我正在寻找一种算法或 C++/Matlab 库，可用于分离两个相乘的图像。下面给出了这个问题的可视化示例。
图像 1 可以是任何东西（例如相对复杂的场景）。图 2 非常简单，可以用数学方法生成。图像 2 总是具有相似的形态（即下降趋势）。通过将图像 1 乘以图像 2（使用逐点乘法），我们得到转换后的图像。
只给定转换后的图像，我想估计图像 1 或图像 2。有没有算法可以做到这一点？
以下是 Matlab 代码和图像：
load('trans.mat');
imageA = imread('room.jpg');
imageB = abs(response); % loaded from MAT file
[m,n] = size(imageA);
image1 = rgb2gray( imresize(im2double(imageA), [m n]) );
image2 = imresize(im2double(imageB), [m n]);
figure; imagesc(image1); colormap gray; title('Image 1 of Room')
colorbar
figure; imagesc(image2); colormap gray; title('Image 2 of Response')
colorbar
% This is image1 and image2 multiplied together (point-by-point)
trans = image1 .* image2;
figure; imagesc(trans); colormap gray; title('Transformed Image')
colorbar
更新
有很多方法可以解决这个问题。这是我的实验结果。感谢所有回答我问题的人！
1、图像的低通滤波
正如duskwuff 所指出的，对变换后的图像进行低通滤波器会返回图像2 的近似值。在这种情况下，低通滤波器是高斯的。您可以看到使用低通滤波器可以识别图像中的乘性噪声。
2.同态滤波
正如 EitenT 所建议的，我检查了同态滤波。知道这种类型的图像过滤的名称后，我设法找到了一些我认为有助于解决类似问题的参考资料。
SP 银行、信号处理、图像处理和模式识别。纽约：普伦蒂斯霍尔，1990 年。
A. Oppenheim, R. Schafer 和 J. Stockham, T.，“乘法和卷积信号的非线性滤波”，IEEE Transactions on Audio and Electroacoustics，第一卷。16，没有。3，第 437 – 466 页，1968 年 9 月。
盲图像反卷积：理论与应用。博卡拉顿：CRC 出版社，2007 年。
Blind image deconvolution一书的第5章特别好，里面有很多同态滤波的参考资料。这可能是最通用的方法，可以在许多不同的应用程序中很好地工作。
3.优化使用fminsearch
正如 Serg 所建议的，我使用了一个目标函数fminsearch。由于我知道噪声的数学模型，因此我能够将其用作优化算法的输入。 这种方法完全是针对特定问题的，并且可能并不总是在所有情况下都有用。
这是图像2的重建：
这是图像 1 的重建，通过除以图像 2 的重建形成：
这是包含噪声的图像：
源代码
这是我的问题的源代码。如代码所示，这是一个非常具体的应用程序，并非在所有情况下都能正常工作。
N = 1001;
q = zeros(N, 1);
q(1:200) = 55;
q(201:300) = 120;
q(301:400) = 70;
q(401:600) = 40;
q(601:800) = 100;
q(801:1001) = 70;
dt = 0.0042;
fs = 1 / dt;
wSize = 101;
Glim = 20;
ginv = 0;
[R, ~, ~] = get_response(N, q, dt, wSize, Glim, ginv);
rows = wSize;
cols = N;
cut_val = 200;
figure; imagesc(abs(R)); title('Matrix output of algorithm')
colorbar
figure;
imagesc(abs(R)); title('abs(response)')
figure;
imagesc(imag(R)); title('imag(response)')
imageA = imread('room.jpg');
% images should be of the same size
[m,n] = size(R);
image1 = rgb2gray( imresize(im2double(imageA), [m n]) );
% here is the multiplication (with the image in complex space)
trans = ((image1.*1i)) .* (R(end:-1:1, :));
figure;
imagesc(abs(trans)); colormap(gray);
% take the imaginary part of the response
imagLogR = imag(log(trans));
% The beginning and end points are not usable
Mderiv = zeros(rows, cols-2);
for k = 1:rows
val = deriv_3pt(imagLogR(k,:), dt);
val(val > cut_val) = 0;
Mderiv(k,:) = val(1:end-1);
end
% This is the derivative of the imaginary part of R
% d/dtau(imag((log(R)))
% Do we need to remove spurious values from the matrix?
figure;
imagesc(abs(log(Mderiv)));
disp('Running iteration');
% Apply curve-fitting to get back the values
% by cycling over the cols
q0 = 10;
q1 = 500;
NN = cols - 2;
qout = zeros(NN, 1);
for k = 1:NN
data = Mderiv(:,k);
qout(k) = fminbnd(@(q) curve_fit_to_get_q(q, dt, rows, data),q0,q1);
end
figure; plot(q); title('q value input as vector');
ylim([0 200]); xlim([0 1001])
figure;
plot(qout); title('Reconstructed q')
ylim([0 200]); xlim([0 1001])
% make the vector the same size as the other
qout2 = [qout(1); qout; qout(end)];
% get the reconstructed response
[RR, ~, ~] = get_response(N, qout2, dt, wSize, Glim, ginv);
RR = RR(end:-1:1,:);
figure; imagesc(abs(RR)); colormap gray
title('Reconstructed Image 2')
colorbar;
% here is the reconstructed image of the room
% NOTE the division in the imagesc function
check0 = image1 .* abs(R(end:-1:1, :));
figure; imagesc(check0./abs(RR)); colormap gray
title('Reconstructed Image 1')
colorbar;
figure; imagesc(check0); colormap gray
title('Original image with noise pattern')
colorbar;
function [response, L, inte] = get_response(N, Q, dt, wSize, Glim, ginv)
fs = 1 / dt;
Npad = wSize - 1;
N1 = wSize + Npad;
N2 = floor(N1 / 2 + 1);
f = (fs/2)*linspace(0,1,N2);
omega = 2 * pi .* f';
omegah = 2 * pi * f(end);
sigma2 = exp(-(0.23*Glim + 1.63));
sign = 1;
if(ginv == 1)
sign = -1;
end
ratio = omega ./ omegah;
rs_r = zeros(N2, 1);
rs_i = zeros(N2, 1);
termr = zeros(N2, 1);
termi = zeros(N2, 1);
termr_sub1 = zeros(N2, 1);
termi_sub1 = zeros(N2, 1);
response = zeros(N2, N);
L = zeros(N2, N);
inte = zeros(N2, N);
% cycle over cols of matrix
for ti = 1:N
term0 = omega ./ (2 .* Q(ti));
gamma = 1 / (pi * Q(ti));
% calculate for the real part
if(ti == 1)
Lambda = ones(N2, 1);
termr_sub1(1) = 0;
termr_sub1(2:end) = term0(2:end) .* (ratio(2:end).^-gamma);
else
termr(1) = 0;
termr(2:end) = term0(2:end) .* (ratio(2:end).^-gamma);
rs_r = rs_r - dt.*(termr + termr_sub1);
termr_sub1 = termr;
Beta = exp( -1 .* -0.5 .* rs_r );
Lambda = (Beta + sigma2) ./ (Beta.^2 + sigma2); % vector
end
% calculate for the complex part
if(ginv == 1)
termi(1) = 0;
termi(2:end) = (ratio(2:end).^(sign .* gamma) - 1) .* omega(2:end);
else
termi = (ratio.^(sign .* gamma) - 1) .* omega;
end
rs_i = rs_i - dt.*(termi + termi_sub1);
termi_sub1 = termi;
integrand = exp( 1i .* -0.5 .* rs_i );
L(:,ti) = Lambda;
inte(:,ti) = integrand;
if(ginv == 1)
response(:,ti) = Lambda .* integrand;
else
response(:,ti) = (1 ./ Lambda) .* integrand;
end
end % ti loop
function sse = curve_fit_to_get_q(q, dt, rows, data)
% q = trial q value
% dt = timestep
% rows = number of rows
% data = actual dataset
fs = 1 / dt;
N2 = rows;
f = (fs/2)*linspace(0,1,N2); % vector for frequency along cols
omega = 2 * pi .* f';
omegah = 2 * pi * f(end);
ratio = omega ./ omegah;
gamma = 1 / (pi * q);
% calculate for the complex part
termi = ((ratio.^(gamma)) - 1) .* omega;
% for now, just reverse termi
termi = termi(end:-1:1);
%
% Do non-linear curve-fitting
% termi is a column-vector with the generated noise pattern
% data is the log-transformed image
% sse is the value that is returned to fminsearchbnd
Error_Vector = termi - data;
sse = sum(Error_Vector.^2);
function output = deriv_3pt(x, dt)
N = length(x);
N0 = N - 1;
output = zeros(N0, 1);
denom = 2 * dt;
for k = 2:N0
output(k - 1) = (x(k+1) - x(k-1)) / denom;
end