkelhood = = Regularzaton term (a ror nformaton) = htt://as-lab.ds.unm.t 5 / 46 I. Froso, M. Lucchese,. A. Borghese

16 A ror term Let us call x and y the two comonents of the gradent of the mage. These are easly comuted, for nstance as: x =(,j) (-,j); y = (,j) (,j-); The gradent gaden (a vector!) wll be ndcated as ; ndcates the norm of the gradent. htt://as-lab.ds.unm.t 6 / 46 I. Froso, M. Lucchese,. A. Borghese

17 A ror term mage gradents (no nose) x = (,j) (-,j) y = (,j) (,j-) y htt://as-lab.ds.unm.t 7 / 46 I. Froso, M. Lucchese,. A. Borghese

18 A ror term mage gradents (nose) x = (,j) (-,j) y = (,j) (,j-) y htt://as-lab.ds.unm.t 8 / 46 I. Froso, M. Lucchese,. A. Borghese

19 A ror term norm of mage gradent o nose ose In the real mage, most of the areas are characterzed by an (almost) null gradent norm; We can for nstance suose that s a random varable wth Gaussan dstrbuton, zero mean and varance equal to β. [ote that, n the nosy mage, the norm of the gradent assume hgher values low means low nose!] htt://as-lab.ds.unm.t 9 / 46 I. Froso, M. Lucchese,. A. Borghese

20 MAP and regularzaton theory Posson nose, normal dstrbutonb for the norm of the gradent: f ( ) = ln[ L( ) ] ln ( ) n n = [ ( )] = n, ln ln ex = = = β π β = [ ( )] + ( ) n, ln ln β π + ββ = = Const!!! egatve log lkelhood Regularzaton term (a ror nformaton) = htt://as-lab.ds.unm.t 0 / 46 I. Froso, M. Lucchese,. A. Borghese

21 MAP and regularzaton theory We look for the mnmum of f The lkelhood lh s maxmzed m (data afttng term) At the same tme, the squared norm of the gradent s mnmzed (regularzaton term) The regularzaton arameter (/β ) balances between a erfect data fttng and very regular mage f [ n, ] + ( ) = ln ( ) n = β = htt://as-lab.ds.unm.t / 46 I. Froso, M. Lucchese,. A. Borghese

22 MAP and regularzaton theory For (/β ) = 0 we get the maxmum lkelhood soluton; Increasng (/β ) we get a more regular (less nosy) soluton; For (/β ) ->, a comletely smooth mage s acheved. (/β ) = 0 ose reducton. (/β ) = ose and edge reducton htt://as-lab.ds.unm.t / 46 (/β ) = 0. I. Froso, M. Lucchese,. A. Borghese

23 Fx the deas A statstcal based denosng flter s acheved mnmzng: f=-ln[l( n )]-λ ln[()] The data fttng term s derved from the nose statstcal dstrbuton (lkelhood of the data); generally, the choce for ths term s unquestonable. The regularzaton term s derved from a-ror knowledge regardng some roertes of the soluton; ths term s generally user defned. Deendng on the regularzaton arameter λ, the frst or the second term assume more or less mortance. For λ->0, the maxmum lkelhood soluton s obtaned. htt://as-lab.ds.unm.t 3 / 46 I. Froso, M. Lucchese,. A. Borghese

24 Gbbs ror U to now, we assumed a normal dstrbutonbuton for the norm of the gradent, Tkhonov regularzaton (quadratc enalzaton). A more general framework s obtaned consderng: () =ex[-r()] (Gbb s ror) R() -> Energy functon ~ regularzaton term (note that -ln ex[-r()] = R()!) Tkhonov assumes R() = -½ (ll ll/β) htt://as-lab.ds.unm.t 4 / 46 I. Froso, M. Lucchese,. A. Borghese

25 Edge reservng denosng? Tkhonov term enalzes the mage edges (hgh gradent) more than the nose 6 4 gradents. It s well known that 0 Tkhonov regularzaton does not 8 reserve edges. 6 An edge reservng 4 algorthm s obtaned consderng R()=-ll ll [Total varaton, TV]. R( Grad ) Tkhonov Total varaton Grad htt://as-lab.ds.unm.t 5 / 46 I. Froso, M. Lucchese,. A. Borghese

26 Tkhonov vs. TV (revew) Fltered mage Dfference Tkhonov => Orgnal mage TV => htt://as-lab.ds.unm.t 6 / 46 I. Froso, M. Lucchese,. A. Borghese

27 TV n dgtal radograhy: startng ont and roblems n, nosy mage affected by Posson nose (negatve log lkelhood => KL);, nose free mage (unknown); R() = (Total varaton); Mnmze Mnm f( n ) = KL( n,) + λ Σ =... How to comute? => A comromse between comutatonal effcency and accuracy has to be acheved. How to mnmze f( n )? => An teratve otmzaton technque s requred. htt://as-lab.ds.unm.t 7 / 46 I. Froso, M. Lucchese,. A. Borghese

28 How to comute? x = (u,v) (u-,v) y = (u,v) (u,v-) = x + y L norm = [ x + y ] ½ L norm = x + y + xy + yx = [ x + y + xy + yx ] ½ = x + y + xy + yx + = [ x + y + xy + yx + ] ½ Comu utatona al cost The comutatonal cost ncreases wth the number of neghbours consdered for comutng the gradent; The comutatonal cost s hgher for L norm wth resect to L norm; What about accuracy? => See exermental results! htt://as-lab.ds.unm.t 8 / 46 I. Froso, M. Lucchese,. A. Borghese

29 How to mnmze f( n )? f( n ) s strongly non lnear; solvng df( n )/d=0 drectly s not ossble => teratve otmzaton z methods. ) Steeest descent + lne search (SD+LS) ) Exectaton Maxmzaton (damed wth lne search - EM) 3) Scaled gradent (SG) htt://as-lab.ds.unm.t 9 / 46 I. Froso, M. Lucchese,. A. Borghese

30 Steeest descent + lne search (SD+LS) P k+ = k -α df( n )/d => => P k+ = k -α df( n )/d Thedamng arameter α s estmated at each teraton to assure convergence (f k+ <f k ); +: easy mlementaton; -: slow convergence, the method has been damed (lne search) to mrove convergence (α>). htt://as-lab.ds.unm.t 30 / 46 I. Froso, M. Lucchese,. A. Borghese

31 EM + lne search (EM) Consder the xel, then: df( n )/d =0 => => dkl( n )/d + dr/d =0 => β dr/d + - n, =0=> => = n, /(β dr/d + ) [Fxed on teraton] Damed formula: = (-α) +α n,i /(β dr/d +) The damng arameter α s estmated at each teraton to assure convergence (f k+ <f k ); +: easy mlementaton, fast convergence; -: the method has been damed to assure convergence (α<, what haens when β dr/d R + -> 0???). htt://as-lab.ds.unm.t 3 / 46 I. Froso, M. Lucchese,. A. Borghese

32 Scaled gradent (SG) Consder the gradent method formula; Each comonent of the gradent s scaled to mrove convergence (S s a dagonal matrx contanng the scalng arameters): P k+ = k -α S df( n )/d The matrx S s comuted from an oortune gradent decomoston and KKT condtons; +: easy mlementaton, fastest convergence; t can also be demonstrated that, for ostve ntal values, the estmated soluton remans ostve at each teraton! -:???. htt://as-lab.ds.unm.t 3 / 46 I. Froso, M. Lucchese,. A. Borghese

33 Problems wth dr/d Indeendentlyendently fom from the otmzaton method, the term dr/d has to be comuted at each teraton far any ; Wehave: dr/d = d[σ =.. (ll ll )]/d XOR dr/d = d[σ =.. (ll ll )]/d htt://as-lab.ds.unm.t 33 / 46 I. Froso, M. Lucchese,. A. Borghese

34 Problems wth dr/d L R/d Let us comute t for ll.ll (R/d = d[σ =.. (ll ll )]/d ).. ( ) ( ) [ ] ( ) ( ) [ ]...,,,,,, = + + = + = = d v u v u v u v u d d d d dr y x ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ( )...,...,,,,,,,,,, + + = = v u v u v u v u v u v u v u v u v u d d d y x To avod dvson by zero: ( ) ( ) ( ) [ ] ( ) ( ) [ ]...,,,,...,,,,, = δ v u v u v u v u v u d dr y x y x htt://as-lab.ds.unm.t I. Froso, M. Lucchese,. A. Borghese 34 / 46

35 Problems wth dr/d Let us comute t for ll.ll (R/d = d[σ =.. (ll ll )]/d ) dr d = = d ( + ) d ( u, v) ( u, v) ( u, v) ( u, v ) + ( u, v ) ( u, v ) ( u, v ) ( u, v ) x, y, = = +... = d = d [ sgn( x, ) + sgn( y, )] = +... Here dvsons by zero are automatcally avoded only sgn s requred -> comutatonally effcent! htt://as-lab.ds.unm.t 35 / 46 I. Froso, M. Lucchese,. A. Borghese

36 Questons How many neghbor hb xels do wehave to consder to acheve a satsfyng accuracy at low comutatonal cost? Best norm, ll.ll vs ll.ll? Best otmzaton method (SD+LS, EM, SG)? htt://as-lab.ds.unm.t 36 / 46 I. Froso, M. Lucchese,. A. Borghese

37 TV n dgtal radograhy Research n rogress htt://as-lab.ds.unm.t 37 / 46 I. Froso, M. Lucchese,. A. Borghese

38 Results (answers) 75smulated radograhs wth hdfferent frequency content, corruted by Posson nose (max 5,000 hotons). For any fltered mage, measure: MAE = /Σ =.. l,nosefree -,fltered l RMSE = [/Σ =.. (,nosefree -,fltered ) ] ½ KL = Σ =.. [,nosefree ln(,nosefree /,fltered )+,nosefree-,fltered ] htt://as-lab.ds.unm.t 38 / 46 I. Froso, M. Lucchese,. A. Borghese

39 neghbors (000000) vs. 4 neghbors (0000) htt://as-lab.ds.unm.t 39 / 46 I. Froso, M. Lucchese,. A. Borghese

40 ll.ll vs. ll.ll htt://as-lab.ds.unm.t 40 / 46 I. Froso, M. Lucchese,. A. Borghese

41 EM vs. SD+LS htt://as-lab.ds.unm.t 4 / 46 I. Froso, M. Lucchese,. A. Borghese

42 EM vs. SG htt://as-lab.ds.unm.t 4 / 46 I. Froso, M. Lucchese,. A. Borghese

43 Convergence and teratons htt://as-lab.ds.unm.t 43 / 46 I. Froso, M. Lucchese,. A. Borghese

44 Flter effect Orgnal Fltered htt://as-lab.ds.unm.t 44 / 46 I. Froso, M. Lucchese,. A. Borghese

45 Flter effect: before flterng htt://as-lab.ds.unm.t 45 / 46 I. Froso, M. Lucchese,. A. Borghese

46 Flter effect: after flterng htt://as-lab.ds.unm.t 46 / 46 I. Froso, M. Lucchese,. A. Borghese

47 Concluson Effectve edge reservng flter; neghbors, ll.ll and EM acheve the best comromse between accuracy and comutatonal cost; SD acheves results better then EM when the regularzaton arameter s not correctly selected. Adatve regularzaton arameter; GPU (CUDA) mlementaton; Exandng the lkelhood model Mxture of Posson, Gaussan and Imulsve nose; Include the sensor ont sread functon. htt://as-lab.ds.unm.t 47 / 47 I. Froso, M. Lucchese,. A. Borghese