frc

frc is a Python library for calculating Fourier Ring Correlation (FRC) curves and their associated resolution.

This is particularly useful for determining the resolution of images taken with super-resolution microscopy techniques.

Its most computationally intensive functions are all implemented either in NumPy or Rust, making this library quite performant.

Fourier Ring Correlation

Introduced in 1982, Fourier Ring Correlation compares two 2D images, which are assumed to be noise-independent. In Fourier space, mage structure dominates the lower spatial frequencies (in the Fourier domain), while noise dominates at the higher frequencies.

In Fourier space, there are rings of constant spatial frequency around the origin. By calculating the correlation between the rings in the two images (varying the spatial frequency), you can compute an FRC curve (y-axis: correlation, x-axis: spatial frequency).

At some spatial frequency, the signal cannot be separated from the noise. What spatial frequency exactly depends on what threshold function is used. The standard 0.143 and 1/2-bit thresholds are both available, as well as an algorithm to compute the intersection and resulting resolution.

For additional information about Fourier Ring Correlation, see R.P.J. Nieuwenhuizen et al. (2013).

1FRC

FRC requires two noise-independent images to work. However, modern cameras are often shot noise-limited (Poisson noise). This library includes a new method due to Bernd Rieger and Sjoerd Stallinga (Department of Imaging Physics, TU Delft), called 1FRC, which uses a technique called binomial splitting to derive two images from a single one, where the pixel counts of the derived images are independently Poisson noise distributed.

They have an upcoming paper detailing the method and providing additional theoretical and experimental justification. This library was produced for my bachelor thesis (which focused on 1FRC) and was supervised by Bernd Rieger. The bachelor thesis can be found here..

Installation

You can download this library from PyPI:

pip install frc

This library depends on tmtenbrink/rustfrc, DIPlib, scipy and of course, numpy.

Usage

The code snippet below (which for illustration purposes assumes you have a numpy array or DIP image representing your input, its associated scale and also matplotlib to plot the result) will calculate the FRC curve and the associated resolution using the standard 1/7 threshold.

import frc
import numpy as np
import matplotlib.pyplot as plt

... # get an image and scale

img = np.array(img)
# Input can be a numpy array or DIP image
img = frc.util.square_image(img, add_padding=False)
img = frc.util.apply_tukey(img)
# Apply 1FRC technique
frc_curve = frc.one_frc(img)

img_size = img.shape[0]
xs_pix = np.arange(len(frc_curve)) / img_size
# scale has units [pixels <length unit>^-1] corresponding to original image
xs_nm_freq = xs_pix * scale
frc_res, res_y, thres = frc.frc_res(xs_nm_freq, frc_curve, img_size)
plt.plot(xs_nm_freq, thres(xs_nm_freq))
plt.plot(xs_nm_freq, frc_curve)
plt.show()

Troubleshooting

If you cannot find an intersection (NoIntersectionException), be sure to plot the curve without using the frc_res method and see if there even is an intersection. For example:

import frc
import numpy as np
import matplotlib.pyplot as plt

img = ... # get an image and scale

# Apply 1FRC technique
frc_curve = frc.one_frc(img)
img_size = img.shape[0]
xs_pix = np.arange(len(frc_curve)) / img_size
# scale has units [pixels <length unit>^-1] corresponding to original image
xs_nm_freq = xs_pix * scale
plt.plot(xs_nm_freq, frc_curve)
plt.show()

For images with a significant amount of non-Poisson noise, the 1FRC method has been shown to fail (adjustments are possible).