###### tags: `BioI-maging`

# SPA

#### Earlier Hack on Algebraic Reconstruction
https://hackmd.io/Sx2R0o8gRZyDSxsP0KIBRw?both

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Def: homogeneous structure: a structure with similar components
heterogeneous structure: a structure with dissimilar components, causing irregular attentuation of beam intensity (for e.g. CT).
"Heterogenous refers to a structure having a foreign origin. For example, heterogenous bone formation is bone where bone should not exist. To make matters worse, heterogenous bone formation is often also heterogeneous! Heterogenous is, therefore, the antonym of homogenous, meaning the tissue or structure is located or originates from an expected location. "(https://radiopaedia.org/articles/heterogeneous-vs-heterogenous?lang=us#:~:text=Heterogeneous%20refers%20to%20a%20structure,structure%20having%20a%20foreign%20origin.)

##### question:
1. Do iterative methods only work on 3. 3-D reconstruction, or all of 1, 2, and 3 (i.e. classification, orientation, 3-D reconstruction)?
2. What is the difference between single particle reconstruction and single particle analysis?

## SPA
A micrograph contains information about millions of particles in the sample. If the sample is homogeneous, then all particles are essentially the same kind of molecule but in different orientation. Thus, this micrograph shows the 2-D projections of a particle in different angles. Thus, we can do a 3-D reconstruction of this partcle.

Main steps of SPA:
1. Sample preparation (Biochemistry)
2. Sample Preparation for SPA
3. Imaging : Electron Microscopy
4. Data quality assessment
5. CTF determination and correction
6. Particle selection
7. 2D Classification
8. 3D Classification
9. 3D reconstruction
10. Model building

Answer to question 2: It seems single-particle analysis more focus on step 6 to 8, whereas single-particle recosntruction more focus on step 8 to 10.

##### 1. & 2. Sample Preparation:
Use negative staining to check the suitability of the sample. The sample should be pure and stable.

"Before embarking on a high resolution (cryo EM) time consuming SPA project, almost all of the samples are analyzed with negative staining." (Nanyang Tech PPT)

##### 3. Imaging
EM imaging: Recording images of a sample on an EM.

"
Several thousands (some times millions) of images are required for high resolution SPA

Automated data acquisition enables to image large number of images"(Nanyang Tech PPT)

##### 5. CTF
CTF: mathematically describe how information is tranferred to EM images.

"Without CTF correction an EM image is not a true(accurate) projection representation of a 3D object."(Nanyang Tech PPT)

##### 6. Particle Selection
A micrograph has many particles on it.
Particle selection: Selecting and windowing out individual particle on a micrograph.

##### 7. 2-D Classification
2-D Classification: Infer the orientations of images from a set of noisy particle images and group similar iamges together. (S. Chockchowwat & C. Bajaj, Probabilistic PolarGMM) Then obtain clean class averages by averaging aligned and clustered noisy images

Fourier-Bessel steerable principal component analysis (FBsPCA) : Used for computing rotation.
Motivation of the paper: Make FBsPCA also handle translations (S. Chockchowwat & C. Bajaj, Probabilistic PolarGMM)

MSA(multivariate stat analysis) $\to$ think about multiple random variables and their correlation.(Nanyang Tech PPT)
Movtivation: Partition the set of all images into equivalence classes. Then create a class average which has the best representativeness of all the elements in that class for each class.

Alignment: align images by shifting(translation) & rotating.

2-D classification $\to$ Class averages have better contrast and higher signal to noise ratio. It is a unbiased process with less user interference.(Nanyang Tech PPT)

2-D classification is to "group these images into multiple distinct clusters with a translation and rotational alignment for each image such that the average of aligned images in each cluster reflects an improved signal-to-noise ratio projection of the particle from each specific orientation." (S. Chockchowwat & C. Bajaj, Probabilistic PolarGMM)

Cryo-EM users often rely on SPA toolkits to implement the clustering and averaging algorithms. The most common toolkits are EMAN2 and RELION.

EMAN2: Use autocorrelation to find translational and rotational invariant features. Then it performs $k-means$ clustering in MSA bases.

RELION(ML2D): Use expectation maximization algorithm to optimize a log likelihood function.

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##### 7.1 Probabilistic PolarGMM: Unsupervised Cluster Learning of Very Noisy Projection Images of Unknown Pose
~by Supawit Chockchowwat & Chandrajit L. Bajaj

Achivement: FBsPCA representation was extended to handle both rotation and translation. A polarGMM based on FBsPCA was introduced to solve the 2-D particle clustering problem.

FBsPCA can handle rotational alignment originally (3.1). It is extended with translation operators to handle translation alignment in the article (3.2).

It turns out that the translation operator is expensive. Thus, the authors introduce a pre-centering algorithm to reduce the translation search space (3.3).

A comprehensive 2-D classification algorithm is constructed with the consideration of clustering and alignment imperfection (3.4).


$\pmb {2-D}$ $\pmb {Classification}$ $\pmb {Result}:$
Accuracy, AMI, & Homogeneity: PolarGMM is consistently better than EMAN2 and RELION.

Completeness: PolarGMM is better. RELION sometimes reports better completeness measure due to the collapse in its number of clusters (This is a very unfavored situation).

Translation Alignment Error: EMAN2 is slightly better than polarGMM.

$\pmb {Conclusion}$:
Compare with EMAN2 and RELION, polarGMM has comparable clustering accuracy and better efficiency.

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##### 9. 3-D Reconstruction:

Many ideas of backprojection&FBP are applied here.(Nanyang Tech PPT)

Unlike Cyro-ET, we don't know the angle of both 2-D images and their class averages.

Angular Reconstruction: "two different 2D projections of a 3D object always have a common one dimensional (1D) line in their projections where amplitude and phase are similar in two images" (Nanyang Tech PPT)
$\to$ We can find a common line, and align 2-D projections along that common line.

We take the Fourier transform for 2 images and look at their sinogram to find the common line.

3-D refinement (Iterative methods):
The reconstruction besed on class averages (Call this model A) can be improved using original 2-D projection images.

Steps:
1. Reproject model A in all directions. Call the images of reprojection reprojection images.
2. Create a partition of all original images with reprojection images as the template for each class (correlations).
3. Adjust the reprojection images based on original images.
4. Recreate the 3-D reconstruction
5. Repeat until the resolution of the reconstruction stop improving.

Resolution of the recosntruction:
Fourier Shell Correlation (FSC):
1. Create a partition of the set of all images where the partition only has two classes with equal cardinality.
2. Create a 3-D reconstruction for each subset (all elements in a equivalence class).
3. Calculate the correlation of these 2 3-D recosntructions using Fourier shell.
(Nanyang Tech PPT)