AnalytixTricks and Thoughts
https://ahmetcecen.com/blog/
Sat, 18 Mar 2023 07:08:42 +0000Sat, 18 Mar 2023 07:08:42 +0000Jekyll v3.9.3Regular 2pt Statistics PCA on Half VortexThu, 09 Mar 2017 00:00:00 +0000
https://ahmetcecen.com/blog/rotstats/2017/03/09/Half-Vortex-PCA-Regular/
https://ahmetcecen.com/blog/rotstats/2017/03/09/Half-Vortex-PCA-Regular/projectorvisualizationrotstatsRegular 2pt Statistics PCA on Full VortexThu, 09 Mar 2017 00:00:00 +0000
https://ahmetcecen.com/blog/rotstats/2017/03/09/Full-Vortex-PCA-Regular/
https://ahmetcecen.com/blog/rotstats/2017/03/09/Full-Vortex-PCA-Regular/projectorvisualizationrotstatsRegular 2pt Statistics PCA on 20 Class DatasetTue, 07 Mar 2017 00:00:00 +0000
https://ahmetcecen.com/blog/rotstats/2017/03/07/RotStats-PCA-Regular/
https://ahmetcecen.com/blog/rotstats/2017/03/07/RotStats-PCA-Regular/projectorvisualizationrotstatsPartial Fourier Basis Complex Polar 2pt Statistics PCA on 20 Class DatasetTue, 07 Mar 2017 00:00:00 +0000
https://ahmetcecen.com/blog/rotstats/2017/03/07/RotStats-PCA-Polar-Regular/
https://ahmetcecen.com/blog/rotstats/2017/03/07/RotStats-PCA-Polar-Regular/projectorvisualizationrotstatsPartial Fourier Basis Absolute Polar 2pt Statistics PCA on 20 Class DatasetTue, 07 Mar 2017 00:00:00 +0000
https://ahmetcecen.com/blog/rotstats/2017/03/07/RotStats-PCA-Polar-Absolute/
https://ahmetcecen.com/blog/rotstats/2017/03/07/RotStats-PCA-Polar-Absolute/projectorvisualizationrotstatsBrief Visual Explanation of PCAWhy do we use PCA?
Objective and hierarchical identification of most characteristic features.
Features are independent and uninformed.
PCA is an orthogonal coordinate system transformation that prioritizes maximum variance.
Here what that statement looks like in 2D.
The following is a 3D example showing the same effect in a different sense.
Tue, 04 Oct 2016 11:00:00 +0000
https://ahmetcecen.com/blog/class/2016/10/04/PCA/
https://ahmetcecen.com/blog/class/2016/10/04/PCA/classmaterials-informaticsclassFiltering and Fourier TransformFourier Space Filtering
Filtering in real space can be represented by the following relationship:
\[Filtered Image = Cx\]
Where $x$ is the filter, and $C$ is a circulant matrix constructed using the input image. For a simple example let us assume the input image and filter are 1D. The matrix $C$ is constructed by shifting the image periodically once at each row. Notice below the 1D image starts with the green pixel and ends with the red, and repeatedly shifted to produce the matrix $C$. The vectors corresponding to these shifts are illustrated to the left. Convince yourself the multiplication in the image below is equivalent to filtering of the 1D image at the first row of the matrix with the yellow filter.
Any circulant matrix
can be diagonalized by the Fourier matrix F as:
Read DFT Matrix and convolution theorem from Wikipedia if you are interested in understanding why this is so.
This fact can be exploited to achieve multiplication of a vector with
in O(nlogn) time using FFT. The following scheme can be used for this purpose, where
is the first column
and
is a vector containing the main diagonal of
:
where
is element-wise multiplication. Notice the above algortihm is actually the Convolution Theorem in disguise, where
is the Fourier Transform operator:
The mirror mismatch between convolution and correlation operations can be accounted for by conjugation of the filter in Fourier space.
Tue, 04 Oct 2016 10:50:00 +0000
https://ahmetcecen.com/blog/class/2016/10/04/FiltandFFT/
https://ahmetcecen.com/blog/class/2016/10/04/FiltandFFT/classmaterials-informaticsclass2-Point Statistics & FilteringGraphical Explanation of Filtering
Let us assume we have the following matrix, or image.
We are interested using this filter on our image.
A filter is applied on an image by placing it centered at every possible location on an image, multiplying the values of the filter and the current subsection of the image corresponding to the location of the filter. The multiplied values are then summed up and the resulting value is placed at the pixel the filter was centered on.
Here is a detailed graphic of a single step.
Repeat this for every pixel.
How do we deal with the boundaries? Many ways, but for our purposes, we will either assume that one end of the image is connected to the other (periodic boundary) or there is nothing (zeros) beyond the boundary (non-periodic boundary).
Filtering for 2-Point Statistics
For efficient calculation of 2-point statistics, we use the image itself (phase of interest marked with 1s, all other phases marked with 0s) as both the input and the filter. Graphical examples of how this would work are given below for 2 vectors.
One step.
Another step.
It follows that the resulting matrix will give us the counts of succesful trials in the frequentist definition of 2-point statistics, for each vector. Normalization will yield the actual frequencies.
Tue, 04 Oct 2016 10:40:00 +0000
https://ahmetcecen.com/blog/class/2016/10/04/Filtering/
https://ahmetcecen.com/blog/class/2016/10/04/Filtering/classmaterials-informaticsclassUnderstanding 2-Point StatisticsFrequentist Approach
If I threw a particular vector at every possible location in this image, what fraction of them will land both on white pixels.
Bayesian Approach
I have a paper with 2 holes in it seperated by a particular vector. If I put this paper on a random location on top of the image, what is the probability that I will see a white pixel on both holes, given both holes reside within the image.
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Topological Approach
Normally, the location of each pixel in the image is described by a vector from the x axis, and a vector from the y axis. If I instead describe each white point, by vectors from every other white point in the image, how many times would I observe a particular vector (normalized by the number of points/vectors).
Tue, 04 Oct 2016 10:30:00 +0000
https://ahmetcecen.com/blog/class/2016/10/04/2-point-tutorial/
https://ahmetcecen.com/blog/class/2016/10/04/2-point-tutorial/classmaterials-informaticsclassExample PatternsCarbon Fiber
Crystalline Polymer
Amorphous Polymer
Kikuchi Pattern
Wood
Circular Mesh
Fri, 26 Aug 2016 00:00:00 +0000
https://ahmetcecen.com/blog/class/2016/08/26/example-patterns/
https://ahmetcecen.com/blog/class/2016/08/26/example-patterns/exampledemoclass