Sampling on graph signals is one of the fundamental topics in graph signal processing. In this letter, we consider the random sampling of$k$-bandlimited signals from the local measurements and show that no more than$O(k\log k)$measurements with replacement are sufficient for the accurate and stable recovery of any$k$-bandlimited graph signals. We propose two random sampling strategies based on the minimum measurements, i.e., the optimal sampling and the estimated sampling. The geodesic distance between vertices is introduced to design the sampling probability distribution. Numerical experiments are included to show the effectiveness of the proposed methods.

Magnetic resonance imaging (MRI) can provide multi-modality MR images by setting task-specific scan parameters, and has been widely used in various disease diagnosis and planned treatments. However, in practical clinical applications, it is often difficult to obtain multi-modality MR images simultaneously due to patient discomfort, and scanning costs, etc. Therefore, how to effectively utilize the existing modality images to synthesize missing modality image has become a hot research topic. In this paper, we propose a novel confidence-guided aggregation and cross-modality refinement network (CACR-Net) for multi-modality MR image synthesis, which effectively utilizes complementary and correlative information of multiple modalities to synthesize high-quality target-modality images. Specifically, to effectively utilize the complementary modality-specific characteristics, a confidence-guided aggregation module is proposed to adaptively aggregate the multiple target-modality images generated from multiple source-modality images by using the corresponding confidence maps. Based on the aggregated target-modality image, a cross-modality refinement module is presented to further refine the target-modality image by mining correlative information among the multiple source-modality images and aggregated target-modality image. By training the proposed CACR-Net in an end-to-end manner, high-quality and sharp target-modality MR images are effectively synthesized. Experimental results on the widely used benchmark demonstrate that the proposed method outperforms state-of-the-art methods.