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Why is the cross-correlation coefficient between two identical images equal to one?

  Every image can be represented by a matrix. For example, a simple image with a diagonal line can be represented by an identity matrix, A1: Suppose we want to calculate the correlation coefficient of A1 and itself, we will use the following formula: Based on the matrix multiplication rule, we can only multiply a m x n matrix A with a n x p matrix B in which the column number of A is equal to the row number of B. In our case, after vectorizing A1, it becomes a 1x4 matrix [1001]. To multiply A1 and A1, we have to first transpose A1 into A1 T which is a 4x1 matrix, then we can calculate the dot product and the coefficient:
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How to calculate total dose and dose per frame with a given image condition?

Assuming we already know the following parameters of our image condition on a TEM: Voltage=300 Pixel size = 1 Å Dose rate = 8 (electron/pixel/second) Exposure time = 10 s Number of frames per movie= 40 How to calculate total dose per movie? Total dose (electron/Å^2)= Dose rate (electron/Å^2/second) x exposure time (s)                                        = 8 (electron/pixel/second)/ (1 Å x 1 Å ) x 10 s                                        = 80 (electron/Å^2) How to calculate dose per frame? Dose per frame (electron/Å^2)= Total dose (electron/Å^2) / number of frames                                                = 80 (electron/Å^2) / 40                                                = 2 (electron/Å^2)
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What Does Fourier Transform Do to Cryo-EM Images?

Cryo-EM images are formed by pixels with different intensity in grayscale. Each pixel is represented as a number in the image file. Below is a table simulating the pixel intensity of an image. If we plot the numbers in the last row with column indexes, we may get a curve look like this: Here we need to introduce a theorem that every curve can be approximated by the sum of simple sine waves (y = A sin(wx +b)) as presented in the figure below. Therefore, if we apply Fourier transform to the curve, we will get a set of sine waves with different amplitude (A), phase (b) and frequency (1/w). In this way, we convert the information in the real image into three parameters in the Fourier space. And the process of converting a curve into several sine waves is called Fourier transform. The red curve on the left is able to be approximated by the sum of several blue curves on the right. Image from https://www.ritchievink.com/blog/2017/04/23/understanding-the-fourier-transform-by-exampl
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What is the Preferred Orientation Problem in Cryo-EM And How to Deal with It?

1.        What is the preferred orientation problem? Preferred orientation problem occurs when most of the particles presented the same view in the EM image. It has been a tricky problem for high-resolution structure determination with cryo-EM, especially for samples without symmetry. It is caused by the adherence of the sample to the air-water interface or to the substrate of the grid (carbon film or graphene oxide). Let’s take 20S Proteasome as an example. Figure 1. The side view and top view (A) of the 20S proteasome (EMD-8726). The simulated image with the preferred side view and top view (B). When most of the proteasome particles prefer to “lie on the grid”, we will see a lot of side views like showing in Fig.1B (left panel). And when most of them “stand on the grid”, only top views are observed (Fig1B, right panel).  2.        How does the preferred orientation affect the resolution? To answer this question, we will need to first understand the “central slice
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