Algorithm 3. PSWFs-based steerable PCA.

Required: PSWFs expansion coefficients of M images ${\{{\widehat{a}}_{N,n}^m\}}_{m=0}^{M-1}$ for (N, n) ∈ ΩT and a bandlimit c. Precomputation: Evaluate the PSWFs ${\psi }_{N,n}^c(\frac{k}{L})$, $\frac{k}{L}\in \mathbf{D}$ and their eigenvalues ${\alpha }_{N,n}^c$ for (N, n) ∈ ΩT according to [31]. Compute the expansion coefficients of the mean image for ${\widehat{\mu }}_{0,n}=\frac{1}{M}{\sum }_{m=0}^{M-1}{\widehat{a}}_{0,n}^m$ for n = 0, …, n0 where n0 is the largest index n such that (0, n) ∈ ΩT. Update ${\widehat{a}}_{0,n}^m\leftarrow {\widehat{a}}_{0,n}^m-{\widehat{\mu }}_{0,n}$. Compute the eigenvalues λ̂1, …, λ̂|ΩT| and eigenvectors ĝ1, …, ĝ|ΩT| of the matrix C (from (30)) by diagonalizing each of its blocks separately. Compute the sampled basis functions $g_{\ell }(\frac{k}{L})={\sum }_{n=0}^{n_{\ell }}{\widehat{g}}_{N_{\ell },n}^{\ell }{\psi }_{N_{\ell },n}^{\ell }(\frac{k}{L})$, where nℓ stands for the largest index n such that (Nℓ, n) ∈ ΩT, and ${\widehat{g}}_{N_{\ell },n}^{\ell }$ are the entries of the eigenvector ĝℓ corresponding to the pair (Nℓ, n). Compute the coefficients of Im in the steerable-PCA basis by $c_{\ell ,m}={\sum }_{n=0}^{n_{\ell }}{\widehat{a}}_{N_{\ell },n}^m{\left({\widehat{g}}_{N_{\ell },n}^{\ell }\right)}^{\ast }$.