  Implicit surface representations such as the signed distance function (SDF)
have emerged as a promising approach for image-based surface reconstruction.
However, existing optimization methods assume solid surfaces and are therefore
unable to properly reconstruct semi-transparent surfaces and thin structures,
which also exhibit low opacity due to the blending effect with the background.
While neural radiance field (NeRF) based methods can model semi-transparency
and achieve photo-realistic quality in synthesized novel views, their
volumetric geometry representation tightly couples geometry and opacity, and
therefore cannot be easily converted into surfaces without introducing
artifacts. We present $\alpha$Surf, a novel surface representation with
decoupled geometry and opacity for the reconstruction of semi-transparent and
thin surfaces where the colors mix. Ray-surface intersections on our
representation can be found in closed-form via analytical solutions of cubic
polynomials, avoiding Monte-Carlo sampling and is fully differentiable by
construction. Our qualitative and quantitative evaluations show that our
approach can accurately reconstruct surfaces with semi-transparent and thin
parts with fewer artifacts, achieving better reconstruction quality than
state-of-the-art SDF and NeRF methods. Website: https://alphasurf.netlify.app/
