Nonconvex prior image constrained compressed sensing (NCPICCS): Theory and simulations on perfusion CT

Purpose: To present and evaluate a new image reconstruction method for dynamic CT based on a nonconvex prior image constrained compressed sensing (NCPICCS) algorithm. The authors systematically compared the undersampling potential, functional information recovery, and solution convergence speed of four compressed sensing (CS) based image reconstruction methods using perfusion CT data: Standard ℓ₁ -based CS, nonconvex CS (NCCS), and ℓ₁ -based and nonconvex CS, including an additional constraint based on a prior image (PICCS and NCPICCS, respectively). Methods: The Shepp-Logan phantom was modified such that its uppermost ellipses changed attenuation through time, simulating both an arterial input function (AIF) and a homogeneous tissue perfusion region. Data were simulated with and without Poisson noise added to the projection data and subsequently reconstructed with all four CS-based methods at four levels of undersampling: 20, 12, 6, and 4 projections. Root mean squared (RMS) error of reconstructed images and recovered time attenuation curves (TACs) were assessed as well as convergence speed. The performance of both PICCS and NCPICCS methods were also evaluated using a kidney perfusion animal experiment data set. Results: All four CS-based methods were able to reconstruct the phantoms with 20 projections, with similar results on the RMS error of the recovered TACs. NCCS allowed accurate reconstructions with as few as 12 projections, PICCS with as few as six projections, and NCPICCS with as few as four projections. These results were consistent for noise-free and noisy data. NCPICCS required the fewest iterations to converge across all simulation conditions, followed by PICCS, NCCS, and then CS. On animal data, at the lowest level of undersampling tested (16 projections), the image quality of NCPICCS was better than PICCS with fewer streaking artifacts, while the TAC accuracy on the selected region of interest was comparable. Conclusions: The authors have presented a novel method for image reconstruction using highly undersampled dynamic CT data. The NCPICCS method takes advantage of the information provided by a prior image, as in PICCS, but employs a more general nonconvex sparsity measure [such as the ℓ_p -norm (0<p≤1)] rather than the conventional convex ℓ₁-norm. Despite the lack of guarantees of a globally optimal solution, the proposed nonconvex extension of PICCS consistently allowed for image reconstruction from fewer samples than the analogous ℓ₁-based PICCS method. Both nonconvex sparsity measures as well as prior image information (when available) significantly reduced the number of iterations required for convergence, potentially providing computational advantages for practical implementation of CS-based image reconstruction techniques.