医学影像物理学课件：07yj-X-ray CT02

1、X-CT Imaging Method,Equation Group Back Projection Fourier Transform Filtered Back Projection,The key of X-CT is get the distribution of attenuation coefficients from projections.,Equation Group,Horizontal direction:,Vertical direction:,Effective,Diagonal direction:,Character: There are so much equa

2、tions and operand that computer can not adopt this way.,Back (reverse) projection,Back projection method is a faster imaging method of CT. It project the P(projection, the sum of attenuation coefficients, come from detector array) to the pixel space in the reverse direction. By this way and some mat

3、hematical operation, we can get a image of distribution of attenuation coefficients.,Simple Model,Mathematic Expression of Back Projection,投影值：,某一角度的反投影,反投影获得的图像,Example,Back Projection Imaging System,取投影,反投影,图像重建,反投影成像系统,Dirac函数,冲激函数h(x,y),扫描方式：旋转+平移 射线先平行移动，从物体的一侧到另一侧；然后旋转一个角度，再作平行移动，之后又旋转一个角度，直到完

4、成半周。,Character: Blur column (star) shadow,输入信号经系统处理后产生影响，卷积的物理意义。,坐标系：,为离散值,根据反投影重建定义式，,连续投影,将 代入,Dirac函数特性：,为,相应于反投影重建算法的系统，其点扩展函数不是Dirac函数，不是完美系统。,解析法,Fourier Transform Filtered Back Projection,Projection Theorem,图像在某一角度上投影的傅立叶变换正好等于该图像吸收系数 f(x,y) 在相对应角度（频域）上二维傅立叶变换的形式。,Mathematic demonstration of

5、 Projection Theorem,先证 即,而 f(x,y) 的二维傅立叶变换为,当v = 0 时,傅立叶算子,旋转坐标系,若投影的视角为 ，利用旋转坐标系xr,yr，使积分路径沿着yr, 此时xr与x 轴成 角。有,以上说明u,v不是独立的，变换等式成立的条件是u,v满足角度关系，即投影定理。,Fourier Transform Method,采集不同视角下的投影； 求出各投影的一维傅立叶变换，即图像二维图像傅立叶变换过原点的各切片； 将上述切片汇集成图像的二维傅立叶变换； 求傅立叶反变换获得重建图像,待建图像是f(x,y), 其二维傅立叶变换为F(u,v)，极坐标下为,根据投影定理，

6、可在不同视角 下的投影 的一维 傅立叶变换求得。,步骤：,数学表达,求逆傅立叶变换：,在x,y坐标系,Filtered Back Projection (Convolution Method/ Convolution Back Projection),Filtered Back Projection is a improvement of Back projection. On the base of the latter, the digital filter is adopt before CT image is formed. By filtering high frequency noise, there is no star shadow any longer On the image from this method. We can obtain good images without pay too much. So it is the most typical method in X-ray CT.,Mathematic Base of FBP,在扫描过程中的实际位移用s表示，,交换积分次序：,定义,方括号内为p与q的卷积，是下面反投影法没有的，需要增加滤波处理，所以叫滤波反投影。,无卷积时：,滤波函数,R-L滤