Computed tomography

As explained in Geometry for Programmers

An imaging technique that reconstructs cross-sectional or volumetric representations of internal structures from measurements taken around an object, typically x-ray measurements.

Computed tomography (CT) reconstructs internal structures from many x-ray measurements taken around an object. A scanner compares emitted radiation with the lower energy detected after passage through the body, then combines measurements from different angles into a 3D image whose voxels store radiodensity. section 12.1

The reconstruction process connects physical measurements with a small mathematical model:

• A single x-ray view records total radiodensity along each ray, but cannot uniquely locate individual bones, tissues, or airways. section 12.1
• Many views provide enough constraints to compute the unknown voxel values by solving a series of linear systems. section 12.1
• A four-pixel 2D slice makes the geometry concrete: two two-pixel x-ray scans from different angles help determine values a11, a12, a21, and a22. section 12.1

From a scanned part to a replacement

CT can capture a train, ship, or airplane part whose original blueprints are unavailable. The scan can support a boundary representation, which can then be milled, converted into a triangle mesh for 3D printing, or retained as smooth parametric surfaces for manufacturing. section 12.5

Figure 12.2 A CT scan scheme (FDA, public domain, via Wikimedia Commons), and our simplified example. We take two 1D x-ray pictures at a different angle (first b₁₁, b₁₂ and then b₂₁, b₂₂) to compute a single 2D scan ([[a₁₁, a₁₂], [a₂₁, a₂₂]]).

Use the scan scheme to relate angled 1D x-ray measurements to the reconstructed 2D pixel grid. section 12.1

How Geometry for Programmers covers it

How deep it goes

Explains detector measurements, radiodensity, a four-pixel reconstruction, and linear systems, stopping short of full reconstruction algorithms.

Examples

Uses a four-pixel slice, Hounsfield-unit tissue values, and replacement-part scanning for milling or 3D printing.

What it assumes

Assumes familiarity with pixels and basic geometry while introducing voxels, radiodensity, ray paths, and linear-system reconstruction.

What you can do after

Evaluate how angled x-ray measurements constrain voxel values and apply CT data to reproduce manufactured parts.

How it connects

Connects x-ray imaging, voxels, radiodensity, linear systems, boundary representation, parametric surfaces, triangle meshes, and SDFs.

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Also explained in

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  Deep Learning with PyTorch, Second EditionComputed tomography imaging data consisting of cross-sectional body slices.

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Related concepts

• voxelpart of · A voxel is a discrete volumetric element, typically cubical, used to represent spatial data or occupancy in a three-dimensional grid.
• 3D imageproduces · A 3D image is a three-dimensional sampled representation of geometry or spatial data arranged as image-like values.
• radiodensitymeasures · Radiodensity is a measure of a material’s attenuation or absorption of X-rays, particularly as represented in medical imaging.
• Hounsfield unituses · A Hounsfield unit is a standardized measure of x-ray radiodensity in computed tomography relative to water.
• lung tumor detectionused by · The task of automatically identifying suspected malignant lung tumors from chest CT scans.

This explainer was generated by Manning's knowledge graph from the text of Geometry for Programmers. The section links point to the parts of the book it draws on.