Consistently and unconditionally energy-stable linear method for the diffuse-interface model of narrow volume reconstruction

In this paper, we propose an unconditionally energy-stable linear method with improved consistency for the diffuse-interface (phase-field) model of three-dimensional narrow volume reconstruction based on point clouds. To detect the unorganized point set, a control function is added to the original Allen–Cahn (AC) equation. This modified AC equation is an extension of the image segmentation model in two-dimensional space. By introducing an appropriate time-dependent variable, we first transform the governing equation into an equivalent form. Based on the Crank–Nicolson type discretization in time, the linear time-marching scheme is developed. To improve the consistency between the original and modified energies, we apply a simple and effective correction algorithm after updating the phase-field variable in each time iteration. We can analytically prove the unconditional energy stability of the proposed method. We also describe the fully discrete implementation with spatial discretization using the finite difference method. Computational experiments validate that the proposed scheme is practical for reconstructing narrow volumes based on point clouds.