Summary

We reconstruct dynamic (spatio-temporal) PET data using regularization based on exponential-spline wavelets (E-spline wavelets) that are specially tailored to model time activity curves (TACs) in PET.

Introduction

Tomographic reconstruction from positron emission tomography (PET) data is an ill-posed problem that requires regularization. An attractive approach is to impose an ℓ ₁ regularization constraint, which favors sparse solutions in the wavelet domain. This can be achieved quite efficiently thanks to the iterative algorithm developed by Daubechies et al. , 2004.

Main Contribution

In this work, we apply the iterated-thresholding technique and extend it for the reconstruction of dynamic (spatio-temporal) PET data.

• Instead of using classical wavelets in the temporal dimension, we introduce exponential-spline wavelets (E-spline wavelets) that are specially tailored to model time-activity curves (TACs) in PET. We show that the exponential-spline wavelets naturally arise from the compartmental description of the dynamics of the tracer distribution.
• We address the issue of the selection of the "optimal" E-spline parameters (poles and zeros) and we investigate their effect on reconstruction quality.
• We demonstrate the usefulness of spatio-temporal regularization and the superior performance of E-spline wavelets over conventional Battle-Lemarié wavelets in a series of experiments.

We find that the E-spline wavelets outperform the conventional wavelets in terms of the reconstructed signal-to-noise ratio (SNR) and the sparsity of the wavelet coefficients. Based on our simulations, we conclude that replacing the conventional wavelets with E-spline wavelets leads to equal reconstruction quality for a 40% reduction of detected coincidences, meaning an improved image quality for the same number of counts or, equivalently, a reduced exposure to the patient for the same image quality.