Clustering Vertex-Weighted Graphs by Spectral Methods

Spectral techniques are often used to partition the set of vertices of a graph, or to form clusters. They are based on the Laplacian matrix. These techniques allow easily to integrate weights on the edges. In this work, we introduce a p-Laplacian, or a generalized Laplacian matrix with potential, which also allows us to take into account weights on the vertices. These vertex weights are independent of the edge weights. In this way, we can cluster with the importance of vertices, assigning more weight to some vertices than to others, not considering only the number of vertices. We also provide some bounds, similar to those of Chegeer, for the value of the minimal cut cost with weights at the vertices, as a function of the first non-zero eigenvalue of the p-Laplacian (an analog of the Fiedler eigenvalue).

An Improved Version of the Generalized Laplacian Pyramid Algorithm for Pansharpening

The spatial resolution of multispectral data can be synthetically improved by exploiting the spatial content of a companion panchromatic image. This process, named pansharpening, is widely employed by data providers to augment the quality of images made available for many applications. The huge demand requires the utilization of efficient fusion algorithms that do not require specific training phases, but rather exploit physical considerations to combine the available data. For this reason, classical model-based approaches are still widely used in practice. We created and assessed a method for improving a widespread approach, based on the generalized Laplacian pyramid decomposition, by combining two different cost-effective upgrades: the estimation of the detail-extraction filter from data and the utilization of an improved injection scheme based on multilinear regression. The proposed method was compared with several existing efficient pansharpening algorithms, employing the most credited performance evaluation protocols. The capability of achieving optimal results in very different scenarios was demonstrated by employing data acquired by the IKONOS and WorldView-3 satellites.