Topology Preserving Marching Cubes-like Algorithms on the Face-Centered Cubic Grid (original) (raw)

A Generalized Marching Cubes Algorithm Based On Non-Binary Classifications

We present a new technique for generating surface meshes from a uniform set of discrete samples. Our method extends the well-known marching cubes algorithm used for computing polygonal isosurfaces. While in marching cubes each vertex of a cubic grid cell is binary classified as lying above or below an isosurface, in our approach an arbitrary number of vertex classes can be specified. Consequently the resulting surfaces consist of patches separating volumes of two different classes each.

Cubical Marching Squares: Adaptive Feature Preserving Surface Extraction from Volume Data

Computer Graphics Forum, 2005

In this paper, we present a new method for surface extraction from volume data which preserves sharp features, maintains consistent topology and generates surface adaptively without crack patching. Our approach is based on the marching cubes algorithm, a popular method to convert volumetric data to polygonal meshes. The original marching cubes algorithm suffers from problems of topological inconsistency, cracks in adaptive resolution and inability to preserve sharp features. Most of marching cubes variants only focus on one or some of these problems. Although these techniques could be combined to solve these problems altogether, such a combination might not be straightforward. Moreover, some feature-preserving variants introduce an additional problem, inter-cell dependency. Our method provides a relatively simple and easy-to-implement solution to all these problems by converting 3D marching cubes into 2D cubical marching squares, resolving topology ambiguity with sharp features and eliminating inter-cell dependency by sampling face sharp features. We compare our algorithm with other marching cubes variants and demonstrate its effectiveness on various applications. into polygonal meshes. These methods allow us to accurately represent geometric objects as volumetric data, manipulate them volumetrically, and efficiently display them by converting on the fly the volumetric data into polygonal meshes. However, although the original marching cubes algorithm is generally effective, it has problems with topological inconsistency, cracks in adaptive resolution and inability to preserve sharp features.

Marching cubes: A high resolution 3D surface construction algorithm

ACM Siggraph Computer Graphics, 1987

We present a new algorithm, called marching cubes, that creates triangle models of constant density surfaces from 3D medical data. Using a divide-and-conquer approach to generate inter-slice connectivity, we create a case table that defines triangle topology. The algorithm processes the 3D medical data in scan-line order and calculates triangle vertices using linear interpolation. We find the gradient of the original data, normalize it, and use it as a basis for shading the models. The detail in images produced from the generated surface models is the result of maintaining the inter-slice connectivity, surface data, and gradient information present in the original 3D data. Results from computed tomography (CT), magnetic resonance (MR), and single-photon emission computed tomography (SPECT) illustrate the quality and functionality of marching cubes. We also discuss improvements that decrease processing time and add solid modeling capabilities.

A novel and efficient implementation of the marching cubes algorithm

Computerized Medical Imaging and Graphics, 2001

In this paper, a novel and ef®cient implementation of the marching cubes (MC) algorithm is presented for the reconstruction of anatomical structures from real three-dimensional medical data. The proposed approach is based on a generic rule, able to triangulate all 15 standard cube con®gurations used in the classical MC algorithm as well as additional cases presented in the literature. The proposed implementation of the MC algorithm can handle the Type A`hole problem' which occurs when at least one cube face has an intersection point in each of its four edges. Theoretical and experimental results demonstrate the ability of the new implementation to reproduce standard MC results, resolving Type A`hole problem'. Finally, the proposed implementation was applied to real medical date to reconstruct anatomical structures. The output of the proposed technique is in WWW compliant format. q

A "Group Marching Cube" (GMC) Algorithm for Speeding up the Marching Cube Algorithm

IEICE Transactions on Information and Systems, 2011

Although the Marching Cube (MC) algorithm is very popular for displaying images of voxel-based objects, its slow surface extraction process is usually considered to be one of its major disadvantages. It was pointed out that for the original MC algorithm, we can limit vertex calculations to once per vertex to speed up the surface extraction process, however, it did not mention how this process could be done efficiently. Neither was the reuse of these MC vertices looked into seriously in the literature. In this paper, we propose a "Group Marching Cube" (GMC) algorithm, to reduce the time needed for the vertex identification process, which is part of the surface extraction process. Since most of the triangle-vertices of an isosurface are shared by many MC triangles, the vertex identification process can avoid the duplication of the vertices in the vertex array of the resultant triangle data. The MC algorithm is usually done through a hash table mechanism proposed in the literature and used by many software systems. Our proposed GMC algorithm considers a group of voxels simultaneously for the application of the MC algorithm to explore interesting features of the original MC algorithm that have not been discussed in the literature. Based on our experiments, for an object with more than 1 million vertices, the GMC algorithm is 3 to more than 10 times faster than the algorithm using a hash table. Another significant advantage of GMC is its compatibility with other algorithms that accelerate the MC algorithm. Together, the overall performance of the original MC algorithm is promoted even further.

Topology Preserving Digitization with FCC and BCC Grids

Lecture Notes in Computer Science, 2006

In digitizing 3D objects one wants as much as possible object properties to be preserved in its digital reconstruction. One of the most fundamental properties is topology. Only recently a sampling theorem for cubic grids could be proved which guarantees topology preservation . The drawback of this theorem is that it requires more complicated reconstruction methods than the direct representation with voxels. In this paper we show that face centered cubic (fcc) and body centered cubic (bcc) grids can be used as an alternative. The fcc and bcc voxel representations can directly be used for a topologically correct reconstruction. Moreover this is possible with coarser grid resolutions than in the case of a cubic grid. The new sampling theorems for fcc and bcc grids also give absolute bounds for the geometric error.

Extending marching cubes with adaptative methods to obtain more accurate iso-surfaces

2010

This work proposes an extension of the Marching Cubes algorithm, where the goal is to represent implicit functions with higher accuracy using the same grid size. The proposed algorithm displaces the vertices of the cubes iteratively until the stop condition is achieved. After each iteration, the difference between the implicit and the explicit representations is reduced, and when the algorithm finishes, the implicit surface representation using the modified cubical grid is more accurate, as the results shall confirm. The proposed algorithm corrects some topological problems that may appear in the discretization process using the original grid.

Marching Cubes in an Unsigned Distance Field for Surface Reconstruction from Unorganized Point Sets

Proceedings of the International Conference on Computer Graphics Theory and Applications, 2010

Surface reconstruction from unorganized point set is a common problem in computer graphics. Generation of the signed distance field from the point set is a common methodology for the surface reconstruction. The reconstruction of implicit surfaces is made with the algorithm of marching cubes, but the distance field of a point set can not be processed with marching cubes because the unsigned nature of the distance. We propose an extension to the marching cubes algorithm allowing the reconstruction of 0-level iso-surfaces in an unsigned distance field. We calculate more information inside each cell of the marching cubes lattice and then we extract the intersection points of the surface within the cell then we identify the marching cubes case for the triangulation. Our algorithm generates good surfaces but the presence of ambiguities in the case selection generates some topological mistakes.

Topological equivalence between a 3D object and the reconstruction of its digital image

IEEE transactions on pattern …, 2007

Digitization is not as easy as it looks. If one digitizes a 3D object even with a dense sampling grid, the reconstructed digital object may have topological distortions and, in general, there exists no upper bound for the Hausdorff distance. This explains why so far no algorithm has been known which guarantees topology preservation. However, as we will show, it is possible to repair the obtained digital image in a locally bounded way so that it is homeomorphic and close to the 3D object. The resulting digital object is always wellcomposed, which has nice implications for a lot of image analysis problems. Moreover, we will show that the surface of the original object is homeomorphic to the result of the marching cubes algorithm. This is really surprising since it means that the well-known topological problems of the marching cubes reconstruction simply do not occur for digital images of r-regular objects. Based on the trilinear interpolation, we also construct a smooth isosurface from the digital image that has the same topology as the original surface. Finally, we IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 29, NO. 1, JANUARY 2007 1 . P. Stelldinger is with the