Algorithm 1. Evaluating PSWFs expansion coefficients (direct method).

Required: An image $\{I(\frac{k}{L})\}$ sampled on a Cartesian grid of size (2L + 1) × (2L + 1) with $\frac{k}{L}\in \mathbf{Q}$, and Q = [−1, 1] × [−1, 1]. Precomputation: Choose a bandlimit c (≤ πL) and a truncation parameter T. Evaluate the PSWFs ${\psi }_{N,n}^c(\frac{k}{L})$ and their eigenvalues ${\alpha }_{N,n}^c$ according to [31] for (N, n) ∈ ΩT (see (8)), where $\frac{k}{L}\in \mathbf{D}$, and compute the normalized eigenvalues ${\lambda }_{N,n}^c=\frac{c}{2\pi }{\alpha }_{N,n}^c$. For all (N, n) ∈ ΩT, compute the expansion coefficients ${\widehat{a}}_{N,n}=\frac{{|{\lambda }_{N,n}^c|}^2}{L^2}\sum _{\frac{k}{L}\in \mathbf{D}}I(\frac{k}{L}){({\psi }_{N,n}^c(\frac{k}{L}))}^{\ast }$.