International Journal of Digital Multimedia Broadcasting

Volume 2017, Article ID 2724184, 15 pages

https://doi.org/10.1155/2017/2724184

## Adaptive Geometry Images for Remeshing

^{1}Beijing Key Laboratory of Multimedia & Intelligent Software Technology, Faculty of Information Technology, Beijing University of Technology, Beijing 100124, China^{2}Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Shaofan Wang; nc.ude.tujb@nafoahsgnaw

Received 6 March 2017; Accepted 30 April 2017; Published 2 August 2017

Academic Editor: Yifeng He

Copyright © 2017 Lina Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Geometry images are a kind of completely regular remeshing methods for mesh representation. Traditional geometry images have difficulties in achieving optimal reconstruction errors and preserving manually selected geometric details, due to the limitations of parametrization methods. To solve two issues, we propose two adaptive geometry images for remeshing triangular meshes. The first scheme produces geometry images with the minimum Hausdorff error by finding the optimization direction for sampling points based on the Hausdorff distance between the original mesh and the reconstructed mesh. The second scheme produces geometry images with higher reconstruction precision over the manually selected region-of-interest of the input mesh, by increasing the number of sampling points over the region-of-interest. Experimental results show that both schemes give promising results compared with traditional parametrization-based geometry images.

#### 1. Introduction

Triangular meshes are important tools for representing geometric data in computer graphics, due to the ease of generation procedure from point clouds and simple manipulation. However, many applications of meshes such as mesh morphing and mesh compression tend to use triangular meshes with regular structure. Traditional triangular meshes, which have irregular connectivity due to the generation procedure, need to be remeshed. This technique is referred to as* remeshing*. Geometry images are a completely regular remeshing method, which represents a triangular mesh as an image array, where the vertex-set of the mesh is stored as the pixels of the image and where the connectivity of the mesh is intrinsically embedded in the image array. Such a regular structure of meshes is helpful for reducing the representation of geometric data and coworking well with many image-based applications such as image compression and rendering process.

In general, geometry images include three steps: mesh parametrization, resampling, and quantification. The first step maps 3D vertices of the input mesh to regular parametrization domain (square, rectangle, or sphere), the second step imposes sampling over the parametrization domain via interpolation methods, and the third step transforms the coordinates of sampling points to pixel values of an image array. To reconstruct a mesh for geometry images, the vertex-set is obtained from the pixels of the image array, and the edge-set is obtained from the connectivity of the adjacent pixels of the array.

Although fruitful research work was proposed for geometry images, many of them focuses on the mesh parametrization and ignores the importance of the resampling step, which increases the burden of parametrization technique, as the parametrization leads to a complicated and nonconvex optimization which heavily depends on the connectivity of the input mesh (while the resampling scheme depends more on the connectivity of the regular sampling fashion instead of the input mesh). In particular, traditional geometry images have difficulties in achieving optimal reconstruction errors, or in preserving manually selected geometric details. To solve such two issues, we propose two adaptive geometry images. The first scheme produces geometry images with the minimum Hausdorff error, by finding the optimization direction for sampling points based on the Hausdorff distance between the original mesh and the reconstructed mesh. The second scheme produces geometry images with higher reconstruction precision over the manually selected region-of-interest of the input mesh, by increasing the number of sampling points over the region-of-interest. We compare our schemes with traditional geometry images using state-of-the-art mesh parametrization scheme and adaptive sampling scheme in terms of both reconstruction error and mesh compression. Experimental results on both qualitative comparison and quantitative comparison show that our schemes outperform traditional geometry images.

#### 2. Related Work

##### 2.1. Geometry Images

Gu et al. [1] propose the pioneering work of geometry images, which maps a triangular mesh onto a square domain by using a minimizing-geometric-stretch parametrization and gives a regular sampling for surface geometry. Praun and Hoppe [2] propose spherical parametrization for geometry images, which facilitates the representation of genus-zero closed meshes. Gauthier and Poulin [3] fill nonzero genus meshes and propose spherical parametrization for treating meshes of arbitrary genus. Zhou et al. [4] propose an adaptive sampling scheme for geometry images, which keeps most details of models. Gauthier and Poulin [5] propose another sampling scheme for geometry images to maintain both edge features and sharp features. Meng et al. [6] adopt differential coordinates to correct the vector direction of the reconstruction model, which makes the reconstruction model accurately preserve the detailed features of the original model.

The aforementioned work of geometry images maps models into single-chart geometry images, which tends to produce high geometric stretch and ignore details of models. Alternatively, Tewari et al. [7] propose multichart geometry images by cutting the model into some irregular subslices, but it required a lot of space to store the information of subslices. Carr et al. [8] convert the irregular subslices to quadrilateral subslices. Yao and Lee [9] decompose a mesh into square GIM charts with different resolutions, each of which is adaptively determined by a local reconstruction error. Feng et al. [10] propose geometry images for generating triangular patches based on a curvilinear feature. The feature preserves salient features and supports GPU-based LOD representation of meshes.

##### 2.2. Other Remeshing Methods

Alliez et al. [11] propose an interactive remeshing of irregular geometry, which represents the original mesh as a series of 2D parametrization maps. The algorithm facilitates the real-time interaction and intricate control using a map which controls the sampling density over the surface patch. Alliez et al. [12] propose a polygonal remeshing method using the intrinsic anisotropy of natural or man-made geometry. The authors use curvature directions to drive the remeshing process and determine appropriate edges for the remeshed version in anisotropic regions. The method provides the flexibility to produce meshes ranging from isotropic to anisotropic, from coarse to dense, and from uniform to curvature adapted. Dong et al. [13] propose a new quadrilateral remeshing method for manifolds of arbitrary genus. The method computes the gradient of smooth harmonic scalar fields defined over the mesh and forms the polygons of the output mesh using two nets of integral lines. Huang et al. [14] propose a quadrangulation method, by extending the spectral surface quadrangulation approach with the coarse quadrangular structure derived from the Morse-Smale complex of an eigenfunction of the Laplacian operator on the input mesh. The quadrilateral mesh is reconstructed from the Morse-Smale complex by computing a globally smooth parametrization. Zhang et al. [15] propose a new method for remeshing a surface into a quadrangle, by constructing a special standing wave on the surface to generate the global quadrilateral structure, which controls the quad size in two directions and precisely aligning the quads with feature lines.

#### 3. Hausdorff Error Driven Geometry Images

We propose Hausdorff error driven geometry images in this section. The key step of our scheme is to find the points, edges, or faces of the original mesh and reconstruction mesh which achieve the maximum Hausdorff distance and then compute the gradient direction of the Hausdorff distance. Our scheme consists of three phases: an initial adaptive sampling, approximate representation of Hausdorff distance, and the adjustment of sampling vertices, which are described in the following three subsections and illustrated in Figure 1. We shall use the calligraphy letter for representing a mesh and denote , , to be the vertex-set, the face-set, and the point-set (i.e., all points within each face) of a mesh , respectively, and denote , , to be the edge-set of a face , the set of two end-vertices of an edge , and the set of three end-vertices of a face , respectively.