Self Fourier shell correlation: properties and application to cryo-ET

The Fourier shell correlation (FSC) is a measure of the similarity between two signals computed over corresponding shells in the frequency domain and has broad applications in microscopy. In structural biology, the FSC is ubiquitous in methods for validation, resolution determination, and signal enhancement. Computing the FSC usually requires two independent measurements of the same underlying signal, which can be limiting for some applications. Here, we analyze and extend on an approach to estimate the FSC from a single measurement. In particular, we derive the necessary conditions required to estimate the FSC from downsampled versions of a single noisy measurement. These conditions reveal additional corrections which we implement to increase the applicability of the method. We then illustrate two applications of our approach, first as an estimate of the global resolution from a single 3-D structure and second as a data-driven method for denoising tomographic reconstructions in electron cryo-tomography. These results provide general guidelines for computing the FSC from a single measurement and suggest new applications of the FSC in microscopy.

Microscopy images are typically corrupted by imaging artifacts.In cryo-EM, this is modeled by the contrast transfer function (CTF) which is the Fourier transform of the point spread function of the microscope.We therefore discuss the e↵ects of these modifications on the estimation of the FSC by SFSC.Often, the forward model may be written as: where x is the Fourier transform of the underlying signal, ĉ is the CTF e↵ect, and ŷ is the Fourier transform of the corrupted observation.Given two images of the same signal a↵ected by the same CTF, the FSC can be readily used to estimate the SSNR, but it should be noted that it will be an estimate of the SSNR for the corrupted signal x = F 1 {ĉ • x}, and not the clean signal x.Similarly, the SFSC can also be used to estimate the SSNR of the corrupted signal assuming the measurement follows the properties assumption 1 and assumption 2. We illustrate this with two simulated examples.The first example is for a typical CTF in cryo-EM images, described by: with w = 0.1, = 2.51 pm, z = 2.8 µm, C s = 2.0 µm, and ↵ = 0.87.The second example is for a CTF that resembles a tilt series image in cryo-ET (see Supplementary Note 7), described by: with ↵ = 3/4 and z = 5.We generate noisy, CTF corrupted images following eq.( 17).The FSC, SFSC and spherically averaged power spectrum are then computed for each image Figure S1.As expected, the oscillations of the CTF modulate the correlation profile in both the FSC and SFSC.Importantly, the SFSC still approximates the FSC well.In these scenarios, the traditional resolution value obtained from a threshold may not be meaningful as we expect the FSC to oscillate.Most importantly, while the CTF may change the reported resolution value, the Wiener filter computed using the SFSC of a CTF modified signal is still a statistically optimal filter for denoising.We note that if the CTF is not radially symmetric, then the CTF corrupted signal should not be expected to satisfy radially symmetric assumptions.In this case, both the FSC and SFSC are poor estimators of the SSNR.However, when applied to reconstructions from a collection of CTF corrupted images with a random and uniform distribution of poses, the reconstructions will have approximately radially symmetric noise and variance even if the images do not.18).(b) Results for image formed using CTF in eq. ( 19).Also plotted is the absolute value of the radial CTF.
pairs to be compared.There are two main disadvantages of splitting in a checkerboard-like pattern compared to only splitting along one dimension at a time, as proposed in this work.First, the variance of the noise in the downsampled measurements is scaled by 2 dim , where dim is the number of dimensions split across.This relation is derived in the following section.Second, the Nyquist frequency will be reduced to half of the original.We demonstrate both of these e↵ects using the 2-D case of an image in Figure S2.

Supplementary Note 3 General relation between FSC and SFSC
To relate the SFSC to the FSC, we need to relate the DFT of the decimated signals to the DFT of the original signal.We first show this relation for a signal split along one dimension, as advocated in this work, and then provide an example for a signal split along two dimensions, as done for images in the original splitting scheme by Koho et al. [8], to demonstrate how it extends to a signal split along multiple dimensions.Let x be a 1-D signal of length N .The DFT of x is defined as x , where !N = exp( 2⇡i/N ).The signal x can be split into even index terms x e [n] = x[2n] and odd index terms x o [n] = x[2n + 1] for n 2 {0, . . ., (N/2) 1}.We want to relate xe [k] and xo [k], the DFTs of x e and x o , to the DFT of the original signal x[k].The relation can be seen by splitting the DFT of x into the sum of the even and odd terms: Next, we use that ! 2 N = exp( 2⇡i/(N/2)) = !N/2 to get: Applying the definition of the DFT to the right side of the equation yields: x Importantly, k = 0, . . ., N 1, and xe and xo are N/2 periodic1 .We can then independently relate xe [k] and xo [k] to x[k] as follows: Equation ( 23) and eq.( 24) form the framework of our analysis presented in Section 2.2, and describe the relation between the DFTs of a 2-D or 3-D measurement that has been split into alternating voxels along one dimension.The analysis above holds for a signal split along multiple dimensions as well, since the DFT can be applied along subsequent dimensions.Specifically, we are referring here to the checkerboard-like splitting pattern from [8].Suppose now that x is a 2-D signal (i.e., an image).If the signal x is split into even and odd index terms along both dimensions, then we have that: where e denotes even and o denotes odd indexing for each dimension of x, and k 1 and k 2 are the frequency indexes of x.In the case of a noisy measurement, we are interested in the relation between the SFSC from the downsampled measurements using this splitting scheme and the SSNR.Following the arguments presented in Section 2.2, if both assumptions on the signal and noise are met, then: For the general case of decimating into even and odd terms over multiple dimensions, we see that: Thus, when splitting along multiple dimensions, there is a scaling of 2 dim on the noise variance compared to a scaling of 2 from splitting once as proposed in this work.From eq. ( 28), the relation between the EFSC and ESFSC split over multiple dimensions is: We show in Figure 3A that the SFSC will yield an underestimate of the FSC if the correction in eq. ( 29) is not applied.For the 2-D case in Figure 3A, the SFSC was computed using our splitting scheme.Thus eq. ( 29) is equal to eq. ( 12).
Noise estimation from reconstructed volumes

Figure S1 :
Figure S1: E↵ect of the CTF on the FSC and SFSC.Clean image is a projection of EMD-11657 (N ⇥ N = 360 ⇥ 360, pixel size = 0.812 Å).Additive Gaussian noise was generated with SNR = 15 and B noise = 10 Å2 .(a) Results for image formed using CTF in eq.(18).(b) Results for image formed using CTF in eq.(19).Also plotted is the absolute value of the radial CTF.

Figure S2 :
FigureS2: SFSC computed from an image downsampled by splitting along 1 and 2 dimensions.The pixel diagram depicts one pair out of two for the 1⇥ split and one pair out of six for the 2⇥ split.The input image used for the FSC and SFSC is from Figure3B.The SFSC from the 1⇥ split image is reported as the average of both pairs.The SFSC from the 2⇥ split image is reported as the average of the six combinations and is plotted with an error envelope representing the standard deviation of the SFSC from all combinations.

Figure S4 :Figure S6 :
Figure S4: Central slices of the raw half maps, the regions containing only noise, and their corresponding power spectrum (below each image) for the four structures in Figure 4.