One-pass filtered reconstruction from finite Born data

Inputs and admissible conditions. Fix the domain $D$, wavenumber $k>0$, and library Fourier convention from the fixed-frequency data definition. Give nonempty finite lists of incident directions $d_1,\dots ,d_s\in S^2$ and observation directions ${\theta }_1,\dots ,{\theta }_r\in S^2$. Give a finite measurable partition $D={⨆}_{n=1}^N{E}_n$ with $\mathrm{\mid }{E}_n\mathrm{\mid }>0$, complex Born-data samples $y=(y_{ba})\in {\mathbb{C}}^{rs}$, a discrepancy threshold $\delta \ge 0$, and an iteration budget $B\in {\mathbb{N}}_0:=\{0,1,2,\dots \}$. The sample frequency for pair $({\theta }_b,d_a)$ is ${\xi }_{ba}:=k({\theta }_b-d_a)\mathrm{/}(2\pi )$. The declared finite discrete Born Fourier matrix $F\in {\mathbb{C}}^{rs\times N}$ has entries $F_{(b,a),n}:=\frac{1}{4\pi }{\int }_{E_n}{e}^{-2\pi ix\cdot {\xi }_{ba}}dx\mathrm{.}$ For a piecewise-constant contrast $q_c:={\sum }_{n=1}^N{c}_n{1}_{E_n}$, its sampled first Born data vector is exactly $Fc$. The input also declares a reduced singular-value decomposition $Fc={\sum }_{j=1}^p{\sigma }_j⟨c,v_j⟩u_j(c\in {\mathbb{C}}^N),$ where $p=\mathrm{rank}F$, the positive singular values ${\sigma }_j$ have orthonormal right vectors $v_j$ and orthonormal left vectors $u_j$, and the empty sum is used when $F=0$. Give a scalar filter $g:[0,\mathrm{\infty })\to \mathbb{C}$ whose values at the ${\sigma }_j$ are finite and with $g(0)=0$. The threshold filter $g_{\tau }(\sigma )=1\mathrm{/}\sigma$ for $\sigma >\tau$ and $g_{\tau }(\sigma )=0$ for $0\le \sigma \le \tau$ is the truncated singular-value pseudoinverse at threshold $\tau \ge 0$.

Instructions and outputs.

1. If $B=0$, return no estimate, no residual, and status budget-exhausted, with zero filter applications.
2. If $B\ge 1$, form the coefficient estimate $\widehat{c}:={\sum }_{j:{\sigma }_j>0}g({\sigma }_j)⟨y,u_j⟩v_j,$ where $⟨z,w⟩={\sum }_{\mathrm{\ell }}{z}_{\mathrm{\ell }}\overline{w_{\mathrm{\ell }}}$ is linear in its first argument. Return the piecewise-constant reconstruction $\widehat{q}:={\sum }_{n=1}^N{\widehat{c}}_n{1}_{E_n}$ after exactly one filter application.
3. Compute the finite data residual $R:=\mathrm{\parallel }F\widehat{c}-y{\mathrm{\parallel }}_2$, where $\mathrm{\parallel }z{\mathrm{\parallel }}_2=({\sum }_{\mathrm{\ell }}\mathrm{\mid }{z}_{\mathrm{\ell }}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. Return status discrepancy when $R\le \delta$, and residual-above-threshold when $R>\delta$; in either case also return $\widehat{q}$, $R$, and the application count $1$.

This procedure only defines one filtered finite-dimensional Born reconstruction and a residual status. The residual is not an exact-range test for an infinite-dimensional compact forward operator and does not establish a nonlinear solution, identifiability, stability, convergence, or correctness of the reconstructed contrast.

The linearized far-field samples are $\widehat{q}(k(\theta -d)\mathrm{/}(2\pi ))\mathrm{/}(4\pi )$.

The first Born family is the local Fréchet derivative of the nonlinear scattering data at zero contrast.

The special filter $g_{\tau }$ is the finite-matrix truncated singular-value pseudoinverse.

For a compact Hilbert-space map, exact range membership also requires orthogonality to the adjoint kernel and a finite Picard sum.

Countable Choice is inherited from the fixed-frequency data, Born-linearization, and contextual Picard suppliers.

The library Fourier transform uses the stated $2\pi$ normalization.