數(shù)字圖像處理 圖像重建

第5章圖像復原與重建

ImageRestorationandReconstruction醫(yī)學影像

(d)T2加權MRI成像(e)PET成像解剖照片CT成像T1加權MRI成像T2加權MRI成像PET成像由投影重建圖像投影數(shù)據(jù)獲取反投影法重建圖像暈環(huán)效應反投影法重建圖像計算機斷層（ComputedTomography，CT）基本原理以不同工作方式工作的CT掃描器投影與Radon變換直線的法線表示Radon變換中的幾何Radon變換中的幾何Inputspacecoordinatesx,yInputfunctionf(x,y)Outputspacecoordinatesa,sOutputfunctionF(a,s)MathematicalDescriptionofProjections:RadonTransformyy′xx′qx′g(x′,q)g=RfRadonTransformRadonTransformOperatororProjectionOperatorTheprojection(orsumray)inthedirectiondefinedbytheangleqiswhere=sTherefore,LetSince圖像和Radon變換正弦圖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從正弦圖到圖像：反投影法原始圖像正弦圖（投影數(shù)據(jù)）反投影重建傅里葉切片定理（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斜坡濾波器。不可積？從正弦圖到圖像：濾波反投影法濾波反投影法：濾波器設計加窗限制帶寬的斜坡濾波器（降低模糊和振鈴效應）濾波反投影重建圖像使用斜坡濾波器使用漢明窗斜坡濾波器濾波反投影重建圖像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投影濾波重建使用