Tricks and Thoughts https://ahmetcecen.com/blog/ Sat, 18 Mar 2023 07:08:42 +0000 Sat, 18 Mar 2023 07:08:42 +0000 Jekyll v3.9.3 Thu, 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/ projector visualization rotstats Thu, 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/ projector visualization rotstats Tue, 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/ projector visualization rotstats Tue, 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/ projector visualization rotstats Tue, 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/ projector visualization rotstats Why 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/ class materials-informatics class Fourier 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/ class materials-informatics class Graphical 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/ class materials-informatics class Frequentist 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. Your browser does not support the video tag. 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/ class materials-informatics class Carbon 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/ example demo class