Algorithm 2. Evaluating PSWFs expansion coefficients (efficient 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. Choose the number of radial nodes 𝒩r and angular ${\mathcal{N}}_{\theta }^{\ell }$ nodes for 1 ≤ ℓ < 𝒩r (according to (18) and (19) or similar relaxed conditions). Compute the quadrature nodes ${\omega }_{\ell ,j}^{2c}=(r_{\omega }^{\ell },{\theta }_{\omega }^{\ell ,j})$ and weights for ${\mathcal{W}}_{\ell ,j}^{2c}$ for ℓ = 1, …, 𝒩r and $j=1,\dots ,{\mathcal{N}}_{\theta }^{\ell }$, as described in [31]. Evaluate the radial part of the PSWFs $R_{N,n}^c(r)$ at the radial quadrature nodes for $r_{\omega }^{\ell }$ (N, n) ∈ ΩT (see (8)). Compute ${\varphi }^c({\omega }_{\ell ,j}^{2c})$ from (14) by NFFT. For (N, n) ∈ ΩT, compute the expansion coefficients ãN,n via equations (26) and (27).