Diffeomorphic Reconstruction of a 2D-Simple Non-Parametric Manifold Via Shape Gradient

We present a variational method for reconstructing 2D simple manifolds from a given fixed level set data using shape gradients. Starting from an initial triangulated surface (mesh), iteratively evolves the mesh by minimizing a shape energy functional that reflects local geometric properties of the surface. The evolution is performed via a gradient descent scheme, guiding the mesh toward accurate alignment with the object's boundary while ensuring smoothness. This approach ensures both geometric fidelity and regularity of the reconstructed shape. The effectiveness of the method is demonstrated through experiments on various synthetic shapes and also for human brain white matter MRI T1 data. AMS Subject Classification (2020): 49Q10, 49Q20, 65D18, 68U10, 53A05, 65K10