Safe Corridors (a series of overlapping convex shapes) have been used recently in multiple state-of-the-art motion planning methods. They allow to represent the free space in the environment in an efficient way for collision avoidance. In this paper, we propose a new framework for generating Safe Corridors. We assume that we have a voxel grid representation of the environment. The proposed framework improves on a previous state-of-the-art voxel grid based Safe Corridor generation method. It also creates a connectivity graph between polyhedra of a given Safe Corridor that allows to know which polyhedra intersect with each other. The connectivity graph can be used in planning methods to reduce computation time. The method is compared to other state-of-the-art methods in simulations in terms of computation time, volume covered, safety, number of polyhedron per Safe Corridor and number of constraints per polyhedron.