Optimization for Blob-Based Image Reconstruction With Generalized Kaiser–Bessel Basis Functions

Generalized Kaiser-Bessel radial basis functions are frequently used in tomographic reconstructions, and the parameter selection can significantly impact both the qualitative and quantitative image characteristics. Currently, the blob parameters are simply selected based on the first-order approximation of a uniform object. To obtain optimal images, a detailed investigation and optimization of blob parameters is needed. In this paper, we aim at optimizing the parameters of the basis functions for optimal image representation and image reconstruction. We first represent the unknown radiotracer activity distribution as the coefficients of blob basis functions on a Bravais lattice from crystallography. We point out that the optimal sampling lattice is just the reciprocal of the lattice that achieves highest packing of equally sized spheres. Based on multidimensional sampling theorem and the Kepler conjecture (proven by Thomas C. Hales), we obtain that the body-centered cubic (BCC) and hexagonal close-packing lattices are optimal in terms of either reducing computational cost at similar image quality or improving image quality at the same computational cost. To optimize the blob parameters, we formulate two new image quality metrics-total harmonic distortion (THD) and minimum approximation error (MAE) based on the theory of approximation-to analytically characterize the blob representation errors of a nonzero uniform object and an arbitrary object, respectively. The MAE is formulated in terms of the power spectral density of an imaging object, and we use it for object-dependent blob parameter optimization. We validate the THD and MAE through numerical examples using homogeneous and heterogeneous objects, respectively. We also present 3-D blob-based image reconstructions using BCC lattice with different shape parameters to show the usefulness of the proposed optimization.