利用MATLAB进行图像重建的算法研究
摘要
[目的]：建立实验系统获得物体横剖面的投影数据，然后利用具有强大的数学计算和图像处理功能的MATLAB工具软件，完成对该剖面的图像重建。
[方法]：取一理想X射线源，测出其射线强度I_o和其经过物体某一剖面后的射线强度I。若衰减系数为μ=μ(x，y)，则在某一方向l沿路径L的总衰减为∫_Lμdl=ln(I_o／I)此即射线投影。因此测得I_o和I，可推出∫μdl，从而推出被积函数μ，即获得了μ分布(密度分布)的图像。这个过程用卷积一逆投影算法通过Matlab程序语言来实现。
[结果]：在Matlab语言环境里，完成卷积一逆投影算法设计，并通过用Matlab编程实现等角间隔扇形投影数据模拟采样，同时用该模拟采样得到的扇形投影数据通过Matlab伪彩图和灰度图绘图指令程序实现图像重建。
[结论]：1．由于Matlab工具软件具有强大的数值分析、矩阵运算、信号处理、图形显示和图像处理功能，并且还提供了良好的用户界面，因此使该语言环境下的图像重建可以比在其它语言(如C语言)环境下的图像重建更简便更快速设计质量也更高。
2．卷积一逆投影算法较之其它算法如迭代法、平行投影的傅立叶变换法有更高精度且能快速实现。
3．重建图像的质量受混叠效应和噪声干扰的影响。射线数越少混叠
[Objective]: Establish an experimental system to obtain the fan-shaped projection data and then realize the image reconstruction with Matlab, which has powerful function in math calculation, plot and image display.
[Methods]: 1. Establish an ideal X-ray source and a detecting instrument. 2. Detect the intensity of the X-ray source before and after passing though the cross section of the object , i.e. Io and I. Assuming declining coefficient μ = μ (x,y), the total declining along with L is ∫Lμdl=ln(I_o/I) which named as projection. With the I_o and I known, ∫ μ dl can be deduced and therefore the image of the intensity distribution can be obtained. 3. Realize the algorithm of filtering-back projection with Matlab.
[Results]: 1. Achieve the algorithm design for convolution-back projection method with matlab. 2. Obtain the analogue sampling data of equiangle interval fan-shaped projection by programming with matlab. 3. Draw the reconstruction image using pcolor and mat2gray in matlab.
[方法]：取一理想X射线源，测出其射线强度I_o和其经过物体某一剖面后的射线强度I。若衰减系数为μ=μ(x，y)，则在某一方向l沿路径L的总衰减为∫_Lμdl=ln(I_o／I)此即射线投影。因此测得I_o和I，可推出∫μdl，从而推出被积函数μ，即获得了μ分布(密度分布)的图像。这个过程用卷积一逆投影算法通过Matlab程序语言来实现。
[结果]：在Matlab语言环境里，完成卷积一逆投影算法设计，并通过用Matlab编程实现等角间隔扇形投影数据模拟采样，同时用该模拟采样得到的扇形投影数据通过Matlab伪彩图和灰度图绘图指令程序实现图像重建。
[结论]：1．由于Matlab工具软件具有强大的数值分析、矩阵运算、信号处理、图形显示和图像处理功能，并且还提供了良好的用户界面，因此使该语言环境下的图像重建可以比在其它语言(如C语言)环境下的图像重建更简便更快速设计质量也更高。
2．卷积一逆投影算法较之其它算法如迭代法、平行投影的傅立叶变换法有更高精度且能快速实现。
3．重建图像的质量受混叠效应和噪声干扰的影响。射线数越少混叠
[Objective]: Establish an experimental system to obtain the fan-shaped projection data and then realize the image reconstruction with Matlab, which has powerful function in math calculation, plot and image display.
[Methods]: 1. Establish an ideal X-ray source and a detecting instrument. 2. Detect the intensity of the X-ray source before and after passing though the cross section of the object , i.e. Io and I. Assuming declining coefficient μ = μ (x,y), the total declining along with L is ∫Lμdl=ln(I_o/I) which named as projection. With the I_o and I known, ∫ μ dl can be deduced and therefore the image of the intensity distribution can be obtained. 3. Realize the algorithm of filtering-back projection with Matlab.
[Results]: 1. Achieve the algorithm design for convolution-back projection method with matlab. 2. Obtain the analogue sampling data of equiangle interval fan-shaped projection by programming with matlab. 3. Draw the reconstruction image using pcolor and mat2gray in matlab.
引文
1．陈桂明 张明照等 应用MATLAB建模与仿真 科学出版社 北京
2．刘豹主编 现代控制理论 机械工业出版社 北京
3. The Math Works Inc.. Matlab User's Guide the Mathworks inc. NewYork
1. G. T. Herman, Image Reconstruction from Projections, Academic Press. 1980
2. P. Gilbert, iterative Methods for the Reconstruction of Three Dimentional Objects from There Projections, J. Theoret. Biol,, 36, 1972, 105-117.
3. A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall Inc. 1989
4．数字图像处理导论 赵荣椿 西北工业大学出版社 西安 1995
5．MATLAB6.0程序设计与实例应用 中国铁道出版社 北京 2001
6. The MathWorks Inc.. Matlab User's Guide the Mathworks inc. New York
7. E. L. Hall, Computer Image Processing and Recognition, Academic Press 1980
8．W．韦尔科威茨，S．多伊奇著，林劲\郭营川译，《生物医学仪器——原理与设计》，上海科学技术出版社，1981．12
2．刘豹主编 现代控制理论 机械工业出版社 北京
3. The Math Works Inc.. Matlab User's Guide the Mathworks inc. NewYork
1. G. T. Herman, Image Reconstruction from Projections, Academic Press. 1980
2. P. Gilbert, iterative Methods for the Reconstruction of Three Dimentional Objects from There Projections, J. Theoret. Biol,, 36, 1972, 105-117.
3. A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall Inc. 1989
4．数字图像处理导论 赵荣椿 西北工业大学出版社 西安 1995
5．MATLAB6.0程序设计与实例应用 中国铁道出版社 北京 2001
6. The MathWorks Inc.. Matlab User's Guide the Mathworks inc. New York
7. E. L. Hall, Computer Image Processing and Recognition, Academic Press 1980
8．W．韦尔科威茨，S．多伊奇著，林劲\郭营川译，《生物医学仪器——原理与设计》，上海科学技术出版社，1981．12