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,反投影獲得的圖像,Example,Back Projection Imaging System,取投影,反投影,圖像重建,反投影成像系統(tǒng),Dirac函數(shù),沖激函數(shù)h(x,y),掃描方式：旋轉(zhuǎn)+平移 射線先平行移動(dòng)，從物體的一側(cè)到另一側(cè)；然后旋轉(zhuǎn)一個(gè)角度，再作平行移動(dòng)，之后又旋轉(zhuǎn)一個(gè)角度，直到完

4、成半周。,Character: Blur column (star) shadow,輸入信號(hào)經(jīng)系統(tǒng)處理后產(chǎn)生影響，卷積的物理意義。,坐標(biāo)系：,為離散值,根據(jù)反投影重建定義式，,連續(xù)投影,將 代入,Dirac函數(shù)特性：,為,相應(yīng)于反投影重建算法的系統(tǒng)，其點(diǎn)擴(kuò)展函數(shù)不是Dirac函數(shù)，不是完美系統(tǒng)。,解析法,Fourier Transform Filtered Back Projection,Projection Theorem,圖像在某一角度上投影的傅立葉變換正好等于該圖像吸收系數(shù) f(x,y) 在相對(duì)應(yīng)角度（頻域）上二維傅立葉變換的形式。,Mathematic demonstration of

5、 Projection Theorem,先證 即,而 f(x,y) 的二維傅立葉變換為,當(dāng)v = 0 時(shí),傅立葉算子,旋轉(zhuǎn)坐標(biāo)系,若投影的視角為 ，利用旋轉(zhuǎn)坐標(biāo)系xr,yr，使積分路徑沿著yr, 此時(shí)xr與x 軸成 角。有,以上說(shuō)明u,v不是獨(dú)立的，變換等式成立的條件是u,v滿足角度關(guān)系，即投影定理。,Fourier Transform Method,采集不同視角下的投影； 求出各投影的一維傅立葉變換，即圖像二維圖像傅立葉變換過(guò)原點(diǎn)的各切片； 將上述切片匯集成圖像的二維傅立葉變換； 求傅立葉反變換獲得重建圖像,待建圖像是f(x,y), 其二維傅立葉變換為F(u,v)，極坐標(biāo)下為,根據(jù)投影定理，

6、可在不同視角 下的投影 的一維 傅立葉變換求得。,步驟：,數(shù)學(xué)表達(dá),求逆傅立葉變換：,在x,y坐標(biāo)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,在掃描過(guò)程中的實(shí)際位移用s表示，,交換積分次序：,定義,方括號(hào)內(nèi)為p與q的卷積，是下面反投影法沒有的，需要增加濾波處理，所以叫濾波反投影。,無(wú)卷積時(shí)：,濾波函數(shù),R-L濾