压缩感知理论改变了香农采样定理的信号处理思路,具有十分重要的科研应用价值。压缩感知框架下信号重构是获取数字终端产品的关键性环节,典型的重构方法是以基追踪(BP)算法为代表,核心是解决L₁范数最小化问题,但是BP算法在高维的信号重构中表现不佳。因此,本文提出一种基于分形维度的压缩感知高维信号重构方法,采用分形中的Minkowski维度代替L₁范数作为重构问题的目标函数。实验的可视化结果和信噪比均表明,分形压缩感知信号重构方法既保持了BP算法的优点又改善了其维度的广延性。

In the research field of digital signal processing,compressive sensing (CS) becomes more and more important because it changes the traditional signal processing method based on Shannon’s sampling theorem.Under the CS framework,signal recovery is a key point to obtain the digital termination product.The basis pursuit (BP) algorithm seems the most fundamental method of CS recovery,which is essentially an L₁-norm minimization problem.However,BP can not be used for the signals with more than one dimension.Therefore,this paper presents a new high-dimension CS recovery method based on fractal dimension theory.The Minkowski dimension is used to replace the L₁-norm as an object function in CS recovery.The visualization and SNR of our experimental results show that fractal CS recovery not only inherits the advantage of BP but also improves the dimensional extensive property.