分数阶傅里叶域中图形信号的最优采样与重构
Optimal Sampling and Reconstruction of Graph Signals in the Fractional Fourier Domain
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中文总结 AI 辅助
研究图形信号在分数阶傅里叶域的采样与重构，提出基于GFRFT域的框架，引入分数阶作为参数，通过统一公式结合多种先验，得出多种方法，理论分析表明其能提供更优表示，实验显示该域采样重构性能优于GFT域方法。
中文摘要 AI 辅助
图形信号采样与重构通常在图形傅里叶变换（GFT）域中进行。然而，当实际图形信号在GFT频谱中集中度不足时，重构性能可能受限。本文提出基于图形分数阶傅里叶变换（GFRFT）域的图形信号采样与重构框架。引入分数阶作为可调整的频谱域参数，选择最优阶为给定图形信号和采样模型提供更合适的表示域。在统一采样重构公式下，纳入子空间、平滑度和随机先验，考虑无约束和预定义重构机制，得出多种分数域采样与重构方法。理论分析表明，最优GFRFT域可通过提高能量集中度和降低投影残差提供更合适的低维频谱表示。还考虑了残余泄漏和噪声放大的影响以解释这种表示优势如何转化为重构误差降低。实验结果表明，GFRFT域采样与重构通常比GFT域方法具有更好的恢复性能。
英文摘要
Graph signal sampling and reconstruction are commonly formulated in the graph Fourier transform (GFT) domain. However, the reconstruction performance may be limited when practical graph signals are not sufficiently concentrated in the GFT spectrum. To address this issue, this paper proposes a graph signal sampling and reconstruction framework based on the graph fractional Fourier transform (GFRFT) domain. The fractional order is introduced as an adjustable spectral domain parameter, and the optimal order is selected to provide a more suitable representation domain for a given graph signal and sampling model. Under a unified sampling reconstruction formulation, subspace, smoothness, and stochastic priors are incorporated, and both unconstrained and predefined reconstruction mechanisms are considered, leading to several fractional domain sampling and reconstruction methods. Furthermore, the theoretical analysis shows that the optimal GFRFT domain can provide a more suitable low-dimensional spectral representation by improving energy concentration and reducing projection residual. The effects of residual leakage and noise amplification are further considered to explain how this representation advantage is translated into reconstruction error reduction. Experimental results show that, GFRFT domain sampling and reconstruction generally achieve better recovery performance than GFT domain methods.
