Diffeomorphic Reconstruction Of A 2D Simple Non Parametric Manifold From Level Set Data Via Shape Gradients

TL;DR

The paper introduces a variational framework based on shape gradients to reconstruct a 2D simple manifold from level-set data, formulating an energy E(Γ) = ∫Γ (α φ^2 + β |∇Γ φ|^2) dΓ that promotes alignment with the zero level set while enforcing smoothness. By evolving an initial triangulated sphere via gradient descent along the shape gradient, the method achieves a diffeomorphic, smooth triangulated surface that accurately represents the target boundary. Numerical experiments on Sphere, Ellipsoid, Fused Sphere, and Cylinder phantoms demonstrate robust reconstruction under noise and superior smoothness compared to Marching Cubes and DistMesh, with tangential gradient terms enhancing boundary regularity. The approach holds potential for applications such as brain cortex reconstruction from MRI data, and future work aims to handle highly concave regions and non-simple shapes with improved initializers.

Abstract

A variational approach to the reconstruction of a shape (2D simple manifolds) as triangulated surface from given level set using shape gradients is presented. It involves an energy functional that depends on the local shape characteristics of the surface. Minimization of the energy through an iterative procedure using the gradient descent method yields a triangulated surface mesh which matches the boundary of the object of interest and this model ensures the smoothness of the boundary.

Figures (3)

• Figure I: Signed Distance Function $\phi$
• Figure II: Rotation of patch: Red patch (before rotation) and green patch (after rotation).
• Figure III: Flowchart for Surface Evolution