understand that computed tomography (CT) scanning produces a 3D image of an internal structure by first combining multiple X-ray images taken in the same section from different angles to obtain a 2D image of the section, then repeating this process a

Production of X‑rays

X‑rays are produced when high‑energy electrons are decelerated in the vicinity of a metal target. The two principal mechanisms are:

• Bremsstrahlung (braking radiation) – a continuous spectrum produced when electrons are deflected by the electric field of nuclei.
• Characteristic radiation – discrete lines emitted when an inner‑shell electron is ejected and an outer‑shell electron fills the vacancy.

The energy of an X‑ray photon is given by

\$E = h\nu\$

where \$h\$ is Planck’s constant and \$\nu\$ is the frequency of the photon.

X‑ray tube components

Use of X‑rays in Imaging

When X‑rays pass through matter, their intensity is reduced according to the exponential attenuation law:

\$I = I_0 e^{-\mu x}\$

\$I_0\$ is the incident intensity, \$\mu\$ the linear attenuation coefficient, and \$x\$ the thickness of the material.

Differences in \$\mu\$ for various tissues produce contrast on a radiograph. Conventional radiography records a single projection, whereas computed tomography (CT) reconstructs cross‑sectional images.

Computed Tomography (CT) Scanning

Conceptual Overview

CT creates a three‑dimensional (3‑D) representation of an object by:

1. Acquiring many X‑ray projections of a thin slice from different angles around the object.
2. Reconstructing a two‑dimensional (2‑D) cross‑sectional image of that slice using mathematical algorithms.
3. Repeating the acquisition and reconstruction for adjacent slices along the longitudinal axis.
4. Stacking the series of 2‑D images to form a volumetric data set, which can be displayed as 3‑D renderings.

Acquisition of Projections

During a single rotation (typically 0.5–1 s), the X‑ray source and detector rotate synchronously around the patient. For each angular position \$\theta\$, the measured intensity \$I(\theta, s)\$ is recorded, where \$s\$ denotes the detector coordinate perpendicular to the beam.

Image Reconstruction

The most common algorithm is filtered back‑projection. The steps are:

1. Compute the Radon transform \$R(\theta, s)\$ of the object from the measured intensities.
2. Apply a frequency‑domain filter (e.g., Ram-Lak) to each projection to correct for the blurring inherent in simple back‑projection.
3. Back‑project the filtered data over all angles to obtain the attenuation map \$\mu(x,y)\$ for the slice.

Mathematically, the reconstructed value at point \$(x,y)\$ is

\$\mu(x,y) = \int{0}^{\pi} \left[ \int{-\infty}^{\infty} R(\theta,s) \, | \omega | \, e^{i\omega (s - x\cos\theta - y\sin\theta)} \, d\omega \right] d\theta\$

where \$|\omega|\$ represents the filter in the frequency domain.

CT Numbers (Hounsfield Units)

Reconstructed attenuation values are expressed as Hounsfield Units (HU):

\$\text{CT} = 1000 \times \frac{\mu - \mu{\text{water}}}{\mu{\text{water}}}\$

Typical values: air ≈ –1000 HU, water = 0 HU, bone ≈ +1000 HU.

Formation of the 3‑D Image

After reconstruction, each slice is stored as a matrix of pixels (voxels when combined with depth). By stacking the matrices, a volumetric data set is created. Software can then:

• Display axial, coronal, and sagittal views.
• Generate multiplanar reconstructions (MPR).
• Render volume‑rendered images for anatomical visualization.

Key Points to Remember

• CT combines many 2‑D X‑ray projections taken around a single slice to reconstruct that slice.
• Repeating the slice acquisition along the body axis yields a stack of 2‑D images.
• The stack is interpreted as a 3‑D representation of the internal structure.
• Mathematical reconstruction (filtered back‑projection or iterative methods) converts raw projection data into quantitative attenuation maps expressed in Hounsfield Units.