MATLAB 中的滤波反投影和设计滤波器

Question

我正在尝试编写自己的 MATLAB 代码来计算逆氡变换 (iradon)，到目前为止，我已经成功地使用斜坡滤波器、汉明 window 以及根据 Kak 和 Shakey 的教科书，在我的代码中使用空间域中的一维投影与 window h 的卷积。但是，我认为如果我将 window h 的 FFT 乘以一维投影的 FFT，我应该能够获得相同的重建。不幸的是，我得到的是一团糟。

function [out] = myfbp4(arg2);


ph = phantom();
sino = radon(phantom,0:0.1:179);

rho2 = 183; % max rho
rho1 = -183; % min rho;
step = 1;
npos = 367;
dtheta = 0.1;
angles = deg2rad(0:dtheta:179);
nproj = length(angles);



n1 = ceil(log(npos)/log(2)); 
N = 2^n1;                  % smallest power of 2 greater than npos (for fft efficiency)
N = 1024;   % for zero padding
fs = 1; % sampling frequency
T = 1;  % sample spacing

% grid for reconstructed image
x1 = rho1:step:rho2;
y1 = rho1:step:rho2;

[yyy, xxx] = ndgrid(-y1, x1);


% build filter h according to Kak and Shakey 
h = -floor(npos/2):T:floor(npos/2);

for i = 1:length(h)
    if (mod(h(i),2) == 1) 
        h(i) = -1./(h(i)^2 * pi^2 * T^2);
    else
        h(i) = 0;
    end    
    h(ceil(npos/2)) = 1/(4*T^2);
end

%%%%%%%%%%%   RAMP FILTER   %%%%%%%%%%%%%%%%
%%%%%%%%%% this is un needed when using h %%
%%%%%%%%%% see below... %%%%%%%%%%%%%%%%%%%%
rampFilterNum = [0:1:N/2-1 -N/2:1:-1];
rampFilterAbs = abs(rampFilterNum);
rampFilter = rampFilterAbs *fs/N; 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% positions made to correspond to N point fft
drho = (rho2 - rho1) / (npos-1);
rho  = rho1 + (0:(npos-1)) * drho;
positions = zeros(nproj, length(rho));
for i = 1:nproj,
  positions(i, :) = rho;
end

if (strcmp(arg2,'filter')) 
    % compute FT of h and multiply by fft projections
    fftProj = fft(sino, N);
    hfft = fft(h,N);
    fftProjFiltered = bsxfun(@times, hfft', fftProj);
    ifftProj = real(ifft(fftProjFiltered));
    filteredProjections = ifftProj;
end

if (strcmp(arg2,'conv')) 
    % make image my convolution of projections with h
    for iproj = 1:nproj
        sino(:, iproj) = conv(sino(:,iproj), h, 'same');
    end
    filteredProjections = sino;
end


% display the image through backprojection
fdata = zeros(size(xxx));
for iproj = 1:nproj
    theta = angles(iproj);
    rho1 = xxx*cos(theta) + yyy*sin(theta); % rotate coordinate system by theta
    %r = x1;
    r = positions(iproj,:);

    fdata1 = filteredProjections(1:npos,iproj); % filtered projections
    %fdata1  interp1(
    fdata2 = interp1(r, fdata1, rho1, 'linear', 0); 
    fdata = fdata + deg2rad(dtheta) * fdata2; %theta*fdata2;
end

out = fdata;
end

只需运行 out = myfbp4('conv') 或 myfbp4('filter') 就会显示不同的结果。似乎卷积工作正常，但过滤方法并没有像我希望的那样工作。

谁能看出问题所在？（抱歉，如果有任何多余的代码，我试图删除大部分......我还应该提到这段代码是从某个地方借来的并做了一些修改，但我不记得我是在哪里找到的）。

提前致谢

编辑：问题已解决。问题是我没有采用 window h 的傅里叶变换的绝对值来获得频率 window。对于那些发现这个的人， hfft = abs(fft(h,N)) 应该替换 hfft = fft(h, N).

Answer 1

问题已解决。问题是我没有采用 window h 的傅立叶变换的绝对值来获得频率 window。对于那些发现这个的人， hfft = abs(fft(h,N)) 应该替换 hfft = fft(h, N).

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