I am using PCA and PHATE for dimensionality reduction and manifolds visualization and I am trying to retrieve the loadings of PHATE vectors onto the different features. For PCA, is straight-forward because I get the eigenvectors from the .components_ attribute from sklearn. With PHATE, I am getting only (?) the PHATE data, i.e. projected data on the PHATE space.
I thought that I could try to retrieve the loadings (or eigenvectors) by projecting back using the data, i.e. X.T # PHATEdata, which I am not sure about because there is supposed to be nonlinear operations for producing the PHATEdata on a lower dimensional space.
However, after trying it, I ended up with the following result which shows correspondence between the PCA loadings and PHATE loadings.
PC1 vs PHATE1 loadings
However, there is not such correspondence between the PC2 and PHATE2 loadings.
Thanks in advance.
Related
I have stereo pairs from two orbits of a satellite. I am trying to generate dense match points between these pairs so that I can estimate 3D world coordinates using elements of Satellite-Earth Geometry.
Question: Which method/approach can be used for this arbitrary matching of the stereo pairs (which are not rectified so that x-only parallax is present)?
So far:
1) I am successful at using Hierarchical image matching based on normalized cross correlation which gives good results but takes a lot of time.
2) I have tried Semi Global Matching on images obtained after uncalibrated - rectification. But this method doesnot give good correspondence, though the disparity map generated is nice and smooth. Also, uncalibrated-rectification is very sensitive process.
Assume I do not have any camera parameters with me.
Edit:
1) The images are taken by pushbroom model with a line-sensor.
2) The images are neither georeferenced nor orthorectified, they are just radiometrically corrected.
I have vectors in 4096 VLAD codes, each one of them representing an image.
I have to run PCA on them without reducing their dimension on learning datasets with less than 4096 images (e.g., the Holiday dataset with <2k images.) to obtain the rotation matrix A.
In this paper (which also explains why I need to run this PCA without dimensionality reduction) they solved the problem with this approach:
For efficient computation of A, we compute at most the first 1,024 eigenvectors by eigendecomposition of the covariance matrix, and
the remaining orthogonal complements up to D-dimensional are filled using Gram-Schmidt orthogonalization.
Now, how do I implement this using C++ libraries? I'm using OpenCV, but cv::PCA seems not to offer such a strategy. Is there any way to do this?
I am currently facing a problem, to depict you what my programm does and should do, here is the copy/paste of the beginning of a previous post I've made.
This program lies on the classic "structure from motion" method.
The basic idea is to take a pair of images, detect their keypoints and compute the descriptors of those keypoints. Then, the keypoints matching is done, with a certain number of tests to insure the result is good. That part works perfectly.
Once this is done, the following computations are performed : fundamental matrix, essential matrix, SVD decomposition of the essential matrix, camera projection matrices computation and finally, triangulation.
The result for a pair of images is a set of 3D coordinates, giving us points to be drawn in a 3D viewer. This works perfectly, for a pair.
However, I have to perform a step manually, and this is not acceptable if I want my program to efficiently work with more than two images.
Indeed, I compute my projection matrices according the classic method, as follows, at paragraph "Determining R and t from E" : https://en.wikipedia.org/wiki/Essential_matrix
I have then 4 possible solutions for my projection matrix.
I think I have understood the geometrical point of view of the problem, portrayded in this Hartley and Zisserman paper extract (chapters 9.6.3 and 9.7.1) : http://www.robots.ox.ac.uk/~vgg/hzbook/hzbook2/HZepipolar.pdf
Nonetheless, my question is : Given the four possible projection matrices computed and the 3D points computed by the OpenCV function triangulatePoints() (for each projection matrix), how can I elect the "true" projection matrix, automatically ? (without having to draw 4 times my points in my 3D viewer, in order to see if they are consistent)
Thanks for reading.
I'm trying to design a line detector in opencv, and to do that, I need to get the Gaussian matrix with variance σs.
The final formula should be
H=Gσs∗(Gσd')T, and H is the detector that I'm going to create, but I have no idea how am I supposed to create the matrix with the variance and furthermore calculate H finally.
Update
This is the full formula.where “T” is the transpose operation.Gσd' is the first-order derivative of a 1-D Gaussian function Gσd with varianceσd in this direction
****Update****
These are the two formulas that I want, I need H for further use so please tell me how to generate the matrix. thx!
As a Gaussian filter is quite common, OpenCV has a built-in operation for it: GaussianBlur.
When you use that function you can set the ksize argument to 0/0 to automatically compute the pixel size of the kernel from the given sigmas.
A Gaussian 2D filter kernel is separable. That means you can first apply a 1D filter along the x axis and then a 1D filter along the y axis. That is the reason for having two 1D filters in the equation above. It is much faster to do two 1D filter operations instead of one 2D.
I have 2D data (I have a zero mean normalized data). I know the covariance matrix, eigenvalues and eigenvectors of it. I want to decide whether to reduce the dimension to 1 or not (I use principal component analysis, PCA). How can I decide? Is there any methodology for it?
I am looking sth. like if you look at this ratio and if this ratio is high than it is logical to go on with dimensionality reduction.
PS 1: Does PoV (Proportion of variation) stands for it?
PS 2: Here is an answer: https://stats.stackexchange.com/questions/22569/pca-and-proportion-of-variance-explained does it a criteria to test it?
PoV (Proportion of variation) represents how much information of data will remain relatively to using all of them. It may be used for that purpose. If POV is high than less information will be lose.
You want to sort your eigenvalues by magnitude then pick the highest 1 or 2 values. Eigenvalues with a very small relative value can be considered for exclusion. You can then translate data values and using only the top 1 or 2 eigenvectors you'll get dimensions for plotting results. This will give a visual representation of the PCA split. Also check out scikit-learn for more on PCA. Precisions, recalls, F1-scores will tell you how well it works
from http://sebastianraschka.com/Articles/2014_pca_step_by_step.html...
Step 1: 3D Example
"For our simple example, where we are reducing a 3-dimensional feature space to a 2-dimensional feature subspace, we are combining the two eigenvectors with the highest eigenvalues to construct our d×kd×k-dimensional eigenvector matrix WW.
matrix_w = np.hstack((eig_pairs[0][1].reshape(3,1),
eig_pairs[1][1].reshape(3,1)))
print('Matrix W:\n', matrix_w)
>>>Matrix W:
[[-0.49210223 -0.64670286]
[-0.47927902 -0.35756937]
[-0.72672348 0.67373552]]"
Step 2: 3D Example
"
In the last step, we use the 2×32×3-dimensional matrix WW that we just computed to transform our samples onto the new subspace via the equation
y=W^T×x
transformed = matrix_w.T.dot(all_samples)
assert transformed.shape == (2,40), "The matrix is not 2x40 dimensional."