Fourier Analyzer — User Manual

The Fourier transform decomposes an image f(x,y) into a sum of plane waves. For a 2D continuous image its forward transform is

F(u,v) = ∫∫ f(x,y) · e^{−i 2π(ux + vy)} dx dy.

Using Euler's identity e^−iθ = cos θ − i sin θ, the integral splits into a cosine part (the real component) and a sine part (the imaginary component): F(u,v) = R(u,v) − i·I(u,v). The complex value F therefore carries two real numbers at every frequency (u,v): how much of a cosine wave and how much of a sine wave at that spatial frequency is needed to reconstruct the image.

The same information can be re-expressed in polar form. The amplitude is the modulus |F| = √(R² + I²); it tells how strong the wave at frequency (u,v) is. The phase is the angle φ = atan2(I, R); it tells where that wave is positioned on the image. Amplitude alone discards location; phase alone keeps structure but loses contrast.

After centring (which the app does automatically), the zero frequency — the DC term, equal to the mean of the image — sits at the centre of displayed Fourier transform. Low spatial frequencies (large features, slow variation) lie close to that centre; high spatial frequencies (fine detail, sharp edges) lie near the corners. The highest frequency representable on a sampled grid is the Nyquist frequency: along each axis it equals 1 / (2 · pixel size), so a 1 Å pixel resolves down to 2 Å along the row / column directions and √2 · pixel size at the corners.

For any real-valued input image the transform obeys Friedel symmetry: F(−u,−v) = F*(u,v), where * denotes the complex conjugate. As a consequence the amplitude display is point-symmetric about the centre, and the phase display is anti-symmetric (opposite frequencies carry opposite-sign phases). This means the Fourier transform display is fully described by one half of its area; the other half is a redundant mirror image.

The two panels below show these conventions on a concrete example: a small patch of a 2D crystal, and its Fourier transform. The lattice of blobs in real space produces a reciprocal lattice of discrete spots in Fourier space — rotated by the same angle as the real-space lattice, and the wider the blob spacing, the closer the spots sit to the centre.

Left: real space. The image is indexed by x and y in pixels, with the origin (0,0) in the bottom-left corner. Right: Fourier space. The axes u and v are spatial frequencies in 1/pixel; the origin (0,0) sits in the centre and carries the DC term, and the edges of the display are the Nyquist frequency, 1 / (2 · pixel size) — the finest detail a sampled image can hold. Note that the transform is point-symmetric about its centre (Friedel symmetry), and that its lattice of spots is rotated by the same angle as the lattice in the image.

These two theorems are the reason why so much of image processing is done in Fourier space. Both describe an operation that is expensive and hard to picture in real space — sliding one image over another and integrating the overlap at every possible shift — and both state that in Fourier space the very same operation becomes a simple point-by-point multiplication of the two transforms. In one sentence:

convolution and correlation in real space  ↔  multiplication in Fourier space.

The Fourier Analyzer lets you see both sides of that equivalence: the real-space images on the left, their transforms on the right.

1. Convolution — "stamping" one image onto another

The everyday picture. Take the object f and replace every single point of it by a small blurred blob h, then add all those blobs up. That is a convolution. If the object is a molecule made of atoms and h is the blur that the microscope gives to one atom (its point spread function, PSF), then the recorded image is exactly the object convoluted with the PSF. Blurring, smoothing, band-pass filtering, applying a CTF — all of these are convolutions.

In real space the convolution of an image f(x,y) with a kernel h(x,y) is written with the symbol ⊛ and defined as

(f ⊛ h)(x,y) = ∫∫ f(x′,y′) · h(x − x′, y − y′) dx′ dy′.

Read it as an instruction: for each output position (x,y), shift the kernel h so that its origin lands there, multiply it with the image underneath, and sum everything up. Doing this for all output pixels of an N×N image costs of the order of N⁴ multiplications.

In Fourier space the same result is obtained by a single multiplication — this is the convolution theorem:

FT{ f ⊛ h } = F(u,v) · H(u,v),     i.e.   f ⊛ h = FT⁻¹{ F · H }.

Each frequency of the image is simply scaled (in amplitude) and shifted (in phase) by the value of H at that same frequency. H is called the transfer function of the operation — in an electron microscope it is the CTF. Because the fast Fourier transform costs only about N²·log N operations, the Fourier route is dramatically faster than the real-space sum, and it is also far easier to interpret: "which frequencies survive, and with what sign?" is answered by simply looking at H.

The theorem also works the other way round (the dual): a multiplication in real space is a convolution in Fourier space,

FT{ f · h } = F ⊛ H.

This is why masking matters: cutting an image out with a sharp-edged mask (a multiplication in real space) convolutes its Fourier transform with the transform of the mask, and the ripples of that transform smear every spot in Fourier space — the familiar "ringing" artefacts. A soft-edged (cosine-tapered) mask has a compact transform and therefore does much less damage.

Example. The object below consists of five copies of a seven-atom motif. Convoluting it with a ring-shaped point spread function replaces every atom by that ring. In Fourier space (bottom row) nothing is smeared at all — the transform of the object is simply multiplied, frequency by frequency, with the transform of the PSF, which suppresses some resolution shells and keeps others.

Top: real space — the object f, the point spread function h, and the recorded image f ⊛ h. Bottom: the corresponding Fourier amplitudes — the very same image is obtained by multiplying F with H pixel by pixel. The dark rings in |F·H| are the frequencies that the PSF does not transmit.

2. Correlation — "how well do these two match, at every shift?"

The everyday picture. Slide a reference image g (a template) across the image f, and at every possible shift ask: how similar are the two here? The answer, plotted as a function of the shift, is the cross-correlation function. Wherever the template matches, the product of the two adds up constructively and a bright peak appears; the position of the peak is exactly the shift at which the match occurs. This is how particles are picked, how images are aligned to a reference, and how drift between movie frames is measured.

In real space the cross-correlation (symbol ⊗) is

(f ⊗ g)(x,y) = ∫∫ f(x′ + x, y′ + y) · g*(x′,y′) dx′ dy′,

where * denotes the complex conjugate (for a real image, simply g itself). Again an N⁴-type cost if done literally.

In Fourier space, the correlation theorem states

FT{ f ⊗ g } = F(u,v) · G*(u,v),     i.e.   f ⊗ g = FT⁻¹{ F · G* }.

The only difference to the convolution theorem is the complex conjugate on G. Since conjugation means "flip the sign of all phases", the correlation keeps the amplitudes of both transforms but subtracts the phases of the reference from those of the image: |F|·|G| with phase (φ_F − φ_G). Where the two images agree, the phase difference is zero for all frequencies, all waves add up in step, and a sharp peak forms at that position — that is the whole magic of correlation, seen from Fourier space.

Example. Five identical particles are buried in noise so strong that they are hard to see by eye. Correlating the image with a single noise-free copy of the particle produces one clear peak per particle, at the particle's position.

Top: real space — a noisy image, the reference, and their cross-correlation with one peak per particle. Bottom: the same result is the inverse transform of F·G*, a plain point-by-point multiplication.

Every value in a Fourier transform is a complex number. That sounds forbidding, but the reason is simple and very physical: to describe a wave you need two numbers — how strong it is, and where its crests sit. A single real number cannot carry both. A complex number is exactly the book-keeping device that holds two numbers in one symbol, with multiplication rules that happen to be precisely the rules that waves obey.

1. Real and imaginary parts

The imaginary unit i is defined by the one property

i² = −1,

i.e. i = √−1, a number that no real number can be (any real number squared is positive). Nothing deeper is needed. A complex number is then simply a pair of ordinary real numbers, one plain and one carrying the label i:

z = a + i·b,

where a = Re(z) is the real part and b = Im(z) the imaginary part. The i is best read not as "imaginary" in the sense of unreal, but as a bookkeeping tag that keeps the second number from mixing with the first — just as the "y" in a map coordinate does not mix with the "x".

The complex conjugate of z, written z*, is obtained by flipping the sign of the imaginary part: z* = a − i·b. It will turn out to be the mathematical equivalent of "mirror" or "run the wave backwards", and it is the only difference between the convolution and the correlation theorem.

2. Cosine and sine: why a wave needs two numbers

Consider one of the plane waves that a Fourier transform decomposes an image into. It has a given wavelength and direction, but it may sit anywhere: its crests can be shifted along the wave direction. There are two "reference" waves for a given frequency: the cosine, which has a crest exactly at the origin, and the sine, which is the same wave shifted by a quarter of a wavelength. Any shifted version of that wave can be written as a weighted sum of these two:

a·cos(2πx) + b·sin(2πx) = A·cos(2πx − φ),   with  A = √(a² + b²)  and  φ = atan2(b, a).

So the two numbers (a, b) — how much cosine, how much sine — carry exactly the same information as the two numbers (A, φ) — how strong, and how shifted. This is the whole content of the complex number z = a + i·b that the Fourier transform stores at each frequency.

Adding 0.6 of a cosine and 0.8 of a sine gives one single wave of the same frequency, with amplitude √(0.6² + 0.8²) = 1, shifted by φ = 53°. The pair (cos-weight, sin-weight) and the pair (amplitude, phase) are two descriptions of the same wave.

3. The Argand diagram: amplitude and phase

Because a complex number is a pair of real numbers, it can be drawn as a point in a 2D plane, or better, as an arrow from the origin in a 2D plane. This plane is called the Argand diagram (or complex plane): the horizontal axis carries the real part (the cosine content), the vertical axis the imaginary part (the sine content).

The Argand diagram. The same complex number is described either by its Cartesian components (a, b) — the cos/sin view — or by the length and direction of the arrow (|z|, φ) — the amplitude/phase view.

The last line of the figure is Euler's identity, e^iφ = cos φ + i·sin φ, which lets us write any complex number in the compact polar form

z = |z| · e^iφ.

Read it as: "an arrow of length |z| pointing in the direction φ". It is exactly the same object as a + i·b, only written in polar instead of Cartesian coordinates. This is why the exponential e^{−i2π(ux+vy)} in the Fourier integral is nothing but a wave: it is an arrow of length 1, spinning as you walk across the image.

4. Adding and subtracting: use the Cartesian form

Addition is done component by component — real with real, imaginary with imaginary — exactly like adding two vectors head-to-tail:

(a₁ + i·b₁) + (a₂ + i·b₂) = (a₁ + a₂) + i·(b₁ + b₂),    and likewise with − for subtraction.

Note that amplitudes do not simply add: two waves of equal amplitude and opposite phase cancel completely (1 + (−1) = 0), while two in-phase waves reinforce. This is interference, and it is why adding is natural in the cos/sin (Cartesian) form and awkward in the amplitude/phase (polar) form. It is also the reason why the Fourier transform is linear: FT{f + g} = F + G, frequency by frequency.

5. Multiplying and dividing: use the polar form

Multiplication in Cartesian form follows from i² = −1:

(a₁ + i·b₁)·(a₂ + i·b₂) = (a₁a₂ − b₁b₂) + i·(a₁b₂ + b₁a₂),

which looks messy — but in polar form it becomes beautifully simple. Multiplying two complex numbers multiplies their amplitudes and adds their phases; dividing divides the amplitudes and subtracts the phases:

z₁·z₂ = |z₁||z₂| · e^{i(φ₁+φ₂)}        z₁/z₂ = (|z₁|/|z₂|) · e^{i(φ₁−φ₂)}.

Two consequences are worth memorising, because they explain most of what happens in Fourier space:

• Multiplying by a pure phase factor e^iφ is a rotation in the Argand diagram: it leaves the amplitude untouched and only moves the wave. Applying a phase ramp (a phase that grows linearly with frequency) to every Fourier component therefore shifts the whole image in real space without changing a single amplitude — the shift theorem. The Phase ramp tool of the Fourier Analyzer does exactly this.
• Multiplying by z* subtracts phases. That is why the cross-correlation is F·G* (see Convolution and Correlation Theorems above): where image and reference agree, all phase differences become zero, every wave adds up in step, and a peak forms. The extreme case is a number multiplied by its own conjugate, z·z* = |z|², in which the phase cancels completely — applied to a whole transform, F·F* = |F|² is the power spectrum, which is why it contains no phase information at all.

Division is used less often, but it is the formal basis of deconvolution: if an image is a convolution, its transform is a product (F·H), so dividing by H undoes it — which fails wherever H is near zero (dividing by almost nothing amplifies noise without limit), and this is precisely why CTF correction does not simply divide by the CTF.

6. How the Fourier Analyzer displays complex numbers

A complex-valued image cannot be shown as one grey-scale picture — there are two numbers per pixel. The app therefore offers four ways to look at the very same Fourier transform; cycle through them with the Mode button (see FT display modes below). Nothing is recomputed when you switch: only the way the complex numbers are rendered changes.

The Contrast Transfer Function (CTF) of an electron microscope defines, how contrast from the structure is transfered into the image. This can be studied in real space by investigating the point spread function (PSF) of the microscope, which is the ripple function that is convoluted with every atomic detail of the sample to give the image. But this can also, and much more efficiently, be studied in Fourier space, where the Fourier transform of the projection of the sample is simply multiplied by the CTF, to give the Fourier transform of the image. This multiplication is much easier to understand and to interpret than the convolution in real space. This is one of the many examples, where Fourier space is helpful.

Below, you will find a short introduction to the CTF, and how it can be calculated and visualized in the Fourier Analyzer.

The CTF equation (Scherzer wave-aberration function). The heart of the CTF is the wave-aberration function χ, also called the Scherzer phase-shift equation. It gives the phase error that the objective lens imparts to an electron wave scattered to spatial frequency q, relative to the unscattered beam:

    χ(q) = ½·π·C_s·λ³·q⁴ − π·Δf·λ·q²,

which is often written in the factored form χ(q) = π·λ·q²·(½·C_s·λ²·q² − Δf). The individual components are:

• q — the spatial frequency (in 1/Å), i.e. the radial distance from the centre of the Fourier transform. Higher q means finer detail. The scattering angle is β = λq.
• λ — the electron wavelength, fixed by the acceleration voltage (relativistically corrected). Higher voltage → shorter λ → weaker aberrations. For example 300 kV gives λ ≈ 0.0197 Å.
• Δf — the defocus (positive = underfocus). This is the term you tune most often. It scales with q², so it dominates at low and medium spatial frequency and sets the spacing of the inner Thon rings.
• C_s — the spherical aberration coefficient of the objective lens. Its term scales with q⁴, so it only becomes important at high spatial frequency, where it bends the phase over ever more rapidly and crowds the outer Thon rings together.
• ½ and π — geometric constants from the path-length / phase bookkeeping; the ½ belongs specifically to the spherical-aberration term.

The two terms have opposite sign: defocus and spherical aberration partly cancel. Choosing the defocus so that this cancellation flattens χ over the widest possible frequency band gives the classic Scherzer defocus, Δf_Sch ≈ 1.2·√(C_s·λ), at which the CTF stays close to its optimum (−1) over the broadest range of q before the first zero crossing.

The final transfer function combines the wave-aberration phase χ with the contrast as

    CTF(q) = A·sin(−χ) + B·cos(−χ)

Here, A = sqrt(1 − B²), where A is the fraction of phase contrast and B is the fraction of amplitude contrast.

As a result, the CTF is a sinusoidal function of the wave-aberration phase χ, which itself is a polynomial function of the spatial frequency q. The CTF oscillates between positive and negative values, with zero crossings at the Thon rings. The amplitude contrast term B shifts the Astigmatism, beam tilt and other effects add further terms as discussed below.

A typical CTF

The plot below is the radial CTF for a realistic set of parameters: 300 kV (λ ≈ 0.0197 Å), C_s = 2.7 mm, 0.8 µm underfocus, and 7% amplitude contrast. It is a 1D cut of the 2D CTF, plotted against spatial frequency q.

Radial CTF for 300 kV, C_s = 2.7 mm, 0.8 µm underfocus, 7% amplitude contrast. Blue: the CTF; red dashed: the ± decay envelope.

Reading the plot from left to right:

• At q = 0 the curve starts near +0.07, not at zero. This small offset is the amplitude-contrast term B: without it a pure phase-contrast CTF would be exactly 0 at the origin, transferring no low-resolution information at all.
• First passband. The CTF rises to its maximum (≈ 1) and stays high over a broad low-frequency band before its first zero crossing (here near q ≈ 0.08 1/Å, i.e. ≈ 12 Å). Features in this band are transferred with full, single-sign contrast — this is the useful low-resolution information.
• Zero crossings (Thon rings). Every point where the curve touches 0 is a spatial frequency where no information is transferred. In a 2D power spectrum these zeros trace out the dark Thon rings. Their spacing is set mainly by the defocus: more underfocus → more, closer-spaced rings.
• Contrast reversals. Between successive zeros the CTF alternates sign (positive lobe, then negative lobe, …). Each sign flip is a contrast reversal: black features become white and vice versa. Correcting for these sign flips is exactly what CTF correction does before averaging particles.
• Rings crowd together at high q. The oscillations get faster toward the right because the C_s·q⁴ term in χ grows rapidly — the same effect that limits the point-resolution of the microscope.
• The envelope (red dashed). The oscillation amplitude decays toward high frequency. This envelope comes from partial spatial and temporal coherence (finite gun-opening angle, energy spread, defocus spread) plus specimen/detector effects, and it sets the effective resolution limit: beyond the point where the envelope falls into the noise, the Thon rings are no longer visible and no information survives.

What makes up the envelope?

The red dashed envelope is not a single physical effect but the product of several damping terms, each suppressing high spatial frequencies for a different reason:

E(q) = E_tc(q)·E_sc(q)·MTF(q)·E_drift(q)·E_B(q).

• Temporal coherence (focal spread). The accelerating voltage, the objective-lens current and the electrons’ own energy spread ΔE all fluctuate, and through the chromatic aberration C_c these blur the focal length by a focal (defocus) spread Δ. Averaging the CTF over that Gaussian defocus spread damps it as
    E_tc(q) = exp[ −½(πλΔq²)² ],   with   Δ = C_c √((ΔE/E)² + (2ΔI/I)² + (ΔV/V)²).
  In the tool this is the Energy spread and Defocus spread parameters.
• Spatial coherence (beam convergence). A finite illumination semi-angle α (the gun-opening angle) lights the specimen from a range of directions; each direction gives a slightly shifted CTF, and their sum damps as
    E_sc(q) = exp[ −(πα)²(C_sλ²q³ − Δz q)² ].
  The damping is strongest where the CTF oscillates fastest (large dχ/dq), so it can even be reduced near the defocus that flattens χ. In the tool this is the Open angle gun parameter.
• Detector MTF. The camera does not transfer high frequencies perfectly; its modulation transfer function MTF(q) multiplies the signal and falls off toward the Nyquist frequency. (There is no simple closed form; it is not simulated here.)
• Specimen motion and drift. Beam-induced motion or stage drift over the exposure smears the image. For a linear drift of total length d the envelope is a sinc,
    E_drift(q) = |sinc(πdq)| = |sin(πdq) / (πdq)|. (Not simulated here.)
• Overall B-factor. Radiation damage and all residual, hard-to-model damping are usually lumped into a single empirical Gaussian,
    E_B(q) = exp(−Bq²/4),
  the B-factor that is later removed when sharpening a map.

The 2D version of the plot above, for the same typical parameters (300 kV, C_s = 2.7 mm, 0.8 µm underfocus, 7% amplitude contrast, no beam tilt). Right: the 2D CTF, whose radial cut is exactly the 1D curve above — each dark ring is a zero crossing. Left: its inverse Fourier transform, the real-space PSF that CTF SIM writes to panel 1 — the image of a single atom.

Note that the width of this real-space image is 128 nm. This shows that the ripples of the PSF reach out rather wide in the image. That is the reason why in single particle cryo-EM, the CTF has to be corrected first, before particles are cropped, so that the fine ripples that encode the high-resolution details of the image are preserved.

Both panels are point-symmetric: rotating either by 180° about its centre leaves it unchanged. This is the normal, aberration-symmetric case — with no beam tilt the wave-aberration χ is even in k, so the CTF is even, its power spectrum is centrosymmetric, and the PSF is a symmetric ring pattern that does not shift the apparent position of the atom. The concentric rings in the PSF are the delocalisation caused by defocus: information at each spatial frequency is spread out into a ring whose radius grows with defocus. Turning on beam tilt breaks this symmetry — see Beam tilt and coma below, where the PSF becomes a one-sided comet.

Astigmatism

Astigmatism. Real lenses are rarely perfectly round: the focal length differs slightly between two perpendicular directions, so the defocus becomes a function of azimuth, Δf(θ) = Δf + Δf_a·cos(2(θ − θ_a)). Here Δf_a is the astigmatism amplitude and θ_a its direction, and the otherwise circular Thon rings become ellipses. The example below adds an astigmatism amplitude of Δf_a = 300 nm at θ_a = 30° to the same CTF as above (so the defocus swings between 0.5 and 1.1 µm across the two axes):

Notice the two ellipses are perpendicular. Along the maximum-defocus direction the CTF oscillates fastest, so its rings are pulled in toward the centre — the ring ellipse is short along that axis. In real space the defocus delocalisation is largest along that same direction, so the PSF is stretched along it. The long axis of the PSF therefore lines up with the short axis of the ring ellipse.

Unlike beam-tilt coma, astigmatism keeps both images point-symmetric: the cos(2θ) term is even under k → −k (it has two-fold symmetry), so the CTF stays even, the power spectrum stays centrosymmetric, and the atom is blurred but not shifted. Astigmatism is exactly what CTF FIT measures alongside the mean defocus: the ellipticity of the rings gives the astigmatism magnitude and the tilt of the ellipse gives its angle. (Note the CTF SIM Astigmatism field is the amplitude Δf_a — half of the full peak-to-peak difference between the two axes.)

Beam tilt and coma.

Beam tilt. When the illuminating beam is not exactly parallel to the optical axis but tilted by a small angle τ, the scattering angle is shifted (the spatial frequency is effectively displaced, k → k + τ/λ). To first order in the tilt this adds a coma-like term to the wave-aberration function:

  Δχ = 2π·C_s·λ²·q³·τ·cos(θ − τ_dir),

where q is the spatial frequency, θ its azimuth in Fourier space, λ the electron wavelength, C_s the spherical aberration, τ the tilt magnitude (in radians; entered in mrad), and τ_dir the tilt direction (CCW from +x, the same convention as the astigmatism direction). This extra phase scales with C_s·q³ and is odd in θ: the cos(θ − τ_dir) factor changes sign across the origin, so Δχ(−k) = −Δχ(k). Setting the beam tilt to 0 mrad removes it and recovers the usual centrosymmetric case.

Two symmetries — keep them apart. Everything that follows is forced by two exact Fourier facts. They are easy to conflate, and conflating them is what makes beam tilt confusing:

• Hermitian, F(−k) = F*(k)  ⇔  the real-space image is real.
• Even, F(−k) = F(k)  ⇔  the real-space image is point-symmetric.

Coma is a real image that is one-sided. So it demands a transfer function that is Hermitian but not even. Note what this rules out: a real-valued function in Fourier space can never deliver a comet, because if C is real then h(−r) = h*(r), and the real part of the back-transform is then exactly point-symmetric — all of the asymmetry has moved into the imaginary part. “Asymmetric rings” and “asymmetric PSF” are therefore two different wishes, and no single real-valued function grants both at once.

The three CTF SIM buttons. Because of that trade-off, CTF SIM does not have one “Compute”. It offers three genuinely different models of the same microscope — different physics, not different display options. Each writes a different array into Fourier space (panel 2) and turns it into a real-space image (panel 1) in a different way. All figures below use 300 kV, C_s = 2.7 mm, 350 nm defocus and 8 mrad of beam tilt along the horizontal axis, straight from the program.

1. Pupil Function — P(q) = E(q)·e^−iχ(q), the wave-optics view

The pupil is the aberrated lens acting on the electron wave: its phase is the full wave aberration, its modulus only the partial-coherence envelope. There are no Thon rings, because rings belong to the intensity CTF below, not to the pupil. Under tilt the pupil is genuinely non-Hermitian, so h = FT⁻¹{P} is a complex wave and panel 1 shows the intensity |h|² that a detector would record — the classic comet. This is the aberration as an optician sees it: the shape of the focused spot. (Amplitude contrast enters only as a constant phase here, so it factors out of |h|² and does not change the PSF.)

2. Complex CTF — T(q) = E·(A·sin(−χ_even) + B·cos(−χ_even))·e^−iχ_odd

This is the linear image-intensity transfer function for a weak-phase object, and the physically correct model of a real micrograph. The tilted aberration splits into an even part χ_even (defocus, C_s, astigmatism, and the defocus/astigmatism the tilt itself induces), which makes the oscillating Thon rings, and an odd part χ_odd (coma), which enters purely as a phase and never touches the modulus. T is Hermitian — not by approximation but by necessity, since image intensity is real — so the rings stay symmetric and the PSF comes out real. And because T is Hermitian without being even, that real PSF is still one-sided. This is the one mode where both panels are simultaneously what a microscope would give you.

3. Real-valued CTF — C(q) = E·(A·sin(−χ_tilt) + B·cos(−χ_tilt))

Here the transfer function is evaluated at the full tilted aberration and kept purely real — no even/odd split, no phase factor. Since χ_tilt(k) ≠ χ_tilt(−k), C is real but not even, hence not Hermitian, and the Thon rings themselves go one-sided. This is the picture people often have in mind when they say “beam tilt makes the CTF asymmetric”, and it is a useful didactic model — but it is not a microscope, and it costs you the comet exactly as the symmetry rules predict. Use it to see the trade-off, not to predict data.

Mode	Fourier space (panel 2)	Real space (panel 1)	Use it for
Pupil Function	Non-Hermitian; modulus is the bare envelope, no rings	|h|² — the comet	Seeing coma as the shape of the focused spot; wave optics
Complex CTF	Hermitian; symmetric Thon rings, tilt hidden in the phase	Real and one-sided — the comet	What a real micrograph and its power spectrum actually look like
Real-valued CTF	Real but not even; lopsided Thon rings	Point-symmetric — no coma	Demonstrating why asymmetric rings and an asymmetric PSF exclude each other

The practical punchline. The power spectrum of any real micrograph is centrosymmetric — Friedel’s law — so you can never detect beam tilt by hunting for lopsided Thon rings in your data. Only the didactic third mode shows those, and only because it is not the transform of any real image. That is also why coma is hard to tell from astigmatism by eye. Coma’s asymmetry lives in the phases and in the coherent intensity PSF, so it has to be found by other means: coma-free alignment on the microscope, Zemlin tableaux of deliberately tilted spectra, phase-based methods, and beam-tilt refinement during high-resolution single-particle reconstruction. Residual tilt shifts apparent positions and corrupts phases — which is what limits resolution — even while the Thon rings look perfectly innocent. Set the tilt back to 0 mrad in any mode and the comet disappears.

Figure note: the real-space panels are zoomed to the central ~140 px and intensity-stretched (γ = 0.45) so the faint tail is visible; the Fourier panels are shown as the program’s power-spectrum display, unmodified. Clicking the FT⁻¹ arrow reproduces panel 1 in every mode — each buffer remembers whether its Fourier data inverts to an intensity or to a real part.

The window is split into four panels: the real-space image, its Fourier transform, a history of the images you have loaded or produced, and the matching power spectra. Edits in either panel update the other in real time, so you can directly observe how a change in real space affects Fourier space and vice versa.

1. Click Load in the top toolbar and select an image (PNG, JPG, MRC, …). It appears in panel 1 (top-left); the FFT appears in panel 2 (top-right).
2. Cycle the FT display with the Mode button to see cos/sin, amplitude/phase, complex (HSV), or power-spectrum representations.
3. Pan with click-and-drag, zoom with the scroll wheel or pinch gesture, in either panel.
4. Pick a tool from the icon column on the right of panel 2 (or the left of panel 1). A parameter window appears at the bottom of the panel; adjust the values and click Apply.
5. Mistakes are cheap: ⌘Z / Ctrl-Z undoes the last operation. Each result is also stored in a history slot (panel 3) so you can return to it later.

The window is divided into four resizable panels:

The header bar at the top contains two clickable fields: the Fourier Analyzer title (opens an About dialog with contact info) and Manual (this page).

The mask center for display toggle that used to live in this toolbar now sits next to the panel-2 histogram, stacked above the freeze display contrast toggle — see the Mouse & trackpad section below.

The panel-2 display can show the Fourier transform in four ways. Cycle through them with the Mode button in the top toolbar.

The Fourier transform is always centred: the DC term (zero frequency) sits at the centre of panel 2, and high frequencies are at the corners.

Panel 3 holds sixteen image slots labelled a through p. The currently active slot is highlighted. Each slot keeps:

• The real-space image and its raw pixel values.
• The file path or generated label (e.g. “phase ramp: N=1024 dir=30° step=10°”).
• The pixel size in Ångström.
• A power-spectrum thumbnail (shown in panel 4).

In the desktop versions of the program, the History is persisted between sessions — you can close the app and pick up where you left off. Use the New button to start a fresh slot, or click an empty thumbnail to begin one.