A differentiable Shepp-Logan phantom and its applications in exact cone-beam CT.

Recently, several exact cone-beam reconstruction algorithms, such as the generalized filtered-backprojection (FBP) and backprojection-filtration (BPF) methods, have been developed to solve the long object problem. Although the well-known 3D Shepp-Logan phantom (SLP) is often used to validate these algorithms, it is deficient due to the discontinuity of the SLP. In this paper, we first construct a differentiable polynomial function to approximate the unit rectangular function on [-1, 1]. Then, we use this function to obtain a differentiable ellipsoid phantom, whose x-ray transform is differentiable for any smooth scanning trajectory. Finally, we propose a differentiable Shepp-Logan phantom (DSLP) for numerical simulation of the exact cone-beam CT algorithms. Our numerical simulation shows that the reconstructed DSLP has a better image quality than the reconstructed SLP, and is complementary to the traditional SLP for evaluation of the exact cone-beam CT algorithms.