ℓ1优化在网格去噪中的应用（HHME2013）

网格去噪是计算机图形学中的经典问题，而如何在去除噪声的同时保持网格的特征结构是这一研究方向所面临的最大挑战。提出一种具有稀疏性的全局网格去噪方法，该方法源于信号处理理论中稀疏表示的基本思想，通过优化全局能量函数来去除网格模型的噪声，同时能够保持网格模型的特征结构。该方法共分为两个步骤，第一步为网格面法向量的滤波，首先建立全局优化模型，对噪声网格的面法向量进行滤波优化，其中引入ℓ1范数来保证解的稀疏性，使得优化后新的面法向量能够保持网格的特征结构；第二步为网格曲面的重建，根据第一步得到的新的面法向量，按照面法向量的定义，建立最小二乘意义下的网格顶点的重建模型，求解得到新的网格曲面。另外，由于该模型是全局方法，避免了现有滤波方法可能出现的不收敛等问题，能够取得比较满意的去噪效果。最后，通过大量实验表明，在去除噪声的同时，能较好地保持网格的特征结构，尤其对于CAD模型有很好的实验效果。

The application of ℓ1-optimization in mesh denoising

Mesh denoising is a typical problem in Computer Graphics. The key challenge we face in this field is to denoise the mesh and maintain the structure of the mesh at the same time. And it is becoming the hottest topic in this area. We propose a global mesh denoising method using ℓ1-sparsity. This method is motivated by the fundamental theory of sparse representation in the field of signal processing. The global optimization of an energy function is employed to remove noises from the mesh while the features are preserved. There are two steps in our method. The first step is the filtering of the face normals. We formulate a global optimization model to optimize the face normals of the noised mesh. Then we use the ℓ1-norm to ensure the sparsity of the solution, which preserves the structures of mesh features. The second step is the reconstruction of the dnoised mesh. Given the new filtered face normals, we create a vertex reconstruction model under the least-square sense according to the definition of the face normal. The denoised mesh is updated by the solution of the reconstruction model. Furthermore, our model solves the denoising problem globally, which avoids the problems appeared in existing methods, such as the convergence problem. A large number of experiments demonstrate that our method is able to remove noises, at the same time, preserve the features of the mesh, especially for the CAD models.