text:
\section{Introduction}
\endgroup

In recent years, single particle cryo-electron microscopy (cryo-EM) has emerged as a leading tool for resolving
the 3D structure of macromolecules from multiple 2D projections of a specimen~\citep{cheng2017cryo}.
In this technique, multiple copies of a particle are embedded in vitreous ice and imaged in an electron microscope.
This yields a set of micrographs, each containing several 2D particle projections.

The micrograph does not contain clean particle projections but is contaminated by several factors, including noise, ice aggregates and carbon film projection.
The noise stems from an inherent limitation on the number of imaging electrons that can be applied to the specimen.
The interference from carbon film and ice aggregates are due to the particular sample preparation techniques used.

The 2D projections in the micrograph are also distorted by convolution with a point spread function.
This point spread function is due to the electron microscope configuration. It attenuates certain frequencies and flips the sign of certain frequency bands.
A 3D density map reconstructed from distorted projections yields an unreliable representation of the particle \citep{frank2996book}.
It is therefore important to estimate the point spread function and account for it during reconstruction.

To estimate these parameters, it is convenient to consider the Fourier transform of the point spread function, known as the \emph{contrast transfer function} (CTF).
This is due to two factors.
First, the CTF has a simple expression in the polar coordinates of the spatial frequency.
Second, its effect is directly visible in the frequency domain where the CTF acts as a pointwise multiplication rather than a convolution \citep{erickson1971ctf}.

CTF estimation is one of the first steps in the single particle cryo-EM pipeline.
Indeed, accounting for the CTF is needed in a variety of tasks, such as particle picking \citep{heimowitz2018apple}, denoising \citep{bhamre2016denoising}, class averaging \citep{scheres2012relion}, ab initio reconstruction \citep{punjani2017cryosparc}, refinement \citep{scheres2012relion, punjani2017cryosparc,tang2007eman2,grant2018cistem} and heterogeneity analysis \citep{scheres2012relion}.

The CTF is typically modeled as a sine function whose argument depends on the spatial frequency and several parameters of the objective lens of the microscope \citep{frank2996book}.
The parameters we focus on in this paper are the defocus and astigmatism of the objective lens as these are unknown and must be estimated from the data. Additionally, the CTF is multiplied by a damping envelope, which suppresses the information in high frequencies~\citep{sorzano2007damping}.

When estimating the CTF parameters, it is common to first estimate the power spectrum of the micrograph.
The observed micrograph image is typically modeled as a CTF-dependent term plus a noise term unaffected by CTF.
The first term corresponds to a noiseless micrograph, that is, the tomographic projection of the sample filtered by the CTF \citet{zhu19977gaussian}.
Modeling the unfiltered and filtered micrographs as 2D random fields, we find that their power spectra are closely related: the latter equals the former multiplied by the squared CTF. This multiplication induces concentric rings, known as Thon rings~\citep{thon1971book}, in the power spectrum of the filtered micrograph (see Fig.~\ref{fig:astigmatism}).
Estimating the CTF therefore reduces to fitting the parameters of the CTF to the estimated power spectrum.

The vast majority of CTF estimation methods use a variant of the periodogram when estimating the power spectrum of the micrograph. This is due to its speed and simplicity.
Unfortunately, the periodogram produces a biased and inconsistent estimate of the micrograph power spectrum.

Beyond these issues with the spectral estimators, fitting the CTF model to the estimated power spectrum is complicated by factors such as  the high levels of noise present in the micrograph, {coincidence loss at the detector and more.
The expected power spectrum equals the power spectrum of the clean, filtered micrograph plus a background term caused by the aforementioned complications.
The background  masks the true oscillations of the power spectrum of the particle projection. }
It is therefore important to estimate and remove the background from the estimated power spectrum~\citep{zhu19977gaussian, rohou2015ctffind4, mindell2003ctffind3, kai2016gctf}.

Assuming that the micrograph power spectrum and the background were both estimated perfectly, the background-subtracted power spectrum equals the power spectrum of the filtered, clean micrograph.
One way to estimate the CTF parameters is then to maximize the correlation of the background-subtracted power spectrum estimate and a squared CTF (or some monotonic function thereof)~\citep{rohou2015ctffind4, mindell2003ctffind3,kai2016gctf, tani1996correlation}.
Optimizing the correlation then provides an estimate of the defocus and astigmatism.
Another approach identifies a single ring in the estimated power spectrum and uses it to derive a closed-form solution of the CTF parameters~\citet{yan2017single_ring}. In order to formulate this solution, all prior knowledge regarding the spherical aberration must be ignored.

In this paper, we present ASPIRE-CTF, which is a new method for CTF estimation, available as part of the ASPIRE package.\footnote{\url{https://github.com/ComputationalCryoEM/ASPIRE-Python}}
We first estimate the power spectrum using a multitaper estimator \citep{babadi2014multitaper}, further reducing the variance by averaging estimates from multiple regions of the micrograph.
Using this estimated power spectrum, we estimate the background noise spectrum using linear programming (LP).
Instead of using an approximate background model, our scheme ensures that the background-subtracted power spectrum estimate is non-negative and convex.
We also show that the CTF is contained in the span of a small number of steerable basis functions.
Thus, we further reduce the variance in our power spectrum estimate by projecting onto this span.

Given the power spectrum estimate, we provide two solutions for estimating the CTF parameters.
Our first solution is similar to~\citep{rohou2015ctffind4,mindell2003ctffind3, kai2016gctf, tani1996correlation}, where CTF parameters are estimated by maximizing the correlation of the square root of the power spectrum estimate with the absolute value of simulated CTFs.
The second solution uses the spatial frequencies of several zero-crossings.
Since we expect these zero-crossings to coincide with those of the squared CTF, we use them to define an overdetermined system of equations over the CTF parameters that we then solve. We note that, while our first solution is more robust, our second solution is faster to compute.

Our method is experimentally verified in Section~\ref{sec:experiment}. This is done via a
comparison of the defocus estimates with that of~\citet{rohou2015ctffind4, kai2016gctf} on several datasets from the CTF challenge~\citep{marabini2015challenge}.
We show that our power spectrum estimation method is usually in agreement with one of the state-of-the-art methods~\citet{rohou2015ctffind4, kai2016gctf}.

The main contribution of this paper, appearing in Sections \ref{subsubsec:variance1}-\ref{sec:bk}, is our method for estimating the power spectrum of a micrograph. We reduce the variability of the power spectrum estimate, and are therefore  the first to obtain an estimate where several zero-crossing rings of the CTF are easily recovered without additional assumptions.

We present the pipeline of our method in Fig. \ref{fig:pipeline}. For each step of our suggested framework, we refer the reader to the appropriate section of the paper.

\begin{figure}[t]
\centering
{\includegraphics[width=0.78\linewidth]{heimo1}}
\caption{Pipeline of ASPIRE-CTF. The input is a movie and the outputs are the estimated defocus parameters that define the CTF. In the bottom portion of the graph, all actions done on the 1D radial profile of the power spectrum are presented on the left. Additionally, all actions performed on the 2D power spectrum are presented on the right.}
\label{fig:pipeline}
\end{figure}

\subsection*{Notation}

Given a 2D stationary random field $\mathbf{x}$ defined over $\mathbb{Z}^2$, we denote its autocovariance function by $R_\mathbf{x}$.
The Fourier transform of $R_\mathbf{x}$ is known as the \emph{power spectrum} of $\mathbf{x}$ and is given by
\begin{equation}
    S_\mathbf{x}(\mathbf{g}) = \sum_{n_1=-\infty}^\infty \sum_{n_2=-\infty}^\infty R_\mathbf{x}[m, n] \, \mathrm{e}^{\mathrm{j} 2\pi (g_1 n_1 + g_2 n_2)},
\end{equation}
for $\mathbf{g} = (g_1, g_2) \in [-1/2, 1/2]^2$ and $\mathrm{j} = \sqrt{-1}$.
We denote magnitude of the spatial frequency vector $\mathbf{g}$ by ${r}$ and its counterclockwise angle with the positive x-axis by ${\alpha}$.

\section{Materials and Methods}
\subsection{Problem formulation}
\label{sec:formation}

In the sample preparation stage of the single particle cryo-EM pipeline, many copies of a particle are embedded in vitreous ice.
The imaging process uses an electron microscope to obtain a micrograph containing 2D projections of each instance.
Under the weak-phase object approximation, we may describe this process by the linear model~\citep{frank2996book,thon1971book,mindell2003ctffind3}
\begin{equation}
\label{equ:image_formation}
\mathbf{y} = h_{\phi} * \mathbf{x} + \mathbf{e},
\end{equation}
where the clean tomographic projection $\mathbf{x}$ and the additive noise $\mathbf{e}$ are modeled as 2D stationary random fields \citep{frank2996book}.
Since convolution preserves stationarity, the observed micrograph $\mathbf{y}$ is also a stationary random field.
In this model, the clean projection $\mathbf{x}$ is convolved with the point spread function of the microscope $h_\phi$ which depends on a parameter vector $\phi$.
We will at times denote this clean, but filtered, micrograph by $\mathbf{z} = h_\phi * \mathbf{x}$.

The CTF is the Fourier transform of the point spread function $H_{\phi}$ and may be modeled by~\citep{rohou2015ctffind4}
\begin{equation}
H_{\phi}( \mathbf{g} ) = - \sin( \chi_{\phi} ( \mathbf{g} ) ),
\label{equ:ctf_sine}
\end{equation}
where $\mathbf{g}$ is the spatial frequency.
Its phase is given by
\begin{equation}
\chi_{\phi} ( \mathbf{g} ) = \frac{1}{2 p^2} \pi \lambda  {r}^2 \Delta f_\phi({\alpha}) - \frac{1}{2 p^4} \pi \lambda^3  {r}^4 C_s + w,
\end{equation}
where $\lambda$ is the electron wavelength, $C_s$ is the spherical aberration, $w$ is the amplitude contrast, and $p$ is the pixel size.
We also have the astigmatic defocus depth
\begin{equation}
\Delta f_\phi({\alpha}) = \Delta f_1 + \Delta f_2 + ( \Delta f_1 - \Delta f_2 ) \cos ( 2 {\alpha} - 2 \alpha_f ),
\label{equ:astigmatism}
\end{equation}
where ${\alpha}$ is the polar angle of $\mathbf{g}$ and $\Delta f_1$, $\Delta f_2$, and $\alpha_f$ are the major and minor defocus depths and the defocus angle, respectively.
These together form the defocus vector $\phi = (\Delta f_1, \Delta f_2, \alpha_f)$, which parametrizes the CTF.
The values $\Delta f_1$ and $\Delta f_2$ determine the amount of defocus along two perpendicular axes, while $\alpha_f$ specifies the counterclockwise angle between the major defocus axis and the positive x-axis.
The difference $\Delta f_1 - \Delta f_2$ measures the amount of \emph{astigmatism} in the CTF. A visualization of the effect of astigmatism is provided in Fig. \ref{fig:astigmatism}.

\begin{figure}[t]
\centering
\subfigure[]{
{
\setlength{\fboxsep}{0pt}%
\setlength{\fboxrule}{0.3pt}%
\fbox{\includegraphics[width=0.4\linewidth]{heimo2a}}}
}
\hspace{0.3cm}
\subfigure[]{
{
\setlength{\fboxsep}{0pt}%
\setlength{\fboxrule}{0.3pt}%
\fbox{\includegraphics[width=0.4\linewidth]{heimo2b}}}
}
\caption{Absolute value of example CTFs. (a) Radially symmetric CTF $\sfrac{(\Delta f_1 - \Delta f_2)}{(\Delta f_1+\Delta f_2)}=0$.
(b) Highly astigmatic CTF $\sfrac{(\Delta f_1 - \Delta f_2)}{(\Delta f_1+\Delta f_2)} = \sfrac{1}{2}$).\vspace{-4ex}}
\label{fig:astigmatism}
\end{figure}

The model~\eqref{equ:ctf_sine} allows us to discern several properties of the CTF.
First, $H_{\phi}$ is real and oscillates between positive and negative values.
As a result, it has several zero crossings.
Second, the CTF is radially symmetric when $\Delta f_1 = \Delta f_2$ (the non-astigmatic case).
Third, with no spherical aberration (i.e., $C_s = 0$) the level sets of the CTF consist of ellipses centered at the origin.
The spherical aberration $C_s$ thus accounts for small deviations from the elliptical shape.

While the parameters $\lambda$, $C_s$, and $w$ are typically known from the microscope configuration, the defocus parameters $\phi$ vary widely between experiments.
We must therefore estimate them to obtain an accurate model of the CTF.

To estimate $\phi$, we turn to the power spectrum of the micrograph.
The power spectra $S_\mathbf{x}$, $S_\mathbf{y}$, and $S_\mathbf{e}$ of $\mathbf{x}$, $\mathbf{y}$, and $\mathbf{e}$, respectively, are related by
\begin{equation}
\label{equ:true_psd}
S_\mathbf{y} ( \mathbf{g} ) = \vert H_{\phi} ( \mathbf{g} ) \vert^2 \, S_\mathbf{x} ( \mathbf{g} ) + S_\mathbf{e} ( \mathbf{g} ).
\end{equation}
This follows from \eqref{equ:image_formation} and the fact that convolving a stationary random field with $h_\phi$ multiplies its power spectrum by the square Fourier transform magnitude $|H_{\phi}|^2$.

Equation~\eqref{equ:true_psd} suggests that estimates of the power spectra $S_\mathbf{y}$, $S_\mathbf{x}$, and $S_\mathbf{e}$ can be useful in resolving the CTF.
We note that $S_\mathbf{x}$ and $S_\mathbf{e}$ are slowly decaying while $\vert H_{\phi} \vert^2$ oscillates rapidly in comparison. The background subtracted power spectrum is therefore approximately proportional to $|H_{\phi}|^2$. It follows that in order to
estimate the defocus parameters $\phi$, we may estimate $S_\mathbf{y}-S_\mathbf{e}$ and maximize its correlation with $|H_{\phi}|^2$.
This approach is used in \citep{rohou2015ctffind4, mindell2003ctffind3, kai2016gctf}.

Another approach is to estimate $\phi$ from zero-crossings of $S_\mathbf{y} - S_\mathbf{e}$ \citep{yan2017single_ring, tani1996correlation}.
Specifically, for spatial frequencies where $H_{\phi}(\mathbf{g}) = 0$, we have $S_\mathbf{y} ( \mathbf{g} ) - S_\mathbf{e} ( \mathbf{g} ) = 0$.
Identifying these zero-crossings from estimates of $S_\mathbf{y} - S_\mathbf{e}$ thus constrains the zeros of $H_{\phi}$ and lets us estimate its defocus parameters $\phi$.

For both approaches, the first step is to estimate the background-subtracted power spectrum $S_\mathbf{y} - S_\mathbf{e}$.
In the following, we propose an estimation method and show how the resulting estimate may be used to estimate $\phi$ by either maximizing correlation or matching zero-crossings.

As mentioned above, the CTF is also multiplied by an exponentially decreasing envelope function \citep{sorzano2007damping}, which effectively acts as a low-pass filter on $h_{\phi} * \mathbf{x}$.
In this paper, rather than include the envelope function in our analysis, we ignore high frequencies as they are strongly attenuated by the envelope. We also reduce the effect of the envelope function by estimating the CTF using the square root of our power spectrum estimate as in \citep{rohou2015ctffind4, mindell2003ctffind3}.
 In this way, the effect of the envelope function on the two methods discussed  is smaller.
\subsection{Power spectrum estimation}
\label{subsec:estimate}

In this section we present several methods for estimating the power spectrum of the micrograph. We first present the periodogram estimator and then show different methods for reducing its bias and variance.

\subsubsection{Periodogram estimator}
\label{subsec:periodogram}

In an experimental setting, we only have access to an $N \times N$ sample of $\mathbf{y}$, given by the values $\mathbf{y}[k_1, k_2]$ for $(k_1, k_2) \in \{0, 1, \ldots, N-1\}^2$.
Given these values, a common power spectrum estimator is provided by the periodogram \citep{oppenheim1989discrete}
\begin{equation} \label{equ:periodogram}
\hpsd^{(\mathrm{p})}_{\mathbf{y}} (\mathbf{g}) = \frac{1}{N^2}  \left\vert \sum_{k_1,k_2=0}^{N-1} \mathbf{y} \left[ k_1, k_2 \right] \, \mathrm{e}^{-\mathrm{j} 2 \pi ( g_1 k_1 + g_2 k_2 ) }  \right\vert^2,
\end{equation}
for $\mathbf{g} \in [-1/2, 1/2]^2$.
While $\hpsd^{(\mathrm{p})}_\mathbf{y}(\mathbf{g})$ may be calculated for any $\mathbf{g}$, it is typically calculated on the $N \times N$ grid
\begin{equation}
M_N = \left\{ -\frac{1}{2}, -\frac{1}{2} + \frac{2}{N}, \ldots, \frac{1}{2} - \frac{2}{N} \right\}^2.
\end{equation}
This enables the use of fast Fourier transforms (FFTs) for computing the periodogram with $O(N^2 \log N)$ computational complexity.
Due to this and its ease of implementation, the periodogram is a popular spectral estimator in cryo-EM.

Since our goal is to estimate $S_\mathbf{y}$, let us consider how well it is estimated by the periodogram.
The mean square error (MSE) of $\hpsd^{(\mathrm{p})}_\mathbf{y}$ at $\mathbf{g}$ is given by
\begin{equation}
\operatorname{MSE}(\hpsd^{(\mathrm{p})}_{\mathbf{y}}(\mathbf{g})) = \mathbb{E}\left[\vert \hpsd^{(\mathrm{p})}_\mathbf{y}(\mathbf{g}) - S_\mathbf{y}(\mathbf{g}) \vert^2\right].
\end{equation}
To analyze the source of error, it is useful to define the \emph{bias} and \emph{variance} of the periodogram.
The bias is defined as
\begin{equation}
\operatorname{Bias}(\hpsd^{(\mathrm{p})}_{\mathbf{y}}(\mathbf{g})) = \mathbb{E}\left[\hpsd^{(\mathrm{p})}_{\mathbf{y}}(\mathbf{g})\right] - S_{\mathbf{y}}(\mathbf{g})
\label{equ:bias}
\end{equation}
and measures the deviation of the expectation from the true value, while the variance
\begin{equation}
\operatorname{Var}(\hpsd^{(\mathrm{p})}_{\mathbf{y}}(\mathbf{g})) = \mathbb{E}\left[\left\vert \hpsd^{(\mathrm{p})}_\mathbf{y}(\mathbf{g}) - \mathbb{E}\left[\hpsd^{(\mathrm{p})}_\mathbf{y}(\mathbf{g})\right] \right\vert^2 \right]
\end{equation}
measures the average deviation of the periodogram from its expectation.
Both contribute to the MSE through the identity
\begin{equation}
\operatorname{MSE}(\hpsd^{(\mathrm{p})}_\mathbf{y}(\mathbf{g})) = \operatorname{Bias}^2(\hpsd^{(\mathrm{p})}_\mathbf{y}(\mathbf{g})) + \operatorname{Var}(\hpsd^{(\mathrm{p})}_\mathbf{y}(\mathbf{g})).
\end{equation}
A low MSE therefore requires low bias and low variance.

The periodogram, however, fails on both counts.
First, while the periodogram is asymptotically unbiased~\citep{percival1993multitapers,thompson1982spectral}, its bias remains large for small samples.
Second, {the periodogram is an \emph{inconsistent} estimator, that is, its variance does not decrease with an increase in sample size. Therefore,
a periodogram that extends over the entire micrograph will have variance approximately equal to that of a periodogram that extends over some section of the micrograph.}

In the following sections, we will therefore consider different approaches to reducing both the bias and variance of the periodogram.

\subsubsection{Bartlett's method}
\label{subsubsec:variance1}

We first consider an approach for reducing variance called \emph{Bartlett's method} \citep{oppenheim1989discrete}.
In this approach, the periodogram estimate is computed for several non-overlapping regions of the image.
These estimates are then averaged, reducing the variance by a factor approximately equal to the number of regions used.
It may therefore be tempting to drastically reduce the size of these regions.
However, in experimental data, averaging over regions that are too small will increase the bias.
Among other things, this would prevent us from properly estimating the low spatial frequencies.

We thus divide our image into $B$ non-overlapping blocks $\mathbf{y}_0, \ldots, \mathbf{y}_{B-1}$ of size $K \times K$.
The averaged periodogram is
\begin{equation} \label{equ:periodogram_blocks}
\hpsd^{(\mathrm{b})}_\mathbf{y}(\mathbf{g}) = \frac{1}{B} \sum_{b=0}^{B-1} \hpsd^{(\mathrm{p})}_{\mathbf{y}_b} (\mathbf{g}).
\end{equation}
If each block $\mathbf{y}_b$ is independent of the others, we have $\operatorname{Var}(\hpsd^{(\mathrm{b})}_\mathbf{y}(\mathbf{g})) = B^{-1} \operatorname{Var}(\hpsd^{(\mathrm{p})}_\mathbf{y}(\mathbf{g}))$.
Note that, since the block size is now $K \times K$, we sample $\mathbf{g}$ on $M_K$.

\subsubsection{Welch's method}
\label{subsec:bias}

The expected value of the periodogram estimator is known to be a convolution between the true power spectrum of the micrograph and a 2D Fej\'{e}r kernel~\citep{percival1993multitapers}. As the Fej\'{e}r kernel has high sidelobes, this convolution leads to frequency leakage and therefore a high bias.

One method of lowering the bias of the periodogram estimation is tapering \citep{percival1993multitapers}.
This multiplies the data $\mathbf{y}$ by a data taper $\mathbf{w}$ prior to computing the periodogram, resulting in a \emph{modified periodogram}.
While many options for data tapers exist, such as the Hann window \citep{vulovic2012taper}, Babadi and Brown \citep{babadi2014multitaper} suggest the use of the zeroth-order discrete prolate spheroidal sequence (DPSS) \citep{slepian1978prolates}.
The expected value of this modified periodogram is  a convolution between the true power spectrum of the micrograph and a kernel with smaller sidelobes than those of the Fej\'{e}r kernel~\citep{percival1993multitapers}. This reduces the frequency leakage, and, therefore, the bias of the estimator.

While the taper may be applied to the entire micrograph, it is also possible to apply it to each block in Bartlett's method \eqref{equ:periodogram_blocks}.
The resulting approach is known as \emph{Welch's method} \citep{welch1967wosa}.
Welch also showed that further variance reduction is possible using overlapping (typically half-overlapping) blocks~\citep{frenandez19977periodogram, huang2003env, frank2996book, zhu19977gaussian}.
This yields the \emph{modified averaged periodogram},
\begin{equation}
\hpsd^{(\mathrm{w})}_{\mathbf{y}}(\mathbf{g}) = \frac{1}{B} \sum_{b=1}^{B} \hpsd^{(\mathrm{p})}_{\mathbf{y}_b \cdot \mathbf{w}}(\mathbf{g})
\end{equation}
where $\mathbf{y}_b \cdot \mathbf{w}$ is the pointwise multiplication of $\mathbf{y}_b$ and $\mathbf{w}$.

\subsubsection{Multitaper estimators}
\label{subsec:multi}
As discussed in Section~\ref{subsubsec:variance1}, one way to lower the variance in the periodogram is to average several estimates.
For this reason, Thomson \citep{thompson1982spectral} suggested combining the estimates obtained from multiple tapers.
Each taper yields a different estimate of the power spectrum, and averaging them significantly reduces the variance.
A large number of tapers, however, results in significant smoothing of the power spectrum estimate, so the variance reduction needs to be balanced with an increase in bias for non-smooth power spectra.
Thomson found that higher-order DPSSs were well-suited to this task and called the resulting power spectrum estimator the \emph{multitaper estimator}.
These estimators have recently demonstrated their usefulness for noise power spectrum estimation in cryo-EM \citep{anden2017factor,anden2019multitaper}.  For details of the DPSS data tapers we refer the reader to Appendix A.

Combining all the above methods for variance and bias reduction, we arrive at the multitaper estimator
\begin{equation} \label{equ:multitaper}
\hpsd^{(\mathrm{mt})}_{\mathbf{y}}(\mathbf{g}) = \frac{1}{L B} \sum_{b=0}^{B-1} \sum_{\ell=0}^{L-1} \hpsd^{(\mathrm{p})}_{{\mathbf{y}}_b\cdot\mathbf{w}_\ell}(\mathbf{g}),
\end{equation}
where $\mathbf{w}_\ell$ is the $\ell$th out $L$ DPSSs for grids of size $K \times K$.

 Figs. \ref{fig:compare}-\ref{fig:compare2}  present a comparison between $\hpsd^{(\mathrm{p})}_{\mathbf{y}}$, $\hpsd^{(\mathrm{b})}_\mathbf{y}$, $\hpsd^{(\mathrm{w})}_\mathbf{y}$, and $\hpsd^{(\mathrm{mt})}_\mathbf{y}$.
The CTF oscillations are best resolved by the multitaper estimator $\hpsd^{(\mathrm{mt})}_\mathbf{y}$.

\begin{figure*}
\begin{tabular}{cccc}
\centering
Periodogram & Bartlett's method & Welch's method & Multitaper method\\
{\includegraphics[width=0.22\linewidth]{heimo3a}} &
{\includegraphics[width=0.22\linewidth]{heimo3b}} &
{\includegraphics[width=0.22\linewidth]{heimo3c}} &
{\includegraphics[width=0.22\linewidth]{heimo3d}}\\
{\includegraphics[width=0.22\linewidth]{heimo3a_zc}} &
{\includegraphics[width=0.22\linewidth]{heimo3b_zc}} &
{\includegraphics[width=0.22\linewidth]{heimo3c_zc}} &
{\includegraphics[width=0.22\linewidth]{heimo3d_zc}}
\end{tabular}
\caption{
Power spectrum estimation and zero-crossings of the estimation for a $\beta$-galactosidase micrograph from the EMPIAR-10017 dataset \citep{scheres2015semi}. In the top row we present several power spectrum estimates (plotted on a logarithmic scale). The estimates are produced using the Periodogram estimator (leftmost column), Bartlett's method (center-left column), Welch's method (center-right column) and the Multitaper method with $L=9$ data tapers (rightmost column). Below each estimate we present its zero-crossings (which are determined as specified in Section~\ref{subsec:zero_cross}).
Blocks of size $512 \times 512$ were used.
The zero-crossings of the CTF are most easily identifiable in the multitaper estimates.}
\label{fig:compare}
\end{figure*}

\begin{figure*}
\begin{tabular}{cccc}
\centering
Periodogram & Bartlett's method & Welch's method & Multitaper method\\
{\includegraphics[width=0.22\linewidth]{heimo3e}} &
{\includegraphics[width=0.22\linewidth]{heimo3f}} &
{\includegraphics[width=0.22\linewidth]{heimo3g}} &
{\includegraphics[width=0.22\linewidth]{heimo3h}}\\
{\includegraphics[width=0.22\linewidth]{heimo3e_zc}} &
{\includegraphics[width=0.22\linewidth]{heimo3f_zc}} &
{\includegraphics[width=0.22\linewidth]{heimo3g_zc}} &
{\includegraphics[width=0.22\linewidth]{heimo3h_zc}}
\end{tabular}
\caption{
Power spectrum estimation and zero-crossings of the estimation for an 80S ribosome micrograph from the EMPIAR-10028 dataset \citep{wong2014empiar}. In the top row we present several power spectrum estimates (plotted on a logarithmic scale). The estimates are produced using the Periodogram estimator (leftmost column), Bartlett's method (center-left column), Welch's method (center-right column) and the Multitaper method with $L=9$ data tapers (rightmost column). Below each estimate we present its zero-crossings (which are determined as specified in Section~\ref{subsec:zero_cross}).
Blocks of size $512 \times 512$ were used.
Once again, the zero-crossings of the CTF are most easily identifiable in the multitaper estimates.}
\label{fig:compare2}
\end{figure*}

\subsection{Background subtraction}
\label{sec:bk}

In this section we present a method for removing the background spectrum and further reducing the variability of the estimator $\hpsd^{(\mathrm{mt})}_\mathbf{y}$.
We do this by first estimating the radial profile of the background spectrum and removing this estimation from $\hpsd^{(\mathrm{mt})}_\mathbf{y}$.
Further, we show that the set of squared CTFs is contained in the span of a steerable basis and project our estimate onto that basis.

\subsubsection{Estimating the  background }
\label{subsec:bksub1}

We saw in~\eqref{equ:true_psd} that the micrograph power spectrum can be expressed as the sum of two spectra: the clean, filtered power spectrum $\vert H_{\phi}(\mathbf{g}) \vert^2\,S_\mathbf{x}(\mathbf{g})$ and the background  $S_\mathbf{e}$.
The background-subtracted power spectrum is therefore
\begin{equation}
S_{\mathbf{y}} (\mathbf{g}) - S_{\mathbf{e}} (\mathbf{g}) = \vert H_{\phi} (\mathbf{g}) \vert^2 \, S_{\mathbf{x}} (\mathbf{g}).
\end{equation}
An estimate of $S_\mathbf{y} - S_\mathbf{e}$ is used by many methods to estimate the CTF parameters $\phi$ \citep{rohou2015ctffind4, zhu19977gaussian}.
Their success therefore depends on accurate estimation of the background $S_\mathbf{e}$.

The background is influenced by many factors and accurately modeling these factors is an open challenge.
Many background estimation methods instead treat the background as a radially symmetric and slowly varying function
\citep{frank2996book}.

The background estimation problem can be formulated as a curve fitting problem \citep{frank2996book}. We note that the background should coincide with $\hpsd^{(\mathrm{mt})}_\mathbf{y}$ at the zero-crossings of $H_{\phi}$. Furthermore, the background should be strictly smaller than $\hpsd^{(\mathrm{mt})}_\mathbf{y}$ at spatial frequencies where $H_{\phi}$ does not have a zero-crossing (since $S_\mathbf{x}$ is strictly  positive). We therefore estimate the background by minimizing the difference between $\hpsd^{(\mathrm{mt})}_\mathbf{y}$  and $S_\mathbf{e}$.

While the radial profile of the background is monotonically decreasing in most settings,  this is not the case when a Gatan K2 direct detector is used in counting mode with a high dose rate. Rather, the background will be monotonically decreasing in the lower frequencies and monotonically increasing in higher frequencies \citep{li2013background}. Since a monotonically decreasing function, as well as a function that is at first monotonically decreasing and later monotonically increasing, must be convex, we model the background as the non-negative, convex function that is closest to, and no larger than, $\hpsd^{(\mathrm{mt})}_\mathbf{y}$.

We propose estimating the background $S_\mathbf{e}$ through LP.
Specifically, we minimize the $\ell_1$ norm of the background-subtracted power spectrum estimate subject to several linear constraints.
The first constraint ensures that the background-subtracted power spectrum estimate is non-negative, while the remaining constraints ensure that $\hat{S}_\mathbf{e}$ is a non-negative and convex.

Since we assume the background is radially symmetric, we consider its radial profile. To this end, we calculate the radial average of $\hpsd^{(\mathrm{mt})}_\mathbf{y}(\mathbf{g})$, which we denote, by a slight abuse of notation, $\hpsd^{(\mathrm{mt})}_\mathbf{y}(r)$. The radial averaging is performed by projecting $\hpsd^{(\mathrm{mt})}_\mathbf{y}(\mathbf{g})$ on the circularly symmetric (i.e., purely radial) elements of a steerable basis (see Section \ref{subset:steerable}).

The resulting linear program, whose result we denote by $\hpsd_{\mathbf{e}}^{(\mathrm{lp})}$, is then
\begin{equation}
\begin{array}{ll}
\displaystyle \operatorname*{minimize}_{\hpsd_{\mathbf{e}}} &  \underset{r=0,\frac{1}{K},\dots,\frac{m}{K}}{\sum} {\hpsd^{(\mathrm{mt})}_{\mathbf{y}} \left( r \right)} - \hpsd_{\mathbf{e}}\left( r \right)\\
 \text{subject to}  &
\hpsd_{\mathbf{e}} ( r ) \leq {\hpsd^{(\mathrm{mt})}_{\mathbf{y}} \left( r \right)}, \quad r=0,\dots,\frac{m}{K}\\
 & \hpsd_{\mathbf{e}} \left( r + 1\right)  + \hpsd_{\mathbf{e}} \left( r -1\right)  \ge 2 \hpsd_{\mathbf{e}} \left( r \right), r=1,\dots,\frac{m}{K}\\
 &  \hpsd_{\mathbf{e}} \left( r \right) \ge 0,  \quad r=0,\dots,\frac{m}{K},
\end{array}
\end{equation}
where
$\hpsd_{\mathbf{e}} = \begin{bmatrix} \hat{S}(0), \ldots, \hat{S}(m/K)) \end{bmatrix}^T$, and
  $0<m/K \le 0.5$  is the  spatial frequency above which $\hpsd^{(\mathrm{mt})}_{\mathbf{y}} \left( r \right)$ is typically dominated by noise.  As its default, ASPIRE-CTF sets $m/K=3/8$.

 We present the result of our linear program in Fig. \ref{fig:background}.
\begin{figure}[t]
\centering
\subfigure[] {\label{fig:back_est} \includegraphics[width=0.44\linewidth]{heimo4a}}
\hspace{0.3cm}
\subfigure[] {\label{fig:back_sub} \includegraphics[width=0.44\linewidth]{heimo4b}}
\caption{Background estimation for a $\beta$-galactosidase micrograph from the EMPIAR-10017 dataset \citep{scheres2015semi}.
On the left is the 1D radial profile of the multitaper power spectrum estimate $\hpsd^{(\mathrm{mt})}_{\mathbf{y}}\left(r\right)$ and the estimated background $\hpsd_{\mathbf{e}}^{(\mathrm{lp})} \left(r \right)$.
On the right is the background-subtracted power spectrum.}
\label{fig:background}
\end{figure}
Expanding the 1D background spectrum to a 2D function, we again abuse notation slightly and set $\hpsd_{\mathbf{e}}^{(\mathrm{lp})} (\mathbf{g}) = \hpsd_{\mathbf{e}}^{(\mathrm{lp})} (r)$ for all $\mathbf{g} \in [-1/2, 1/2]^2$.
We denote the background-subtracted power spectrum estimate $\hpsd^{(\mathrm{mt})}_\mathbf{y}(\mathbf{g})-\hpsd_{\mathbf{e}}^{(\mathrm{lp})}(\mathbf{g})$ by $\hpsd^{(\mathrm{lp})}_\mathbf{z}(\mathbf{g})$ where $\mathbf{z} = h_{\phi} * \mathbf{x}$.

 A different LP that can be used for background estimation was suggested in \citep{huang2003env}. However, contrary to our non-parametric approach which assumes convexity alone,  \citep{huang2003env} suggests a LP based on parametric estimation.

\subsubsection{Expansion over a steerable basis}
\label{subset:steerable}

In this section, we show that any function of the form~\eqref{equ:ctf_sine}-\eqref{equ:astigmatism}  is contained in a low-dimensional subspace spanned by a set of steerable basis functions, such as a Fourier--Bessel basis \citep{zhao2013fourier,Zhao2016fb} or prolate spheroidal wave functions (PSWFs) \citep{landa2017prolates, landa2018prolates}.
We will use this property to further reduce the variability of the power spectrum estimator by projecting the background-subtracted power spectrum estimate $\hpsd^{(\mathrm{lp})}_\mathbf{z}(\mathbf{g})$ onto this subspace.

A steerable basis consists of functions $f_{k,q}({r}) \, \mathrm{e}^{\mathrm{j} k {\alpha}}$, where $k \in \mathbb{Z}$ and $q = 0, \ldots, p_k-1$ for some $p_k \ge 0$.
The radial part $f_{k,q}(r)$ depends on the specific choice of basis (e.g., in a Fourier--Bessel basis, it is a scaled Bessel function of order $q$) and does not enter explicitly into our analysis.
We shall therefore leave it unspecified.
A given function in polar coordinates may be decomposed in the basis as
\begin{equation}
\label{equ:steerable}
x({r}, {\alpha}) = \sum_{k=-\infty}^\infty \sum_{q=0}^{\infty} a_{k,q} \, f_{k,q}({r}) \, \mathrm{e}^{\mathrm{j} k {\alpha}},
\end{equation}
where $a_{k,q} \in \mathbb{C}$ is the coefficient corresponding to angular frequency $k$ and radial frequency $q$.

To determine the steerable basis expansion of the CTF~\eqref{equ:ctf_sine}, we consider its Taylor expansion around $\Delta f_1 - \Delta f_2 = 0$,
\begin{multline}
H_{\phi} ( \mathbf{g} ) = \sum_{\substack{n = 0 \\ n~\text{even}}}^{P} \frac{\left( -1 \right)^{\frac{n}{2}+1}}{n!} \sin ( \chi_{\phi}^0 ( {r} )) C_{n,\phi}(\mathbf{g}) \\+  \sum_{\substack{n=1 \\ n~\text{odd}}}^{P} \frac{\left( -1 \right)^{\frac{n+1}{2}}}{n!} \cos ( \chi_{\phi}^0 ( {r} )) C_{n,\phi}(\mathbf{g}) + R_P(\mathbf{g}),
\end{multline}
where $R_P(\mathbf{g})$ is the remainder term,
\begin{equation*}
C_{n,\phi}(\mathbf{g}) = \left( \frac{1}{2}\pi \lambda \left( \Delta f_1 - \Delta f_2 \right) \cos \left( 2 ( {\alpha} - \alpha_f) \right) \frac{ {r} ^2}{p^2} \right)^n,
\end{equation*}
and
\begin{equation*}
\chi_{\phi}^0 ( {r} ) = \frac{1}{2} \pi \lambda  {r}^2 (\Delta f_1+\Delta f_2) - \frac{1}{2} \pi \lambda^3  {r}^4 C_s + w
\end{equation*}
is the non-astigmatic phase function.

The remainder term is bounded by a function of
\begin{equation*}
\left(\frac{\Delta f_1 - \Delta f_2}{\Delta f_1+\Delta f_2}\right)^{(P+1)}
\end{equation*}
and is therefore small when astigmatism is small, which is the case for experimental cryo-EM data.
We therefore conclude that
\begin{equation}
\label{equ:ctf_approx}
H_{\phi}(\mathbf{g}) \approx -\sin(\chi_{\phi}^0 ({r})) - \cos(\chi_{\phi}^0({r})) C_{1,\phi}(\mathbf{g})
\end{equation}
is a good approximation of the CTF.

Since $\cos(\alpha) = \frac{1}{2}(\mathrm{e}^{-\mathrm{j} \alpha} + \mathrm{e}^{-\mathrm{j} \alpha})$, we rewrite \eqref{equ:ctf_approx} as
\begin{multline}
\label{equ:taylor_ctf}
H_{\phi}(\mathbf{g}) \approx -\sin(\chi_{\phi}^0({r})) \\
-\frac{1}{4p^2} \cos(\chi_{\phi}^0({r})) \pi \lambda (\Delta f_1-\Delta f_2) \mathrm{e}^{-\mathrm{j} 2\alpha_f} r^2 \, \mathrm{e}^{\mathrm{j} 2{\alpha}} \\
-\frac{1}{4p^2} \cos(\chi_{\phi}^0({r})) \pi \lambda (\Delta f_1-\Delta f_2) \mathrm{e}^{\mathrm{j} 2\alpha_f} r^2 \, \mathrm{e}^{-\mathrm{j} 2{\alpha}}.
\end{multline}
Comparing \eqref{equ:taylor_ctf} and \eqref{equ:steerable}, we see that only terms corresponding to $k = -2$, $0$, and $2$ are present.

Concretely, we compute the coefficients $a_{k,q}$ of the expansion of $\sqrt{\hpsd^{(\mathrm{lp})}_\mathbf{z}}$ over the steerable basis functions with radial frequencies to $k = 0, \pm 2$. The coefficients are computed through an inner product on a $K \times K$ grid:
\begin{equation}
a_{k,q} = \frac{1}{K^2} \sum_{\mathbf{g} \in M_K} \sqrt{\hpsd^{(\mathrm{lp})}_\mathbf{z}(\mathbf{g})} f_{k, q}({r}) \mathrm{e}^{\mathrm{j} k {\alpha}},
\end{equation}
where $r$ and ${\alpha}$ are the polar coordinates of $\mathbf{g}$.
Evaluating \eqref{equ:steerable} for these $a_{k,q}$ and squaring the result then gives a new power spectrum estimate, which we denote as $\hat{S}_\mathbf{z}$.

In Fig. \ref{fig:PSD_complete}, we present the results of our power spectrum estimation method on a micrograph from the EMPIAR-10028 dataset \citet{wong2014empiar}.
This includes multitaper estimate $\hpsd^{(\mathrm{mt})}_\mathbf{y}$ as well as the background-subtracted estimate $\hpsd^{(\mathrm{lp})}_\mathbf{z}$ and the projection onto the steerable basis with $k = 0, \pm 2$.
The result is smooth enough that many of the zero-crossings of the power spectrum can be easily resolved.

\begin{figure}[t]
\centering
\subfigure[]{\label{subfig:psd}\includegraphics[width=0.3\linewidth]{heimo5a}}
\hspace{0.1cm}
\subfigure[]{\includegraphics[width=0.3\linewidth]{heimo5b}}
\hspace{0.1cm}
\subfigure[]{\includegraphics[width=0.3\linewidth]{heimo5c}}
\caption{Power spectral density estimates. (a) Background-subtracted estimate $\hpsd^{(\mathrm{lp})}_\mathbf{z}$. (b) Projection onto steerable basis functions for $k = 0, \pm 2$. (c) Zero-crossings of panel b, determined as specified in Section~\ref{subsec:zero_cross}.}
\label{fig:PSD_complete}
\end{figure}

\subsection{CTF Parameter Estimation}

In the previous sections we have introduced our method for estimating the background-subtracted power spectrum.
In this section we will discuss two methods that use this estimate to recover the defocus and astigmatism of the micrograph.

\subsubsection{CTF estimation through correlation}
\label{subsec:cross_corr}

The power spectrum $S_\mathbf{x}$ is a slowly-varying function of radial frequency. It follows that the oscillations in $|H_{\phi}(\mathbf{g})|^2 \, S_\mathbf{x}(\mathbf{g})$ are due to those of $|H_{\phi} (\mathbf{g})|^2$.
As a consequence, the square root of the background-subtracted power spectrum $|H_{\phi}(\mathbf{g})| \, S_\mathbf{x}^{1/2}(\mathbf{g})$ is proportional to the absolute value of the CTF (we note that using the square root instead of the actual power spectrum estimate reduces the influence of large values). It follows that the correlation test is a useful tool in estimating CTF parameters. Indeed,
many CTF estimation methods therefore estimate the defocus parameters $\phi$ through correlation with a simulated CTF magnitude \citep{rohou2015ctffind4,mindell2003ctffind3,kai2016gctf}.

The Pearson correlation of $|H_{\phi}(\mathbf{g})|$ and $\hat{S}_\mathbf{z}^{1/2}(\mathbf{g})$ is
\begin{equation}
\label{equ:corr_def}
P_{cc}(\phi) = \frac{\sum_{\mathbf{g} \in R} |H_{\phi}(\mathbf{g})| \, \hat{S}_\mathbf{z}^{1/2}(\mathbf{g})}{\left( \sum_{\mathbf{g} \in R} |H_{\phi}(\mathbf{g})|^2 \, \sum_{\mathbf{g} \in R} \hat{S}_\mathbf{z}(\mathbf{g}) \right)^{1/2} },
\end{equation}
where $R$ is the set of frequencies over which correlation is computed, and will be defined below.
To optimize $P_{cc}(\phi)$, we first need an initial guess for the parameters $\phi$.
For this, we follow \citet{kai2016gctf} and first consider non-astigmatic CTFs where $\Delta f_1 = \Delta f_2$, which renders the value of $\alpha_f$ irrelevant.
We thus calculate $P_{cc}(\phi)$ for $\phi = (\Delta f, \Delta f, 0)$ with $\Delta f$ on a 1D grid from $\Delta f_{\mathrm{min}}$ to $\Delta f_{\mathrm{max}}$ with a step of $\Delta f_{\mathrm{step}}$.
Since $H_{\phi}$ is considered (at this stage) to be radially symmetric, we define the set of frequencies over which correlation is computed as
\begin{equation}
R = \left\{ m_1, m_1+ \frac{2}{N}, \ldots, \frac{3}{8} \right\} \times \{0\},
\end{equation}
where $m_1$ is the first maximum of the radial profile of $\hpsd^{(\mathrm{lp})}_\mathbf{z}$. That is, we only consider frequencies $\mathbf{g}$ along a 1D radial profile and, furthermore, ignore the very low and very high frequencies (since the very low frequencies may dominate the cross-correlation result and the very high frequencies are strongly effected by the envelope function). The $\Delta f$ which maximizes $P_{cc}(\phi)$ on this grid is denoted $\Delta f_\star$.

To estimate the astigmatism of the CTF, we compute the principal directions of the second-order moments of $\hat{S}_\mathbf{z}^{1/2}$.
Specifically, we form the $2 \times 2$ matrix $\mathbf{M}$ given by
\begin{multline}
\begin{aligned}
& M_{1,1} = {\textstyle \sum_{\mathbf{g} \in M_K}} \, g_1^2  \, \hat{S}_\mathbf{z}^{1/2}(\mathbf{g}),\\
& M_{1,2} = M_{2,1} = {\textstyle \sum_{\mathbf{g} \in M_K}} \, g_{1} g_{2} \, \hat{S}_\mathbf{z}^{1/2}(\mathbf{g}),\\
& M_{2,2} = {\textstyle \sum_{\mathbf{g} \in M_K}} g_2^2 \, \hat{S}_\mathbf{z}^{1/2}(\mathbf{g}).
\end{aligned}
\end{multline}
The eigenvalues $\mu_1$ and $\mu_2$ of $\mathbf{M}$ estimate the size of the major and minor axes in $\hat{S}_\mathbf{z}^{1/2}$.
We therefore expect the ratio $\mu_1/\mu_2$ to approximate $\Delta f_1/\Delta f_2$.
Combining this with the estimated mean defocus $\Delta f_\star$, we get
\begin{align}
\frac{1}{2}(\Delta f_1 + \Delta f_2) &= \Delta f_\star \\
\frac{\Delta f_1}{\Delta f_2} &= \frac{\mu_1}{\mu_2},
\end{align}
which has the solution
\begin{align}
\Delta f_{1,\star} &= \frac{2\mu_1}{\mu_1+\mu_2} \Delta f_\star, \quad
\Delta f_{2,\star} &= \frac{2\mu_2}{\mu_1+\mu_2} \Delta f_\star.
\end{align}

In order to improve our estimation of the defocus parameters, we run gradient descent on  $P_{cc}(\phi)$. As we no longer approximate the image as non-astigmatic, we define the set of frequencies over which correlation is computed as $R = M_K$.  We now initialize our gradient descent at $\phi = (\Delta f_{1,\star}, \Delta f_{2,\star}, \alpha_f)$, where $\alpha_f$ is set as detailed in \citep{kai2016gctf} to an arbitrarily selected $0\le a < \pi/6$ (e.g. $a=\pi/12$) and $(a+\pi/6)$, $(a-\pi/6)$, $(a+\pi/3)$, $(a-\pi/3)$ or $(a-\pi/2)$.
One run of gradient descent is performed for each value of $\alpha_f$ and the result with the highest value of $P_{cc}(\phi)$ is kept.
The resulting $\phi$ is our defocus estimate for the micrograph.

We note that, as is done in \citep{rohou2015ctffind4,mindell2003ctffind3,kai2016gctf}, we discard information in the lower and higher frequencies of $ \hat{S}_\mathbf{z}$. These frequencies can be determined by the user. As default values we use those suggested in \citep{rohou2015ctffind4}.

\subsubsection{CTF estimation through zero-crossings}
\label{subsec:zero_cross}

We have seen in the previous sections that the true background-subtracted power spectrum is $\vert H_{\phi} (\mathbf{g}) \vert^2 S_\mathbf{x} (\mathbf{g})$.
Under the assumption that $S_\mathbf{x} (\mathbf{g})$ is slowly-varying, it follows that at any frequency where the background-subtracted power spectrum reaches a minimal value of zero, the CTF must reach a zero-crossing.

Furthermore, even if the aforementioned assumption did not hold true, we could still infer the frequencies where the CTF reaches a zero-crossing.
This is due to the fact that the zero-crossings of the CTF are known to create concentric, nearly elliptical rings, centered around the origin (see Section~\ref{sec:formation}).
Therefore, this can be used a cue to differentiate between any minima of $\vert H_{\phi} (\mathbf{g}) \vert^2 \, S_\mathbf{x} (\mathbf{g})$ that stem from the zero-crossings of $H_{\phi}$ and any minima that stem from $S_\mathbf{x}$.

As we show in Fig.~\ref{fig:PSD_complete}(c), our estimation of the background-subtracted power spectrum enables easy detection of several elliptical rings where $\hat{S}_\mathbf{z}$ reaches its minima. To do this, we define any pixel with a value smaller than that of at least six out of its eight neighbors as a zero-crossing.
Once the minima of $\hat{S}_\mathbf{z}$ are found, we discard any frequency that is not on a closed ring. Furthermore, we verify that the spatial frequencies of pixels residing on closed rings representing the minima of $\hat{S}_\mathbf{z}$ form ellipses approximately centered at the origin.
We are then left with frequencies of several zero-crossings of the CTF.

Since we have $H_{\phi}(\mathbf{g}) = - \sin(\chi_{\phi}(\mathbf{g}))$, we reach a zero-crossing of the CTF when $\chi_{\phi}(\mathbf{g})$ is an integer multiple of $\pi$.
Formally, the set of spatial frequencies on the $\ell$th ring of zero-crossings, denoted by  $G_\ell$, satisfies
\begin{equation}
\chi_{\phi}(\mathbf{g}_\ell) = \pi \ell.
\label{equ:zero_constraint}
\end{equation}
Empirically, we are typically able to identify at least three rings, that is, three different values of $\ell$.

Combining \eqref{equ:zero_constraint} for all $\mathbf{g}$ in $\hat{G}_\ell$ and combining these for different values of $\ell$, we obtain an overdetermined system of equations.
Solving it yields an estimate for the defocus vector $\phi$.
To solve the system, we use the \textsf{trust-region-dogleg} method implemented in MATLAB (a variant of \citep{powel1970fsolve}).

 We note that the estimated positions of CTF zeroes are extremely sensitive to the method of background subtraction. This method should therefore be used to obtain an initial estimate of the defocus. Refinement of the solution can be done as in Section~\ref{subsec:cross_corr} using gradient-based optimization.

We note that typically both methods suggested in this section achieve similar results. However, while the zero-crossings-based method has lower computational complexity, the correlation-based method is more robust to noise. Therefore, for micrographs with very low SNR we recommend using the correlation-based method, while for cleaner micrographs we suggest using the zero-crossings method.

\section{Results}
\label{sec:experiment}

We present experimental results for the ASPIRE-CTF framework presented in this paper. We apply our framework to
datasets that are publicly available from the EMPIAR database~\citep{iudin2016empiar} or the CTF challenge \citep{marabini2015challenge}. Unless otherwise stated, in the experiments below we use $L=4$ tapers and project the power spectrum onto the steerable PSWF basis.

\subsection{Estimating CTF from movie frames}

The CTF may be estimated either from motion-corrected micrographs,  from several frames averaged in real space  or, alternatively, directly from the frames. This is done by estimating $\hpsd^{(\mathrm{mt})}$ individually from each frame and averaging the estimates (see Fig.~\ref{fig:pipeline}). A benefit of estimating the CTF directly from the frames is that this practice enables us to correct for motion and estimate the CTF concurrently, thus speeding up the pipeline. Furthermore, any errors added by motion-correction will have no effect on the CTF estimation~\citep{bartesaghi2014frames}.

In this section, we compare the CTF estimates produced from motion-corrected micrographs to the estimation produced directly from the frames. We do this over several publicly available datasets, namely, EMPIAR-10002~\citep{bai2013empiar}, EMPIAR-10028~\citep{wong2014empiar}, EMPIAR-10242~\citep{zhang2019empiar}, and EMPIAR-10249~\citep{herzik2019dataset}. A summary of these datasets appears in Table~\ref{table:movies}.

While the EMPIAR-10028 and EMPIAR-10242 datasets contain both movies and motion-corrected micrographs, EMPIAR-10002 and EMPIAR-10249 contain movies alone. We therefore use MotionCor2 \citep{Zheng2017motion} to produce the motion-corrected micrograph for these two datasets. We present in Fig.~\ref{fig:movies}  a comparison of the astigmatism ($\Delta f_1-\Delta f_2$), average defocus ($\Delta f_1/2+\Delta f_2/2$), and astigmatism angle ($\alpha_f$) as estimated from a motion-corrected micrograph with an estimate produced from the raw movie frames. We note that, as expected, the parameters estimated from each of these methods are nearly identical.

\begin{table*}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
  \hline
 Dataset & Molecule & Pixel size (\AA) & Spherical & Voltage (kV) & Microscope & Detector \\
              &           &      & aberration &           &  & \\
 \hline
EMPIAR-10002 & 80S ribosome & 1.77 & 2.0 & 300& Polara & Falcon  \rom{1}\\
EMPIAR-10028 & 80S ribosome & 1.34 & 2.0 & 300 & Polara & Falcon  \rom{2}\\
EMPIAR-10242 & 2N3R tau filaments & 1.04 & 2.7 & 300 & Titan Krios & Gatan K2 Summit\\
EMPIAR-10249 & HLA dehydrogenase & 0.56 & 2.7 & 200 & Talos Arctica & Gatan K2 Summit\\
\hline
\end{tabular}
\end{center}
\caption{Description of the EMPIAR-10002, EMPIAR-10028, EMPIAR-10242 and EMPIAR-10249 datasets.}
\label{table:movies}
\end{table*}

\begin{figure*}[t]
\begin{center}
\subfigure[]{\label{subfig:psd}\includegraphics[height=0.22\linewidth]{heimo6a}}
\subfigure[]{\label{subfig:psd}\includegraphics[height=0.22\linewidth]{heimo6b}}
\subfigure[]{\label{subfig:psd}\includegraphics[height=0.22\linewidth]{heimo6c}}
\caption{Defocus estimation on sample micrographs of the EMPIAR-10028, EMPIAR-10002, EMPIAR-10042 and EMPIAR-10049 datasets. We compare the estimation of each parameter when using the average power spectrum of the frames to the estimation when using the power spectrum of the motion-corrected micrograph. (a) Average defocus (in nm). (b) Astigmatism (in nm). (c) Angle $\alpha_f$ (in radians).}
\label{fig:movies}
\end{center}
\end{figure*}

\subsection{CTF Challenge}
\label{subsec:exp1}
The CTF challenge \citep{marabini2015challenge} consists of nearly $200$ micrographs of GroEL, 60S ribosome, apoferritin and TMV virus.
These micrographs are taken from eight experimental datasets and one synthetic dataset, each referred to by a  number ranging from $001$ to $009$. In the following, we restrict our attention to the experimental datasets, that is, sets $001$ through $008$.

The advantage of the CTF challenge is that each dataset is acquired using a different combination of microscope and camera, allowing for a qualitative comparison of CTF estimation methods for a variety of experimental setups.
Notably, datasets $003$ and $004$ use a Gatan K2 direct detector in counting mode with a high electron dose, causing $S_\mathbf{e}$ to increase at high frequencies \citep{li2013background}.
Additionally, dataset $008$ has an especially low signal-to-noise ratio (SNR), rendering CTF estimation difficult. A summary of these datasets is presented in \citep{marabini2015challenge}.

The estimate of each micrograph's power spectrum is computed as detailed in Section~\ref{subsec:multi}. Specifically, we divide the micrograph into half-overlapping blocks of size $K \times K$, where $K=512$, and use $L=4$ tapers in the estimation. We then estimate the background spectrum as detailed in Section~\ref{subsec:bksub1} and expand the background-subtracted power spectrum over the PSWF basis in order to reduce variability in the power spectrum estimate (Section~\ref{subset:steerable}). We use the correlation-based method (Section~\ref{subsec:cross_corr}) to estimate defocus parameters, and denote the resulting vector of defocus parameters as $\phi_{a}^{(512)}$.

In some cases, a power spectrum of size $512 \times 512$ may not capture the oscillations of the power spectrum with sufficient accuracy \citep{rohou2015ctffind4}. We therefore compute a second estimate of the power spectrum using half-overlapping blocks of size $1024 \times 1024$. As this reduces the number of blocks, the variance of the estimator will grow. We therefore use $L=16$ data tapers in this case. We use this estimate of the power spectrum to estimate a vector of  defocus parameters which we denote by $\phi_{a}^{(1024)}$.

We compare our results to the estimates produced by CTFFIND4 (version 4.1.13) and Gctf (version 1.06).
We denote the vector of estimated defocus parameters produced by CTFFIND4 when using block of size $512 \times 512$ and $1024 \times 1024$ as $\phi_{c}^{(512)}$ and  $\phi_{c}^{(1024)}$, respectively. We further denote the vector of estimated defocus parameters produced by Gctf when using block of size $512 \times 512$ and $1024 \times 1024$ as $\phi_{g}^{(512)}$ and  $\phi_{g}^{(1024)}$, respectively. For each estimation method we select the vector of estimated defocus parameters that leads to highest correlation with the estimated power spectrum, that is
\begin{equation}
\phi_j^* = \arg \underset{\phi_j \in \{\phi_j^{(512)},  \phi_j^{(1024)}  \}}{\max} \left( \frac{1}{s_t} \sum_{m=1}^{s_t} P_{cc}^{m} (\phi_j ) \right),
\end{equation}
where $j \in \{ a, c, g\}$, $s_t$ is the number of micrographs in the $t$th dataset and $P_{cc}^{m}$ is the correlation for the $m$th micrograph in the dataset, computed as in~\eqref{equ:corr_def}. We note that the correlation is computed with the power spectrum estimate suggested in \citep{mindell2003ctffind3}.
 That is, we compute the Pearson correlation coefficient between the background subtracted power spectrum computed as in~\citep{mindell2003ctffind3} and using blocks of size $512 \times 512$ with $H_{\phi_j^{(512)}}$, and between the background subtracted power spectrum computed using blocks of size $1024 \times 1024$ with $H_{\phi_j^{(1024)}}$. In this manner, we choose the block size that best captures the oscillations of each dataset.

In order to compare the consistency of these $3$ methods, we present the differences between $\phi_a$, $\phi_c$ and $\phi_g$ in Tables~\ref{tab:mean}-\ref{tab:std}. That is, for each micrograph $m$ we compute
\begin{equation}
\begin{aligned}
& \epsilon_{j,k}(\Delta f_1) = (\phi_j(1) - \phi_k(1)) / \phi_j(1)\\
& \epsilon_{j,k}(\Delta f_2) = (\phi_j(2) - \phi_k(2)) / \phi_j(2)\\
& \epsilon_{j,k}(\alpha_f) = (\phi_j(3) - \phi_k(3)) / \phi_j(3)
\end{aligned}
\label{equ:dif_comparison}
\end{equation}
where $j$ and $k$ are two estimation methods (ASPIRE-CTF, CTFFIND4 or Gctf). We report the mean and variance of $\epsilon$.  We note that either $\epsilon_{a,g}$ or  $\epsilon_{a,c}$ are often smaller than $\epsilon_{c,g}$, thus showing the ASPIRE-CTF estimate to be in the consensus of the three estimation vectors.

\begin{table*}
\begin{center}
\scalebox{0.8}{
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c |}
  \hline
 Dataset & Molecule & $\epsilon_{a,g} (\Delta f_1)$ & $\epsilon_{a,c} (\Delta f_1)$ & $\epsilon_{c,g} (\Delta f_1)$ & $\epsilon_{a,g} (\Delta f_2)$ &    $\epsilon_{a,c} (\Delta f_2)$ &   $\epsilon_{c,g} (\Delta f_2)$ &   $\epsilon_{a,g} (\alpha_f)$ & $\epsilon_{a,c} (\alpha_f)$ &  $\epsilon_{c,g} (\alpha_f)$ \\
 \hline
001 & GroEL & 0.0157 & 0.2370 & 0.2364 & 0.0155 &0.1282 & 0.6081 & 1.6599 &  2.1429 & 1.3848\\
002 & GroEL  & 1.3236 & 1.6500 & 0.1515 & 1.4876 & 0.9746 & 0.2374 & 3.6137 & 5.3979 & 1.9592\\
003 & 60S ribosome & 0.0029 & 0.0049 & 0.0029 & 0.0030 & 0.0032 & 0.0017 &  6.1042 & 6.6618 & 2.9159 \\
004 & 60S ribosome & 0.0051 & 0.0130 & 0.0098 &  0.0057 &  0.0406 & 0.2990 &  4.2653 & 2.3667 & 2.6050\\
005 & apoferritin & 0.0036 &  0.1744 & 0.8809 & 0.0036 & 0.1997 & 1.5230 & 0.6597 & 0.8927 &  0.7413\\
006 & apoferritin & 0.0099 & 0.2094 & 0.6392 & 0.0069 & 0.3409 & 1.5217 & 0.2670 & 1.3014 & 1.6142\\
007 & TMV virus & 0.0224 &  0.2144 & 0.5163 & 0.0265 & 0.2674 & 1.1320 & 0.4254 & 0.9891 & 0.8985 \\
008 & TMV virus & 0.1961 & 0.6043 & 0.3988 & 0.2163 & 0.3016 & 0.5707 & 3.0253 & 1.8514 & 2.8394 \\
\hline
\end{tabular} }
\end{center}
\caption{Comparison between parameters estimated by ASPIRE-CTF, CTFFIND4 and Gctf. We present the mean (over each dataset) of normalized differences between each two CTF estimation methods as detailed in~\eqref{equ:dif_comparison}. The subscripts $a$, $g$ and $c$ indicate ASPIRE-CTF, Gctf and CTFFIND4, respectively.}
\label{tab:mean}
\end{table*}

\begin{table*}
\begin{center}
\scalebox{0.8}{
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c |}
  \hline
 Dataset & Molecule & $\epsilon_{a,g} (\Delta f_1)$ & $\epsilon_{a,c} (\Delta f_1)$ & $\epsilon_{c,g} (\Delta f_1)$ & $\epsilon_{a,g} (\Delta f_2)$ &    $\epsilon_{a,c} (\Delta f_2)$ &   $\epsilon_{c,g} (\Delta f_2)$ &   $\epsilon_{a,g} (\alpha_f)$ & $\epsilon_{a,c} (\alpha_f)$ &  $\epsilon_{c,g} (\alpha_f)$ \\
 \hline
001 & GroEL  & 0.0144 &  0.4710 & 0.5276 & 0.0267 &  0.2489 & 1.9206  &  3.1585 & 2.0435 & 2.2632\\
002 & GroEL  & 2.9445 & 2.2763 & 0.2655 & 3.3142 & 1.9597 & 0.5196 & 9.2965 & 15.0486 & 2.7796\\
003 & 60S ribosome & 0.0022 &  0.0031 & 0.0036 & 0.0031 &  0.0047 & 0.0020 & 20.8434 & 22.3327 &  15.8681\\
004 & 60S ribosome & 0.0028 & 0.0328 & 0.0387 & 0.0039 & 0.1778 & 1.4504 & 9.4786 & 3.5234 & 3.9324\\
005 & apoferritin & 0.0037 & 0.3265 & 1.8456 & 0.0029 & 0.3591 &  3.1834 & 0.9381 & 0.7127 & 0.6918\\
006 & apoferritin & 0.0126 & 0.2902 & 1.9539 & 0.0064 & 0.3330 &  2.7270 & 0.3485 & 1.1711 & 2.7480 \\
007 & TMV virus & 0.0192 & 0.2698 & 0.8581 & 0.0172  & 0.3335 & 2.0805 & 0.6437 & 0.9296 & 0.9966 \\
008 & TMV virus & 0.8337 & 0.7936 & 0.8543 & 0.8947 & 0.4088 &  0.9907 & 4.7943 & 2.6911 &  6.1583 \\
\hline
\end{tabular}
}
\end{center}
\caption{Comparison between parameters estimated by ASPIRE-CTF, CTFFIND4 and Gctf. We present the standard deviation (over each dataset) of normalized differences between each two CTF estimation methods as detailed in~\eqref{equ:dif_comparison}. The subscripts $a$, $g$ and $c$ indicate ASPIRE-CTF, Gctf and CTFFIND4, respectively.
}
\label{tab:std}
\end{table*}

Fig.~\ref{fig:vis_compare} contains a visual comparison between the power spectrum computed by our suggested framework and the power spectra computed by Gctf and CTFFIND4. We present the comparison over a micrograph from the eighth set of the CTF challenge as this set is known to be difficult. We note that the oscillations of the ASPIRE-CTF power spectrum are highly noticeable. In comparison, the variability of the power spectra computed by Gctf and CTFFIND4 make visual detection of oscillations challenging.

\begin{centering}
\begin{figure}[t]
\centering
{\label{subfig:psd}\includegraphics[width=0.6\linewidth]{heimo7c}}
\caption{%
Visual comparison between the power spectra computed by ASPIRE-CTF (top row), Gctf (bottom left) and CTFFIND4 (bottom right) on a sample micrograph of dataset 008. On the top row we present $\hpsd^{(\mathrm{lp})}_\mathbf{z}$ (right) and $\hat{S}_\mathbf{z}$ (left).}
\label{fig:vis_compare}
\end{figure}
\end{centering}

\subsection{Runtime}

We compute runtime
of ASPIRE-CTF and  CTFFIND4 over dataset $001$ of the CTF challenge. For both methods, we partition the micrograph into blocks of size  $512 \times 512$. When running ASPIRE-CTF we employ $L=4$ data tapers. Furthermore, we use the exhaustive search option for CTFFIND4, and perform an exhaustive 1D search in ASPIRE-CTF. While the CTF estimation results are comparable, there is a significant speedup when using ASPIRE-CTF. Runtime for ASPIRE-CTF is 22.5 seconds on average per micrograph, while runtime for CTFFIND4 is 541 seconds.

These experiments are run on a $2.6$ GHz Intel Core i7 CPU with four cores and $16$ GB of memory. We do not compare to the runtime of Gctf as it must be run on a GPU.

\subsection{Consistency in low SNR}

To test consistency of results with changing SNR, we turn to the EMPIAR-10249 dataset \citet{herzik2019dataset}.
This dataset consists of movies with $44$ frames per movie.
Usually, all these frames, except for a few frames at the beginning and a few at the end, are motion-corrected and summed to create a micrograph.
This is due to the fact that a micrograph created from as many motion-corrected frames as possible will have the best SNR.

We disregard the first frame and use MotionCor2 \citep{Zheng2017motion} to create $9$ motion-corrected micrographs.
These consist of summing $5$, $8$, $13$, $18$, $23$, $28$, $33$, $38$, and $43$ motion-corrected frames, respectively.
This gives us a sequence of micrographs with increasing SNR.

We estimated the CTF parameters independently from each micrograph in the manner detailed in Section~\ref{subsec:exp1}.
Fig. \ref{fig:astigmatism_vs_defocus} shows the astigmatism $|\Delta f_1-\Delta f_2|$ vs.\ mean defocus $(\Delta f_1 + \Delta f_2)/2$ of the CTF estimation for each method and over each micrograph. We see that while the average defocus values remain similar for all three methods, Gctf incurs a larger error in the astigmatism when 23 frames are used. On the other hand, our method and CTFFIND4 achieve consistent estimates regardless of the amount of frames averaged.

\begin{figure}[t]
\centering
{\label{subfig:psd}\includegraphics[width=0.65\linewidth]{heimo9}}
\caption{Estimated astigmatism vs. defocus of the CTF parameters. The circular markers present the defocus and astigmatism estimated from a micrograph with $43$ summed frames.}
\label{fig:astigmatism_vs_defocus}
\end{figure}

\section{Conclusion}

In this paper we have presented a novel approach for power spectrum estimation of cryo-EM experimental data. Our approach uses the multitaper estimator, which often leads to reduced
mean square error over Bartlett's and Welch's methods.
Additionally, we presented a method for error reduction that is driven directly by the mathematical model of the contrast transfer function. We did this by projecting the power spectrum estimate onto a steerable basis and discarding any basis function where the CTF must be negligible. We showed that the combination of these two contributions leads to greatly reduced variability in our estimator.

We presented experimental results on twelve datasets, and showed that our method is well
suited to both motion-corrected micrographs and raw movies data.

\section*{Acknowledgments}
This work was partially supported by the Simons Foundation Math+X Investigator Award and the Moore Foundation Data-Driven Discovery Investigator Award.
The authors
thank B. Landa and I. Sason for help optimizing the PSWF code.
The authors are also indebted to B. Landa, Y. Shkolnisky and A. Rohou for helpful comments and discussions.
The Flatiron Institute is a division of the Simons Foundation.

\section*{Appendix A}

Zeroth-order discrete prolate spheroidal sequence (DPSS) \citep{slepian1978prolates} were proposed as data tapers in (\citealp{babadi2014multitaper}; Section~\ref{subsec:bias}). Here we describe their generation.

The zeroth-order discrete prolate spheroidal sequence is a sequence of $d$ 1-D vectors, determined as the $d$ leading eigenvectors of the matrix $\mathbf{\mathcal{L}} \in \mathbb{R}^{K \times K}$, where
\begin{equation}
\mathcal{L}[{k,m}] = \frac{\sin (\pi R (k-m)))}{\pi (k-m)},
\end{equation}
and $R = \frac{2d}{N}$. We denote the resulting data tapers by $\mathbf{t}_1,\dots,\mathbf{t}_N$.

As the blocks $\mathbf{y}_b$ are 2D, that is $\mathbf{y}_b \in \mathbb{R}^{K \times K}$, the data tapers we use are defined as
$$\mathbf{w}_{pq} = \mathbf{t}_p^T \mathbf{t}_q, \quad 0 \le p,q < d.$$
$$w_{dq+p} [k_1, k_2] = t_p[k_1] t_q[k_2].$$

Lastly, we note that $d$ is selected such that
$$ (d-1)^2 < L \le d^2.$$

\section*{References}

redpajama_set_name: RedPajamaArXiv


It is difficult to decide as you are insecure about the future outcome of your decision.
For this you try to reduce your insecurity in order to make the best possible decision/the best decision.
Reaching agreement requires the same from the other party./ When reaching agreement it is the same situation for the other party.
So if you would like to have a positive/favourable decision for yourself, reduce your partner's insecurity about the decision.


text:
Journalism and Realism
Paper Text – $24.95
Thomas Connery (Author)
David Abrahamson (General Editor)
Roy Clark (Foreword by)
Medill School of Journalism Visions of the American Press
Journalism and Communications
5.25 x 8
Rendering American Life
Thomas B Connery
Both newspaper and magazine journalism in the nineteenth century fully participated in the development and emergence of American Realism in the arts, which attempted to accurately portray everyday life, especially in fiction. Magazines and newspapers provided the raw material for American Realism, but were also its early and vocal advocates. This symbiotic relationship reached its peak from 1890 to 1910, when writers who might be called the first literary journalists (or, much later, "new journalists") closed the circle by more fully adopting the fiction writer's style of attempting to "show the reader real life," as their literary progeny Tom Wolfe would put it many years later. Journalism and Realism fills a much-needed gap in the scholarship of American Realism.
Thomas Connery is a professor of Communication and Journalism and former dean of the College of Arts & Sciences at the University of St. Thomas in St. Paul, Minnesota, the editor of A Sourcebook of American Literary Journalism, the co-author of Writing Across the Media, and one of the founding editors of the scholarly journal Literary Journalism Studies.

redpajama_set_name: RedPajamaCommonCrawl


Description: The Trans-NIH GeroScience Interest Group (GSIG) cordially invites you to its winter seminar, featuring Dr. George Hajishengallis. Dr. Hajishengallis is a Professor of Microbiology at Penn Dental Medicine. His field of interest lies at the interface of microbial pathogenesis and immunity, and his work has illuminated important mechanisms of immunopathogenesis of well-adapted pathogens that exploit the human immune system and subvert it to their own advantage. His interests also include the impact of aging on inflammatory diseases.
The GeroScience Interest Group (GSIG) was formed to enhance opportunities for discussion of the intersection between the biology of aging and the biology of disease and conditions that are of interest across ICs. It is focused on basic biology, but with a longer view towards translation. If you are interested in learning more, please visit the GSIG web site http://sigs.nih.gov/geroscience/Pages/default.aspx.
NLM Title: Age-related alterations in neutrophil recruitment and impact on chronic inflammation / Dr. George Hajishengallis ; Trans-NIH GeroScience Interest Group.
Abstract: (CIT): The Trans-NIH GeroScience Interest Group (GSIG) cordially invites you to its winter seminar, featuring Dr. George Hajishengallis. Dr. Hajishengallis is a Professor of Microbiology at Penn Dental Medicine. His field of interest lies at the interface of microbial pathogenesis and immunity, and his work has illuminated important mechanisms of immunopathogenesis of well-adapted pathogens that exploit the human immune system and subvert it to their own advantage. His interests also include the impact of aging on inflammatory diseases. The GeroScience Interest Group (GSIG) was formed to enhance opportunities for discussion of the intersection between the biology of aging and the biology of disease and conditions that are of interest across ICs. It is focused on basic biology, but with a longer view towards translation.


text:
Home > Posts tagged "stefan everts"
Tag: stefan everts
Stefan Everts talks Yamaha
Only the top motocross riders will make it to world championship level, and just a small number of them will win a Grand Prix. An elite few have been crowned champion, but only one has managed three GP victories on a single day. And it comes as no surprise that
Stefan Everts discusses surviving Malaria
This is a special episode with 10-time World Motocross Champion, Stefan Everts and his son Liam. Great stuff Jeff Emig and Ricky Carmichael.
Stefan Everts & Joel Robert illness updates
There has been great Christmas Day news coming out of Belgium as the country's two most decorated motocross stars have issued updates on their illnesses and both look positive. Joel Robert, the six-time world 250 champion, has revealed he has now left hospital after an operation to amputate part of his
Everts off the critical list
Stefan Everts is no longer on the critical list after being struck down by Malaria recently. He is no longer in an induced coma but still remains very poorly at the University Hospital in Leuven. Only family members are permitted to see him and they are asking for privacy at this
Everts in coma with Malaria
Ten-times world champion Stefan Everts is in an induced coma in a Belgian hospital fighting off Malaria, local press reports have revealed. The 46-year-old was admitted to hospital near his home in Leuven during the week, where he is currently in intensive care. Around a month ago he took part in a
Thousands pay tribute to Geboers
Around 2000 mourners, including former world champions Stefan Every, Harry Everts and Joel Smets, waited patiently in line to say a final farewell to five-times world champ Eric Geboers at the Lommel GP track in Belgium yesterday. Local news media and TV channels were on hand to record the event as
Liam Everts – chip off the old block
When your dad is the only ten-time world motocross champion and your grandad is a four-time champ, it's almost set in stone that you're going to be a successful racer. Liam Everts, son of Stefan and grandson of Harry, is on the way to stardom. He spent many years in
Suzuki confirms axing all factory teams
Suzuki has finally officially confirmed it is to pull its factory teams out of the world championship MXGP and MX2 classes and the All-Japan series. And as it will also no longer have an official factory team in the USA, replaced by the private JGR effort, it means there will
Suzuki axes factory MXGP team
Suzuki will pull the plug on its official MXGP team as well as its MX2 team for 2018. Although there has not been an official announcement yet, several sources have revealed that Suzuki GP team boss told his employees of the news on Thursday. The Belgian-based Suzuki team, run by Sylvain
Stefan Everts launches his own booze!
Ten times world champion Stefan Everts has launched his own range of premium gin! Called S72 Gin, it costs 43 Euros a bottle and you can order it HERE! The website explains how it all came about: https://s72gin.com/ After a chance encounter with Belgian mountainbike world champion Filip Meirhaeghe on the 'Eternal Fame'
Time for the British Grand Spree!
Herlings dominant in France qualifying
The GOAT rejoins Kawasaki!


text:
require 'test_helper'

class PagesControllerTest < ActionController::TestCase

  test "view should have a flash div" do
    get :home
    assert_select ".alert", "success"
  end

end

redpajama_set_name: RedPajamaGithub


Electric patient hoist for indoors usage.
Electric patient hoist for indoors usage. It is composed of two parts for easy transfer. The legs can spread or get closed in 3 positions, that can be performed even while the patient are already on it, offering great comfort during use.


Competition – Hello Korea !
The other day, because I have no working computer and so had no powerpoints, which are the staple of my lessons, I had to read out of the dry text book. 2/3rds of the class had no books, (this is the 2nd of the highest level students in my school – gah) and I made a face and said, "how can you call yourself students when you have no books?" 1/3 of the class got the bright idea to have an excuse to run in the halls and they made a mad dash out of the classroom. At that time I said, "while those jokers are out finding their books, let's tell some jokes," and I pulled out the Konglish puns my friend Joyce gave me two years ago.
You know, almost three years here and pretty much inured of my culture shock, yet this competition thing still drives me crazy.
Take, for instance, A week ago after midterms and all the teachers had to go bowling as their mandatory morale activity. Usually they don't make me go to these things, but I thought bowling with my colleagues would be fun, as there would be less bitching as when they have to hike or play intramural sports like their last outings. And it WAS fun! Until halfway through it became evident it was a competition. So half of the teachers had to sit and pick their noses while they had a playoff, which took forever.
The concept of personal best is something that just escapes all my students and their teachers. It always has to be a competition. And then, on the other spectrum, nobody in this society can lose. So there are no consequences for not putting in any effort. It's a sea of mediocrity and resignation that one will never get ahead and a persistent focus on material gain with a resentment towards those who do win.
Speaking of competition, SuperStar K 3 is almost un-watchable for me now, as a foreigner. There's just too much talking by the M.C. Towards the end, there's almost a half hour of the finalists just standing there on stage in agony. Last week, the judges were a little too liberal with their judging at the beginning of the show and then just got tougher and tougher. Most notable for me was this girl This girl, Lee Jeon Ah, who got a low score, but I thought her arrangement of The Eagle's Desperado was really artistic and fortunately, Korea voted higher for her than the judges did, placing her as 4th for the evening.
Not competition related, but not having my technology crutch related, today I had the students write a bucket list. Let me just say that if those same lists appeared in America, the police would be called and the school would be locked down. The kids said it was just a joke, but man…scary stuff! I told them I wouldn't be able to sleep tonight. One boy put down as item #10 (#1-9 being blank) that he wanted to see his family before he died. I asked him where his family was, and he said Guri (about an hour away). "And you never see them?" Nope. I didn't pry because there were other students around, but of course I wondered if he was adopted.
Next week I go to the Vice Principal's daughter's wedding. Nothing like having to find something to wear and cough up money for a present to someone you've never met in your whole life. Fortunately, I found a disco-era gray pantsuit reminiscent of Tom Ford that looks fabulous and only cost $8. Just need to find some shiny chemise to wear under it. So exciting to find a vintage store that's actually affordable. Got three other fabulous things and the total was $24. Would have cost over $100 in Seoul.
This week is the school festival. This school talent show/carnival is the biggest event of the school year – much bigger than exams. It's everyone's chance to be like their pop idols. (There were no school events at Baekyoung, except teachers day – there is no teacher's day here) I would have put together some dance routine, but nobody told me when the auditions were again, and it's too late now. In reality, it seems like there's always a whole lot of reasons for not studying going on at the technical school level. And Saturday I went to school to see if my computer might be fixed and saw a whole lot of activity going on, as club activities on Saturdays are also a whole lot of not studying. I sometimes wonder if these things I read about Korean student stress is even real. It just doesn't seem like they have it so bad. It seems like every other day they compensate for the pressure – maybe even too much.
I've stopped eating rice this past week and I FEEL SO MUCH BETTER. Of course, every Korean has to comment and they all think I'm crazy. They can't comprehend that rice = carbohydrate = stored fat, or that I could possibly enjoy a meal without it or feel full without it. Most times I walk away from a meal feeling like a slug, but by eliminating the rice the portions become perfect, though the rice does cut the huge amount of salt and red pepper, so the dishes seem even more heavy-handed on the seasoning than ever before.
Just edited my closet of emergency clothes that don't work. They're in little piles all over the kitchen floor. There's not enough time to put them away, so I imagine when I get home Momo will have dragged them all over the place. Little miss is kind of funny, though, as I've seen her running around the house wearing my underwear in the past.
Feeling really anxious about the future, as nothing's set and if Vegas doesn't transpire I've got no compass. Next month is the dreaded "open classroom" again where the Native English Teacher has to prepare a showcase lesson and pretend they slave that hard every day and engage the kids as if they were best buddies and include the Korean English Teacher in the lesson as if they cared to help every day and make them look good. I'm told this one is really really important. I get the sense that this year will be the deciding factor whether the school district will even continue the high school program…This entire school year is just treading water, every day. Not a good feeling, despite being pretty at peace with myself. I'm just longing to be settled and have some security after so many decades in limbo. It's not like I don't recognize that even with Vegas I'll become settled – but it's a darn sight better than having the reduction axe hanging over your head, in a job where your hands are tied, the rewards are few, and which separates you from everything you care about. Still, it's easy and pays relatively well.
Tread water. Try to appreciate the small blessings.


text:
How To Sell A Domain Name: 8-Step Framework (with Insights)
Doron Vermaat
Doron Vermaat Doron Vermaat is the co-founder of domain name investor platform, Efty. He is a prolific domain investor who owns a porfolio of domains and also blogs about it. Prior to Efty, he served 10+ years in senior sales and business development roles. Check out Doron's LinkedIn and follow him on Twitter.
Selling a domain name can be tough. I explain a 9-step framework on how to sell a domain name at the maximum sale price. Read the guide and insights.
by Doron Vermaat
Many of us have the habit of registering a domain name whenever we have a great (or not so great) idea for a new startup company, product, project, blog, or side-business. But those ideas often remain just that, an idea.
And the domain name will end up sitting dormant in your registrar account for months, years, or even decades alongside an often growing family of other, unused domain names. These domain names don't have to stay dormant; starting up a business is incredibly scary, but achievable.
Other reasons for owning unused domain names could be that you retired from or closed a business, decided to stop updating your blog or personal site, or acquired other companies over the years, with their domains.
If so, you're probably wondering if these domains are worth anything to someone else and if yes, how you can sell your domain names?
How to Sell a Domain Name Fast: My 8-Step Process
Step 1: Determine the value of the domain name.
1. Start with studying comparable sales
2. Use Domain Appraisal Tools
3. Get An Expert Opinion
Step 2: Price your domain name
Step 3: List Domain Name with a For-Sale landing page
Step 4: List your domain name for sale on domain marketplaces
Sedo
Afternic
BrandBucket
Step 5: Sell your domain name at auction (fastest way)
GoDaddy Auctions
Flippa
Namejet
Step 6: Find a buyer with outbound sales
Step 7: Hire a domain name broker
Step 8: Use an escrow service for the safe transfer of the funds and domain
In this post, I will guide you through all the different steps in the process of successfully selling a domain name.
From valuation to pricing, marketing, listing, negotiating, and closing a sale with a safe and secure way to transfer the domain name to its new owner and the seller (that will be you) receiving payment.
You can use the links below to jump to a specific section.
Determine the value of the domain name.
Price your domain name.
Make it clear that the domain name is for sale with a For-Sale landing page.
List your domain for sale on domain marketplaces.
Sell your domain at auction.
Find a buyer with outbound sales.
Hire a domain name broker.
Use an escrow service for the safe transfer of the funds and the domain name.
Before selling a domain name, you first need to figure out how much your domain name is worth.
This is easier said than done and the value of a domain name can range anywhere from a few dollars to numbers in the six or even seven-figure (!) range. The only 100% correct answer is that a domain name is worth whatever a buyer is willing to pay for it but luckily there are some techniques for appraising a domain name that can give you a good idea about the potential value of the domain name if any.
A great resource to find recent domain names sales and research sales of domain names similar to the ones you own is NameBio.com.
This awesome site offers a historical database containing $1.5 billion dollars in domain name sales and is updated daily to show how much domain names have recently sold across a number of different marketplaces. Use it to your advantage to see how your domain name compares to recent sales.
Use filters for length, keywords and of course extension (such as .com, .net or .io) to narrow down a list of domain names that sold that are similar to yours.
Another method to get an indicator of the value of your domain is to use an automated domain name appraisal tool such as Estibot or GoDaddy's GoValue appraisal tool.
Both sites rely on complex algorithms to calculate the value of a domain name based on many different attributes such as domain length, extension, word count, pronunciation as well as SEO-related factors like keywords, the number of searches, monthly searches, and cost per click. Keep in mind that these appraisals are automated and will never be able to account for everything because a domain's value is not a scientific absolute.
A third option you have is to ask for an expert opinion.
If the previous two methods both give you a high value for your domain name you might want to consider asking for an expert appraisal by a trusted domain name broker to make sure you price the domain name right.
You also can consider sharing your domain name in the free domain name appraisal section on NamePros, a forum for those that actively buy and sell domain names with the goal to turn a profit. Be prepared for some candid feedback but also keep in mind that not necessarily everyone that leaves a comment is an expert.
Now that you have an indicator of what your domain could potentially sell for it's time to decide on the right pricing strategy.
Most aftermarket domain names sell in the $2,000 range, and the vast majority of the rest of the domain names that are being purchased on the secondary market go for under $5,000.
So unless you have your hands on a premium domain name that should sell for well into the five or perhaps even six-figures it's highly recommended you market the domain name with a realistic, "priced to sell", Buy-It-Now price.
Domains with a Buy Now price are times more likely to sell than the standard "make offer" domains. Another reason to choose this strategy is that a Buy-It-Now pricing creates a sense of urgency for buyers; if they don't buy now, someone else might and they may lose out on the opportunity to own the domain name forever.
If you own a highly sought-after domain name and enjoy the process of negotiating a deal, there are several reasons to go with the "Make an Offer" approach instead of putting a fixed price on your domain. Receiving offers on your domain name will allow you to do some due diligence on the interested party and find out what their plans with the domain name are and how much they are potentially able and willing to pay to acquire your domain name.
When someone thinks of a domain name and wants to see if it's available, what do they do? They type it into their browser. And if you don't have a crystal clear "this domain is for sale" message on your domain name and an easy way for a potential buyer to contact you, you're losing out on the biggest opportunity to sell your domain name.
A For-Sale landing page will let any visitor immediately know that your domain name is available for purchase and how to get in touch. There are several ways to create a For-Sale landing page for your domain name.
If you have an existing website you can simply redirect each domain name you own and want to sell to a landing page on that site with an inquiry form. Or you can use a For-Sale landing page from one of the domain marketplace sites such as Sedo or Afternic.
The disadvantage of using those pages is that they often display advertising that can confuse visitors. And when you sell your domain name through a marketplace you have to pay them a commission for the sale, ranging anywhere from 10 up to 30% of the total sales price, – even when your buyer came directly through the domain's own landing page.
Last but not least, you can use a commission-free, domain name sales platform like Efty. Efty does all the updates, keeps your For-Sale landing pages up and running, and their pages are optimized to help sell domain names.
If you made it all the way here you've already learned about the importance of capturing direct visitors to the domain name with a landing page first and foremost. But this does not mean that listing your domain name for sale on domain marketplaces won't add value.
Many of the large marketplaces have repeat buyers and a so-called distribution network in which they partner with other websites such as registrars to put your domain name in front of potential buyers across the globe.
Sedo has been operating since 2001 and with over 18 million domains listed, Sedo is one of the world's largest domain marketplaces. The site itself reaches around 2 million visitors per month but they also promote your domain across a large number of partner sites. Listing your domain name for sale with Sedo is free but you pay a commission between 10 and 20% when your domain name sells.
This platform was founded in 1999 and listing more than five million domain names available for sale and handling more than 75 million domain searches each month – Afternic is a force to be reckoned with. This marketplace is owned by GoDaddy which is where it gets its big advantage over similar sites. When you list your domain name for sale at Afternic it will also show up as a premium domain when people will search for it on the GoDaddy.com website. Listing a domain name on Afternic is free but you pay 20% commission for a successful sale.
Do you own a domain name that isn't necessarily very descriptive but would make a great brand name for a new company, product or service?
If so, BrandBucket might have the right audience to help you land a buyer. This marketplace is curated which means each domain name has to go through a submission and approval process. Once approved, your domain gets listed with a custom designed logo and description which helps potential buyers envision. This marketplace charges a $10 fee to list your domain and requires exclusivity which means you can not sell your domain name elsewhere while you have it listed in the marketplace. With 30% the commission is steep compared to other marketplaces.
If you're in the market to sell your domain name quickly, there are a few auction sites waiting to help make it happen. These sites operate in a similar fashion to online auction sites such as eBay. You create a listing, set a reserve price or min openings bid and in some cases a Buy It Now price.
The majority of buyers on domain name auctions sites are domain name investors (individuals who purchase domain names with the sole purpose to resell them for a profit) so you have to be prepared to sell your domain name at so-called "wholesale" or reseller price levels which usually is between 5 to 15% of the "retail" price of a domain name.
As an example, I sold TechTuna.com, a domain name I purchased for $10 in 2016 for $69 to someone at an auction site who then listed the domain name for sale at BrandBucket.com where it sold for around $2,000 shortly after.
Godaddy Auctions is GoDaddy's aftermarket site for secondary domains. The marketplace is very popular with individuals and companies that are looking for exclusive, soon-to-expire domain names to add to their domain name portfolios. The marketplace is visited by 900,000 users every month and with GoDaddy's status as the world's #1 domain registrar you are ensured plenty of exposure for your auction listing if your domain name is valuable.
There are several ways to list your domain names including Buy-It-Now with a fixed price, Make Offer or as a 7-day auction. This is your typical online auction. The seller sets a minimum price, bidders battle it out and, when time runs out, the high bidder wins. There is a small ($5) annual membership fee to join the site and when you sell a domain name the commision is anywhere between 10 and 20%.
Flippa is primarily a marketplace for buying and selling online businesses but they also have a section for domain name auctions. The site is not as popular as it once was but with 2 million monthly visitors it still is a popular destination and many domain names are sold every day on the site. Listing your domain name for auction cost $9 and listings that are successfully sold will result in a 15% success fee.
One thing that makes Flippa different than other marketplaces is that they allow you to write a detailed description for your auction listing. If you're interested in learning some of the best practices on how to list, write and promote your listings on Flippa, I recommend this video interview with superseller Ali Zandi. Viewers beware – the video is a couple of years old and much has changed at but some of Ali's advice around marketing and promoting your auction listings is timeless and more relevant than ever.
NameJet is another domain name aftermarket company, operating an auction platform mainly popular among domain name investors. The site has a large number of buyers and moves a huge number of secondary domain names every day. Not everyone can list domain names for sale on NameJet but the site provides an online option to submit your domains for consideration for an upcoming auction.
In order to provide the highest quality of domains to their buyers, they do not auction all names submitted; your names are forwarded and reviewed by the NamesJet team. To submit your domain name you will need to have an active account first. Commission on a successful sale is 15%.
Do you have a descriptive domain name in a niche with many potential end-users? Something like AustinBaker.com or AtlantaLawyer.com.
If the answer is yes, you might want to consider putting in a bit of effort to increase the likelihood of selling your domain name by actively reaching out to relevant businesses that could benefit from owning your domain name.
Outbound sales are typically done via email or phone to a targeted group of leads that have been gathered after some research. The best place to start your research is by doing a Google search for businesses that match the keywords of your domain name.
Once you have a list of businesses gathered you can use tools such as LinkedIn or Clearbit Connect to find the names and email addresses of the person in charge of marketing or technology such as the Chief Marketing Officer or CTO. If a business is small I recommend emailing the owner directly. Make sure to send emails individually, keeping each of them personalized and short.
Such as in the example below.
Subject: Selling AtlantaLawyer.com
I own the AtlantaLawyer.com domain name and I am reaching out to a few companies in the legal sector here in Atlanta to explore a possible sale of the domain name. Would you be interested in owning this domain name for your business?
Your contact details.
By keeping things simple and to the point and finish your email with a question you are most likely to generate responses. These can range from "no thank you" to sometimes slightly less polite replies but, if your domain is valuable you will see that you are likely to get a couple of emails back asking you "How much?".
Need some more tops on outbound techniques from an experienced domain name investor? Check out this blog post by Ali Zandi which is packed with great information to help you get started. And while you're at it, make sure to read part two in which Ali shares everything you want to know about negotiating and closing the sale.
Does all of the above sound difficult or tiring? Well, I didn't say this was easy, did I?
If you own a domain name with a significant value you can, and probably should, consider getting a professional domain name broker involved to help you find a buyer and facilitate the sale on your behalf. Most domain brokers work on a commission basis, meaning they are paid a percentage of the sale price.
This no-cure, no-pay structure means most established brokers will only be interested in working with you on finding a buyer for your domain name if it has a value that is at least in the high five-figure range. It is very important that you make yourself familiar with the broker's policies and pricing structure prior to engaging the services. A written agreement clearly outlining all the nitty-gritty details such as commission and exclusivity is highly recommended.
Below is a list of domain name brokers and brokerage firms with a good reputation and track record:
Buckley Media
Media Options
Starfire Holdings
NameCorp
Guta.com (specialized in the Chinese market)
Make sure to do your own due diligence before working with a domain name broker and always use a licensed escrow service to facilitate the sale of the domain name and the transfer of funds.
When buying or selling a domain online, using a licensed and bonded escrow service will reduce the risk for both the buyer and the seller.
Using an escrow service to handle your domain name sale guarantees you get paid and the buyer gets their domain. All of the domain name marketplaces and auction sites referenced throughout this guide offer their own, free escrow service as part of their platform so when you sell a domain name through one of these sites, all you have to do is make sure you are able to transfer your domain name the moment it sells.
But if you sell your domain name through direct negotiation with a buyer or you are using the service of a brokerage firm you should always transact through a licensed escrow company.
The way a domain name escrow works is that both parties agree to the terms of the transaction, which include the domain name(s) and sale price. The buyer will then have to submit a payment for the agreed amount into the escrow account. Once the escrow service has verified and secured the payment, you are instructed to transfer the domain name to the buyer. Once the buyer confirms they have possession of the domain the funds will be released to you.
The most commonly used escrow companies for domain names sales are Escrow.com and Payoneer Escrow.
Analyzed by Doron Vermaat
Doron Vermaat is the co-founder of domain name investor platform, Efty. He is a prolific domain investor who owns a porfolio of domains and also blogs about it. Prior to Efty, he served 10+ years in senior sales and business development roles. Check out Doron's LinkedIn and follow him on Twitter.


The core of the Diving Instructor course! The PADI Instructor Development Course (IDC) is the process of preparing instructor trainees to serve as PADI assistant instructors and open water scuba instructors. Instructor candidates will develop instructor-level dive theory knowledge, role modeling skills, rescue skills, and leadership skills by learning how to teach students in confined waters, classrooms, and open waters. Also during the IDC, candidates will have a complete understanding of PADI's educational goals and philosophy. As a professional dive instructor, you will be able to upgrade your diver training skills.
To become an Open Water Scuba Instructor, you must pass the PADI Instructor Exam (IE) and the PADI Instructor exam is conducted fairly by a specially trained PADI Supervisor with a worldwide standardized PADI exam Assessment System. During the instructor test which lasts 2days, we evaluate the instructor candidate's diving theory, the understanding of the PADI system, the PADI course education ability, the attitude and professionalism as a lecturer.
This course is for those who want to become a PADI Open Water Scuba instructor, those who want to work two jobs, individuals who lead diving trips overseas regularly, and those who want to live and work overseas.
2. How long is the total duration of this course?
PADI Open Water Scuba Instructor Course takes at least 9 days. (At least 7 days for crossover course for instructors from other groups). So, it usually takes two weeks from the instructor development course to the instructor exam.
2) Once you have completed this course, you will be able to take the PADI Instructor Exam. Once you have passed the instructor exam, you will then become a PADI Open Water Scuba Instructor. After two weeks of passing the exam, you will receive an e-mail of your teaching status, and from then on you will be able to begin your PADI course as a PADI instructor. Your Instructor certification will be mailed to your residence within one or two months.


text:
EMGS – Change of CFO
CFO of Electromagnetic Geoservices ASA ("EMGS" or the "Company"), Hege Veiseth, has resigned her position with the Company to pursue new challenges as CFO of Polaris Media ASA. Ms Veiseth joined EMGS in 2007 and has served as CFO since 2016.
Anders Eimstad has been appointed interim CFO of EMGS. Mr Eimstad holds a Master of Business Administration from Baylor University. He joined EMGS in 2006 and has held a number of positions within the Company, most recently as VP Business Control.
Mr Eimstad's appointment will be effective as of 1 January 2020. Ms Veiseth will continue to serve as an employee of the Company for a transitional period.
CEO of EMGS, Bjørn Petter Lindhom, comments:
"I would like to take this opportunity to thank Hege for her outstanding dedication to EMGS over a number of years, and whish her the best of luck in her new position."
Hege Veiseth, CFO, +47 99 21 67 43


text:
Q: kong request transformer invoke multiple microservice APIs I am exploring Kong. I am wondering if I can achieve the following scenario using request transformer plugin.

*

*A request is reached at the request transformer plugin.

*Split the request into three microservice API requests.

*Call the three APIs sequentially.

*Merge the response of all three APIs.

*Send the response back.

A: I think that is not possible as Kong is not an orchestration/business process/ESB tool. I see two solutions for this use case:

*

*Deploy a BPM engine like Camunda, which takes care of orchestrating services

*Create your own service which does the calls for you and merges the responses

redpajama_set_name: RedPajamaStackExchange