PET Slides

Introduction to Image Reconstruction

  • Topic Title: Image Reconstruction: Data Models and Reconstruction Methods

  • Presenter: Prof Andrew J. Reader, School of Biomedical Engineering and Imaging Sciences

  • Focus on Positron Emission Tomography (PET) as a primary example, applicable to general medical imaging.

PET Image Reconstruction Basics

  • Image Acquisition: Fundamental process in PET imaging, involving 3 data formats:

    • Backprojected images

    • Sinograms

    • List-mode data

  • Data Models: Two primary models in use:

    • Convolution Radon Transform (or x-ray transform)

Key Components in PET Imaging

  • Radiotracer: Commonly used radioactive isotopes, e.g., 18F, injected into the patient.

  • Detection Mechanism: PET scanners utilize high-density crystals to detect annihilation photons at 511 keV, which allows for imaging of various organs including the brain and heart, and diagnostic applications such as dementia and cancer.

  • Medical Imaging Process: Steps involved from data acquisition to image reconstruction, delineating the measured data and resulting radiotracer distribution.

The Image Reconstruction Problem

  • The goal is to reconstruct the object function f(r) from detected signals.

  • Image Dimension: Standard pixel dimension is 256x256 with over 65,000 parameters.

  • PET scanner configuration includes a detector ring that captures photon pairs, defining a line of response (LOR).

Data Representation Models

  • Point Source Representation:

    • Convolution Model: Converts each point source into a point spread function (PSF), modeled mathematically as:

      • f(r) = ∫ f(r')δ(r - r')dr'

      • g(r) = ∫ f(r')h(r - r')dr'

    • Assumptions: The convolution assumes a linear shift-invariant (LSI) system, essential for modeling point sources effectively.

Types of Data Used

  • List-mode Data: Includes the parameters:

    • D1, D2 (detector positions)

    • t (time)

    • E1, E2 (energy)

    • ∆t (time differences)

  • These parameters assist in detailed modeling of radiotracer information.

Advanced Reconstruction Methods

  • First Reconstruction Method: Based on backprojected images with the equation g(r) = ∫ f(r')h(r - r')dr' involving deconvolution for improving accuracy.

  • Second Reconstruction Method: Utilizing sinogram data, this method is termed Filtered Backprojection (FBP) which involves:

    1. Fourier transformation of sinograms

    2. Application of a ramp filter

    3. Inverse Fourier transformation followed by backprojection.

Sinogram and Radon Transform

  • Sinograms: Modelled using 2D Radon transform, characterizing that counts in a sinogram are proportional to summed activity along a line.

  • Important equations:

    • m(s,φ) ≈ ∫ f(x,y)dy

    • Relationships illustrating conversions between angle and position:

      • x = s cos φ - l sin φ

      • y = s sin φ + l cos φ

Corrections in Medical Imaging

  • Data Corrections Needed: Essential steps include subtracting randoms and scatter from sinograms, followed by multiplicative correction for attenuation factors and normalisation respectively.

  • Steps defined:

    1. Subtract randoms (r) and scatter (s)

    2. Apply attenuation correction factors (ACF)

    3. Correct for non-uniform detector efficiency.

Iterative Reconstruction Techniques

  • Maximum Likelihood Expectation Maximization (ML-EM): A statistical approach designed for accurate modeling, utilizing a Poisson noise model.

  • Need for Iterative Reconstruction: More accurate compared to FBP, it allows for enhanced modeling of noise and signal behavior in PET images.

Conclusion

  • Overview of reconstruction methods, including improvements made over the years from basic FBP to advanced ML-EM techniques, emphasizing iterative approaches that lead to greater accuracy in medical imaging.

  • Closing remarks on ongoing innovations in the field and the importance of statistical models in PET.

  • End of Lecture: Review of all topics covered, leading to a deeper understanding of image reconstruction techniques in biomedical imaging.