Circular-Arc Modal Migration for Ultrasound Imaging--Part I: Finite-Arc Spectral Theory and Stolt-Pullback Factorization

Abstract

Circular-arc ultrasound imaging requires a spectral construction that accommodates radius-dependent propagation and the spatially varying phase of offset-source transmission. This first of two articles develops Circular-Arc Modal Migration (CAMM), a direct Stolt-type imaging framework in angular modal coordinates. Starting from a first-Born acoustic model, angular decomposition and local radial Stolt mappings convert channel spectra into fields for coherent image synthesis. The offset transmit phase is represented by a phase-screen factorization through reusable spectral factors and spatially weighted fields. A scalar-pullback approximation simplifies this correction to reading one reference image at a geometrically fitted radial coordinate. Both realizations share the same modal construction. Error controls address local radial synthesis, transmit-path expansion, scalar approximation, and phase-screen truncation, with direct integration verifying the separated numerical evaluation. Same-RF point-target comparisons show that scalar pullback closely preserves the principal peak positions and widths of phase-screen imaging in the compact semicircular configuration, while phase screens better control surrounding responses in the shallow convex examples. The retained modal spectrum, frequency band, image radius, and source displacement guide this accuracy--computation choice. CAMM thereby extends direct spectral image formation to circular arcs and supplies the theoretical and numerical basis for the image-quality and performance studies in Part~II.