1、 解剖照片解剖照片CT成像成像T1加权加权MRI成像成像T2加权加权MRI成像成像PET成像成像x-ray sourcecollimatorobjectdetector晕环效应晕环效应以不同工作方式工作的以不同工作方式工作的CT扫描器扫描器直线的法线表示直线的法线表示Mathematical Description of Projections:Radon Transformyyg(x,)f(x,y)d y xxxg(x,)g(x,)f(x,y)(xcos ysin x)dxdyg(x,)f(x cos y sin,x sin y cos)d y x x cos y siny x sin y
2、cosx sg(s,)f(x,y)(xcos ysin s)dxdyg=RfRadon TransformRadon Transform Operator or Projection OperatorThe projection(or sum ray)in the direction defined by the angle iswhere=sTherefore,LetSincex xcos ysin正弦图正弦图0204060801001201400102030405060708090100f(x,y)0o179oMATLAB Example of the Radon TransformRg(
3、s,)xysg(s,0)sB=radon(A,0:179);Full circle in RTShifted full circle in RTEmpty circle in RTShifted empty circle in RTThin stick in RTShifted thin stick in RTFull triangle in RTShifted full triangle in RTFull square in RTShifted full square in RTEmpty square in RTShifted empty square in RT|x|2/3+|y|2/
4、3 1 in RT|x|+|y|1 in RT|x|3/2+|y|3/2 1 in RT|x|2+|y|2 1 in RT|x|6+|y|6 1 in RT|x|n+|y|n 1 for n in RT2D Gaussian in RTShifted 2D Gaussian in RT0(,)(cossin,)f x yg xyd 原始图像原始图像正弦图（投影数据）正弦图（投影数据）反投影重建反投影重建傅里叶切片定理（傅里叶切片定理（Projection-Slice Theorem）If g(s,)is the Radon transform of f(x,y),then G(ns,)=F(n
5、s cos,ns sin)G(ns,)=1D Fourier transform of g(s,)w.r.t.sF(nx,ny)=2D Fourier transform of f(x,y)xssg(s,)nxnyThe 1D FT of the projection of an image =Slice of the 2D FT of the imageF(nx,ny)f(x,y)yTheory of pure RT and IRT)2020sinsinF(,)fsincos,cossindf(,)F,cossind =|(,)djss xystststx yxyFed Radon tran
6、sformInverse Radon transform斜坡滤波器。斜坡滤波器。不可积？不可积？加窗限制带宽的斜坡滤波器（降低模糊和振铃效应）加窗限制带宽的斜坡滤波器（降低模糊和振铃效应）使用斜坡滤波器使用斜坡滤波器使用汉明窗斜坡滤波器使用汉明窗斜坡滤波器Original Image ARadon Transform B0o179oMATLAB Example of Computed Tomography using Filtered Back-Projection(FBP)Recovered image CB=radon(A,0:179);C=iradon(B,0:179,nearest,H
7、ann);load treesI=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)MATLAB program for Projection&Back ProjectionLoad an existing color imageTransform it into gray imageCrop itDisplay itRadon transform itTransform into gray imageDisplay projections as an imageExtract one projectionPlot one projectionInverse RadonDisplay recovered image投影滤波重建投影滤波重建基本的扇形射线几何原理基本的扇形射线几何原理sinD扇形射线上任意一点的极坐标表示扇形射线上任意一点的极坐标表示2201(,)(,)()mmf rqhddR 21()()2 sinhs(,)(,)cosqpD
