自动分离已经相乘的两个图像
Nic*_*nar 7 c++ matlab image image-processing
我正在寻找一种算法或C++/Matlab库,可用于将两个图像相乘.下面给出了该问题的可视示例.
图像1可以是任何东西(例如相对复杂的场景).图像2非常简单,可以在数学上生成.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),我们得到变换后的图像.
仅给出转换后的图像,我想估计图像1或图像2.是否有算法可以做到这一点？
这是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
UPDATE
有很多方法可以解决这个问题.以下是我的实验结果.感谢所有回答我问题的人!
1.图像的低通滤波
如duskwuff所述,采用变换图像的低通滤波器返回图像2的近似值.在这种情况下,低通滤波器是高斯滤波器.您可以看到可以使用低通滤波器识别图像中的乘法噪声.
2.同态过滤
正如EitenT所建议的,我研究了同态滤波.知道了这种类型的图像过滤的名称,我设法找到了一些我认为在解决类似问题时有用的参考文献.
SP Bank,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.,"非线性滤波乘法和卷积信号",IEEE Transactions on Audio and Electroacoustics,vol.16,不.3,pp.437-466,1968年9月.
盲图解卷积:理论和应用.博卡拉顿:CRC出版社,2007年.
Blind图像解卷积书的第5章特别好,并且包含许多对同态滤波的参考.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所建议的那样,我使用了一个目标函数fminsearch.由于我知道噪声的数学模型,我能够将其用作优化算法的输入. 这种方法完全针对特定问题,在所有情况下可能并不总是有用.
这是图2的重建:
这是图像1的重建,通过除以图像2的重建形成:
这是包含噪音的图像:
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,并不适用于所有情况.
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
这将是一个困难,不可靠的过程,因为你从根本上试图提取已被破坏的信息(两个图像的分离).把它完美地带回来是不可能的; 你能做的最好的就是猜测.
如果第二个图像总是相对"平滑",则可以通过对变换后的图像应用强低通滤波器来重建它(或者至少是它的近似值).有了这个,您可以反转乘法,或等效地使用互补高通滤波器来获得第一个图像.它不会完全相同,但它至少会成为一种东西.
我会尝试约束优化（fmincon在Matlab中）。如果您了解第二张图像的来源/性质，您可能可以定义一个生成类似噪声模式的多元函数。目标函数可以是生成的噪声图像与最后图像之间的相关性。