论文标题
使用梯度图拉普拉斯正规器的歧管图信号恢复
Manifold Graph Signal Restoration using Gradient Graph Laplacian Regularizer
论文作者
论文摘要
在图形信号处理（GSP）文献中，使用图形拉普拉斯正常化程序（GLR）进行信号恢复，以促进相对于基础图的分段平滑 /恒定重建。但是，对于跨图内的信号逐渐变化，GLR遭受了不良的“楼梯”效果。在本文中，专注于歧管图 - 低维连续歧管上的均匀离散样品的集合 - 我们将GLR推广到梯度图拉普拉斯（GGLR），促进平面 /分段平面（PWP）信号重构。具体而言，对于具有采样坐标（例如2D图像，3D点云）的图形，我们首先定义梯度运算符，使用该梯度运算符，我们在采样歧管空间中构造了节点梯度的梯度图。这将映射到梯度诱导的淋巴结图（GNG）和带有平面信号作为0频率的阳性半明确（PSD）laplacian矩阵。对于无显式采样坐标的歧管图，我们提出了一种图形嵌入方法，以通过快速特征向量计算获得节点坐标。我们得出有效的gglr的均值 - 纠正率最小化权重参数，从而消除了信号估计值的偏差和差异。实验结果表明，在一系列图信号恢复任务中，GGLR的表现优于GLR和GRAPH总变化（GTV），例如GGLR先前的图形信号先验。
In the graph signal processing (GSP) literature, graph Laplacian regularizer (GLR) was used for signal restoration to promote piecewise smooth / constant reconstruction with respect to an underlying graph. However, for signals slowly varying across graph kernels, GLR suffers from an undesirable "staircase" effect. In this paper, focusing on manifold graphs -- collections of uniform discrete samples on low-dimensional continuous manifolds -- we generalize GLR to gradient graph Laplacian regularizer (GGLR) that promotes planar / piecewise planar (PWP) signal reconstruction. Specifically, for a graph endowed with sampling coordinates (e.g., 2D images, 3D point clouds), we first define a gradient operator, using which we construct a gradient graph for nodes' gradients in sampling manifold space. This maps to a gradient-induced nodal graph (GNG) and a positive semi-definite (PSD) Laplacian matrix with planar signals as the 0 frequencies. For manifold graphs without explicit sampling coordinates, we propose a graph embedding method to obtain node coordinates via fast eigenvector computation. We derive the means-square-error minimizing weight parameter for GGLR efficiently, trading off bias and variance of the signal estimate. Experimental results show that GGLR outperformed previous graph signal priors like GLR and graph total variation (GTV) in a range of graph signal restoration tasks.
