Repairing 3D binary images using the BCC grid with a 4-valued combinatorial coordinate system

A 3D binary image I is called well-composed if the set of points in the topological boundary of the cubes in I is a 2-manifold. Repairing a 3D binary image is a process which produces a well composed image (or a polyhedral complex) from the non-well-composed image I.We propose here to repair 3D images by associating the Body-Centered Cubic grid (BCC grid) to the cubical grid. The obtained polyhedral complex is well composed, since two voxels in the BCC grid either share an entire face or are disjoint. We show that the obtained complex is homotopy equivalent to the cubical complex naturally associated with the image I.To efficiently encode and manipulate the BCC grid, we present an integer 4-valued combinatorial coordinate system that addresses cells of all dimensions (voxels, faces, edges and vertices), and allows capturing all the topological incidence and adjacency relations between cells by using only integer operations.We illustrate an application of this coordinate system on two tasks related with the repaired image: boundary reconstruction and computation of the Euler characteristic

Crossing-free paths in the square grid

We consider paths in the 2D square grid, composed of grid edges, given as a sequence of moves in the four cardinal compass directions, without U-turns, but possibly passing several times through the same vertex or the same edge (if the path is open, it cannot pass twice through its starting vertex). We propose an algorithm which reports a self-crossing if there is one, or otherwise draws the path without self-crossings. The algorithm follows the intuitive idea naturally applied by humans to draw a curve: at each vertex that has already been visited, it tries to insert two new segments in such a way that they do not cross the existing ones. If this is not possible, a self-crossing is reported. This procedure is supported by a data structure combining a doubly-linked circular list and a skip list. The time and space complexity is linear in the length of the path

Surface-Based Computation of the Euler Characteristic in the BCC Grid

As opposed to the 3D cubic grid, the body-centered cubic (BCC) grid has some favorable topological properties: each set of voxels in the grid is a 3-manifold, with 2-manifold boundary. Thus, the Euler characteristic of an object O in this grid can be computed as half of the Euler characteristic of its boundary ∂O . We propose three new algorithms to compute the Euler characteristic in the BCC grid with this surface-based approach: one based on (critical point) Morse theory and two based on the discrete Gauss–Bonnet theorem. We provide a comparison between the three new algorithms and the classic approach based on counting the number of cells, either of the 3D object or of its 2D boundary surface

Computing a discrete Morse gradient from a watershed decomposition

We consider the problem of segmenting triangle meshes endowed with a discrete scalar function f based on the critical points of f . The watershed transform induces a decomposition of the domain of function f into regions of influence of its minima, called catchment basins. The discrete Morse gradient induced by f allows recovering not only catchment basins but also a complete topological characterization of the function and of the shape on which it is defined through a Morse decomposition. Unfortunately, discrete Morse theory and related algorithms assume that the input scalar function has no flat areas, whereas such areas are common in real data and are easily handled by watershed algorithms. We propose here a new approach for building a discrete Morse gradient on a triangulated 3D shape endowed by a scalar function starting from the decomposition of the shape induced by the watershed transform. This allows for treating flat areas without adding noise to the data. Experimental results show that our approach has significant advantages over existing ones, which eliminate noise through perturbation: it is faster and always precise in extracting the correct number of critical elements

On Hamiltonian cycles in the FCC grid

The face centered cubic (FCC) grid is a space-filling grid, one of the alternatives to the traditional cubic one. We show that there are five Hamiltonian cycles (non-equivalent up to rotation and symmetry), connecting the faces of a voxel in the FCC grid. Each of the five cycles can be used to trace the boundary of a class of objects in the grid, constructed by iteratively attaching voxels so that each new voxel shares exactly one face with the set of already attached voxels

Transparent management of adjacencies in the cubic grid

We propose an integrated data structure which represents, at the same time, an image in the cubic grid and three well-composed images, homotopy equivalent to it with face-, edge- and vertex-adjacency. After providing an algorithm to build the structure, we present examples showing how, thanks to such data structure, image processing algorithms can be written in a transparent way w.r.t. the adjacency type. Applications include rapid prototyping and teaching

Repairing 3D binary images using the FCC grid

A 3D image I is well-composed if it does not contain critical edges or vertices (where the boundary of I is non-manifold). The process of transforming an image into a well composed one is called repairing. We propose to repair 3D images by associating the face-centered cubic grid (FCC grid) with the cubic grid. We show that the polyhedral complex in the FCC grid, obtained by our repairing algorithm, is well composed and homotopy equivalent to the complex naturally associated with the given image I with edge-adjacency (18-adjacency). We illustrate an application on two tasks related to the repaired image: boundary reconstruction and computation of its Euler characteristic

Abstract Visibility Algorithms on Triangulated Digital Terrain Models

In the paper, we address the problem of computing visibility information on digital terrain models. We present first a general introduction to digital terrain models. Visibility problems on terrains are classified, according to the kind of visibility in- formation they compute, into point visibility, line visibility and region visibility. A survey of the state-of-the-art of the algorithms for computing the different kinds of visibility information is presented, according to the previous classification. A new algorithm for computing the horizon on a digital terrain model is described as well. 1 z f x; y D

Computation of 2D Discrete Geometric Moments Through Inclusion-Exclusion

We propose a new formula for computing discrete geometric moments on 2D binary images. The new formula is based on the inclusion-exclusion principle, and is especially tailored for images coming from computer art, characterized by a prevalence of horizontal and vertical lines. On the target class of images, our formula reduces the number of pixels where calculations are to be performed

Surface-based computation of the Euler characteristic in the cubical grid

For well-composed (manifold) objects in the 3D cubical grid, the Euler characteristic is equal to half of the Euler characteristic of the object boundary, which in turn is equal to the number of boundary vertices minus the number of boundary faces. We extend this formula to arbitrary objects, not necessarily well-composed, by adjusting the count of boundary cells both for vertex- and for face-adjacency. We prove the correctness of our approach by constructing two well-composed polyhedral complexes homotopy equivalent to the given object with the two adjacencies. The proposed formulas for the computation of the Euler characteristic are simple, easy to implement and efficient. Experiments show that our formulas are faster to evaluate than the volume-based ones on realistic inputs, and are faster than the classical surface-based formulas