From tilt series to tomogram

How does a stack of 2D projections become a 3D volume? Backprojection, why you need many angles, and where the missing wedge comes from

Intuition

Each exposure captures only a single 2D projection of the specimen — all depth information squashed into one image. Tilt the specimen and take another, and you get a different “squash” from a different direction. Tomographic reconstruction runs this backwards: it takes the stack of projections from many angles and re-spreads them into the original 3D volume. The simplest recipe is backprojection: smear each projection back along the direction it was taken, sum over all directions, and the object emerges.

An analogy: projections are shadows. One shadow of a hand on a wall tells you almost nothing about the hand’s shape. Light it from above, from the side, and from a slant; drag each shadow back along its own light ray into a sheet of brightness, and only where those sheets overlap is the solid hand. More angles make the overlap trace a sharper outline. Backprojection just does this “drag back along the ray, then sum” once per projection.

The object is on the left, its backprojection on the right; the sliders set the number of projections and the maximum tilt angle:

Object (ground truth)

Backprojection

Projections: 40Max tilt: ±90°

Reconstructed from 40 projections over a ±90° range. More projections → sharper, fewer star-streaks; when tilt is capped at ±90° (< 90°) the un-sampled angles stretch and blur the recon along one direction — the missing wedge, seen in real space.

More projections give a sharper reconstruction as the star-shaped streaks (single-angle smear) fade. Below 90° of tilt, structure along the un-sampled direction stretches and blurs — the missing wedge seen in real space.

Pushing the demo to extremes separates the two kinds of artifact by their cause. Keep just one or two projections: each one smears into a bright band along its own direction, and the bands cross into a star — the too-few-angles streak, which more projections erase. Now drag the maximum tilt from 90° down to 60° and then 45°: the star is gone, but the object starts to elongate along one fixed direction with soft edges — the limited-angular-range artifact, which no number of projections can fix, because what is missing is not sampling density but a whole band of directions that never entered the data. The first is a sampling problem; the second is the missing wedge. They are different in nature, and are treated separately below.

The math

Depth

A projection at angle $\theta$ is the line integral of the object along that direction (the Radon transform):

p_\theta(t) = \iint f(x,y)\,\delta\!\big(x\cos\theta + y\sin\theta - t\big)\,dx\,dy

Here $f(x,y)$ is the 2D object (a tomographic slice) to recover, $\theta$ is the projection direction, and $t$ is the position coordinate along the projection line. The Dirac $\delta$ pins the integral onto the line $x\cos\theta + y\sin\theta = t$ , so $p_\theta(t)$ is the object summed along that line — one “squashed” profile. Fixing $\theta$ and sweeping $t$ gives a full 1D projection; this is exactly what one row of a tilt image samples.

Backprojection smears each projection back along its direction and integrates over all angles:

b(x,y) = \int_0^\pi p_\theta\big(x\cos\theta + y\sin\theta\big)\,d\theta

How to read it: to recover the value at point $(x,y)$ , go into each angle’s projection $\theta$ , pick out the value $p_\theta(x\cos\theta + y\sin\theta)$ of “the projection line passing through this point,” and sum those contributions over all angles. Intuitively, each projection paints its value back uniformly along the whole line, and the point where the object actually sits grows bright because every angle paints over it.

But the result is blurred (equivalent to the truth convolved with a $1/r$ kernel): each projection spreads its energy along the entire line, so points far from the object get painted with background too; after summing, the object is bright but a long-range haze remains, equivalent to convolving the truth with a kernel $1/r$ that decays slowly with distance $r$ . Low frequencies are over-amplified, so the whole image is soft. Real reconstruction uses filtered backprojection (FBP): multiply each projection by a ramp filter $|k|$ in the frequency domain first, then backproject, to restore sharpness. The ramp $|k|$ exactly cancels the $1/|k|$ falloff that the $1/r$ blur has in the frequency domain — it suppresses the over-summed low frequencies and lifts the high frequencies that draw edges, the product is flat, and the blur is removed. The demo above is unfiltered backprojection, which makes the accumulation and the directional streaking directly visible.

In Cryo-ET the ±60° tilt limit makes the integral over $\theta$ incomplete; the missing angular range is the wedge-shaped void in Fourier space — no filter can recover it, which is why methods like CryoGEN / CryoWGEN infer it from a learned prior. The $|k|$ filter fixes “summed-up blur,” and it assumes every direction’s frequencies were sampled; when a whole band of directions has no data at all, those frequencies are zero no matter what you multiply by. That is missing sampling, not blur, so the two are independent problems.

Why you need many angles falls straight out of the central-slice theorem. The central-slice theorem says: the 1D Fourier transform of a projection fills in exactly the line through the origin of the object’s 2D Fourier spectrum that is perpendicular to the projection direction. Each extra angle adds one more spoke through the origin in the Fourier plane. To fill the whole 2D spectrum, those spokes must spread evenly across 180°. This also explains why the artifact is a “star”: with only a few spokes, only a few radial lines of the spectrum are filled and the rest is empty, and the inverse transform back to real space becomes a radial streak along those directions.

In a real pipeline

Real Cryo-ET reconstruction is far more than backprojection. The full chain runs roughly as follows, each step setting up the next:

Tilt-series alignment: mechanical backlash in the stage and beam-induced drift give every projection an unknown rotation, translation, and tilt-angle error; gold fiducial beads dusted on before freezing register all projections into one geometry (details in the alignment section). This step is the precondition for backprojection — without it, each angle’s spokes do not share a common origin, and the sum is blurred.
CTF correction: the microscope’s transfer function flips the sign and depresses the contrast of the projection by spatial frequency, so the defocus must be estimated and inverted (see the CTF section).
Dose weighting: later projections carry higher accumulated electron dose and their high-frequency information is already erased by radiation damage, so they are down-weighted in the frequency domain by dose, to keep damaged high frequencies from polluting the reconstruction.
Weighted backprojection (WBP) or iterative reconstruction: finally, the aligned, corrected, weighted projections are backprojected into a volume. WBP is the filtered backprojection above; iterative schemes like SIRT instead repeat “forward-project, compare, correct,” which is steadier against noise and incomplete data at the cost of speed.

The WBP volume is exactly the missing-wedge-corrupted observation $y$ that the CryoGEN / CryoWGEN models take as input — the pipeline ends by handing off prepared data, and actually filling in the never-sampled information inside the missing wedge is the job of those later methods.

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