I've been tasked with reconstructing a 256x256 image from a 367x180 sinogram in MATLAB, using SVD.
I ran the SVD no problem using [U,S,V] = svd(true_sinogram);
This produces U (367 x 367), S (367 x 180), and V (180 x 180). My issue is that I'm not sure how to reconstruct the original image (not the sinogram) from this.
I've attempted the pseudo-inverse by A = V.*S_recip.*U';, where U' is the transpose of U and S_recip is the reciprocal of S, but I'm getting a "Matrix dimensions must agree" error.
Simple multiplcation of the original elements just results in the original true sinogram, which I can only back-project. Truncating the matrices to 180x180 also doesn't work.
Has anyone got any ideas? I feel I'm missing something obvious. Thank you!
Not mere reciprocals of S but instead its transpose should be used to find pseudoinverse.
For example, if A=U*S*V' then the pseudoinverse of A would be equal to V*S_recip'*U', where S_recip' is the transpose of S_recip, and would have its size as 180 x 367.
Related
I have 1x1024 matrix. So I'd like to estimate a polynomial equation.
X= (0:1023)'
Y= acquired data. A 1024 element vector
Then I try this in MATLAB:
polyfit(x,y,5)
But MATLAB makes an abnormal result with warning.
Warning: Polynomial is badly conditioned. Add points with distinct X values, reduce the degree of the ...
I don't understand what am I doing wrong?
Update
I got a bunch of numbers like this.
Y=
-0.0000000150
...
0.00001
...
0
...
0.17
X= 0~255
polyfit(X,Y,4)
I got a polynomial but it does not match to original curve.
Is there any options to match between original curve and polyfit's curve?
The problem can be attributed to the type of coefficient matrix that polyfit builds from the x vector: a Vandermonde matrix.
When
the elements of the x vector vary too much in magnitude, and
the degree of the fitting polynomial is too high,
you get an ill-conditioned matrix, and the associated linear system cannot be solved reliably.
Try to centre and scale your x vector first, before applying polyfit, as advised at the bottom of the polyfit help page:
Since the columns in the Vandermonde matrix are powers of the vector x, the condition number of V is often large for high-order fits, resulting in a singular coefficient matrix. In those cases centering and scaling can improve the numerical properties of the system to produce a more reliable fit.
(my emphasis)
The warning is because the data that you are supplying to polyfit with your desired degree of polynomial isn't suitable. Specifically, there is an insufficient amount of variability in your data so that you can successfully achieve a good fit. Therefore, MATLAB gives you that warning because the data can't be fit properly with your desired degree polynomial.
The solution to this is to either get more points so that you can get the desired fit of the polynomial degree you want or to decrease the degree of polynomial you want.
Try values that are less than 5... 4, 3 or perhaps 2:
coeff = polyfit(x, y, 4);
%// or
%coeff = polyfit(x, y, 3);
%coeff = polyfit(x, y, 2);
Try each degree until you don't get the warning anymore. However, without the actual data, I can only speculate what's wrong, and this is my best guess.
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
I have just started working using CCA in Matlab. I have two vectors X and Y of dimension 60x1920 and 60x1536 with the number of samples being 60 and variables in the different set of vectors being 1920 and 1536 respectively. I want to know do CCA for reducing them to the subspace and then do feature matching.
I am using this commands.
%% DO CCA
[A,B,r,U,V] = canoncorr(X,Y);
The output I get is this :
Name Size Bytes Class Attributes
A 1920x58 890880 double
B 1536x58 712704 double
U 60x58 27840 double
V 60x58 27840 double
r 1x58 464 double
Can anyone please tell me what these variables mean. I have gone over the documentation several times and still is unclear about them. As I understand CCA finds two linear projection matrices Wx and Wy such that the projection of X and Y on Wx and Wy are maximally correlated.
1) Could anyone please tell me which of the following matrices are these?
2) Also how can I find the projected vectors in the learned subspace of CCA?
Any help will be appreciated. Thanks in advance.
As I understand it, with X and Y being your original data matrices, A and B are the sets of coefficients that perform a change of basis to maximally correlate your original data. Your data is represented in the new bases as the matrices U and V.
So to answer your questions:
The projection matrices you are looking for would be A and B since they transform X and Y into the new space.
The resulting projections of X and Y into the new space would be U and V, respectively. (The r vector represents the entries of the correlation matrix between U and V, which is a diagonal matrix.)
The The MATLAB documentation says this transformation can be done with the following formulae, where N is the number of observations:
U = (X-repmat(mean(X),N,1))*A
V = (Y-repmat(mean(Y),N,1))*B
This page lays out the process nicely so you can see what each coefficient means in the transformation process.
In my current analysis, I am trying to multiply a matrix (flm), of dimension nxm, with the inverse of a matrix nxmxp, and then use this result to multiply it by the inverse of the matrix (flm).
I was trying using the following code:
flm = repmat(Data.fm.flm(chan,:),[1 1 morder]); %chan -> is a vector 1by3
A = (flm(:,:,:)/A_inv(:,:,:))/flm(:,:,:);
However. due to the problem of dimensions, I am getting the following error message:
Error using ==> mrdivide
Inputs must be 2-D, or at least one
input must be scalar.
To compute elementwise RDIVIDE, use
RDIVIDE (./) instead.
I have no idea on how to proceed without using a for loop, so anyone as any suggestion?
I think you are looking for a way to conveniently multiply matrices when one is of higher dimensionality than the other. In that case you can use bxsfun to automatically 'expand' the smaller matrix.
x = rand(3,4);
y = rand(3,4,5);
bsxfun(#times,x,y)
It is quite simple, and very efficient.
Make sure to check out doc bsxfun for more examples.
I'm using the following code To get a partial correlation matrix (original code from http://www.fmrib.ox.ac.uk/analysis/netsim/)
ic=-inv(cov(ts1)); % raw negative inverse covariance matrix
r=(ic ./ repmat(sqrt(diag(ic)),1,Nnodes)) ./ repmat(sqrt(diag(ic))',Nnodes,1); % use diagonal to get normalised coefficients
r=r+eye(Nnodes); % remove diagonal
My original matrix (ts1) is a brain activity over time course (X variable) in multiple voxels -volumetric pixel 3X3 (Y variable).
The problem is, I have more dependent variables(y -voxels ) than independent variables(x - time course).
I get the following Warning-
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 4.998365e-022.
Any thoughts on how to fix the code so I'll get the partial correlation between all of the voxels?
The warning is from Matlab having a problem inverting the covariance matrix.
One solution might be to try pinv()
http://www.mathworks.com/help/techdoc/ref/pinv.html