I'm totally confused regarding PCA. I have a 4D image of size 90x60x12x350. That means that each voxel is a vector of size 350 (time series).
Now I divide the 3D image (90x60x12) into cubes. So let's say a cube contains n voxels, so I have n vectors of size 350. I want to reduce this n vectors to only one vector and then calculate the correlations between all vectors of all cubes.
So for a cube I can construct the matrix M where I just put each voxel after each other, i.e. M = [v1 v2 v3 ... vn] and each v is of size 350.
Now I can apply PCA in Matlab by using [coeff, score, latent, ~, explained] = pca(M); and taking the first component. And now my confusion begins.
Should I transpose the matrix M, i.e. PCA(M')?
Should I take the first column of coeff or of score?
This third question is now a bit unrelated. Let's assume we have a
matrix A = rand(30,100) where the rows are the datapoints and the
columns are the features. Now I want to reduce the dimensionality of
the feature vectors but keeping all data points.
How can I do this with PCA?
When I do [coeff, score, latent, ~, explained] = pca(M); then
coeff is of dimension 100x29 and score is of size 30x29. I'm
totally confused.
Yes, according to the pca help, "Rows of X correspond to observations and columns to variables."
score just tells you the representation of M in the principal component space. You want the first column of coeff.
numberOfDimensions = 5;
coeff = pca(A);
reducedDimension = coeff(:,1:numberOfDimensions);
reducedData = A * reducedDimension;
I disagree with the answer above.
[coeff,score]=pca(A)
where A has rows as observations and column as features.
If A has 3 featuers and >3 observations (Let's say 100) and you want the "feature" of 2 dimensions, say matrix B (the size of B is 100X2). What you should do is:
B = score(:,1:2);
Related
The function to perform an N-dimensional convolution of arrays A and B in matlab is shown below:
C = convn(A,B) % returns the N-dimensional convolution of arrays A and B.
I am interested in a 3-D convolution with a Gaussian filter.
If A is a 3 x 5 x 6 matrix, what do the dimensions of B have to be?
The dimensions of B can be anything you want. There is no set restriction in terms of size. For the Gaussian filter, it can be 1D, 2D or 3D. In 1D, what will happen is that each row gets filtered independently. In 2D, what will happen is that each slice gets filtered independently. Finally, in 3D you will be doing what is expected in 3D convolution. I am assuming you would like a full 3D convolution, not just 1D or 2D.
You may be interested in the output size of convn. If you refer to the documentation, given the two N dimensional matrices, for each dimension k of the output and if nak is the size of dimension k for the matrix A and nbk is the size of dimension k for matrix B, the size of dimension of the output matrix C or nck is such that:
nck = max([nak + nbk - 1, nak, nbk])
nak + nbk - 1 is straight from convolution theory. The final output size of a dimension is simply the sum of the two sizes in dimension k subtracted by 1. However should this value be smaller than either of nak or nbk, we need to make sure that the output size is compatible so that any of the input matrices can fit in the final output. This is why you have the final output size and bounded by both A and B.
To make this easier, you can set the size of the filter guided by the standard deviation of the distribution. I would like to refer you to my previous Stack Overflow post: By which measures should I set the size of my Gaussian filter in MATLAB?
This determines what the output size of a Gaussian filter should be given a standard deviation.
In 2D, the dimensions of the filter are N x N, such that N = ceil(6*sigma + 1) with sigma being the desired standard deviation. Therefore, you would allocate a 3D matrix of size N x N x N with N = ceil(6*sigma + 1);.
Therefore, the code you would want to use to create a 3D Gaussian filter would be something like this:
% Example input
A = rand(3, 5, 6);
sigma = 0.5; % Example
% Find size of Gaussian filter
N = ceil(6*sigma + 1);
% Define grid of centered coordinates of size N x N x N
[X, Y, Z] = meshgrid(-N/2 : N/2);
% Compute Gaussian filter - note normalization step
B = exp(-(X.^2 + Y.^2 + Z.^2) / (2.0*sigma^2));
B = B / sum(B(:));
% Convolve
C = convn(A, B);
One final note is that if the filter you provide has any of its dimensions that are beyond the size of the input matrix A, you will get a matrix using the constraints of each nck value, but then the border elements will be zeroed due to zero-padding.
Lets say I have a samples matrix samples (n_samples x n1) and a labels vector labels (n_samples x 1), where the labels are in the range [1:n2]
I am looking for an efficient way to create an empirical joint probability matrix P in the size n2 x n1.
Where for every sample i, we add its row samples(i, :) to P in the location indicated by labels(i).
I.e. (pseudo code)
for i = 1:n_samples
P(l(i), :) += M(i, :)
Is there a killer matlab command for doing that? Rather than a for loop or arrayfun?
Following #BillBokeey comment: Here is the solution
[xx, yy] = ndgrid(labels,1:size(samples,2));
P = accumarray([xx(:) yy(:)],samples(:));
In an attempt to create my own covariance function in MatLab I need to perform matrix multiplication on a row to create a matrix.
Given a matrix D where
D = [-2.2769 0.8746
0.6690 -0.4720
-1.0030 -0.9188
2.6111 0.5162]
Now for each row I need manufacture a matrix. For example the first row R = [-2.2770, 0.8746] I would want the matrix M to be returned where M = [5.1847, -1.9915; -1.9915, 0.7649].
Below is what I have written so far. I am asking for some advice to explain how to use matrix multiplication on a rows to produce matrices?
% Find matrices using matrix multiplication
for i=1:size(D, 1)
P1 = (D(i,:))
P2 = transpose(P1)
M = P1*P2
end
You are trying to compute the outer product of each row with itself stored as individual slices in a 3D matrix.
Your code almost works. What you're doing instead is computing the inner product or the dot product of each row with itself. As such it'll give you a single number instead of a matrix. You need to change the transpose operation so that it's done on P1 not P2 and P2 will now simply be P1. Also you are overwriting the matrix M at each iteration. I'm assuming you'd like to store these as individual slices in a 3D matrix. To do this, allocate a 3D matrix where each 2D slice has an equal number of rows and columns which is the number of columns in D while the total number of slices is equal to the total number of rows in D. Then just index into each slice and place the result accordingly:
M = zeros(size(D,2), size(D,2), size(D,1));
% Find matrices using matrix multiplication
for ii=1:size(D, 1)
P = D(ii,:);
M(:,:,ii) = P.'*P;
end
We get:
>> M
M(:,:,1) =
5.18427361 -1.99137674
-1.99137674 0.76492516
M(:,:,2) =
0.447561 -0.315768
-0.315768 0.222784
M(:,:,3) =
1.006009 0.9215564
0.9215564 0.84419344
M(:,:,4) =
6.81784321 1.34784982
1.34784982 0.26646244
Depending on your taste, I would recommend using bsxfun to help you perform the same operation but perhaps doing it faster:
M = bsxfun(#times, permute(D, [2 3 1]), permute(D, [3 2 1]));
In fact, this solution is related to a similar question I asked in the past: Efficiently compute a 3D matrix of outer products - MATLAB. The only difference is that the question wanted to find the outer product of columns instead of the rows.
The way the code works is that we shift the dimensions with permute of D so that we get two matrices of the sizes 2 x 1 x 4 and 1 x 2 x 4. By performing bsxfun and specifying the times function, this allows you to efficiently compute the matrix of outer products per slice simultaneously.
I'm trying to reconstruct a 3d image from two calibrated cameras. One of the steps involved is to calculate the 3x3 essential matrix E, from two sets of corresponding (homogeneous) points (more than the 8 required) P_a_orig and P_b_orig and the two camera's 3x3 internal calibration matrices K_a and K_b.
We start off by normalizing our points with
P_a = inv(K_a) * p_a_orig
and
P_b = inv(K_b) * p_b_orig
We also know the constraint
P_b' * E * P_a = 0
I'm following it this far, but how do you actually solve that last problem, e.g. finding the nine values of the E matrix? I've read several different lecture notes on this subject, but they all leave out that crucial last step. Likely because it is supposedly trivial math, but I can't remember when I last did this and I haven't been able to find a solution yet.
This equation is actually pretty common in geometry algorithms, essentially, you are trying to calculate the matrix X from the equation AXB=0. To solve this, you vectorise the equation, which means,
vec() means vectorised form of a matrix, i.e., simply stack the coloumns of the matrix one over the another to produce a single coloumn vector. If you don't know the meaning of the scary looking symbol, its called Kronecker product and you can read it from here, its easy, trust me :-)
Now, say I call the matrix obtained by Kronecker product of B^T and A as C.
Then, vec(X) is the null vector of the matrix C and the way to obtain that is by doing the SVD decomposition of C^TC (C transpose multiplied by C) and take the the last coloumn of the matrix V. This last coloumn is nothing but your vec(X). Reshape X to 3 by 3 matrix. This is you Essential matrix.
In case you find this maths too daunting to code, simply use the following code by Y.Ma et.al:
% p are homogenius coordinates of the first image of size 3 by n
% q are homogenius coordinates of the second image of size 3 by n
function [E] = essentialDiscrete(p,q)
n = size(p);
NPOINTS = n(2);
% set up matrix A such that A*[v1,v2,v3,s1,s2,s3,s4,s5,s6]' = 0
A = zeros(NPOINTS, 9);
if NPOINTS < 9
error('Too few mesurements')
return;
end
for i = 1:NPOINTS
A(i,:) = kron(p(:,i),q(:,i))';
end
r = rank(A);
if r < 8
warning('Measurement matrix rank defficient')
T0 = 0; R = [];
end;
[U,S,V] = svd(A);
% pick the eigenvector corresponding to the smallest eigenvalue
e = V(:,9);
e = (round(1.0e+10*e))*(1.0e-10);
% essential matrix
E = reshape(e, 3, 3);
You can do several things:
The Essential matrix can be estimated using the 8-point algorithm, which you can implement yourself.
You can use the estimateFundamentalMatrix function from the Computer Vision System Toolbox, and then get the Essential matrix from the Fundamental matrix.
Alternatively, you can calibrate your stereo camera system using the estimateCameraParameters function in the Computer Vision System Toolbox, which will compute the Essential matrix for you.
i have R 3d matrix,n varies from 1:100.
I have generated 20 such R matrix.
Now i have to average each R for this 20 experiment.
so that I'll get n,100 avg matrix.
How to average this 20, n Matrix?
I want to add(avg) all 20 times generated R for each n .I must have avg 100 R matrix .
Assuming you actually have a 3D matrix R, it is very easy to average:
R = rand(3,4,5); %Suppose this is your matrix
Now you just need to pick the dimension you want to average in:
mean(R,1) %First dimension
mean(R,2) %Second dimension
mean(R,3) %Third dimension
If you are not sure which one you need, just check the size of all three.