Ctfplotter plots noise-subtracted, rotationally averaged power spectra from tilt series data and allows one to determine the position of the zeros of the microscope CTF. The power spectra are obtained by extracting small squares of image (referred to as tiles) that overlap by 50%, taking their 2D Fourier transforms, rotationally averaging the transforms to obtain one-dimensional curves, dividing by the corresponding curves obtained from images consisting only of noise, and summing these 1D curves. Once defocus is determined from such power spectra, the phase inversions of the CTF can be corrected with the program Ctfphaseflip. Defocus can be found in angular subranges of the tilt series. The defocus values are stored in a table so that they can be visualized, deleted, or recomputed if necessary. All of the values in the table are saved to the defocus file, which is provided as input to Ctfphaseflip.

- The plotter shows graphs of image power as a function of spatial
frequency. The X coordinate is in cycles per pixel (also referred to as
reciprocal pixels).
0.5 equals the Nyquist frequency determined by the pixel size.
If the pixel size is in nanometers, any frequency on the X axis can be
converted to 1/nm by dividing by the pixel size. Below each label in
reciprocal pixels is the corresponding periodicity, or resolution, in
Angstroms.

The Y coordinate is based on the logarithm of image power. - The magenta curve is the logarithm of the power spectrum of the selected
views after subtracting the logarithm of the noise floor. If there are no
noise files, or an option is selected to turn off this noise subtraction, a
line will be fit to the portion of the spectrum past an initial falling phase
and that line will be subtracted instead. A baseline from the high
frequency part of the curve is also subtracted, so the curve will end up around
zero at high frequencies.

If you use a fitting method that fits just one curve, the green curve fits the magenta curve from "X1_starts" to "X2_ends", which can be set in the fitting range dialog.

If you use a method that fits two curves, the green curve is fit to the magenta curve from "X1_starts" to "X1_ends" and the blue curve fits the magenta curve from "X2_starts" to" X2_ends", all of which can be set in the fitting range dialog.

In either case, the curves are drawn over the whole frequency range. - To zoom in an area, hold the left mouse button down and
draw a rubber band around that area.

To go back to the previous zoom level, push the Zoom-out icon.

Successive zooms are kept in a list, and you can go forward or backward through the zooms using either the Zoom-in and Zoom-out icons or the standard hot keys for zooming in 3dmod, "-" or "+/=". - Use arrow keys to scroll vertically and horizontally. You can also use the mouse scroll wheel to scroll vertically.
- The text field marked
with "Z" on the top line reports the frequency of the
first zero, either found by fitting to the power spectra, or set by
double-clicking with the left mouse button. The absolute frequency of the
first zero in 1/nm is this frequency divided by the pixel size.

The text field marked with "D" on the top line reports the found defocus in microns.

If you double-click on the position of the second zero with the middle or right mouse button, the text field marked "D2" shows the defocus implied by that second zero position, and the text field marked "D-avg" shows the average of the defocus from the first and second zeros.

When fitting to a CTF-like curve, the text field "Err:" shows the error of the fit. This error is the square root of the mean squared difference between the power spectrum and the fitted curve over the fitting range, in the log units of the Y axis. - Double clicking at a point in the plot with the left mouse button will set
the found frequency of the first zero to the frequency at the point.

The found defocus field will change accordingly. This is useful for manually locating the first zero and finding its corresponding defocus.

Once a first zero position is set either by fitting or manually clicking, double-clicking with the middle or right mouse button can be used to indicate the second zero position. The text fields based on the second zero will be set back to "NA" whenever a new first zero is set. - The
**Alternate colors**checkbox in the lower left corner provide an alternative set of display colors that should work better with some forms of color blindness.

Angles | Open the tilt angle range and tile selection dialog. |

Fitting | Open the Fitting Parameters dialog. |

Zoom in again after zooming back out. | |

Zoom out after zooming in. | |

Adding Non-center Tiles button. If this button is enabled, the current estimation is based only on center tiles; push this button to add non-center tiles to the current estimation. | |

Print the plotted curves. | |

Bring up Qt assistant to display this page. |

This dialog allows three different kinds of fitting, which can be selected by the radio buttons at the top. The second and third methods are much less useful than the fit to a CTF-like curve, although they may still have some value for data with very weak CTF signals. They are described at the end of this section, which only covers the portion of this dialog used for basic fitting. Additional features in this dialog are described in the sections below on finding astigmatism and finding phase shift.

After changing parameters in text boxes in this
dialog, either press
**Apply** or type the "Enter" key to have the fit
recomputed with the current parameters.

Fitting to a CTF-like curve involves finding four to seven parameters,
depending on the extent of the fit and whether the
option **Vary exponent of CTF function** is selected. Varying the exponent
can allow the curve to fit better to the width of the dip around the first
zero, but since it adds a parameter, it can destabilize the fitting and
produce bad results in some cases. One of the parameters of the curve is the
defocus. The other parameters for a basic fit are an additive factor, a
scaling factor, and the decay rate for an exponential that attenuates the
curve. However, when there is detectable signal between the second and third
zeros and the fitting region is extended to at least halfway between those
zeros, the program automatically adds two more parameters, a scaling
factor and decay rate for a second exponential. The benefit of this is
illustrated in one of the examples below.

The range of the curve is set from the entries **X1 Starts** and
**X2 Ends**. When the program first starts, it sets these values to be
about 0.05/pixel before the first zero, and close to the second zero,
respectively, based on the expected defocus. If necessary, you should adjust
the starting value so that it is to the right of where the fitted curve
strongly deviates from the actual. In general, you should adjust the ending value
to go as far out as CTF oscillations can be reliably discerned, making it bigger
to fit past the second zero. On the other hand, if the magenta curve
becomes noisy and falls off before a second zero, you should reduce the ending
point of the fit to exclude that region. See the BPV example below.

The fit does rely on an initial approximate value for the defocus and may fail if the actual defocus is far from that value. It takes this value either from the expected defocus or from the current defocus estimate, depending on whether the option is selected to use the current defocus estimate in the Tilt Angle and Tile Selection Dialog. If the defocus revealed by the power spectrum is far from the expected defocus, you should either change the value of the expected defocus, or select the option to use the current defocus estimate and make sure that estimate is approximately correct.

The **Baseline fitting order** spin box controls the fitting of a constrained
polynomial to the apparent baseline of the power spectrum at
frequencies past the first zero. This polynomial is then subtracted to
make the baseline be flat. After improvements to this fitting in IMOD 4.10, this
fitting has been turned on by default with an order of 4. An order of 0
turns off this fitting. The
order can also be adjusted with hot keys 0, 1, 2, 3, and 4. If the deviation
from baseline is very small in your data, you can try turning off this
fitting to see if it improves the signal.

The option to **Get initial baseline from 2-line fit** can be helpful in
cases where the noise subtraction leaves a spectrum that slopes badly even
around the first zero and makes it difficult to detect the first zero and
include it in the polynomial baseline. The program fits two lines to the
power spectrum, one from near the start of the spectrum to a point in the
middle, and a second line fromthe same middle point to the end of the spectrum.
It tries this fit over a range of middle points around the expected position of
the first zero and finds the point where there is the biggest difference between
the slopes of the low and high-frequency lines. This method is always used
for an initial baseline subtraction when there are no noise images, or when the
option described next is selected; in these cases this option is disabled.

The option **Skip noise subtraction** allows you to turn off the use of
the noise images and rely on the two-line initial fit instead. You can use
this both to find out whether it solves some problems in fitting and to
assess whether you can dispense with noise files in the future.

[The rest of this section still needs revision.]

You can also fit a polynomial to the region around the first zero.
The
The **Order of polynomial** spin button allows you to select an order
between 2 and 6, which involves finding 3 to 7 parameters. The goal here is
to get a smooth curve through the dip; the minimum
of the curve is taken as the location of the first zero.

As for CTF-like fitting, the range of the curve is set from the entries
**X1 Starts** and **X2 Ends**. Since you are just trying to localize
the dip at the first zero, you should restrict the range as necessary to get a
good fit there.

Finding the zero at the intersection of two curves involves fitting two separate curves, before and after the first zero. Each can be fit to either a straight line or a Gaussian over the selected range. The program finds the intersection of the two curves, if possible, and assigns that as the first zero.

X1 sets the fitting range for the curve before the first
zero, drawn in green. Use the **X1 fitting method** radio
button to select whether to fit to a straight line or to a Gaussian.

X2 sets the fitting range for the
curve after the first zero, drawn in blue. Use the **X2 fitting method**
radio button to
fit to a straight line or to a Gaussian.

**Middle tilt angle**

**Number of views to fit**:

The program will include in the power spectrum the number of views specified
in the spin box, with the middle view of that set having the tilt angle
specified the middle angle box. It will adjust the middle angle entry as
necessary to match an existing tilt angle and to be a feasible middle angle
for the selected number of views. The range of tilt angles and view numbers
in the spectrum are summarized in the line below the spin box. Views are
numbered from 1.

**Step view range**:

Pressing **Step Up** or **Step Down** will shift the selected views by
the amount in the **>Step view range by** text box.
If this step would make the starting or ending view
go outside the bounds of the
tilt series, the actual step is reduced.
The power spectrum is computed and fit to at the new
angular range. The step size is initialized to be half the original number of views.

**Autofitting**:

Once fitting parameters have been set optimally and an appropriate number of
views to fit has been chosen, the **Autofit All Steps** button can be used to step
automatically to a series of view ranges, find the defocus and optionally
other parameters, and store the defocus in the table.

- The view range will be that specified in the
**Number of views to fit**spin box. - Ranges will span the angular extent specified in the
**Autofit:**and**to**text boxes. Specifically, the range at one extreme will start at the angle in the**Autofit:**box and the range at the other extreme will end at the value in the**to**box. These boxes are initialized with the starting and ending tilt angles of the series. - The spacing between view ranges will be approximately that specified in the
**Step view range by**entry, adjusted to span the desired angular extent. - However, if the
**Fit each view separately**checkbox is selected, each image in the desired angular extent will be fit separately, regardless of the step size and the size of the current view range. The same thing can be accomplished by setting the step size to 0. Fitting to every view separately may be feasible for data from direct electron detectors, which have better high-frequency response than phosphor-based cameras. - The current settings for
**Which defocus to use**and**Initially use only central tiles**are used in computing and fitting the power spectra. - If the current defocus is selected for use, then the fitting will occur in two directions from the currently selected view range. It saves the current defocus before it starts, and when it reaches the extreme in the the first direction it uses those saved values to resume fitting in the oter direction from the starting location. In this way, it can adapt more easily to progressive changes in tilt angle. In this situation, it will do three iterations of the computation and fitting to a final power spectrum to find defocus and possibly phase shift, in order to reach a consistent defocus value.
- If there are already values in the defocus table, the program will ask whether you want to remove or retain ones that are already present in the extent being analyzed. The message will list the middle of the tilt angle range for the first and last one to be removed. If existing entries get deleted, look carefully at the end of the range for ones that might not have been deleted.
- The
**STOP**button can be used to stop the autofitting when the current angle is done. This can be useful if you see it go astray. - The
**Show wedge fits**checkbox in the Fitting Parameters dialog can be toggled at any time during autofitting. You should watch the wedge spectra during the autofitting over any part of the tilt series where there is doubt about whether the wedge fitting might be reliable.

**Apply:**

This button causes the spectrum and fitting to be done with the current paramaters.

**Store in Table:**

Push this button to store
the defocus and any other paratmeters found for
the selected views in the angles and defocus table. The defocus
indicated by "D" is stored unless a second zero has been clicked, in which
case the average defocus indicated by "D-avg" is stored. Astigmatism amount
and axis, phase shift, and cut-on frequency will be stored if their values as
a result of fitting show in the plotter window. The table will not show
phase or cut-on values if they are the same for all table entries (i.e., if
they were set from their respective fields for expected
values), although the defocus file will have these values.

**Expected defocus**:

This field starts out with the expected defocus value
specified in the parameter file, but you can change it. Initially, this value
is used to compute the expected frequency of the first zero of the power
spectrum and to set
the initial values of the X1 and X2 ranges which determine what segments of
the power spectrum are fitted.
When non-center tiles are being included, Ctfplotter uses a defocus to
compute shifts needed to align the power spectra of non-center
tiles with the CTF curve of center tiles. With the radio button group
**Which defocus to use**, you can specify
whether to use the expected defocus shown in the **Expected defocus
(um)** edit field or the defocus last found by the
program as the defocus for computing the shifts. In the latter case, the
autofitting behavior is also modified as described above.

**Initially use only central tiles**:

Choose whether or exclude the noncenter tiles from the estimation
when computing the power spectrum.

With this checked,
only the center region defined by
**Center defocus tol** will be
included in the next computation, and the "Adding Non-center Tiles" button
will be enabled. You need to push that button to add the left region
defined by **Left defocus tol** and the right region
defined by **Right defocus tol**
to the estimation.

Otherwise,
all regions (center, left, right) will be included when the curve is
recomputed,
and the "Adding Non-center Tiles" button will be disabled.

**Phase shift fields**:

The **Expected phase shift** text box can contain a phase shift in
degrees. If phase shift is not being determined, this constant phase shift will be
included in all CTF computations. If it is being found, then the value here
sets the center of the phase shift search range.

The **Cut-on frequency** text box can contain a cut-on frequency that is
relevant when a non-zero phase shift is included in the CTF computations.
See the extensive explanation of this cut-on in the section on *Finding
phadse shift and cut-on ferquency* If cut-on frequency is not being
determined, the constant value here is included in CTF computations. If it
is being found, this values provides a fallback that is used when searches fail
to find a best cut-on frequency.

When **Use current phase estimate** is checked, phase shift searches will
be centered on the value last found by fitting, and the autofitting behavior
will be modified as described above.

**View skipping and breaking fields**:

These entries control which views are included when spectra are analyzed.
The **Views to skip** field can be used to exclude views either from all
spectra, or only from the spectra used to find astigmatism or phase shift.
If **Skip/break only for astig/phase** is not selected, then the program
will behave as if these views simply do not exist. If it is selected, then
the views will be included in spectra used only for defocus measurements and
their defocus can be determined, but they will excluded from any spectra
used to compute phase or astigmnatism.

When **Break groups at view** is selected and the starting view number of a
bidirectional tilt series is entered in the text box, then groups of views
for computing spectra will be entirely on one side or the other of this
boundary. Again, the **Break groups at view** option will determine
whether this restriction is applied for all spectra, or just for ones used
to determine astigmatism or phase shift. It makes sense never to group views
across this boundary, since the conditions affecting these parameters could
easily change during the time interval between the two halves of the tilt
series. On the other hand, defocus wass probably set by autofocusing at each
view on either side of the boundary, which would have produced no
discontinuity in the values, so there seems to be no problem with combining
views across the boundary when just finding defocus. Note that a tilt
series taken with the dose symmetric ("Hagen") scheme of alternating plus
and minus tilt angles should not be treated as a bidirectonal tilt series
because there is no big discontinuity or long time interval on the two sides
of zero degrees.

**Tile & wedge parameters**:

The **+** pushbutton will open up a section of the dialog for parameters
that are rarely adjusted; they include:

**Center defocus tol**: The image region with defocus difference less than this tolerance is defined as the center region. In addition, tiles within this tolerance of the defocus at the center of an off-center strip are included in the strip and summed together before adjusting their spectra to add into the full sum.**Left defocus tol**and**Right defocus tol**: These two entries set the maximum defocus difference for adding in tiles from image regions to the left and right of the center region, respectively. You can set one to be small and the other large in order to assess the results of adding in tiles from only one side of the image, but this will mix tiles at lower and higher defocuses unless the tilt angles are all positive or all negative.**Analyze spectra cropped to pixel size**can be turned on to spread out the power spectrum by cropping off the high frequencies. This cropping can be helpful if the spacing between zeros is fairly small (less than about 0.02-0.025/pixel) and the higher frequencies do not appear to have usable signal. The text box with the new pixel size is initialized with the actual pixel size of the data set and is constrained to containing higher values. When the cropping is enabled or changed, the meaning of the frequency units changes, and the fitting ranges in the Fitting Parameters dialog, which are in reciprocal pixels, are changed to maintain the same actual frequency in reciprocal nanometers (note that the resolutions in Angstroms next to these fitting ranges stay the same). Spreading out the spectrum will result in less averaging in each frequency bin and thus a noisier spectrum, so at some point the cropping will become counter-productive. Use the minimum increase needed to make the spectrum easier to fit or visualize.-
**Tile size**: the tile size in pixels. Spectra with a large number of closely spaced zeros extending to high frequency may benefit from a tile size larger than the default. **Tilt axis angle**: The amount in degrees that the tilt axis deviates from being vertical (Y axis)**Wedge interval**: The angular step between successive wedges when finding astigmatism. This is constrained to be a multiple of the underlying sector size, which is 5 degrees by default and can be controlled by an option in the parameter file.

**Angles and Defocus Table:**

This table shows the starting and ending tilt angle,
the middle of the angular range, and the defocus value for each range where
you have stored results. The lines are in order by the middle of the angular
range. When you store a value for an angular range matching that of an
existing line in the table, the existing defocus value is replaced with the
new one. When the program starts, this table is loaded with values from an
existing version of the output file. Below the table are three buttons that
operate on the table:

Push the **Delete Row** button to delete the entry for the currently
selected row of the table. If you store
results for an angular range and then want to replace them with results from a
wider or narrower range, you would need to delete the row with the initial
results.

Push the **Set Tilt Angles** button to reset the middle tilt
angle and number of views to fit to the values implied by the currently selected row
of the table and recompute the power spectrum. To conveniently step through a
series of angular ranges,
click the mouse anywhere on a row to select that row of the table.
Then press the **Set Tilt Angles** button with the mouse, or using the
keyboard accelerator Alt-T. Thereafter, you can type an up
or down arrow key to move up or down in the table and press this button again,
without moving the mouse. Alternatively, you can double click in the table to
select a row and have its power spectrum displayed.

Push the **Graph Values** button to open separate windows with graphs of
all the determined values in the table
versus tilt angle. The graphs can be closed with the close buttons in their
title bars. They need not be closed before opening another set of graphs.

Push the **Save to File** button to write the contents of the angles and
defocus table to the output file.

Ctfplotter can find astigmatism by making power spectra that are rotationally averaged over a wedge of 2-D Fourier space. It fits to a series of such spectra at the full range of wedge orientatons. For eaxample, with the default parameters, it computes the spectrum from a 90 degree wedge, which is half of frequency space, with the center of the wedge oriented at -90, -85, -80, etc. up to 85 degrees. It is able to compute these overlapping wedge spectra very efficiently because frequency space is divided into small, non-overlapping sectors (5 degrees by default) and rotationally average spectra are store separately for each sector.

Each wedge spectrum is fit to a CTF-like curve to its defocus. When there is astigmatism, this defocus will vary consistently between the low and high defocus axes of astigmatism. The difference between maximum and minimum measured defocus values will be less that the actual astigmatism because these broad wedges average over a range of angles and defocus values. A search is done to find the underlying astigmatism amount and axis angle that best fit the set of measured defocus values.

The wedge spectra will be computed over the maximum of the number of views being
fit for defocus and the number specified with the **Min views for
astigmatism** spin box. It is generally appropriate to compute the wedge
spectra over more views than being used for defocus, for two reasons: 1)
Each wedge spectrum is based on less data than the full spectra for
measuring defocus (e.g., half as much for 90
degree wedges), so if a certain number of views need to be averaged to get
reliable defocus estimates, more views (twice as many for 90 degree wedges)
are needed to get wedge spectra with the same signal-to-noise ratio. 2)
Even when there appears to be good enough signal to measure defocus and astigmatism on
each view (as in the example below), the view-to-view variability in the
estimated defocus will be reduced by averaging over multiple views. Since
astigmatism would be expected to vary slowly through the tilt series, most
variations from view to view are unlikely to be real and reducing them seems
appropriate.

When the program computes wedge spectra for finding astigmatism, it will
display each spectrum and its fit as long as the **Show wedge fits**
option is turned on. You should always have this on when you start finding
astigmatism so that you can assess how reliable the astigmatism fits are.
Watch out for jumps in the fitted curve; a few jcan be tolerated because
there is robust fitting that can discard them, but there are more than a few
jumps, average the wedge spectra over more views. If necessary, you can put
a longer delay time in the text box after **Show wedge fits for**. Also
watch for dramatic changes in the shape of the spectra, such as the dip at
the first zero becoming much wider at some angles; such changes can make the
astigmatism estimate quite inaccurate.

After finding astigmatism, the plotter window shows a second line of numbers: "Astig:" and the astigmatism in microns; "Axis:" and the angle of the high-defocus axis of astigmatism in degrees; "Wedge err:" and the error from fitting the defocus values, in microns. This error is the square root of mean squared error between the measured defocus values and the curve of values expected from the given wedge size and the fitted astigmatism and defocus.

The dependence of the astigmatism estimate on the wedge size has been evaluated by averaging over sets of 10 views for several sample data sets, and by analyzing some high-exposure images from carbon film. No significant dependence was found. This, athough you could use wedge size below 90 with particularly strong CTF signals, there is no real reason to do so. You can increase the wedge size to 120 if that is needed for more SNR.

After astigmatism is determined from the required number of views, the program then assumes that astigmatism when making the spectrum used to find defocus. It composes this spectrum by adding up data from non-overlapping wedges of frequency space, where each wedge spectrum is adjusted for the defocus at its angle before adding into the sum. The final defocus value is the one determined by fitting to this spectrum.

If astigmatism is large enough to cause blurring of zeros in the wedge spectra within the range of frequencies being fit, then the program will iterate, using the initial value of astigmatism to divide each fixed-width wedge into sub-wedges of sufficiently uniform defocus. Iterating in this way eliminates the potential bias in the astigmatism solution from using broad wedges for the analysis. If you are watching the spectra, you may see several rounds of fitting when astigmatism is large.

Ctfplotter can also find the phase shift from a phase plate. It requires
several well-defined zeros in the spectrum to find both defocus and phase
shift reliably. Phase shift is determined with a separate search;
specifically, the phase shift is varied and at each value of phase shift,
a CTF-like curve is fit to the spectrum; the phase shift is picked that
gives the minimum error in that fit. The search is done over the range of
phases entered into the **Phase search range (deg)** text box, centered
on the value in the **Expected phase shift** text box in the Angle Range
and Tile Selection dialog.

Just as for astigmatism, more views can be used to find phase than to find
defocus; the number of views used to find phase is the maximum of the number
for finding defocus and the number set in the **Min views for phase**
spin control. When more views are used to find phase, a spectrum is
computed first from that number of views, phase is found, the spectrum is
computed for the smaller number of views needed for defocus, assuming that
phase value, and then defocus is found from that spectrum, with phase fixed
at the found value. When the same number of views are used for each, a
single spectrum is computed for finding both parameters. However, when both
astigmatism and phase shift are being found, there may be up to three
rounds of spectra with different numbers of views.

If there are many zeros and the CTF does not closely match what the CTF
equations predict with phase shift, the fit curve may be get off at high frequencies
because the much stronger signal from the first few zeros dominates the
fit. A more balanced solution that fits better at the high frequencies
can be obtained by selecting the **Find and fit to zero positions**
option. With this option, the program finds minimum points in the spectrum by
correlating the spectrum around each point with a segment of the expected
CTF function. It then searches for the defocus and phase that best fit this
set of positions, minimizing an error weighted by the strength of the
correlation at each zero. The fit will be better at the high-frequency
zeroes but the CTF-like curve may deviate noticeably around the first zero.
This method requires at least 3 zeros. If the search fails, or there are
fewer than 3 zeros, the program falls back to finding the phase that
optimizes the usual fit of the CTF-like curve to the spectrum.

After finding phase, the second line at the top of the plotter window will show "Phase:" and the value in degrees, or "FAIL" if the search failed. When fitting to zero positions, the label "Err:" on the top line will be shown as "ZF err:" and the value will be the mean deviation of the fitted zero positions in reciprocal pixels (specifically, the weighted sum of squared deviations in position is divided by the number of zeros, and the square root is taken).

Two problems particular to finding phase shift may arise. One is that the
first zero may rise up above the later zeros at high tilt. If this occurs,
first make sure that the fitting is starting with approximately correct
values of defocus and phase shift. For defocus, either adjust **Expected
defocus** if that is what is being used, or make sure that the last found defocus is
approximately correct if the current defocus is being used. Double click to
set a good defocus value, if necessary. Adjust the **Expected phase
shift** to the last known good value and turn off **Use current phase
shift**. Repeat the fitting. If the zero is still elevated, you can try
toggling **Get initial baseline from 2-line fit**, unless you have no
noise files. If none of this works to bring the first zero down, you can
also try changing the fitting range to start past the hump after first
zero. The position of a minimum on a falling part of the spectrum is
shifted by the underlying slope, so it is better to leave the zero out of the
fitting.

The second problem may become evident with many zeros. If the minima of the
fitted curve are consistently off from the minima of the spectrum at one end
or the other, even when fitting directly to zero positions, you may need to
determine a cut-on frequency
as well. With a cut-on frequency, the phase is modeled as rising
exponentially from zero at zero frequency to its true value at high
frequency. This is difficult to solve for since it produces relatively
small deviations in the spacing between zeros from a model with no cut-on
frequency. The error when fitting either the spectrum or zero positions is
rather insensitive to the cut-on frequency, so the search needs to find the
minimum in a long, narrow trough with a shallowly sloping floor. The
program scans the full range of cut-on frequencies from 0 to the value in
the **Maximum cut-on (1/nm)** text box at a regular interval, for ech
value performing a full search for the best phase and defocus. Then it
refines the search around the minimum, again doing the full search at each
cut-on frequency being tested.

This search is activated by selecting "Find cut-on frequency also". Before
doing so, you can assess whether a cut-on frequency helps the fit by
entering some values in the **Cut-on frequency (1/nm)** text box in the Angle Range
and Tile Selection dialog. Try values like 0.05, 0.1, etc up to the
maximum cut-on shown in the text box. If this does not improve the
correspondence between actual and fitted minima to some degree, there
is no point in trying to search for a value. Also, do not try this search
if there are fewer than 4 zeros to fit.

When fitting to a spectrum, this exhaustive 2-D search is now the most
time-consuming operation in Ctfplotter. Searches with zero positions are
much quicker, but require at least 5 zeros. As for phase alone, if fitting
to zero positions is selected and there are too few zeros or the search with
zeros fails, the program falls back to fitting the spectrum. Which way it
fit is still indicated by whether the error is labeled "Err:" (in log power
units) or "ZF err:" (in fequency units). If both kinds of fitting fail to
find a minimum, the program will then fall back to finding phase only, and
assuming a fallback value for cut-on frequency. This will be either the
last cut-on frequency found, if **Use current phase estimate** is
checked, or the value in the **Cut-on frequency** text box if it is
non-zero, or the value stored for nearby angles in the defocus table, if
any, or simply 0. The value used will be shown afer "Cuton:" in the plotter
window with an "FB" after it to indicate that it is a fallback. However, if
the search for phase only fails as well, both "Phase:" and "Cuton:" will be
followed by "FAIL".

Whenever the cut-on frequency is non-zero, the phase values displayed here and saved in the defocus file will be the phase at one particular frequency, 0.3/nm, rather than the phase actually used in the CTF equation, the phase value at infinite frequency. This convention was introduced in order to prevent the phase values from varying widely from view to view, which they will otherwise do because the cut-on frequency is not found very accurately and the phase at infinity would covary with it.

The first example shows the appearance of the plotting window with a tilt series rom the ventral disk of Giardia, taken with a Gatan K2 camera in an electron-counting mode at a nominal defocus of 6 microns. This is the initial spectrum of -21 to 21 degrees.

The range of the display in Y is dominated by the power at low frequencies, so it is necessary to zoom the display. This graph was zoomed by pressing the left mouse button with the cursor under the hump in the magenta curve (at ~0.12 in X, and ~0.4 in Y, midway between the grid lines at 0.2 and 0.6), then dragging the mouse to just below the curves and out to near the right edge to set the lower right corner of the rubber band. Then the window looks like this:

The top line shows that the fitted defocus is fairly far from the expected value of
6.0. As a result, the start of the fitting range, initially set to 0.14, is
at a lower frequency than appropriate. Also, the third zero is fairly well
defined in this plot, so the end of the fitting range, initially 0.275, is much
too low. After changing to **Current defocus estimate** and setting the
fitting range to 0.175 to 0.40, the graph looks like this:

The next step is to see how many views need to be fit at once. This is done by
clicking the **Number of views to fit** spinner repeatedly. When it gets
to 1, the spectrum
is still fairly clear and fits well:

Still fitting to a single view, one can then step through and sample some
higher tilt angles easily with the **Step Up** and **Step Down**
buttons. The step size was initially set to 11 views, and this
is adequate for sampling. At 61 degrees the fitting is still good:

Autofitting could now be done, but it is worth assessing the ability to find
astigmatism first. After returning to the initial tilt angle (-1.1 degrees),
setting **Min views for astigmatism** to 3 to preserve SNR in the wedge
fits, and turning on **Find astigmatism**, the final graph looks like this:

The astigmatism turns out to be quite large, 0.87 microns, and accounting for
it gives a spectrum where the CTF signal goes all the way out to 0.5. The end
of the fitting range can even be set to 0.45. Now autofitting to every view
is quite appropriate.
This is accomplished by turning on
the **Fit each view separately** check box and then pressing the button now
labeled **Autofit All Single Views**. The table is filled in as the
autofitting proceeds, first from low tilt to negative angles then from low tilt to
positive angles. Pressing **Graph Values** opens separate graphs for
defocus, astigmatism amount, and astigmatism axis.

Astigmatism is being found fairly accurately, as indicated most strongly by
the consistency in the axis angle. There are clearly some trends in both the
amount and the angle of astigmatism, but the view-to-view changes are probably
mostly measurement error. This variability is larger with **Min views for
astigmatism** set to 1 and can be reduced further with it set to 5, but
then it is unclear whether some real variations are being reduced as well.
Averaging 3 views is a compromise in the face of this uncertainty.

[The rest of this document still needs revision.]

The second example is from microtubules decorated with Eg5, taken with a DE-12 direct detector camera at a nominal defocus of 6 microns. This shows the spectrum and CTF-like fit for an angular range of -20 to 20 degrees, after switching to including all tiles included and using the current defocus estimate.

The fitting range here was 0.085 to 0.19, and it needs to be changed because the defocus is different from the nominal value, and also because there is good signal between the second and third zeros. Changing the fitting range to 0.1 to 0.25 and varying the exponent of the CTF curve gives the following:

The curve fits the second peak much better now because the fit includes two exponential decays, one dominated by the falloff of the power spectrum before the first zero and the other to accommodate the very different decay from the first to the second peak. With this data set, it was possible to fit ranges of 10 degrees, although ranges of 20 degrees gave cleaner results at one end of the tilt range, but the data were too noisy to allow reliable fitting to every individual image.

The third example shows results with a tilt series from bovine papilloma virus, taken at a nominal defocus of 3 microns and a pixel size of 0.76 nm. These data were taken on a CCD camera and there is little useful signal past the first zero. When the program is started with the default angular range in the command file, the window looks like this:

The range of the display in Y is dominated by the power at low frequencies, so it is essential to zoom the display. This graph was zoomed by pressing the left mouse button with the cursor just above the second low hump in the magenta curve (at ~0.13 in X), then dragging the mouse to just below the curves and just to the left of 0.4 to set the lower right corner of the rubber band. Then the window looks like this:

Now we can see more clearly that the background subtraction is giving a
reasonable power spectrum that is close to flat at high frequencies.
The next step is to add in all of the tiles, either by pushing the "Adding
Non-center Tiles" button or by opening the Angle range dialog and switching to
the **All tiles** option. These two methods will give slightly different
results, and the latter is actually preferable.

The power spectrum is much smoother and it is now easier to see that it has a
discernable dip, but that the green
curve does not fit very well. If we open the Fitting Range & Method dialog, we
see that it is fitting from 0.215 to 0.39, a range that was determined from the
nominal defocus but is not quite appropriate for the actual defocus. The
range was changed to what was shown above in the dialog: the left side was
moved to 0.19 to include more of the falling phase of the power spectrum, and the
right side was reduced to 0.29 because the power spectrum drops off after that
point, well before its second zero. At this point it is also clear that the
actual defocus is different from 3 microns, so we select the **Current
defocus estimate** radio button in the Angle Range dialog so that the
shifting from non-centered tiles will be more accurate.
If the fitting were unstable, a better way to do this would be to
change the **Expected defocus** entry.
After recomputing the curves and zooming the display again, it looks
like this:

Here we see that the
CTF-like fitting works fairly well near the bottom of the dip, and that it
leads to a defocus estimate significantly different from the nominal defocus.
Turning on the **Vary exponent of CTF function** gives a slightly different
fit:

This fitting method can now be used over different angular ranges, and the
results can be stored from each range. If **Autofit All Steps** is
used over the full extent of the tilt series, four ranges have defocus values
near 3.5 but the last one has a value of only 3.1 microns. The curve for this
range lacks a dip, so this value is wrong. This problem is solved by
restricting the high end of the extent analyzed to 57 degrees. After running
with this extent and loading the last range to verify that its spectrum has a
dip, the angle dialog looks like this:

Although there is no need to try the different fitting methods with this data set, they are illustrated next.

This panel shows the power spectrum for the initial angular range with a fourth-order polynomial fit to the frequency range shown in the dialog above. As is typical of polynomials, the green curve goes off wildly outside the fitting range.

This panel shows the power spectrum with determination of the zero from the intersection of two curves, with the frequency range as shown in the dialog above.