CT Scans — Linear Algebra Saves Lives
CT Scans — Linear Algebra Saves Lives
In 1971, Godfrey Hounsfield built the first CT (Computed Tomography) scanner and won the Nobel Prize. The key breakthrough wasn't the X-ray — it was the mathematics of reconstructing a 3D image from flat projections.
The Problem
Imagine slicing a loaf of bread. A regular X-ray gives you a shadow — everything overlaps. A CT scan gives you individual slices. But how do you get a slice without cutting?
The answer: take X-ray measurements from many angles and use math to compute what each tiny cube (voxel) of tissue must look like.
Systems of Linear Equations
Each X-ray beam passes through a row of tissue pixels. The total absorption is the sum of individual absorptions:
$$a_1 x_1 + a_2 x_2 + a_3 x_3 + \cdots = b$$
where each $x_i$ is the unknown density of pixel $i$, and $b$ is the measured absorption.
Shoot beams from 180 angles, each with hundreds of measurement points, and you get a huge system:
$$\begin{cases} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = b_2 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = b_m \end{cases}$$In matrix form: $\mathbf{A}\mathbf{x} = \mathbf{b}$
This is linear algebra — solving systems of linear equations. A modern CT scan solves systems with millions of equations in seconds.
A tiny 2×2 image has unknown densities $x_1, x_2, x_3, x_4$. Three beams measure:
Row 1 (horizontal): $x_1 + x_2 = 7$
Row 2 (horizontal): $x_3 + x_4 = 11$
Column 1 (vertical): $x_1 + x_3 = 8$
Column 2 (vertical): $x_2 + x_4 = 10$
From equation 1: $x_2 = 7 - x_1$
From equation 3: $x_3 = 8 - x_1$
From equation 4: $x_4 = 10 - x_2 = 10 - (7 - x_1) = 3 + x_1$
Substitute into equation 2: $(8 - x_1) + (3 + x_1) = 11 \;\Rightarrow\; 11 = 11$ ✓
We need one more beam (diagonal) to get a unique solution. This is why CT scanners use hundreds of angles!
The Radon Transform
The mathematical foundation of CT is the Radon transform (1917) — it converts a 2D image into a set of line integrals (projections). The inverse Radon transform reconstructs the image.
$$\mathcal{R}f(s, \theta) = \int_{-\infty}^{\infty} f(s\cos\theta - t\sin\theta,\; s\sin\theta + t\cos\theta)\, dt$$
This uses trigonometry ($\sin$, $\cos$) and integration (calculus). Every CT scan you've ever had was reconstructed using trig functions.
CT scans are systems of linear equations solved at massive scale. The linear algebra and trigonometry you study aren't just coursework — they're the mathematics that lets doctors see tumours without surgery.