We propose a novel discrete concept for the total generalized variation (TGV), which has originally been derived to reduce the staircasing effect in classical total variation (TV) regularization, in image denoising problems. We describe discrete, second-order TGV for piecewise constant functions on triangular meshes, thus allowing the TGV functional to be applied to more general data structures than pixel images, and in particular in the context of finite element discretizations. Particular attention is given to the description of the kernel of the TGV functional, which, in the continuous setting, consists of linear polynomials. We discuss how to take advantage of this kernel structure using piecewise constant functions on triangular meshes. Numerical experiments include denoising and inpainting problems for images defined on non-standard grids, including data from a 3D scanner.
翻译:我们提出了一个关于全普遍变异的新颖的离散概念(TGV),最初是用来减少传统全变异(TV)正规化过程中在图像去除问题方面的阶梯效应的。我们用三角间贝的片常数函数描述离散的、二级的TGV,从而使TGV的功能能够适用于比像素图像更一般的数据结构,特别是在有限元素分解的情况下。我们特别注意了TGV功能内核的描述,在连续设置中,该内核由线性多球形组成。我们讨论如何利用三角间贝壳上的片常数函数来利用这一内核结构。数字实验包括非标准网格上界定的图像的分解和油漆问题,包括来自3D扫描仪的数据。