数字图像处理 图像重建

第5章图像复原与重建

ImageRestorationandReconstruction医学影像

(d)T2加权MRI成像(e)PET成像解剖照片CT成像T1加权MRI成像T2加权MRI成像PET成像由投影重建图像投影数据获取反投影法重建图像晕环效应反投影法重建图像计算机断层（ComputedTomography，CT）基本原理以不同工作方式工作的CT扫描器投影与Radon变换直线的法线表示Radon变换中的几何Radon变换中的几何Inputspacecoordinatesx,yInputfunctionf(x,y)Outputspacecoordinatesa,sOutputfunctionF(a,s)MathematicalDescriptionofProjections:RadonTransformyy´xx´qx´g(x´,q)g=RfRadonTransformRadonTransformOperatororProjectionOperatorTheprojection(orsumray)inthedirectiondefinedbytheangleqiswhere=sTherefore,LetSince图像和Radon变换正弦图f(x,y)0o179oMATLABExampleoftheRadonTransformRg(s,q)xyqsg(s,0)sB=radon(A,0:179);FullcircleinRTShiftedfullcircleinRTEmptycircleinRTShiftedemptycircleinRTThinstickinRTShiftedthinstickinRTFulltriangleinRTShiftedfulltriangleinRTFullsquareinRTShiftedfullsquareinRTEmptysquareinRTShiftedemptysquareinRT|x|2/3+|y|2/3

≤1inRT|x|+|y|≤1inRT|x|3/2+|y|3/2

≤1inRT|x|2+|y|2

≤1inRT|x|6+|y|6

≤1inRT|x|n+|y|n

≤1

for

n

inRT2DGaussianinRTShifted2DGaussianinRT从正弦图到图像：反投影法原始图像正弦图（投影数据）反投影重建傅里叶切片定理（Projection-SliceTheorem）Ifg(s,q)istheRadontransformoff(x,y),thenG(ns,q)=F(ns

cosq,ns

sinq)G(ns,q)=1DFouriertransformofg(s,q)w.r.t.sF(nx,ny)=2DFouriertransformoff(x,y)xsqsg(s,q)qnxnyThe1DFToftheprojectionofanimage=Sliceofthe2DFToftheimageF(nx,ny)f(x,y)yTheoryofpureRTandIRT

RadontransformInverseRadontransform斜坡滤波器。不可积？从正弦图到图像：滤波反投影法滤波反投影法：滤波器设计加窗限制带宽的斜坡滤波器（降低模糊和振铃效应）滤波反投影重建图像使用斜坡滤波器使用汉明窗斜坡滤波器滤波反投影重建图像OriginalImageARadonTransformB0o179oMATLABExampleofComputedTomographyusingFilteredBack-Projection(FBP)RecoveredimageCB=radon(A,0:179);C=iradon(B,0:179,'nearest','Hann');loadtreesI=ind2gray(X,map);A=imcrop(I,[1,1,128,128]);figure;imshow(A);B=radon(A,0:179);BI=ind2gray(B,map);figure;imshow(BI,128)BS=B(1:128,1);figure;plot(BS)C=iradon(B,0:179,'nearest','Hann');figure;imshow(C)MATLABprogramforProjection&BackProjectionLoadanexistingcolorimageTransformitintograyimageCropitDisplayitRadontransformitTransformintograyimageDisplayprojectionsasanimageExtractoneprojectionPlotoneprojectionInverseRadonDisplayrecoveredimage投影滤波重建使用