Panel 2 — Fourier tools

In complex and power-spectrum display modes two toggle buttons sit to the right of the Fourier-space histogram, stacked vertically with a small gap:

• mask center for display — masks the central pixels of the FT display so the bright DC component does not dominate the contrast. The mask only affects display, not the underlying Fourier data.
• freeze display contrast — locks the current Fourier-space display range across subsequent Compute operations. When off, contrast auto-scales to the full dynamic range after each operation.

Both buttons use a raised 3D look when off and a sunken look when on. The histogram hint above reads “Click to adjust display contrast” — click the histogram bar to drag the white triangles for manual range adjustment. When a parameter window opens at the bottom-right of the panel, the toggle buttons remain partly visible behind it; clicks on the visible portion still toggle them.

In complex FT mode the colour-coded phase wheel is drawn below the bottom-left corner of the Fourier-space display, with the four cardinal phase labels (0°, ±180°, +90°, −90°) around it.

Multiplies the FFT by (1 − Gaussian) under the cursor, smoothly attenuating Fourier components. Useful for removing isolated diffraction spots or noise peaks.

Writes a chosen real value (with zero imaginary part) into the FFT under a Gaussian footprint. Use a value of 0 to gently push amplitudes to zero; use larger values to boost or inject specific Fourier components.

Have some fun: Create an empty image (value 0 everywhere) and use the paint brush in Fourier space to place one single pixel of value 1 into the Fourier transform. That will back-transform to a sinusoidal wave in real space, because the Fourier transform of a delta function is a sinusoid.
Try placing that single pixel into either the Cosinus or the Sinus half of the Fourier transform, and see how the resulting wave changes.

Synthesise a fresh Fourier transform whose amplitude is uniformly 1 and whose phase increases linearly along a chosen direction. Pure teaching tool: a phase ramp of step s at angle θ back-transforms to a single delta at the corresponding real-space position, illustrating the shift theorem.

Draw a 1D radial / oriented profile of the Fourier amplitude. The line direction is dragged in panel 2; pixels within the chosen integration band perpendicular to the line are averaged.

Keep (or erase) Fourier pixels inside a circular ring centred on the origin. The two ring radii are dragged interactively in panel 2; the parameter window shows their current diameters.

Keep (or erase) Fourier pixels inside an angular wedge centred on the origin. The two wedge edges are dragged interactively. The wedge is mirror-symmetric so that the filter respects the Friedel symmetry of real images.

Keep (or erase) Fourier pixels inside a straight stripe. Useful for isolating a single Fourier direction, or for removing scan-line and grid-pattern artefacts.

Build a reciprocal lattice from two basis vectors u and v (in Fourier pixels) and keep (or erase) circular spots at every lattice position. Used to extract or suppress crystalline content — for example, to denoise a 2D crystal or to remove residual lattice contamination from a non-crystalline target.

Drag in panel 2 to rotate the Fourier transform around the centre. This is mathematically equivalent to rotating the real-space image, but it operates directly on the FFT.

Remember that Fourier space assumes periodic boundaries in real space. When you rotate the FT, then the periodicity in real space may cause some unexpected results, because the rotated FT corresponds to a rotated image that is periodically tiled in the original orientation. You will see many copies of the rotated neighbors from outside of the current image appearing in the back-transformed image, and they will interfere with the rotated version of the current image.

Enforce N-fold rotational symmetry on the Fourier transform around its centre (the DC term). The current FT is averaged with N copies of itself, each rotated by k·360°/N (k = 0…N−1), and the real-space image is updated through the inverse FFT. Because the operation acts directly on the FT, a phase correction is applied so that the rotation centre coincides with the original image centre rather than the array origin.

This is the Fourier-space counterpart of Symmetrize image in panel 1: mathematically the two are equivalent, but symmetrizing here lets you watch the diffraction pattern become N-fold symmetric directly.

This function truncates or expands the FFT by an integer factor N.

When using Fourier cropping, the central 1/N×1/N region of the FFT is kept, and the rest is discarded. During the cropping, the Fourier transform becomes smaller, except if the option "Keep original size" is checked. In that case, the eliminated area is kept but is then only containing zero values.

When using Fourier padding, the FFT is zero-padded to N× its current size. In this case, the Fourier transform becomes larger due to the added zeros around the original FFT. This means that also the real-space image becomes N× larger, but no new information is added. The Fourier transform is now oversampled, and the back-transformed image will be larger in pixel size and smoother.

• Fourier crop keeps only the central 1/N×1/N region. Real-space sampling stays the same (with Keep original size checked) or becomes N× coarser.
• Fourier pad zero-pads the FFT to N× its current size, oversampling the real-space image without adding information.

CTF SIM synthesises a 2D contrast transfer function for an electron microscope with the chosen optical parameters, including astigmatism, defocus spread, amplitude contrast, beam tilt, and the spatial-coherence envelope from a finite gun-opening angle. The result is written to the Fourier side of the currently selected buffer, and its real-space point spread function (PSF) to panel 1. To fit a CTF to an existing image instead of simulating one, use CTF FIT (below).

There is no single “Compute” button. CTF SIM offers three different models of the same microscope — Pupil Function, Complex CTF and Real-valued CTF. They are different physics, not different display options: each writes a different array into panel 2 and turns it into panel 1 in a different way. With no beam tilt they largely agree; under beam tilt they differ fundamentally, and the differences are the point. For the physics behind them, see Beam tilt and coma in the manual; for the CTF itself, see What is the Contrast Transfer Function (CTF) of an Electron Microscope?

1D profile plots (panels 3 & 4). While the CTF SIM tool is active, the two bottom panels show 1D cuts of the complex transfer function C taken along the red direction line in panel 2. You can change the direction of the line by dragging it with the mouse and watch the profiles update live. The two plots show the amplitude and phase of the CTF, respectively.:

• Panel 4 — CTF Amplitude Profile. The modulus |C|, i.e. amplitude only, ignoring phase. The vertical axis runs from 0 to 1. For the two CTF models this is the rectified envelope of the oscillating contrast transfer; for the Pupil Function it is just the coherence envelope E(q), a smooth ringless decay.
• Panel 3 — CTF Phase Profile. The phase arg(C) in radians, from −π to +π, drawn as a continuous curve (a jump at the ±π wrap appears as a vertical connector). What it shows depends on the model:
  • Pupil Function: the full wave aberration itself, winding through many cycles — the aberration lives entirely in this curve.
  • Complex CTF: 0 or ±π (the sign of the contrast transfer) when there is no beam tilt; a tilt adds the odd coma phase χ_odd and tilts the curve away from those levels.
  • Real-valued CTF: always 0 or ±π, tilt or no tilt, because this model is real by construction — the asymmetry shows up in the rings of panel 2 instead.

The horizontal axis of both plots is spatial frequency in 1/Å. The curves are drawn only up to the highest frequency that actually exists in panel 2 along the chosen direction: a ray leaves the square Fourier grid at q = Nyquist when the red line is horizontal or vertical (so the plot stops at ~0.5 cycles/pixel and the rest is left empty), and out to the corner frequency (~0.707) when the line points along a diagonal.

Note that the CTF is computed in Fourier-space with values between -1 and 1. When you back-transform it to real space, the result is the Point Spread Function (PSF) that corresponds to that CTF. The PSF is the impulse response of the imaging system. It shows you, how a single atom would be imaged by the microscope with the given parameters. The PSF has very small values.

If you forward Fourier transform the PSF from real space back to Fourier space, you will get back the original CTF. However, now the values in Fourier space are numerically higher than before, because the Fourier transform is not normalised. As a result, the display of the Powerspectrum as log(1 + amp*amp) will show the CTF differently than before. The different appearance of the back-transformed and again forward-transformed CTF is due to the non-linear scaling for the Powerspectrum display.

For an explanation of CTF theory, see What is the Contrast Transfer Function (CTF) of an Electron Microscope? in the manual, and Beam tilt and coma for what a tilted beam does to it.

CTF FIT estimates the defocus and astigmatism of a real micrograph by fitting a theoretical contrast transfer function to the Thon rings in its power spectrum, in the style of GCtfFind / CTFFIND. You supply only the microscope voltage and C_s; the defocus, astigmatism and its angle are recovered automatically.

How it works. The power spectrum of the input buffer is sampled, a smooth radial background is subtracted (preserving the Thon rings), and the residual is whitened so weak high-frequency rings count as much as strong low-frequency ones. A theoretical |CTF|² is then matched to it by a normalised cross-correlation over the chosen resolution band — first a coarse 1D scan for the mean defocus, then a 2D search over defocus, astigmatism magnitude and angle, followed by iterative refinement.

Output. The fitted CTF is written to the Fourier side of the currently selected buffer as a composite diagnostic: the right half shows the synthesised (fitted) CTF, the top-left quarter the original Fourier transform of the input image, and the bottom-left quarter its elliptically (astigmatism-aware) radially averaged transform. The left half is scaled so it reads at a similar brightness to the fitted model on the right, and because the power spectrum is centrosymmetric the experimental and model rings join continuously across the vertical midline — so a good fit shows rings that line up across the whole image.

Reported values. After Execute the parameter window stays open and reports the fitted defocus (nm) and astigmatism (nm). The astigmatism is reported as the full peak-to-peak difference df₁ − df₂ (the CTFFIND / GCtfFind convention), together with its angle. The same values are also copied into the CTF SIM fields so you can re-simulate or tweak them — note that CTF SIM's astigmatism field is the amplitude (half of the peak-to-peak difference), matching its own definition.

Combine the Fourier transforms of two history slots with an arithmetic operation, optionally complex-conjugating one of the inputs. The result is written to a third slot. Useful for cross-correlation, deconvolution, complex amplitude division, and similar Fourier arithmetic.