So i want to make a source code on matlab that allows me to flip the main diagonal of a gauss matrix to solve a linear equation and make the bottom right triangle into all zeros. But I can't seem to find the correct iterations for the pivot and how to make it so that the bottom right of the matrix is zero. I'm also unable to fine the maximum absolute valueenter image description here
so what is the correct iteration and how do i fine the maximum absolute value?
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To find displacement of a particle, I calculated the cross correlation between two instants (represented by two images with the same size). Then, I padded the images with zeros to see if a translation will have an effect on the displacement.
Thus I found a difference in displacement vector( the difference can reach 1.5 pixel and the size of image is 56x56 pixels)
Is it normal to find a difference after padding?
N.B: To pad the image, I used
new_image(end+1:56,end+1:56)=0;
EDIT
The difference can even be more for some cases (22 px)
Yes, this is weird. The cross-correlation is calculated by multiplying values in both matrices with eachother and taking the sum of these. Adding zeros should not result in a greater sum.
The problem in the code you've posted is that end+1:56 should likely be end+1:end+56, since you pad it with 56 extra zeros below and to the right of the image this way.
Since your goal appears to be to get the cross-correlation of 2 matrices, I recommend you to look at the xcorr2() and xcorr() functions in Matlab. An explanation for xcorr2() and why zero padding should not have any influence (besides searching a larger image) can be found here.
I have a small project on Moving object detection in moving camera in which i have to use negative optical flow vector to minimize ego motion compensation. I have a video and some particular consecutive frames in which average of negative optical flow vector has to be computed. I have already calculated Optical flow between say, (k-1)th and kth frame. Also, I have calculated average of optical flow vector V=[u,v], where v is the average of horizontal optical flow and u is the average of vertical flow. Now, I have to apply inverse of optical flow vector i.e., -V to the (k-1)th frame. I'm new to matlab and i don't know much about it. Please help
I have tried this code segment to do so but the results aren't as expected
function I1=reverseOF(I,V)
R=I(:,:,1);
G=I(:,:,2);
B=I(:,:,3);
[m,n]=size(rgb2gray(I));
for i=1:m
for j=1:n
v1=[j i];
v2=-V;
v3=v1.*v2;
R(floor(1+abs(v3(1,2))),floor(1+abs(v3(1,2))))=R(i,j);
G(floor(1+abs(v3(1,2))),floor(1+abs(v3(1,2))))=G(i,j);
B(floor(1+abs(v3(1,2))),floor(1+abs(v3(1,2))))=B(i,j);
I1(floor(1+abs(v3(1,2))),floor(1+abs(v3(1,2))))=I(i,j);
end
end
I1=cat(3,R,G,B);
enter code here
I have used abs() function because otherwise some error was occuring like "attempted to access negative location; index must be a positive or logical".
Image A and Image B are the images that i have used to estimate the optical flow.enter image description here
This is the result that i am obtaining after applying the above function.
enter image description here
Unfortunately, you cant do this easily. This is a quite advanced research problem, because obtaining the inverse of a vector field on a mesh grid is not an easy problem, actually its quite hard.
Notice that your vector field (optical flow) start in a mesh grid, but it doesnt end in a mesh grid, it ends in random subpixel positions. If you just invert this field, doing -V is not enough! The result wont be the inverse!
This is a open research problem, look for example at this 2010 paper that addresses exactly this issue, and proposes a method to create "pseudoinverses".
Suppose you have that inverse, because you computed it somehow. Your code is quite bad for it, and the solutions (abs!) are showing (no offense) that you are not really understanding what you are doing. For a known vector field {Vx,Vy}, size equals to the image size (if its not, you can figure out easily how to interpolate it unsig interp2 ) the code would look something like:
newimg=zeros(size(I));
[ix,iy]=meshgrid(1:size(I,1),1:size(I,2));
newimg(:,:,1)=interp2(I(:,:,1),ix+Vx,iy+Vy); % this is your whole loop.
newimg(:,:,2)=interp2(I(:,:,3),ix+Vx,iy+Vy); % this is your whole loop.
newimg(:,:,3)=interp2(I(:,:,2),ix+Vx,iy+Vy); % this is your whole loop.
I have an RGB image and I am trying to calculate its Gaussian derivative.
Image is a greyscale image and the Gaussian window is 5x5,st is the standard deviation
This is the code i am using in order to find a 2D Gaussian derivative,in Matlab:
N=2
[X,Y]=meshgrid(-N:N,-N:N)
G=exp(-(x.^2+y.^2)/(2*st^2))/(2*pi*st)
G_x = -x.*G/(st^2);
G_x_s = G_x/sum(G_x(:));
G_y = -y.*G/(st^2);
G_y_s = G_y/sum(G_y(:));
where st is the standard deviation i am using. Before I proceed to the convolution of the Image using G_x_s and G_y_s, i have the following problem. When I use a standard deviation that is an even number(2,4,6,8) the program works and gives results as expected. But when i use an odd number for standard deviation (3 or 5) then the G_y_s value becomes Inf because sum(G_y(:))=0. I do not understand that behavior and I was wondering if there is some problem with the code or if in the above formula the standard deviation can only be an even number. Any help will be greatly appreciated.
Thank you.
Your program doesn't work at all. The results you find when using an even number is just because of some numerical errors.
Your G will be a matrix symmetrical to the center. x and y are both point symmetrical to the center. So the multiplication (G times x or y) will result in a matrix with a sum of zero. So a division by that sum is a division by zero. Everything else you observe is because of some roundoff errors. Here, I see a sum og G_xof about 1.3e-17.
I think your error is in the multiplication x.*G and y.*G. I can not figure out why you would do that.
I assume you want to do edge detection rigth? You can use fspecialto create several edge filters. Laplace of gaussian for instance. You could also create two gaussian filters with different standard deviations and subtract them from another to get an edge filter.
I have a matrix M of size NxP. Every P columns are orthogonal (M is a basis). I also have a vector V of size N.
My objective is to transform the first vector of M into V and to update the others in order to conservate their orthogonality. I know that the origins of V and M are the same, so it is basically a rotation from a certain angle. I assume we can find a matrix T such that T*M = M'. However, I can't figure out the details of how to do it (with MATLAB).
Also, I know there might be an infinite number of transforms doing that, but I'd like to get the simplest one (in which others vectors of M approximately remain the same, i.e no rotation around the first vector).
A small picture to illustrate. In my actual case, N and P can be large integers (not necessarily 3):
Thanks in advance for your help!
[EDIT] Alternative solution to Gram-Schmidt (accepted answer)
I managed to get a correct solution by retrieving a rotation matrix R by solving an optimization problem minimizing the 2-norm between M and R*M, under the constraints:
V is orthogonal to R*M[1] ... R*M[P-1] (i.e V'*(R*M[i]) = 0)
R*M[0] = V
Due to the solver constraints, I couldn't indicate that R*M[0] ... R*M[P-1] are all pairwise orthogonal (i.e (R*M)' * (R*M) = I).
Luckily, it seems that with this problem and with my solver (CVX using SDPT3), the resulting R*M[0] ... R*M[P-1] are also pairwise orthogonal.
I believe you want to use the Gram-Schmidt process here, which finds an orthogonal basis for a set of vectors. If V is not orthogonal to M[0], you can simply change M[0] to V and run Gram-Schmidt, to arrive at an orthogonal basis. If it is orthogonal to M[0], instead change another, non-orthogonal vector such as M[1] to V and swap the columns to make it first.
Mind you, the vector V needs to be in the column space of M, or you will always have a different basis than you had before.
Matlab doesn't have a built-in Gram-Schmidt command, although you can use the qr command to get an orthogonal basis. However, this won't work if you need V to be one of the vectors.
Option # 1 : if you have some vector and after some changes you want to rotate matrix to restore its orthogonality then, I believe, this method should work for you in Matlab
http://www.mathworks.com/help/symbolic/mupad_ref/numeric-rotationmatrix.html
(edit by another user: above link is broken, possible redirect: Matrix Rotations and Transformations)
If it does not, then ...
Option # 2 : I did not do this in Matlab but a part of another task was to find Eigenvalues and Eigenvectors of the matrix. To achieve this I used SVD. Part of SVD algorithm was Jacobi Rotation. It says to rotate the matrix until it is almost diagonalizable with some precision and invertible.
https://math.stackexchange.com/questions/222171/what-is-the-difference-between-diagonalization-and-orthogonal-diagonalization
Approximate algorithm of Jacobi rotation in your case should be similar to this one. I may be wrong at some point so you will need to double check this in relevant docs :
1) change values in existing vector
2) compute angle between actual and new vector
3) create rotation matrix and ...
put Cosine(angle) to diagonal of rotation matrix
put Sin(angle) to the top left corner of the matric
put minus -Sin(angle) to the right bottom corner of the matrix
4) multiple vector or matrix of vectors by rotation matrix in a loop until your vector matrix is invertible and diagonalizable, ability to invert can be calculated by determinant (check for singularity) and orthogonality (matrix is diagonalized) can be tested with this check - if Max value in LU matrix is less then some constant then stop rotation, at this point new matrix should contain only orthogonal vectors.
Unfortunately, I am not able to find exact pseudo code that I was referring to in the past but these links may help you to understand Jacobi Rotation :
http://www.physik.uni-freiburg.de/~severin/fulltext.pdf
http://web.stanford.edu/class/cme335/lecture7.pdf
https://www.nada.kth.se/utbildning/grukth/exjobb/rapportlistor/2003/rapporter03/maleko_mercy_03003.pdf
In every paper i read about encryption they like to show the correlation coefficients of their encrypted image by showing 3 values:
Horizontal correlation coefficient .
vertical correlation coefficient.
diagonal correlation coefficient .
and they show these 3 values for encrypted image and also for plain image(lena).
My question is how to do this in matlab ? and if there is no matlab function for it , what are the equation they are using to get those 3 values ?
Table 2
Correlation coefficients of two adjacent pixels in two images
example:
Plain-image Ciphered image
Horizontal 0.92401 0.01589
Vertical 0.95612 0.06538
Diagonal 0.92659 0.03231
Any lead would be helpful , thanks
If I understood it correctly, you would like to perform a correlation analyis for pairs of pixels within a certain given image.
In principle I would go for the cov function: in your specific case, cov will retrieve a [2*2] symmetric matrix. The diagonal elements will represent your horizontal and vertical correlation coefficients, whereas the lower (upper) triangular elements will stay for the diagonal correlation coefficient.
I hope this will help you.