Image Acquisition Methods Lecture 09: Computerised X-Ray Tomography II

Introduction

  • Presenter: Pascal Peter

  • Copyright Year: ©2025 Pascal Peter

  • Context: Recent discussions on Image Acquisition Methods (IAM).

Overview of Topics

  • Transmission Tomography: Methodology to combine X-ray measurements for creating 3-D images. Significant concepts:

    • Radon Transform: Mathematical tool used for reconstructing images from their projections.

    • Fourier Slice Theorem: An important principle connecting Fourier transforms and image reconstruction.

  • Wavelength and Frequency (in nanometers and hertz): Provides a scale for electromagnetic radiation, including gamma rays and X-rays.

Lecture Themes

  1. Visualisation of CT data

  2. Artefacts and Noise in CT Imaging

  3. Single Photon Emission Computed Tomography (SPECT)

  4. Positron Emission Tomography (PET)

Visualisation of CT Data

Transforming Slice Data to Image Representations

  • Scenario: Post-measurement of CT data and reconstruction leads to an approximate density function ( u ) within the volume, scaled in Hounsfield units (HF).

  • Challenge: How to effectively display this density function?

  • Solutions: Two approaches:

    • Display arbitrary 2-D sections, including tilted ones.

    • Compute 3-D representations to improve visual interpretation.

2-D Visualisation

  • Concept: Display data as a stack of 2-D sections representing entire data arrays.

  • Enhancement: To improve contrast for noteworthy structures:

    • Windowing: Method to display only a selected range of densities.

    • The window is defined by:

    • Window Level: Central point in HF.

    • Window Width: Length of the interval in HF.

3-D Visualisation

  • Concept: Displays complete volume parts of reconstructed data.

  • Advantages: Captures spatial context effectively.

  • Disadvantages: Occlusions/inclusions obstruct a full view of the data.

    • Surface Rendering: Voxels above a threshold appear opaque; others remain transparent. Renders level surfaces based on the reconstructed density function.

    • Volume Rendering: Uses semi-transparency to reveal multiple surface layers simultaneously; necessitates light transport simulation in semi-transparent layers.

    • Denoising: Often a necessary preprocessing step.

Combined Visualisation

  • Workstation Software Layout for CT Inspection: Integration of volume rendering overviews, axial, coronal, and sagittal slices.

Artefacts and Noise in CT Imaging

Discretisation Artefacts

  • Recognition that tomographic reconstruction primarily deals with continuous data; however, only finite data values are actually measured.

  • Results: Discretization leads to:

    • Aliasing Artefacts: Visual distortions when an elliptical structure is reconstructed.

    • Ghost Artefacts: Missing density functions due to all projections being measured as zero; these can be misinterpreted as tumors. Remedy: Regularization (smoothing method).

Aliasing Issues

  • Concept: By sampling at a finite resolution, a limited number of frequencies are accurately represented (sampling theorem).

  • Nyquist’s Frequency: High-frequency signals cannot be accurately represented unless they fall under half the sampling frequency.

  • Problem: Higher frequencies get "wrapped" into lower frequency ranges, leading to aliasing before projections are measured, especially near sharp edges.

Back-Projection Aliasing

  • Observation: Limited number of projections leads to star-shaped patterns indicating angular aliasing.

  • Guideline: Number of projections should match the number of rays for consistent representation.

Partial Volume Effect

  • Reality Check: X-ray beams are not infinitely sharp; they have finite width.

  • Definition: Partial volume effects occur when a beam interacts with structures only partially.

  • Result: Exponential attenuation law may not apply over the entire beam, yielding streak-like artefacts.

Beam Hardening

  • Issue: Non-monochromatic X-ray emissions lead to varying attenuation coefficients across wavelengths.

  • Impact: Attenuation shifts toward harder radiation may violate linearity in assumptions.

Noise in CT Images

  • Photon Noise: Limited photons contribute to individual measurements, introducing stochastic variations (Poisson noise).

  • Calibration Problems: Calibration errors can result in isolated erroneous measurements, termed as bad rays or views, leading to impulse noise.

Computerised Emission Tomography

General Principle

  • Concept: In emission tomography, the object emits gamma rays detected externally.

  • Primary Methods:

    • Single Photon Emission Computed Tomography (SPECT): Measures single gamma photons from injected radionuclides.

    • Positron Emission Tomography (PET): Detects simultaneous pairs of gamma photons.

Single Photon Emission Computed Tomography (SPECT)

Basic Principle

  • Objective: Extend emission radiography by detecting gamma rays after isotopes are injected into the body.

  • Detector: A gamma camera with a parallel-hole collimator rotates around the patient to capture radionuclide distribution.

  • Challenge: Hard gamma rays make attenuation correction more complicated.

Efficacy Comparison

  • Reconstruction: Similar to transmission CT but poses lower image quality.

  • Limitations: Lower resolution and increased effect of scattered photons; however, it offers functional insights into biological activity (e.g., blood flow).

SPECT Example Case Studies

  • Healthy vs. Disease States: Comparison of brain scans between a healthy patient and a patient with early Parkinson's disease (using 123I-FP-CIT).

  • Functional Imaging: SPECT images showing myocardial activity across different axes using Tellurium-201 chloride.

Positron Emission Tomography (PET)

Basic Principle

  • Operational Understanding: Utilizes radionuclides that emit positrons, annihilating with electrons to release dual gamma photons.

  • Detection Framework: Allows for localization of decay events, albeit with some uncertainty due to positron trajectories.

Detectors & Applications

  • Configuration: Ring arrangement of gamma ray detectors to capture coincidences between detected gamma photons.

  • Main Utility: Maps metabolic activity, particularly oxygen consumption within tissues.

Attenuation Compensation in PET

  • Scenario: Consideration of line segments between detectors with the annihilation event's emissions. Probabilities are modeled based on the exponential attenuation law.

  • Compensation Strategy: Use of X-ray scans in the same energy range for attenuation correction enhances PET accuracy.

PET Example Case Studies

  • Alzheimer's Disease Imaging: PET images demonstrating brain metabolic variations with radioactive glucose derivatives.

  • Visual Activity Correlation: PET imaging revealing activity levels in the visual cortex linked with visual stimuli.

Summary of Key Points

  • Common Artefacts in CT: Includes aliasing and ghosting, with suggested guidelines for equalizing projections.

  • Impact of Beam Width: Partial volume effect can lead to significant artefacts.

  • Photonic Noise: Predominantly Poisson characterized noise in imaging.

  • Technical Methods: SPECT and PET are both vital tools in computerized emission tomography, each with strengths in specific biological activity assessments.

Literature References

  • Webb, S. (1988). The Physics of Medical Imaging. Institute of Physics Publishing, Bristol.

  • Kak, A. C., Slaney, M. (2001). Principles of Computerized Tomographic Imaging. SIAM, Philadelphia.

  • Epstein, C. L. (2003). Introduction to the Mathematics of Medical Imaging. Pearson, New Jersey.