dudoNet中的反投影

表達式

在dudo-Net這篇文章中，用到了對反投影的一個表達，這個表達令我眼前一亮。
是這樣的：

其中，

整個過程實際上就是濾波反投影（FBP）的過程。

通過這個表達，我們可以得到CT的圖像域與sino圖域之間的梯度的一個關係：

從此，我們得到了一個關於兩個神經網絡：sinogram domain enhancement net（SE-NET）和image domain enhancement net(IE-NET)的一個關係

matlab實現反投影

我們通過該表達式來實現反投影
整個過程是這樣的：
1.建立圖形
2.投影
3.傅里葉變換，濾波，傅里葉逆變換
4.反投影

%% RadonInversionLayer
% 2020-1-24
close all; clear; clc;
%% Parameters
N = 512;
views = 360;
bins = 512;
dv = 2 * pi / views;
x1 = 0; y1 = 0; r1 = 200; mu1 = 0.5;
x2 = 80; y2 = 80; r2 = 30; mu2 = 1;
%% Generate model
vec = -N/2 + 0.5: N/2 - 0.5;
[xx, yy] = meshgrid(vec, -vec);
img = zeros(N, N);
img((xx - x1).^2 + (yy - y1).^2 <= r1^2) = mu1;
img((xx - x2).^2 + (yy - y2).^2 <= r2^2) = mu2;
figure('name','cirle in circle'); imshow(img, []); title('Original image');
%% Generate projection
dn = 0.5;
inte_vec = -N/2 + dn/2: dn: N/2 - dn/2;
[ox, oy] = meshgrid(vec, -inte_vec);
alpha = dv/2: dv: 2*pi - dv/2;
proj = zeros(views, bins);
for i = 1: views
    theta = alpha(i);
    X = ox .* cos(theta) - oy .* sin(theta);
    Y = ox .* sin(theta) + oy .* cos(theta);
    temp = interp2(xx, yy, img, X, Y, 'linear', 0);
    proj(i, :) = sum(temp);
end
figure; imshow(proj, []); title('Projection image');
%% Filter
% filter = [0: 2/N: 1-2/N, 1: -2/N: 2/N];
filter = 2*[0: (N/2-1), N/2: -1: 1]/N;
proj_filter = zeros(views, bins);
for i = 1: views
    proj_filter(i, :) = fft(proj(i, :));
    proj_filter(i, :) = proj_filter(i, :) .* filter;
    proj_filter(i, :) = ifft(proj_filter(i, :));
end
figure; imshow(proj_filter, []); title('Filter Projection image');
%%BackProjection2
rec = zeros(N, N);
for i = 1: views
    theta = alpha(i);
    pos =  xx .* cos(theta) + yy .* sin(theta);
    temp1 = interp1(vec, proj_filter(i, :), floor(pos), 'linear');
    temp2 = interp1(vec, proj_filter(i, :), ceil(pos), 'linear');
    temp = (ceil(pos)-pos).*temp1+(pos-floor(pos)).*temp2;
    rec = rec + temp/N;
    rec(rec<0) = 0;
end
figure; imshow(rec, []); title('Reconstruction2');