← Back to Week 07 · All Course Demos

PET Sinogram & Tomographic Reconstruction Workbench

Positron Emission Tomography · Radon Transform, Coincidence LORs, Filtered Backprojection (FBP) & Iterative MLEM with Attenuation Correction

Tomographic Principles: An annihilation event sends antiparallel 511 keV photon pairs into opposing detectors, defining a Line of Response (LOR). Sorting millions of LORs by angle \(\theta\) and radial offset \(s\) produces the Sinogram via the Radon transform. Reconstructing the cross-sectional tracer distribution requires inverting the intrinsic \(1/r\) backprojection blurring via the ramp filter, or applying iterative Maximum Likelihood Expectation Maximization (MLEM) to model Poisson noise and tissue photon attenuation.

1. Emission Phantom Click to add Hot Tumor

2. Measured Sinogram p(s, θ) s × θ (Radon Transform)

3. Reconstructed Activity Filtered Backprojection

Reconstruction Algorithm:

Apply 511 keV Attenuation Correction (μ-map)

Tracer Dose & Count Statistics High (Low Noise)

Low Dose (High Poisson Noise) High Dose (Clinical)

Mathematical Foundations of PET Tomography

1. The Radon Transform & Sinogram Geometry:

In 2D PET, each line integral across tracer activity \(f(x, y)\) is parameterized by normal angle \(\theta \in [0, \pi]\) and signed radial displacement \(s\) from the scanner isocenter:

p(s, \theta) = \iint f(x, y) \, \delta(x \cos\theta + y \sin\theta - s) \, dx \, dy

A single isolated hot tumor at polar coordinate \((r_0, \phi_0)\) traces a pure sine wave across the Radon plane: \(s(\theta) = r_0 \sin(\theta - \phi_0)\). The sinogram is the collection of all such intersecting sinusoids.

2. Unfiltered vs. Filtered Backprojection:

Projecting sinogram values back along each LOR without filtering accumulates the image, but introduces an intrinsic \(1/r\) point-spread blurring kernel (\(f_{\text{unfiltered}} = f \ast \frac{1}{r}\)). In Fourier space, the Fourier Slice Theorem proves that 1D projections correspond to radial slices through 2D k-space. The Ram-Lak ramp filter (\(|k|\)) compensates exactly for the high density of radial sampling near the origin. Apodizing with a Hann window (\(H(k) = 0.5 + 0.5\cos(\pi k / k_{\text{max}})\)) prevents high-frequency Poisson noise blow-up.

3. Iterative MLEM Reconstruction:

Filtered Backprojection assumes deterministic, continuous line integrals and easily produces negative artifactual pixel values in low-count regions. Maximum Likelihood Expectation Maximization (MLEM) models the discrete Poisson statistics of radioactive decay:

\hat{f}_j^{(n+1)} = \hat{f}_j^{(n)} \frac{1}{\sum_i c_{ij}} \sum_i c_{ij} \frac{p_i}{\sum_k c_{ik} \hat{f}_k^{(n)}}

Each iteration compares projected estimates against actual measured counts, guaranteeing strictly non-negative activity and superior signal-to-noise ratios.

4. 511 keV Photon Attenuation Correction:

Before escaping the patient's body, 511 keV photons can undergo Compton scattering or photoelectric absorption. The survival probability along an LOR is:

A_i = \exp\left(-\int \mu(x,y) \, dl\right)

independent of where along the LOR the emission took place! Without attenuation correction (CT or MRI-based \(\mu\)-map), deep brain structures appear artificially hypometabolic while the cortical periphery creates false hypermetabolic rims.