灰度图像最小误差阈值分割法的二维推广

354Vol.35,No.420094ACTAAUTOMATICASINICAApril,200911;2.,,,.,.,,.,,TN911.73Two-dimensionalExtensionofMinimumErrorThresholdSegmentationMethodforGray-levelImagesFANJiu-Lun1LEIBo1;2AbstractOne-dimensionalminimumerrorthresholdingmethodassumedthatthehistogramdistributionsofobjectandbackgroundaregovernedbyamixtureGaussiandistribution.Consideringthea®ectsofnoiseandotherfactorsonimagequality,basedontheassumptionofatwo-dimensionalmixtureGaussiandistribution,atwo-dimensionalexpressionoftheminimumerrorthresholdingmethodonthetwo-dimensionalgray-levelhistogramisproposed.Inordertoimprovetherunningspeed,thefastrecursiveformulasarealsogiven.Experimentalresultsshowthatthetwo-dimensionalminimumerrorthresholdingmethodisavaluableimagesegmentationmethod,andcanbewelladaptedtotheimageswithnoisesandlargevariancesbetweenobjectandbackground.KeywordsThresholdsegmentation,minimumerrorthresholdingmethod,two-dimensionalgray-levelhistogram,,[1].,[2](Otsu)[3][4],.KittlerIllingworth,Bayes,,,.[5¡7],,(),\,,[7],.2007-11-232008-04-01ReceivedNovember23,2007;inrevisedformApril1,2008(60572133)SupportedbyNationalNaturalScienceFoundationofChina(60572133)1.7100612.7100711.DepartmentofInformationandControl,Xi0anInstituteofPostandTelecommunications,Xi0an7100612.SchoolofElectronicEngineering,XidianUniversity,Xi0an710071DOI:10.3724/SP.J.1004.2009.00386,,.,,[8¡14].,,[13¡15],,.,,,.1M£N,f(x;y)(x;y),f(x;y)2G=[0;1;¢¢¢;L¡1].h(g),.p(g)=P1i=0Pip(gji),Pi,p(g)p(gji)¹i,¾ip(gji)=1p2¼¾iexpµ¡(g¡¹i)22¾2i¶.,t,4:387P0(t)=tXg=0h(g)(1)P1(t)=L¡1Xg=t+1h(g)(2)¹0(t)=tXg=0h(g)gP0(t)(3)¹1(t)=L¡1Xg=t+1h(g)gP1(t)(4)¾20(t)=tXg=0(g¡¹0(t))2h(g)P0(t)(5)¾21(t)=L¡1Xg=t+1(g¡¹1(t))2h(g)P1(t)(6)t2G=[0;1;¢¢¢;L¡1],KittlerIllingworthJ(t)=1+2[P0(t)ln¾0(t)+P1(t)ln¾1(t)]¡2[P0(t)lnP0(t)+P1(t)lnP1(t)](7)J(t)t=t¤,t¤=argmin0tL¡1J(t)(8)f(x;y)=(0;f(x;y)t¤255;f(x;y)¸t¤(9),f(x;y).,,[16],h(g)p(g),,.,,.,.2M£N,g(x;y)(x;y)K£K,g(x;y)g(x;y)=66641K£KK0Xm=¡K0K0Xn=¡K0f(x+m;y+n)7775(10)b¢c;K,,K0=(K¡1)=2.g(x;y),L,L,f(x;y)g(x;y)(i;j).,(L¡1)£(L¡1),,.Pij,(i;j).Pij=cijM£N(11),cij(i;j),0·i;j·L¡1,PL¡1i=0PL¡1j=0Pij=1.,(s;t),1.,,34,34Pij¼0[8¡10].1,,12,C0(s;t)C1(s;t).1Fig.12Dsegmentationarea(X;Y)p(x;y)=12¼¾1¾2p1¡½2exp·¡12(1¡½2)£µ(x¡¹1)2¾21¡2½(x¡¹1)(y¡¹2)¾1¾2+(y¡¹2)2¾22¶¸(12),¹1¹2XY,¾21¾2238835XY,½XY.,(s;t)P0ij=P0(s;t)p(i;jj0)+P1(s;t)p(i;jj1)(13),P0(s;t),P1(s;t),p(i;jj0)p(i;jj1).p(i;jj0)p(i;jj1)¹¹¹0(s;t)=(¹00(s;t),¹01(s;t))T,¹¹¹1(s;t)=(¹10(s;t),¹11(s;t))T,¾¾¾20(s;t)=(¾200(s;t),¾201(s;t))T,¾¾¾21(s;t)=(¾210(s;t),¾211(s;t))T;p(i;jj0)p(i;jj1)½0½1.P0(s;t)=X(i;j)2C0(s;t)Pij=sXi=0tXj=0Pij(14)P1(s;t)=X(i;j)2C1(s;t)Pij=L¡1Xi=s+1L¡1Xj=t+1Pij(15)P0(s;t)+P1(s;t)¼1(16)¹¹¹0(s;t)=(¹00(s;t);¹01(s;t))T=0BBB@sPi=0tPj=0iPijP0(s;t);sPi=0tPj=0jPijP0(s;t)1CCCAT(17)¹¹¹1(s;t)=(¹10(s;t);¹11(s;t))T=0BBB@L¡1Pi=s+1L¡1Pj=t+1iPijP1(s;t);L¡1Pi=s+1L¡1Pj=t+1jPijP1(s;t)1CCCAT(18)½0(s;t)=sPi=0tPj=0[(i¡¹00(s;t))(j¡¹01(s;t))Pij]P0(s;t)¾00(s;t)¾01(s;t)(21)½1(s;t)=L¡1Pi=s+1L¡1Pj=t+1[(i¡¹10(s;t))(j¡¹11(s;t))Pij]P1(s;t)¾10(s;t)¾11(s;t)(22)¹¹¹T=(¹T0;¹T1)T=ÃL¡1Xi=0L¡1Xj=0iPij;L¡1Xi=0L¡1Xj=0jPij!T(23)PijP0ij,,\,R(s;t)=sXi=0tXj=0PijlnPijP0(s;t)p(i;jj0)+L¡1Xi=s+1L¡1Xj=t+1PijlnPijP1(s;t)p(i;jj1)(24)()12(1¡½20(s;t))sXi=0tXj=0·Pij(i¡¹00(s;t))2¾200(s;t)¡2½0(s;t)Pij(i¡¹00(s;t))(j¡¹01(s;t))¾00(s;t)¾01(s;t)+Pij(j¡¹01(s;t)2)¾201(s;t)¸=P0(s;t)(25)¾¾¾20(s;t)=(¾200(s;t);¾201(s;t))T=0BBB@sPi=0tPj=0(i¡¹00(s;t))2PijP0(s;t);sPi=0tPj=0(j¡¹01(s;t))2PijP0(s;t)1CCCAT(19)¾¾¾21(s;t)=(¾210(s;t);¾211(s;t))T=0BBB@L¡1Pi=s+1L¡1Pj=t+1(i¡¹10(s;t))2PijP1(s;t);L¡1Pi=s+1L¡1Pj=t+1(j¡¹11(s;t))2PijP1(s;t)1CCCAT(20)4:389sXi=0tXj=0Pijlnp(i;jj0)=sXi=0tXj=0Pijln12¼¾00(s;t)¾01(s;t)p1¡½20(s;t)exp½¡12(1¡½20(s;t))·(i¡¹00(s;t))2¾200(s;t)¡2½0(s;t)(i¡¹00(s;t))((j¡¹01(s;t))¾00(s;t)¾01(s;t)+(j¡¹01(s;t))2¾201(s;t)¸¾¸=¡P0(s;t)ln2¼¡P0(s;t)ln¾00(s;t)¾01(s;t)¡P0(s;t)lnp1¡½20(s;t)¡P0(s;t)(26)sXi=0tXj=0PijlnPijP0(s;t)p(i;jj0)=sXi=0tXj=0PijlnPij¡P0(s;t)lnP0(s;t)+P0(s;t)ln2¼+P0(s;t)ln¾00(s;t)¾01(s;t)+P0(s;t)lnp1¡½20(s;t)+P0(s;t)(27),L¡1Xi=s+1L¡1Xj=t+1PijlnPijP1(s;t)p(i;jj1)=L¡1Xi=s+1L¡1Xj=t+1PijlnPij¡P1(s;t)lnP1(s;t)+P1(s;t)ln2¼+P1(s;t)ln¾10(s;t)¾11(s;t)+P1(s;t)lnp1¡½21(s;t)+P1(s;t)(28)1Pij¼0,sXi=0tXj=0PijlnPij+L¡1Xi=s+1L¡1Xj=t+1PijlnPij¼L¡1Xi=0L¡1Xj=0PijlnPij(29)P0(s;t)+P1(s;t)¼1(30)R(s;t)J(s;t)=1¡P0(s;t)lnP0(s;t)¡P1(s;t)lnP1(s;t)+P0(s;t)ln¾00(s;t)¾01(s;t)+P1(s;t)ln¾10(s;t)¾11(s;t)+P0(s;t)lnp1¡½20(s;t)+P1(s;t)lnp1¡½21(s;t)(31).,½0½1,(s;t),.,½0=½1=1=K,K[17],J¤(s;t)=1¡P0(s;t)lnP0(s;t)¡P1(s;t)lnP1(s;t)+P0(s;t)ln¾00(s;t)¾01(s;t)+P1(s;t)ln¾10(s;t)¾11(s;t)(32),J¤(s;t)(s;t)=(s¤;t¤),(s¤;t¤)=argmin0s;tL¡1J¤(s;t)(33)f(x;y)=(0;f(x;y)s¤g(x;y)t¤255;(34)3,.,(0,0),.¹00(s;t)=sXi=0tXj=0iPij¹01(s;t)=sXi=0tXj=0jPij¾200(s;t)=sXi=0tXj=0i2Pij39035¾201(s;t)=sXi=0tXj=0j2Pij¾2T0(s;t)=L¡1Xi=0L¡1Xj=0i2Pij¾2T1(s;t)=L¡1Xi=0L¡1Xj=0j2Pij¹00(s;t)=sXi=0tXj=0iPijP0(s;t)=¹00(s;t)P0(s;t)(35)¹01(s;t)=sXi=0tXj=0jPijP0(s;t)=¹01(s;t)P0(s;t)(36)¹10(s;t)=L¡1Xi=s+1L¡1Xj=t+1iPijP1(s;t)¼¹T0¡¹00(s;t)1¡P0(s;t)(37)¹11(s;t)=L¡1Xi=s+1L¡1Xj=t+1jPijP1(s;t)¼¹T1¡¹01(s;t)1¡P0(s;t)(38)¾200(s;t)=sXi=0tXj=0i2PijP0(s;t)¡¹200(s;t)=¾200(s;t)P0(s;t)¡¹200(s;t)(39)¾201(s;t)=sXi=0