Three-dimensional volume reconstruction using two-dimensional parallel slices

Abstract

In this thesis, we propose a partial differential equation model for the three-dimensional volume reconstruction from 2-D slices. The proposed method is based on the modified Cahn-Hilliard equation for 3-D binary inpainting. In order to satisfy the constraints accurately as well as to obtain a smooth result, we propose a pre-smoothing procedure based on anisotropic diffusion applied to slices. Then we discuss the justification for our inpainting model through $\Gamma$-convergence analysis. After splitting a grayscale image into binary channels, we perform multi-channel Cahn-Hilliard inpainting. Then we adopt smoothing and shock filter as post-processing for combining binary inpainting results. In addition, we add the cartoon-texture decomposition to pre-processing in order to relax a pretty complex slice constraints. We demonstrate how to implement Cahn-Hilliard inpainting with the relaxed constraints using a solver known as convexity splitting. We apply our results for 2-D binary and grayscale images together with 3-D binary and grayscale synthetic images. Furthermore, We apply our method to reconstruct 3-D human body from parallel slices of CT images.