how to combine two same scene images for image registration - image

I try to do image registration on two grayscale images where the images were taken twice with different views. The images were taking by myself using a Lifecam camera.
To register these images, I used template matching method and normalized cross correlation as similarity measure and found the right location. But the result after combination of these two images was not good as I wish. I don't know how to fix it. Do I need to do some rotation or translation first before combine it? If so, I have no idea how to get the real angle for rotation. Or do you have any idea how to fix the image result without applying any rotation?
Input image 1:
Input Image 2:
Result:
This my code:
A = imread('image1.jpg');
B = imread('image2.jpg');
[M1, N1] = size(A); % size imej A n B
[M2, N2] = size(B);
%% finding coordinated of (r2,c2)
r1 = size(A,1)/2; % midpoint of image A as coordinate
c1 = size(A,2
template = imcrop(A,[(c1-20) (r1-20) 40 40]);
[r2, c2] = normcorr(temp,B); % Normalized cross correlation
%% count distance of coordinate (r1,c1) in image A and (r2,c2)in image B
UA = r1; % distance of coordinate (r1,c1) from top in image A
BA = M1 - r1; % distance of coordinate (r1,c1) from bottom
LA = c1; % left distance from (r1,c1)
RA = N1 - c1; % right distance from (r1,c1)
UB = r2; % finding distance of coordinate (r2,c2) from top,
BB = M2 - r2; % bottom, left and right in image B
LB = c2;
RB = N2 - c2;
%% zero padding for both image
if LA > LB
L_diff = LA - LB; % value of columns need to pad with zero on left side
B = [zeros(M2,L_diff),B];
else
L_diff = LB - LA;
A = [zeros(M1,L_diff),A];
end
if RA > RB
R_diff = RA - RB; % value of columns need to pad with zero on right side
B = [B, zeros(M2,R_diff)];
else
R_diff = RB - RA;
A = [A, zeros(M1,R_diff)];
end
N1 = size(A, 2); % renew value column image A and B
N2 = size(B, 2);
if UA > UB
U_diff = UA - UB; % value of rows need to pad with zero on top
B = [zeros(U_diff,N2);B];
else
U_diff = UB - UA;
A = [zeros(U_diff,N1);A];
end
if BA > BB
B_diff = BA - BB; % value of rows need to pad with zero on bottom
B = [B; zeros(B_diff,N2)];
else
B_diff = BB - BA;
A = [A; zeros(B_diff,N1)];
end
%% find coordinate that have double value
if LA > LB
r = r1;
c = c1;
else
r = r2;
c = c2;
end
if UA >= UB
i_Start = r - UB + 1;
else
i_Start = r - UA + 1;
end
if BA >= BB
i_Stop = r + BB ;
else
i_Stop = r + BA;
end
if LA >= LB
j_Start = c - c2 + 1;
else
j_Start = c - c1 + 1;
end
if RA >= RB
j_Stop = c + RB;
else
j_Stop = c + RA;
end
%% add image A and B
A = im2double(A);
B = im2double(B);
final_im = A + B;
for i = i_Start:i_Stop
for j = j_Start:j_Stop
final_im(i,j) = final_im(i,j)/2;
end
end
final_im = im2uint8(final_im);

The answer from rayryeng in Ryan L's first link is quite applicable here. Cross-correlation likely won't provide a close enough match between the two images since the transformation between the two images is more accurately described as a homography than a 2D rigid transform.
Accurate image registration requires that you find this projective transformation. To do so you can find a set of corresponding points in the two images (using SURF, as mentioned above, usually works well) and then use RANSAC to obtain the homography's parameters from the corresponding points. RANSAC does a nice job even when some of the "corresponding" features in your two images are actually not correct matches. Once found, you can use the transformation to move one of your images to the other's point of view and fuse.
Here's a nice explanation of feature matching, RANSAC, and fusing two images with some Matlab code samples. The lecture uses SIFT features, but the idea still works for SURF.

Best published way to perform such a registration is based on fiducial points. You can choose the most clear edges or crossing points as a fiducial and then adjust the smoothness and regularization parameter to register them together.
look at the SlicerRT package. and let me know if you face any problem.

Related

Kaczmarz animation

i am asking for help.. I want to animate the Kaczmarz method on Matlab. It's method allows to find solution of system of equations by the serial projecting solution vector on hyperplanes, which which is given by the eqations of system.
And i want make animation of this vector moving (like the point is going on the projected vectors).
%% System of equations
% 2x + 3y = 4;
% x - y = 2;
% 6x + y = 15;
%%
A = [2 3;1 -1; 6 1];
f = [4; 2; 15];
resh = pinv(A)*f
x = -10:0.1:10;
e1 = (1 - 2*x)/3;
e2 = (x - 2);
e3 = 15 - 6*x;
plot(x,e1)
grid on
%
axis([0 4 -2 2])
hold on
plot(x,e2)
hold on
plot(x,e3)
hold on
precision = 0.001; % точность
iteration = 100; % количество итераций
lambda = 0.75; % лямбда
[m,n] = size(A);
x = zeros(n,1);
%count of norms
for i = 1:m
nrm(i) = norm(A(i,:));
end
for i = 1:1:iteration
j = mod(i-1,m) + 1;
if (nrm(j) <= 0), continue, end;
predx = x;
x = x + ((f(j) - A(j,:)*x)*A(j,:)')/(nrm(j))^2;
p = plot(x);
set(p)
%pause 0.04;
hold on;
if(norm(predx - x) <= precision), break, end
end
I wrote the code for this method, by don't imagine how make the animation, how I can use the set function.
In your code there are a lot of redundant and random pieces. Do not call hold on more than once, it does nothing. Also set(p) does nothing, you want to set some ps properties to something, then you use set.
Also, you are plotting the result, but not the "change". The change is a line between the previous and current, and that is the only reason you'd want to have a variable such as predx, to plot. SO USE IT!
Anyway, this following code plots your algorithm. I added a repeated line to plot in green and then delete, so you can see what the last step does. I also changed the plots in the begging to just plot in red so its more clear what is each of the things.
Change your loop for:
for i = 1:1:iteration
j = mod(i-1,m) + 1;
if (nrm(j) <= 0), continue, end;
predx = x;
x = x + ((f(j) - A(j,:)*x)*A(j,:)')/(nrm(j))^2;
plot([predx(1) x(1)],[predx(2) x(2)],'b'); %plot line
c=plot([predx(1) x(1)],[predx(2) x(2)],'g'); %plot it in green
pause(0.1)
children = get(gca, 'children'); %delete the green line
delete(children(1));
drawnow
% hold on;
if(norm(predx - x) <= precision), break, end
end
This will show:

a program to apply the following transformation function to a grayscale image

I want to apply following transformation function to a grayscale image, i know how to apply it to the following function,
my question is how do i apply a program to the following transformation function,
code so far,
clear;
pollen = imread('Fig3.10(b).jpg');
u = double(pollen);
[nx ny] = size(u)
nshades = 256;
r1 = 80; s1 = 10; % Transformation by piecewise linear function.
r2 = 140; s2 = 245;
for i = 1:nx
for j = 1:ny
if (u(i,j)< r1)
uspread(i,j) = ((s1-0)/(r1-0))*u(i,j)
end
if ((u(i,j)>=r1) & (u(i,j)<= r2))
uspread(i,j) = ((s2 - s1)/(r2 - r1))*(u(i,j) - r1)+ s1;
end
if (u(i,j)>r2)
uspread(i,j) = ((255 - s2)/(255 - r2))*(u(i,j) - r2) + s2;
end
end
end
hist= zeros(nshades,1);
for i=1:nx
for j=1:ny
for k=0:nshades-1
if uspread(i,j)==k
hist(k+1)=hist(k+1)+1;
end
end
end
end
plot(hist);
pollenspreadmat = uint8(uspread);
imwrite(pollenspreadmat, 'pollenspread.jpg');
Thanks in advance
The figure says that for any intensities that are between A and B, they should be set to C. All you have to do is modify your two for loops so that for any values between A and B, set the output location to C. I'll also assume the range is inclusive. You can simply remove the first and last if conditions and use the middle one:
for i = 1:nx
for j = 1:ny
if ((u(i,j)>=r1) && (u(i,j)<= r2))
uspread(i,j) = C;
end
end
end
C is a constant that you would set yourself. Usually for segmentation, this result is very high to distinguish the foreground from the background. You have a uint8 image here, so C = 255; would work.
However, I would recommend you achieve a more vectorized solution. Avoid for loops and use logical indexing instead:
uspread = u;
uspread(u >= r1 & u <= r2) = C;

Implement Adaptive watershed segmentation in Matlab

I will like to implement "Adaptive Watershed Segmentation" in Matlab.
There are six steps in this algorithm. Input is figure(a) and result is figure(d).
Would you please to help me check is there any mistake in my code, and I don't know how to implement the sixth step.
Thank you so much!
Load image:
input_image = imread('test.gif');
Step 1 : Calculate D(x,y) at each (x,y), obtain the Euclidian distance map of the binary image and assign each value of M(x,y) as 0.
DT = bwdist(input_image,'euclidean'); % Trandform distance:Euclidian distance
[h,w]=size(DT);
M = zeros(h,w);
Step 2 : Smooth the distance map using Gaussian filter to merge the adjacent maxima, set M(x,y) as 1 if D(x,y) is a local maximum, and then obtain the marker map of the distance map.
H = fspecial('gaussian');
gfDT = imfilter(DT,H);
M = imregionalmax(gfDT); % maker map, M = local maximum of gfDT
Step3 : Scan the marker map pixel by pixel. If M(x0,y0) is 1, seek the spurious maxima in its neighbourhood with a radius of D(x ,y ).When M(x,y) equals 1 and sqr((x − x0)^2 + (y − y0)^2 ) ≤ D(x0, y0) , set M(x,y) as 0 if D(x,y) < D(x0,y0).
for x0 = 1:h
for y0 = 1:w
if M(x0,y0) == 1
r = ceil(gfDT(x0,y0));
% range begin:(x0-r,y0-r) end:(x0+r,y0+r)
xb = x0-r;
if xb <= 0
xb =1;
end
yb = y0-r;
if yb <= 0
yb =1;
end
xe = x0+r;
if xe > w
xe = w;
end
ye = y0+r;
if ye > h
ye = h;
end
for x = yb:ye
for y = xb:xe
if M(x,y)==1
Pos = [x0,y0 ;x,y];
Dis = pdist(Pos,'euclidean');
IFA = Dis<= (gfDT(x0,y0));
IFB = gfDT(x,y)<gfDT(x0,y0);
if ( IFA && IFB)
M(x,y) = 0;
end
end
end
end
end
end
end
Step 4:
Calculate the inverse of the distance map,and the local maxima turn out to be the local minima.
igfDT = -(gfDT);
STep5:
Segment the distance map according to the markers by the conventional watershed algorithm and obtain the segmentation of binary image.
I2 = imimposemin(igfDT,M);
L = watershed(I2);
igfDT (L==0)=0;
Step 6 : Straighten the watershed lines by linking the ends of the watershed lines with a straight line and reclassifying the pixels along the straight line.
I don't know how to implement this step
Try distance transform and then watershed transform.
im=imread('n6BRI.gif');
imb=bwdist(im);
sigma=3;
kernel = fspecial('gaussian',4*sigma+1,sigma);
im2=imfilter(imb,kernel,'symmetric');
L = watershed(max(im2(:))-im2);
[x,y]=find(L==0);
lblImg = bwlabel(L&~im);
figure,imshow(label2rgb(lblImg,'jet','k','shuffle'));

Vectorized formula for output layer in a neural network

I have a neural network and want to use the trained neural network to solve for a set of test data. What I am struggling with is writing the formula for the hidden layer and for the output layer. I aim to have a vectorized formula but I will also be happy to implement a loop variation.
Now I believe I have the correct formula for the hidden layer and only need one for the output layer, but would appreciate it anyone confirm that it is the vectorized formula.
% Variables
% Xtest test training data
% thetah - trained weights for inputs to hidden layer
% thetao - trained weights for hidden layer to outputs
% ytest - output
htest = (1 ./ (1 + exp(-(thetah * Xtest'))))' ; % FORMULA FOR HIDDEN LAYER
ytest = ones(mtest, num_outputs) ; % FORMULA FOR OUTPUT LAYER
Below you can find both vectorized and loop implementations of the forward propagation. It is possible, that your input data have to be adapted to the code below, because of different notations and the way you store data in your matrices.
You need to add a bias unit to both input and hidden layer.
In order to simplify the work on the implementation and debugging I took some data from the open source machine learning repository and trained the network for the wine classification task.
Xtest - input data [178x13]
y - output class [178x1]
thetah - parameters of the hidden layer [15x14]
thetao -
parameters of the output layer [3x16]
The network separates the input data with rate 97.7%
Here is the code:
function [] = nn_fp()
load('Xtest.mat'); %input data 178x13
load('y.mat'); %output data 178x1
load('thetah.mat'); %Parameters of the hidden layer 15x14
load('thetao.mat'); %Parameters of the output layer 3x16
predict_simple(Xtest, y, thetah, thetao);
predict_vectorized(Xtest, y, thetah, thetao);
end
function predict_simple(Xtest, y, thetah, thetao)
mtest = size(Xtest, 1); %number of input examples
n = size(Xtest, 2); %number of features
hl_size = size(thetah, 1); %size of the hidden layer (without the bias unit)
num_outputs = size(thetao, 1); %size of the output layer
%add a bias unit to the input layer
a1 = [ones(mtest, 1) Xtest]; %[mtest x (n+1)]
%compute activations of the hidden layer
z2 = zeros(mtest, hl_size); %[mtest x hl_size]
a2 = zeros(mtest, hl_size); %[mtest x hl_size]
for i=1:mtest
for j=1:hl_size
for k=1:n+1
z2(i, j) = z2(i, j) + a1(i, k)*thetah(j, k);
end
a2(i, j) = sigmoid_simple(z2(i, j));
end
end
%add a bias unit to the hidden layer
a2 = [ones(mtest, 1) a2]; %[mtest x (hl_size+1)]
%compute activations of the output layer
z3 = zeros(mtest, num_outputs); %[mtest x num_outputs]
h = zeros(mtest, num_outputs); %[mtest x num_outputs]
for i=1:mtest
for j=1:num_outputs
for k=1:hl_size+1
z3(i, j) = z3(i, j) + a2(i, k)*thetao(j, k);
end
h(i, j) = sigmoid_simple(z3(i, j)); %the hypothesis
end
end
%calculate predictions for each input example based on the maximum term
%of the hypothesis h
p = zeros(size(y));
for i=1:mtest
max_ind = 1;
max_value = h(i, 1);
for j=2:num_outputs
if (h(i, j) > max_value)
max_ind = j;
max_value = h(i, j);
end
end
p(i) = max_ind;
end
%calculate the success rate of the prediction
correct_count = 0;
for i=1:mtest
if (p(i) == y(i))
correct_count = correct_count + 1;
end
end
rate = correct_count/mtest*100;
display(['simple version rate:', num2str(rate)]);
end
function predict_vectorized(Xtest, y, thetah, thetao)
mtest = size(Xtest, 1); %number of input examples
%add a bias unit to the input layer
a1 = [ones(mtest, 1) Xtest];
%compute activations of the hidden layer
z2 = a1*thetah';
a2 = sigmoid_universal(z2);
%add a bias unit to the hidden layer
a2 = [ones(mtest, 1) a2];
%compute activations of the output layer
z3 = a2*thetao';
h = sigmoid_universal(z3); %the hypothesis
%calculate predictions for each input example based on the maximum term
%of the hypothesis h
[~,p] = max(h, [], 2);
%calculate the success rate of the prediction
rate = mean(double((p == y))) * 100;
display(['vectorized version rate:', num2str(rate)]);
end
function [ s ] = sigmoid_simple( z )
s = 1/(1+exp(-z));
end
function [ s ] = sigmoid_universal( z )
s = 1./(1+exp(-z));
end
Assuming that your Xtest has dimensions N by M where N is the number of examples and M is the number of features, thetah is a M by H1 matrix where H1 is the number of hidden layers in the first layer and thetao is a H1 by O matrix where O is the number of output classes you do the following:
a1 = Xtest * thetah;
z1 = 1 / (1 + exp(-a1)); %Assuming you are using sigmoid units
a2 = z1 * thetao;
z2 = softmax(a2);
Read more about softmax here.

Fast computation of warp matrices

For a fixed and given tform, the imwarp command in the Image Processing Toolbox
B = imwarp(A,tform)
is linear with respect to A, meaning there exists some sparse matrix W, depending on tform but independent of A, such that the above can be equivalently implemented
B(:)=W*A(:)
for all A of fixed known dimensions [n,n]. My question is whether there are fast/efficient options for computing W. The matrix form is necessary when I need the transpose operation W.'*B(:), or if I need to do W\B(:) or similar linear algebraic things which I can't do directly through imwarp alone.
I know that it is possible to compute W column-by-column by doing
E=zeros(n);
W=spalloc(n^2,n^2,4*n^2);
for i=1:n^2
E(i)=1;
tmp=imwarp(E,tform);
E(i)=0;
W(:,i)=tmp(:);
end
but this is brute force and slow.
The routine FUNC2MAT is somewhat more optimal in that it uses the loop to compute/gather the sparse entry data I,J,S of each column W(:,i). Then, after the loop, it uses this to construct the overall sparse matrix. It also offers the option of using a PARFOR loop. However, this is still slower than I would like.
Can anyone suggest more speed-optimal alternatives?
EDIT:
For those uncomfortable with my claim that imwarp(A,tform) is linear w.r.t. A, I include the demo script below, which tests that the superposition property is satisfied for random input images and tform data. It can be run repeatedly to see that the nonlinearityError is always small, and easily attributable to floating point noise.
tform=affine2d(rand(3,2));
%tform=projective2d(rand(3));
fun=#(A) imwarp(A,tform,'cubic');
I1=rand(100); I2=rand(100);
c1=rand; c2=rand;
LHS=fun(c1*I1+c2*I2); %left hand side
RHS=c1*fun(I1)+c2*fun(I2); %right hand side
linearityError = norm(LHS(:)-RHS(:),'inf')
That's actually pretty simple:
W = sparse(B(:)/A(:));
Note that W is not unique, but this operation probably produces the most sparse result. Another way to calculate it would be
W = sparse( B(:) * pinv(A(:)) );
but that results in a much less sparse (yet still valid) result.
I constructed the warping matrix using the optical flow fields [u,v] and it is working well for my application
% this function computes the warping matrix
% M x N is the size of the image
function [ Fw ] = generateFwi( u,v,M,N )
Fw = zeros(M*N, M*N);
k =1;
for i=1:M
for j= 1:N
newcoord(1) = i+u(i,j);
newcoord(2) = j+v(i,j);
newi = newcoord(1);
newj = newcoord(2);
if newi >0 && newj >0
newi1x = floor(newi);
newi1y = floor(newj);
newi2x = floor(newi);
newi2y = ceil(newj);
newi3x = ceil(newi); % four nearest points to the given point
newi3y = floor(newj);
newi4x = ceil(newi);
newi4y = ceil(newj);
x1 = [newi,newj;newi1x,newi1y];
x2 = [newi,newj;newi2x,newi2y];
x3 = [newi,newj;newi3x,newi3y];
x4 = [newi,newj;newi4x,newi4y];
w1 = pdist(x1,'euclidean');
w2 = pdist(x2,'euclidean');
w3 = pdist(x3,'euclidean');
w4 = pdist(x4,'euclidean');
if ceil(newi) == floor(newi) && ceil(newj)==floor(newj) % both the new coordinates are integers
Fw(k,(newi1x-1)*N+newi1y) = 1;
else if ceil(newi) == floor(newi) % one of the new coordinates is an integer
w = w1+w2;
w1new = w1/w;
w2new = w2/w;
W = w1new*w2new;
y1coord = (newi1x-1)*N+newi1y;
y2coord = (newi2x-1)*N+newi2y;
if y1coord <= M*N && y2coord <=M*N
Fw(k,y1coord) = W/w2new;
Fw(k,y2coord) = W/w1new;
end
else if ceil(newj) == floor(newj) % one of the new coordinates is an integer
w = w1+w3;
w1 = w1/w;
w3 = w3/w;
W = w1*w3;
y1coord = (newi1x-1)*N+newi1y;
y2coord = (newi3x-1)*N+newi3y;
if y1coord <= M*N && y2coord <=M*N
Fw(k,y1coord) = W/w3;
Fw(k,y2coord) = W/w1;
end
else % both the new coordinates are not integers
w = w1+w2+w3+w4;
w1 = w1/w;
w2 = w2/w;
w3 = w3/w;
w4 = w4/w;
W = w1*w2*w3 + w2*w3*w4 + w3*w4*w1 + w4*w1*w2;
y1coord = (newi1x-1)*N+newi1y;
y2coord = (newi2x-1)*N+newi2y;
y3coord = (newi3x-1)*N+newi3y;
y4coord = (newi4x-1)*N+newi4y;
if y1coord <= M*N && y2coord <= M*N && y3coord <= M*N && y4coord <= M*N
Fw(k,y1coord) = w2*w3*w4/W;
Fw(k,y2coord) = w3*w4*w1/W;
Fw(k,y3coord) = w4*w1*w2/W;
Fw(k,y4coord) = w1*w2*w3/W;
end
end
end
end
else
Fw(k,k) = 1;
end
k=k+1;
end
end
end

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