Series-expansion tomography methods that use Natural Basis Functions (NBFs), also called natural pixels, often use iterative solution techniques or solution by Truncated Singular Value Decomposition (TSVD). Here, solution by constrained optimization is proposed. It is shown that significant improvements in the tomographic reconstructions can be obtained, in particular when the coverage by the imaging system is irregular. The analogy between regular NBFs and the filtered backprojection or convolution-backprojection tomography method suggests maximum smoothness in projection space as object function (i.e. a priori information) in the constrained optimization. A further improvement was found by employing NBFs that correspond to a bi-linear interpolation in projection space. The new NBF method is compared with various tomography methods: constrained optimization with local basis functions, NBFs with TSVD, and an iterative projection-space reconstruction method.