Image Segmentation and Classification Lecture 5. Professor Michael Brady FRS FREng Hilary Term 2005

Transcription

1 Image Segmentaton and Cassfcaton Lecture 5 Professor Mchae Brady FRS FREng Hary Term 2005

2 The goa Segmentaton means to dvde up the mage nto a patchwork of regons, each of whch s homogeneous, that s, the same n some sense Intensty, texture, coour, Cassfcaton means to assgn to each pont n the mage a tssue cass, where the casses are agreed n advance Grey matter (GM), Whte matter (WM), cerebrospna fud (CSF), ar, n the case of the bran Note that the probems are nter-nked: a cassfer mpcty segments an mage, and a segmentaton mpes a cassfcaton

3 MRI bran sces WM? GM CSF The nosy MRI mage of the bran sce shown eft s deay pecewse constant, comprsng grey matter, whte matter, ar, ventrces. The rght mage s a segmentaton of the mage at eft. Evdenty, whe t s generay ok, there are severa errors. Bran MRI s as easy as t gets!!

4 Medca mage segmentaton s generay dffcut Nosy mages often Nose-to-sgna s 10% Ths s ten tmes N/S of camera mages Often textured n compex ways Reatvey poory samped Many pxes contan more than one tssue type ths s caed Parta Voume Effect Objects of nterest have compex shapes Sgns of cnca nterest are subte

5 Even MRI mage segmentaton s hard + Exceent contrast between soft tssues + Bran mages are approxmatey pecewse constant; but compex textures n other organs Cassfcaton ought to be easy: GM, WM, CSF, ar There are mage dstortons (eg moton artefact, bas fed, ) Parta voume effect (PVE) structures of nterest (tumours, tempora obes, ) have compex shapes

6 Cassfcaton of bran MRI mages The abes we wsh to assgn to objects are typcay few and known n advance e.g. WM, GM, CSF and ar for bran MRI objects of nterest usuay form coherent contnuous shapes If a pxe has abe c, then ts neghbours are aso key to have abe c Boundares between regons abeed c, d are contnuous Image nose means that the abe to be assgned ntay at any pxe s probabstc, not certan One way to accommodate these consderatons s Hdden Markov Random Feds

7 Segmentaton by cassfcaton of voxes Every pxe s cassfed accordng to ts probabty of beng a member of a partcuar tssue cass. The Maxmum Lkehood (ML) estmator assgns to each voxe x that cass whch maxmses Pr( x C), where C { WM, GM, CSF,...} The Maxmum a Posteror (MAP) estmator P( x y) Normasng ntenstes : PMAP ( x y) = = P( y x) P( x) / P( y) In practce, nether ML nor MAP work we on bran MRI. P( y x) P( x) To understand why, we need to mode the probabty of a pxe, wth a partcuar ntensty, beng a member of a partcuar tssue cass such as GM, WM,

8 Each cass s defned as a probabty densty functon Each cass, say the one wth abe ( = GM, say) has an assocated PDF, wth parameters θ Often θ s a Gaussan θ = ( µ, σ ) p( y ) = f ( y ; θ ) = 1 2πσ exp 2 2 ( y µ ) σ 2 2

9 Tssue Probabty Densty Functons Note the huge overap between the Gaussan pdf for GM and that for WM Ths means that there are many mscassfcatons of GM pxes as WM and vce versa. Even sma amounts of nose can change the ML cassfcaton. Can we do better? (a) take spata nformaton nto account; and (b) mode expected spata mage dstortons. We address both ssues usng a MRF.

10 MRF mode A attce of stes = the pxes of an mage S = { 1,..., N} A famy of random varabes, whose vaues defne a confguraton R = { r, S} ( R 1 = r,..., R 1 N = r N ) R = r The set of possbe ntensty vaues D = { 1,..., d} An mage vewed as a random varabe The set of cass abes for each pxe (eg GM, WM, CSF) A cassfcaton (segmentaton) of the pxes n an mage Y X = { y = ( y1,..., y ) y D, S} N L = { 1,..., } = { x = ( x1,..., x ) x L, S} N

11 Parwse ndependence Cassfcaton s ndependent of mage and neghbourng voxes are ndependent of each other Assume a parametrc (eg Gaussan) form for dstrbuton of ntensty gven abe P ( y, x) = P( y, x ) S p( y ) = f ( y ; θ ) Markovan assumptons: probabty of cass abe depends ony on the oca neghbourhood N P( x) > 0, x X P( x x ) = P( x S { } N x )

12 MRF equvaent to Gbbs dstrbutons Deep theorem of Hammersey & Cfford MRFs are equvaent to systems wth a oca potenta functon Defne cques made of neghbourng pxes Each cque has a oca potenta energy U whch enabes the probabty of the pxe abe to be cacuated effcenty og P( x y) = U ( x y) Equvaenty : P( x y) = exp( U ( x y))

13 MRF-MAP estmaton We seek a abeng of the mage accordng to the MAP crteron: Reca that ) x = arg max x X { P( y x) P( x) } 1 P( x) = exp( U ( x)) Z And, f we have Gaussan dstrbutons of pxe vaues for each cass, and p( y x ) x = 1 exp ( y µ ) = 2 2 2πσ 2σ 2

14 Pxe-wse ndependence & ICM So that P ( y x) = p( y x ) S U ( y ( y x) = 2 S σ µ ) 2 + ogσ The fna step s to use Besag s Iterated Condtona Modes agorthm: Update the cass abe by mnmsngu ( x x k y, x at teraton N {} ) k

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16 Mode estmaton HMRF-EM We have assumed that we know the mode parameters θ ahead of tme. Unfortunatey, ths s rarey = ( µ, σ ) the case. The next dea s to estmate the parameters and do the segmentaton cooperatvey usng the EM agorthm Start : make nta estmatesθ, E - step : cacuate condtona k Q( θ θ ) = x X k p( x y, θ M -step : update estmate of θ = arg maxq( θ θ ) k + 1 k θ = 1... expectaton )og( p( x, cass parameters y θ )

17 Need to correct mage dstorton Orgna mage threshod to fnd whte matter Corrected mage threshod to fnd whte matter

18 Cassfcaton/segmentaton wthout bas correcton Typca MR mages Segmentaton msses ots of grey matter

19 Bas fed affects mage hstogram (pdf) Orgna mage wthout bas fed and ts hstogram Bas fed corrupted mage and ts hstogram Estmatng the bas fed amounts to estmatng the pdf, gven pror assumptons about the bas fed and expected pdf

20 Appyng HMRF to bas correcton E step Compute MAP estmates for the bas fed and cassfcaton B x t ( t) = arg max x X B p( B y, x t = arg mnu ( x y, B ( t 1) t, θ ) t, θ ) M step Compute ML estmates of the parameters of the casses gven current nformaton θ = arg max P( y θ, x, B ( t+ 1) t t θ )

21 Bas fed correcton Orgna mage Estmated bas fed Corrected mage Improved segmentaton

22 Experments on Bran MR Images bas fed segmentaton restoraton hstogram MAP anayss MRF anayss Zhang, Smth, Brady, IEEE Trans. Med. Im. 20(1), 45, 2001

23 Experments on Bran MR Images MAP anayss MRF anayss Zhang, Smth, Brady, IEEE Trans. Med. Im. 20(1), 45, 2001

24 Experments on Bran MR Images

25 L-to-R: orgna mage; estmated bas fed; corrected mage; and segmentaton

26 automatc HMRF segmentaton manua segmentaton by a sked cncan