An experiment in invisible things · No. 12

The X-Ray
Atelier.

Destroy the picture. Keep only its shadows. Then rebuild an object you can no longer see—using nothing but the mathematics of light.

A hands-on inverse problem
Real line integrals + filtered backprojection
Entirely offline · No uploads
One object. Many shadows.
Choose a specimen, start a scan, and watch the invisible reappear.
SCANNER READY TO BEGIN
θ = 0.0°

01 / The original

A hidden object

ƒ
?SPECIMEN CONCEALED
The specimen is a two-dimensional density map. Draw on it to alter the object.

02 / The evidence

Collected shadows

ℛƒ
angle →detector profile ↑
Each narrow column records the X-ray measurements at one angle.

03 / The inference

Reconstructed image

ƒ̂
Every new angle contributes information. Edges appear from filtered shadows.
“You cannot see the object, but its shadows remember its shape.”
00 / 080 PROJECTIONS

Run the experiment

Choose a specimen

Control the machine

80
Ram–Lak

Try switching the filter off. The scanner still collects accurate data, but simply smearing the shadows backward makes a blurred picture. Turn it on to recover sharp boundaries.

Inside the machine

0Views received
—Image similarity
0°Current angle

This is a simplified CT scanner. It does not look inside the object. Instead, at every angle θ it measures how much material lies along each parallel ray.

Rƒ(θ, s) = ∫ ƒ(s nθ + t nθ⊥) dt

These measurements are the Radon transform. The reconstruction reverses them using filtered backprojection. The scanner chooses successive angles with a golden-angle rule, so even its early measurements are spread around the object.

Try this: Set the scanner to 8 angles and compare the image with 80 angles. The object is the same; only the available evidence changes.

The mathematical surprise

Shadows can determine the thing that casts them.

A single shadow erases an entire dimension of information. Yet a sufficiently rich family of projections, taken from different angles, can uniquely determine a suitable two-dimensional density. Tomography turns many incomplete observations into a complete picture.

A small challenge

Can you recognize a shape without seeing it?

Choose Mystery object, run the scan with eight viewing angles, and guess what was hidden. Then raise the count and scan again. Reveal the original only when you've made your guess.