CT reconstruction Simulation

Transcription

1 Lecture IX CT reconstruction Simulation 1

2 Review 1. Direct back projection alg0rithm?. Filtered back projection 3. R-L, S-L filter 4. What are the four kinds of back projection algorithm? 5. What are the processes and features?

3 Contents 1. Shepp Logan phantom. Generation of simulation data 3. Convolution process 4. Back projection 5. Result evaluation 3

4 1. Shepp-Logan phantom 4

5 Shepp-Logan Phantom " Used to evaluate the algorithm. " Shepp-Logan phantom is the most famous one. " Shepp-Logan phantom consists of ten ellipses named a-j, with different center, axis, rotate angle and attenuation coefficient. " Pay attention to the attenuation coefficient accumulation. 5

6 Shepp-Logan phantom " Ellipse function " Matlab program " Coordinate transform 6 ellip = [x0 y0 z0 a b angle mu]; t=0:pi/50:*pi; axis([ ]); hold on; grid on; for m=1:10 x=ellip(m,5)*cos(t); y=ellip(m,4)*sin(t); xx=x*cos(ellip(m,6)*pi/180) -y*sin(ellip(m,6)*pi/180)+ellip(m,1); yy=x*sin(ellip(m,6)*pi /180)+y*cos(ellip(m,6)*pi/180)+ellip(m,); plot(xx,yy); end

7 Shepp-Logan phantom gray image " Discretization 1. 每行每列遍历图像中各点, 根据坐标位置判断处于哪个椭圆的区域, 从而确定灰度. 确定了椭圆, 进行离散, 形成矩阵 然后进行矩阵操作 : 选择和平移 课下自己尝试! P = phantom(def, n); P = phantom(e, n); 7

8 . Parallel beam projection 8

9 Parallel beam projection - for ellipse located at origin point and without rotation 1. Density function. Projection function x y ρ + 1 (, ) f x y = A B gθ ( R) = ρdl = ρmn MN 0 Others 3. Line MN 4. Point of intersection x cosθ + ysinθ = R x y 1, 1, RA = RB = cosθ ± AB sinθ r sinθ ± AB cosθ r r r R R 5. Length of MN MN = r = A cos θ + B AB r r ( x1 x ) + ( y1 y) = 6. Final projection function R sin θ g 9 θ ( R) = ρ AB r r 0 R R r Others

10 Parallel beam projection for arbitrary ellipse 1. Ellipse function [( x x ) cosα + ( y y ) sinα ] [ ( x x ) sinα + ( y y ) cosα ] = AB PQ = r α = A cos A. Length of PQ α α r r R α ( θ α ) + B sin ( θ α ) B 1 θ g θ, α ( R) = ρpq ( R) 0 g θ, α = AB = ρ A cos R α = R x 0 cosθ y0 sinθ 3. Final projection function ( θ α ) + B sin ( θ α ) ( R x0 cosθ y0 sinθ ) A cos ( θ α ) + B sin ( θ α ) 10 Rα r α Others

11 Projection program % P_Projection function function P=P_projection() Y_Length=40; L=3; %Projection times in 180 degree N=18; d=y_length/n; ellip=sub_shepp; P=zeros(L,N,10); k=0; for i=1:size(ellip,1) %size(ellip,1)=10 k=k+1; x_0=ellip(i,1)*y_length/; %Origin point coordinate x0 y_0=ellip(i,)*y_length/; %Origin point coordinate y0 A=ellip(i,4)*Y_Length/; %Long semi-axis B=ellip(i,5)*Y_Length/; %Short semi-axis rf=ellip(i,6)*pi/180; %Alpha,rotate angle of ellipse rol=ellip(i,7); %Attenuation Coefficient p=0; for theta=0:pi/l:pi-pi/l %theta=0~31 p=p+1; n=-n/:n/-1;%n=[-64:1:63] C=*rol*A*B;% constant C=*rol*A*B T=(A^)*(cos(theta-rf))^+(B^)*(sin(theta -rf))^;%r^ I=(n.*d-x_0*cos(theta)-y_0*sin(theta)).^;%Vector G=(C*sqrt(T-I)./T).*(I<=T)+0.*(I>T); %subplot(1); figure(k) plot(g); hold on; grid on; P1(p,:)=G;%P1-Single ellipse' projection, 3 angles,18 points end P(:,:,k)=P1;%10 ellipses' projection hold on figure(10+k); surf(p1); %imshow(p1,[]);%we need change the width end 11 R = radon(i, theta);

12 Projection results " Ellipse a " Ellipse j 1

13 3. R-L filter 13

14 Filter description ( ) = ρ sin ( ρ ) sin ( ρ ) = ( ) h R c R c R C R R L

15 R-L filter Matlab program T=1; %Sample interval L=64; RouZero=1/(*T); N=linspace(-floor(L/),floor(L/)-1,L); %N[64]=[-3,-31,...0,...31] R=N*T; RFFilterConvolutionFunction=RouZero^*(*sinc(*RouZero*R)- sinc(rouzero*r).^); RFFilterFunction=abs(ifftshift(fft(RFFilterConvolutionFunction))); figure(1); subplot(,1,1); plot(rffilterconvolutionfunction); hold on; plot(rffilterconvolutionfunction,'r*'); grid on; xlabel('t (sample =NT)'); ylabel('rffilterconvolutionfunction'); title(['interval=',numstr(t)]); subplot(1); plot(rffilterfunction); grid on; 15

16 4-1. Filter correction 16

17 Filter correction Matlab program function Q=h_projection() close all; clear all; Y_Length=40; L=64; N=18; d=y_length/n; p=0; q=0; H=zeros(L,N,10); h1=ones(l,1);%vector of 64*1 H1=sub_hfilter(N,d);%Matrix of 1*18 H=h1*ifftshift(H1);%Matrix of 64*18 for j=1:10 H(:,:,j)=H;%Matrix of 64*18*10 end load P_projection; S=fftn(P);%Matrix of 64*18*10 Q=ifftn(S.*H); OriginProjection=P(1,:,1); CorrectedProjectoin=Q(1,:,1); figure(); subplot(,1,1); plot(originprojection); hold on; subplot(,1,); plot(correctedprojectoin); filename='h_projection'; save(filename,'q'); FFT -> Multiply -> IFFT 17

18 4- Convolution correction 18

19 Convolution process wrong " 投影函数将向两边分别延伸 N 点 " 卷积函数对称截取 N 点 " 卷积运算后截取 N 点 %%%%%%%%%%%%%%%%%%%%%%% %%%%R_L filter 56 point t=linspace(-18,17,56); filt=0.065*(sinc(t)/-sinc(t/).^/4); figure; plot(filt); title('r L filter); %%%%%%%%%% convoluation process tic shade_temp = zeros(768,180); %56*3 shade_temp(57:51,:) = shade;%middle part 56 element for n = 1:180; shade1_temp(:,n) = conv(shade_temp(:,n),filt');%conv. end shade1 = shade1_temp(385:640,:); %Obtain the result shade1_temp(:,n) = conv(shade_temp(:,n),filt, same ); 19

20 5. Backprojection process 0

21 Back projection R 0 y0 x y 0 0 R 0 = x 0 cosθ + sinθ [ i ( N + 1) / ] Δ j ( N + 1) / = = [ ]Δ N + 1 N + 1 = i cosθ + j sinθ Δ N 1 = = R R 0 0 = ( i 1) cosθ + ( j 1) sinθ ( cosθ + sinθ ) Δ + ( N / ) Δ N 1 ( i 1) cosθ + ( j 1) sinθ ( cosθ + sinθ ) + Δ N Pixel driven or ray driven? N N 1 C θ = ( cosθ + sinθ ) R0 = ( i 1) cosθ + ( j 1) sinθ + C = n + δ 0 ( i, j) = ( 1 δ ) Q ( n ) + δq ( n 1) F θ 0 θ 0 + θ 1

22 Back projection program %%%%%%%%%%%%%%%% Back projection d_th = pi/180; re = zeros(56,56); for view = 1:180 l = ((18.5)*sin((view-1)*d_th))+((18.5)*cos((view-1)*d_th))-18; %-C θ l_temp(view) = l; for x = 1:56 for y = 1:56 n = (x-1)*sin((view-1)*d_th)+(y-1)*cos((view-1)*d_th)-l; if (1<n)&&(n<56) nn = fix(n); n0 = n-nn; re(x,y) = re(x,y)+(1-n0)*shade1(nn,view)+n0*shade1(nn+1,view); %interpolation else re(x,y) = re(x,y); end end end end I = iradon(p, theta, interp, filter, frequency_scaling, output_size);

23 6. Evaluation 3

24 Evaluation of result figure x1 = re(18,:); imshow(re,[]) title( 重建后图像 ') load head.mat figure y1 = head(18,:); imshow(head,[]) title( 重建前图像 '); figure plot(x1,'r') hold on plot(y1); title( 红 --- 重建图像蓝 --- 原始图像 ') 4

25 Evaluation criteria " 归一化均方根距离判据 d = N N ( tu, v ru, v ) u= 1 v= 1 N N ( t ) u, v t u= 1 v= 1 " 归一化平均绝对距离判据 r N N u= 1 v= 1 = N t u, v N u= 1 v= 1 t r u, v " 最坏情况距离 [ N / ] [ N / ] u, v e = max Ti, j Ri, j 1 i 1 j 1 1 T ( t + t + t t ) R ( R + R + R R ) i, j = i, j i+ 1, j i, j+ 1 + i+ 1, j i, j " Any other??? tic,toc = i, j i+ 1, j i, j+ 1 + i+ 1, j+ 1 4

26 Conclusion 1. Parallel beam backprojection reconstruction process?. Filter function 3. Convolution & filter correction 4. Result evaluation 6

27 Questions??? 7