A CT scanner cannot see inside you directly. It fires X-rays straight through the body and measures how much each is absorbed. A single ray records the total absorption along its line — a line integral of the unknown density m(x, y):

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. It is the inverse problem that won a Nobel Prize and reshaped medicine.

From projections back to the image

Each angle gives a 1-D shadow; stacking all angles gives the sinogram, the full data. The classic inversion is filtered backprojection: smear each projection back across the image along its rays, but first apply a sharpening ramp filter in frequency — without it, plain backprojection blurs everything. That ramp is, once again, an inverse filter, and amplifying high frequencies is exactly where noise creeps in.

Tomography is ill-posed in the ways this whole course predicts. With few angles or a limited angular range, whole families of images fit the data (non-uniqueness), and noise in the ramp-filtered data amplifies (instability). 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, cutting the radiation dose by reconstructing well from fewer rays.

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.