A novel O(1)O(1) time algorithm for 3D block-based medial axis transform by peeling corner shells

The block-based medial axis transform (BB_MAT  , for short) of a 3D (2D) binary image, denoted 3D_BB_MAT (2D_BB_MAT), is defined as the problem to find a minimal set of upright 1-cubes (1-squares) whose union corresponds exactly to the 1-voxels (1-pixels) in the binary image. Many parallel algorithms have been proposed for computing the 2D_BB_MAT, almost all proposed approaches are unable to extend to a solution of the 3D_BB_MAT problem. In this paper, we first propose a novel O(1)O(1) time algorithm for solving the 3D_BB_MAT (2D_BB_MAT) problem of a binary image of size N×N×NN×N×N (N×NN×N) on a linear array with a reconfigurable pipelined bus system (LARPBS, for short) by peeling the 3D (2D) corner shell of each 1-voxel (1-pixel) of the image and revealing some properties of the 3D_BB_MAT (2D_BB_MAT). The 3D (2D) chessboard distance transform problem, denoted 3D_CDT (2D_CDT), is the problem for each 0-voxel (0-pixel) of a 3D (2D) binary image to find its nearest 1-voxel (1-pixel) in the binary image based on chessboard distance metric. In this paper, we also solve the 3D_CDT (2D_CDT) based on the computed 3D_BB_MAT (2D_BB_MAT). All the proposed algorithms are very innovative although they look like “straightforward”. To the best of our knowledge, the proposed 2D_BB_MAT (2D_CDT) algorithm is the first algorithm that can extend to a solution of the 3D_BB_MAT (3D_CDT) problem in parallel, and the 3D_BB_MAT (3D_CDT) algorithm is the first parallel algorithm proposed for solving the 3D_BB_MAT (3D_CDT) problem.