无穷可分层叠的分形结构与谱分布理论及其应用研究

The generalized Infinitely Divisible Cascades (IDC) is a kind of very typical and wide random multifractal multiplicative process in nature, and the molding and spectrum analysis of IDC has been the focus and difficulty of current research in the field of fractal and signal processing. Therefore, this project mainly studies the fractal structure, singularity spectrum distribution and spectrum estimation and application of IDC processes. The researches include three parts. Firstly, we study the invariant characterization of IDC in the multi-scale transformation, propose the time-singularity spectrum distribution theory based on the fractal structure and build the complete framework of singularity spectrum analysis of generalized IDC, break the traditional studies, which ignored the different types of singularity and only were confined to differentiability and fractal dimension, and then extend the one-dimension to the multidimensional IDC modeling and spectrum distribution. Secondly, we will study the robust estimation theory and algorithm of IDC spectrum distribution, and propose the wavelet leaders based estimation theory, the Bootstrap resampling technique based algorithm of singularity spectrum analysis and time-singularity spectrum distribution, and furthermore exploit confidence intervals, multi-models hypothesis testing and the time-scale correlation features, and then extend them to the multidimensional IDC. Thirdly, we propose the new target detection and identification algorithms of high-resolution range profile based on the IDC theory, and propose SAR electromagnetic scattering model and new theories and algorithms of SAR imaging based on the two-dimensional IDC surface modeling. The research of this project not only can promote the development of fractal theory and nonlinear theory, but also enriches and develops the theory system of modern signal processing.

广义无穷可分层叠（IDC）是一类自然界广泛存在的随机分形乘性过程，精确高效的IDC建模和谱分析是当前研究的热点和难点。因此，本项目研究IDC的分形结构、奇异谱分布理论及应用，包括：①研究IDC尺度变换下的不变量表征方法，突破传统研究忽略不同奇异类型、仅对可微性和分维分析的局限，提出基于分形结构的时间-奇异性功率谱分布理论，构建广义IDC的完备谱分析框架，并推广到多维IDC建模及其谱分布；②研究稳健的IDC谱估计理论和算法，提出基于小波Leaders和Bootstrap重采样的奇异域谱估计和时间-奇异域二维谱估计方法，研究谱估计的置信区间、多模型假设检验和时间-尺度相关性问题，并推广到多维IDC。③提出基于IDC理论的高分辨距离像检测和识别新算法；提出基于二维IDC建模的SAR电磁散射模型和成像新理论、新算法。本项目的研究不但能推动分形和非线性理论发展，而且会丰富和发展现代信号处理理论体系。

广义无穷可分层叠（IDC）是一类自然界广泛存在的随机分形乘性过程，精确高效的IDC建模和谱分析是随机分形信号处理中的热点和难点问题。本项目研究IDC的分形结构特征、奇异谱分布理论及应用技术，包括①IDC尺度变换下的不变量表征方法，包括基于分形结构的一维和二维信号的时间-奇异性多重分形分布（TSMFSD）理论、多重分形互相关谱（MFCCA）分析理论、奇异性功率谱分布（SPSD）和互奇异性功率谱分布（CSPSD）理论；②研究稳健的IDC谱估计理论和算法，提出基于小波Leaders和去趋势项分析（DFA）的奇异域谱估计和时间-奇异域二维谱估计方法，以及奇异性功率谱和互奇异性功率谱估计算法。③提出基于随机多重分形谱分析的高分辨距离像检测新方法，以及提出基于二维随机分形建模的SAR电磁散射模型和SAR目标检测方法。通过上述研究，取得了重要的理论成果，包括：（1）提出了奇异性功率谱分布（SPSD）分析方法和分数域性SPS分析方法；（2）提出基于SPS和TSPSD的随机多重分形信号重构方法；（3）优化了二维多重分形交叉谱分布理论（二维MFCCA）和算法；（4）提出了奇异域互相关功率谱分析理论（CSPS）和算法；在应用方法，在对雷达海杂波的时间-奇异性功率谱分布和SPS分析的基础上，提出了基于随机乘法层叠模型的雷达海面二维建模的方法，以及多重分形电磁散射建模及多重分形谱分析。进一步，提出了基于CSPS的雷达HRRP/SAR目标检测方法，显著提升了低信噪比、慢速、HRRP及SAR小目标检测能力。上述研究成果不但能推动分形和非线性理论发展，而且会丰富和发展现代信号处理理论体系。