Geometry First, Scattering When Reliable: Sparse-View SAR Target Generation

Sparse-view Synthetic Aperture Radar (SAR) undersamples the azimuth-depression space, making unseen-view generation severely under-constrained as view changes jointly alter coarse image-domain geometry and anisotropic electromagnetic scattering response. Yet coarse geometry admits a relatively stable cross-view approximation, whereas anisotropic scattering remains poorly determined across large angular gaps due to view-specific radar-target interactions. We therefore propose a geometry-first, scattering-when-reliable principle: coarse geometry is made explicit as a conditioning signal, while scattering information is introduced only when sufficient angular evidence supports its query-view variation. We instantiate this principle with $\Phi$-Flow. For geometry, $\Phi$-Flow represents coarse geometry as spatial support, constructing Canonical Spatial Support (CSS) from sparse observations and aligning it to the query view as Query-Aligned Spatial Support (QSS). For scattering, we first investigate which response information remains useful beyond QSS, and accordingly introduce Compatible Scattering Transfer (CST), which transfers relative, query-dependent scattering modulation from a compatible reference with denser angular coverage. QSS, optionally complemented by CST, conditions a flow-matching (FM) generator for unseen-view synthesis. Geometry ablations show that QSS provides query-aligned geometry that model scaling cannot recover. Scattering analyses reveal an inferability-utility trade-off: response information with greater complementary value beyond QSS is harder to infer from sparse observations. CST mitigates this limitation, consistently reducing azimuth error at unseen azimuths and even when the query depression is entirely unobserved for the target. Evaluated on MSTAR and ATRNet-STAR, $\Phi$-Flow achieves state-of-the-art generation quality over all sparsity levels and the best recognition performance under the largest sparsity setting, with only 1.63% of the FM baseline's parameters.