# Name

get_imf_params - Calculate the image formation parameters under extended image formation model

# Usage

params = get_imf_params(1dpw1, 1dpw2, pu, nrank, q, lowf)

## Input

- 1dpw1
- 1D rotationally averaged power spectrum of protain particle images.
- 1dpw2
- 1D rotationally averaged power spectrum of protain noise images.
- pu
- protein model function
- nrank
- the rank of polynomial used to fit background noise
- lowf
low frequencies around (

``1/100 Å``), which contains large amout of protein structure factors and should be excluded from the parameters estimation.

## Output

- params
returned image formation parameters[

``C_1,C_2,B,C_3,C_4``]

## Options

# Description

. 1. **Standard image formation model** and **extended image formation model**

. i. In weak phase approximation, the electron microcopic image can be explained by the standard image formation model[1]:

``O = CTF*E*S+N``- Where CTF and E is the Contrast transfer function, and envelop function. N is the background noise, which, in 1D rotationally averaged power spectrum, is the base line of the power spectrum profile. It used to be heuristically approximated by a linear function in logrithm scale:
``N=e^(C_1*X+C_2)``

. ii. Considering the fact that the baseline subtracted residuals cannot be totally explained as signals(scattering contributed by protein particles), a term regarding solvent scattering is included, so the image fromation is proposed as[3]:

``O = CTF*E*(S+C)+N``- If we consider the solvent scattering, actually is also a sort of noise, then there are two kinds of noises in the image:
``N_1 = CTF*E*C````N_2 = e^(C_1*X+C_2)``.

- If we consider the solvent scattering, actually is also a sort of noise, then there are two kinds of noises in the image:

.2. The **evidence **of solvent noise:

- i . If one simply windows background noise from cryo micrographs, cacluate 1D rotationally averaged power spectrum, one can see weak CTF traces in the power spectrum.

. ii. If the base line background is subtracted, the amplified signal has quite strong CTF imprints and it even can be used to estimate the defocus. [ See reference 2 ]

. iii . If windowing the same number protein of particles, and background noise, and calculate their 1D rotationally averaged power spectra seperately. When comparing them, it is always found that in the plot the power spectrum of noise is higher than the power spectum of protein particles in intermediate frequencies and high frequencies. It indicates that the in the protein particle power spectrum, there appears some missing power in intemediate frequencies and high frequencies, in comparision with the power spectrum of noise background, due to the space occupation of the protein particles. . iv . Both power spectra, calculated from noise images and calculated from particle images have quite similiar envelop function.

.3.** Applications **of estimated paramters in improving 2D alignment, and 3D reconstruction.

- i. Calculate Wiener Filter
- ii. Calculate matched filter
- iii. Power spectrum adjustment

. 4 .** Image formation paramters**

. i . CTF paramters

See filt_ctf[2]. CTF parameters can be estimated automatically.[2] However, currenty, there is no available functions to estimate defocus automatically.

. ii . baseline noise: . ``N_2 = e^(C_1*X+C_2)``.

- Conventionally, it is estimated as a line in logrithm scale ( two parameters ). However, it can be approximated by higher rank polynomials.

. iii . evelop function . The profile of image Fourier factor amplidute is called evelop function. Many factors can affect the shape of envelop function: illumine source size, defocus and defocus spread, astigmatism, unknown experimental artifcats, and so on. However, the envelop function appears mainly due to temporal coherence of electrons. Because ofthe complexity of the origination, the formula is totally empirical. For simplicity, evelop function is approximated roughly as a Gaussian function by many EM researchers[4].

``E= e^(-B*X^2)``

The single parameters in the equation is also called B-factor. See filt_ctf to get more details about B-factor. Because in very low frequencies region(around ``1/100 Å``), there normally appear the protein structural factors. Hence, this area is excluded when performing B-factor estimation.Otherwise, it tends to get a pessimistic B-factor. In this function, the B-factor is estimated from power spetrum of background noise.

. iv . Protein model function.

See pufit

``C_4``

. v . CTF related noise, solvent effect noise.

``C_3``

. 5 . **Attentions** about ``E`` and ``CTF`` in the image formation.

i. 1D power spectrum has a quadric form of both

``E``and``CTF``. So envelop in the image formation should be``E= e^(-1/2*B*X^2)``.

Base line noise should be:

``N_2(X)=sqrt(e^(C_1*X+C_2))``- ii. Power spectrum under the extended image formation is:
``PW2 = CTF(X;dz)^2*e^(-B*X^2)*[C_4*P_u(X)+q*C_3]+e^(C_1*X+C_2)``

# Method

# Reference

.1. R. Wade & J. Frank *Optik* (1974) 49:81.

.2. Z. Huang, P. R. Baldwin, S.Mullapudi, and P .A. Penczek, Automated determination of parameters describing power spectra of micrograph images in electron microscopy. *J. Struct. Biol.* **144** (2003), pp. 79–94.

.3. Z. Huang, P. R. Baldwin, and P .A. Penczek, Power spectrum adjustment and extended image formation. in writing

.4. Ali Saad, Wah Chiu, Pamela, A. Thuman-Commike Multiresolution Approach to Automatic Detection of Spherical Particles from Electron Cryomicroscopy Images. *ICIP* **3** (1998): 846-850

# Author / Maintainer

Zhong Huang

# Keywords

- category 1
- UTILITIES

# Files

utilities.py

# See also

ctf_1d,filt_ctf,pw_extract, and get params cl1

# Maturity

- stable
- works for most people, has been tested; test cases/examples available.

# Bugs

None. It is perfect.