Computed Tomography

Lie still in a CT scanner and it will build a crisp cross-section through your body — and yet the machine never sees inside you directly. All it can physically measure is X-rays going in one side and coming out the other, dimmer than they went in. It fires those beams through you from hundreds of angles, records only how much each one was absorbed, and then solves an inverse problem to reconstruct a 2-D slice from those 1-D shadows. It is arguably the most famous and consequential inverse problem ever posed — and it was worth a Nobel Prize.

A single ray records the total absorption along its line — a line integral of the unknown tissue density m(x, y):

p_\theta(s) = \int_{\text{ray}} m\,(x, y)\,d\ell.

Collecting these for all offsets s and all angles \theta is the Radon transform — the forward operator. Tomography is its inverse: reconstruct the 2-D density from its projections in every direction.

Worked idea 1 — one angle is hopelessly ambiguous

Imagine looking at a filled shape through frosted glass so you see only its shadow from one direction. A tall thin object and a short fat one can cast the very same shadow; countless different images produce an identical single projection. One angle simply cannot pin down where the density sits along each ray — the information is collapsed. This is the non-uniqueness that makes a lone projection useless on its own.

Worked idea 2 — many angles together pin it down

Now add a second angle, then a third, then a hundred. Each new direction constrains the image in a new way, and where all those constraints intersect, only one density can satisfy them at once. Stack every 1-D projection, one row per angle, and you get the sinogram — the full dataset. Reconstructing the image from it is a big linear system Ax = b, exactly the setting of the linear inverse problem: x is the unknown image (one number per pixel), b the measured projections, and A the Radon (line-integral) operator.

Worked idea 3 — filtered backprojection

The classic inversion is filtered backprojection. First you backproject: smear each projection back across the image along its rays, letting every pixel accumulate the shadows that passed through it. But plain smearing over-counts the low frequencies and comes out blurry, so first you apply a sharpening ramp filter in the frequency domain to counteract that blur. That ramp is — once again — an inverse filter, and boosting high frequencies is exactly where noise creeps in. Tomography is therefore ill-posed in the ways this whole course predicts, and the remedy is the same: regularization and priors. Modern iterative CT solves a regularized least-squares problem \min_m \|Rm - p\|^2 + \alpha^2\|Lm\|^2, often with a smoothness or edge-preserving penalty, reconstructing well from fewer rays and so cutting the radiation dose.

One angle at a time

Inside the circular field are two objects (the unknown density). The parallel lines are the X-rays for one projection angle — each measures the total density along its path. Rotate the angle \theta: every direction contributes a new shadow, and only by combining many angles can the two objects be located and separated. Too few angles and the reconstruction stays ambiguous.

The forward operator is the Radon transform: each datum is a line integral of the density.
Classic inversion is filtered backprojection (a ramp filter + smearing back); the ramp is an inverse filter that amplifies noise.
Limited angles → non-uniqueness; noise → instability; the cure is regularization and priors (iterative CT).

Reconstruction quality depends critically on collecting enough projection angles, well distributed all the way around the object. In some real situations you simply cannot scan the full circle — a mammography rig, a scan of a wide flat object, or an industrial part that can only be reached from one side, all give a limited angular range. With too few angles or a restricted range, whole families of images fit the data, the problem becomes severely ill-posed, and reconstructions sprout tell-tale streak artefacts and blur.

Do not be fooled into thinking more computer power alone fixes this — the missing angles mean the data genuinely does not constrain the image, so you must supply the shortfall with a prior (what you expect the image to look like). "Limited-angle" and "sparse-view" CT are active research areas precisely because getting this right lets scanners use fewer X-rays — and so a lower radiation dose for the patient.

The name is a lovely little piece of geometry. Follow a single bright point of density as you sweep through all the angles. Its offset s in the projection at angle \theta is s = x_0\cos\theta + y_0\sin\theta — a sinusoid. So one point in the body traces out a perfect sine curve across the stack of projections, and a whole object paints a superposition of them. Plot the projections with angle running down the page and you see those tell-tale waves: hence sinogram. Reading the image back out of that tangle of sine curves is exactly the inverse Radon transform.

Computed tomography earned Godfrey Hounsfield and Allan Cormack the 1979 Nobel Prize in Physiology or Medicine — a striking honour, since neither was a physician. Cormack, a physicist, worked out much of the underlying mathematics partly in his spare time; Hounsfield, an engineer at EMI (yes, the record label that signed the Beatles), reputedly hit on the core idea while out on a countryside ramble, wondering whether you could reconstruct the contents of a box from readings taken all around it.

The same tomographic inverse-problem mathematics now reconstructs far more than medical slices: it peers inside airport baggage scanners, builds 3-D maps of the Earth's deep interior from earthquake waves (seismic tomography), and even images individual atoms in electron microscopy. Measure shadows from every side, invert the Radon transform, and the hidden interior appears.