压缩感知理论及在成像中的应用
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摘要
军事需求对信息的准确性、时效性和多样性提出了越来越高的要求,作为获取信息的主要手段,成像系统需要不断地提升其分辨率等性能指标。在Shannon采样定理的体系中,提升分辨率意味着减少探测器像素尺寸、增加阵元数量,这将造成系统的复杂度和实现难度非线性地增加。
压缩感知理论提供了一种革命性的解决思路:引入信号的稀疏性,利用少量非相关的压缩采样,通过稀疏优化算法实现信号的高精度重构,避免了盲目追求高分辨率的探测器。本文对压缩感知及其在成像中的应用进行了系统的研究。
理论方面,研究了基于调和变换和数据驱动的信号稀疏表示方法,并利用典型遥感图像库进行实证分析比较。以最小1范数凸优化为主,详细讨论了稀疏重构模型与算法。提出了基于累加互相关性的重构条件,并给出了稀疏重构的误差理论上界。该重构条件综合了已有的相关性和RIP两种准则的优点。同时,利用累加互相关性推广了基于互相关性的测量矩阵优化设计准则。
方法方面,不同于传统从信号匹配滤波的角度解释的成像方法,本文将图像重构视为Fredholm第一类积分方程的求解,利用Hilbert空间的算子理论分析了求解过程病态性的原因,研究通过加入稀疏性约束项将其转为良态的,在此基础上,引入测量矩阵以完成压缩采样,从而建立压缩成像方法。通过系统调制传递函数建模和稀疏重构误差分析,建立了压缩成像性能分析模型,定量地分析了信号稀疏度、测量数据量以及测量噪声等因素对图像重构精度的影响。
应用方面,首先讨论了光学压缩成像的实现模式,包括:1)提出压缩感知量子成像,说明随机热光源符合可重构条件,利用稀疏优化算法显著提高了成像质量。2)研究了焦平面编码的高分辨率成像模式,讨论了基于多路技术和掩膜或DMD编码的压缩采样方式,并提出了变分辨率智能成像模式。3)研究了CMOS低数据率成像模式,通过模拟域向量-矩阵相乘完成投影测量,显著降低了数据率。其次,讨论了压缩成像在雷达系统中的应用,包括1)多测速体制下的高精度雷达目标定位,通过运动轨迹的样条函数表示和节点优化,以及基于稀疏约束的系统误差估计,实现了高精度的弹道解算。2)随机噪声雷达的稀疏重构成像,克服了传统方法背景噪声电平较高的缺陷,在低于Nyquist频率的采样率下,实现无模糊高分辨率成像。3)低数据率ISAR成像,利用随机(0, 1)序列相乘和积分器完成压缩采样,显著降低AD转化的速率,并通过最小1范数方法完成高精度图像重构。
The military applications call for more stringent requirements for the accuracy, timeliness and diversity of information. As a key approach to acquire it, the imaging systems need a substantial performance increase, such as resolution enhancement. In the box of Shannon sampling theorem, raising the resolution means cutting the pixel size of sensor and increasing the number of pixels. This will enlarge the complexity and difficulty of system nonlinearly.
Compressive sensing (CS) has given a revolutionary solution: Based on the sparsity of signal and sub-Nyquist non-coherent compressed sampling, one can recover the original signal by sparse optimization. It avoids the blind pursuit of an excessively high-resolution sensor. This paper investigates the theory of CS and its application in imaging system.
In theoretical aspect, we have studied the Harmonic analysis based and data driven method for sparse representation. A contrast analysis of these methods is given by testing them on a typical database of remote-sensing images. Take the 1 -norm minimization as main factor, the models and algorithms of sparse recovery have been discussed. We have proposed a recovery condition based on cumulative mutual coherence (CMC), and given a theoretical upper bound of the recovery error. This recovery condition combines the advantages the mutual coherence (MC) and restricted isometry property (RIP). Meanwhile, we have extended the optimization criterion of measurement matrices by using CMC.
Unlike the traditional imaging methods which begin with the matched filtering, we take the process of imaging as solving a Fredholm first kind integral equation. The theory of operators Hilbert space in employed to analyze the ill-conditioning of the solution. The constraint of sparsity is introduced to turn it to well-conditioned. After that, the measurement matrix is imported to finish the compressed sampling. Finally, the compressive imaging method is established. We have developed the performance analysis model of compressive imaging via modulation transfer function and error analysis of sparse recovery. The quantitative effects of sparsity of signal, amount of measurements, and noise on the accuracy of image reconstruction are given.
As for practical aspect, firstly, we have considered several realization modes of optical compressive imaging. 1) We have proposed the compressive quantum imaging, and pointed out that the chaotic thermal lights satisfy the recovery condition. So, the quality of imaging can be enhanced thanks to the sparse recovery. 2) We have studied the high-resolution imaging based on focal plan coding. The technology of multiplexing and amplitude mask or (Digital Micro-mirror Device) DMD is used to complete the compressed sampling. A resolution-varied imaging mode is given. 3) We have discussed a low-data-rate imaging mode by using CMOS. The compressed sampling is finished by vector-matrix multiplication in analog domain, and the data rate is significantly reduced. Secondly, we have investigated the application of CS in radar system. 1) The target poisoning in multi-range-rate radar system. By the spline function modeling with knots optimization of the target trajectory, and systematic error estimation via sparse restriction, we have realized a high-precision trajectory solving method. 2) Random noise radar sparse image reconstruction, which can overcome the shortcoming that traditional imaging methods suffer from high background noise floor. Using the new method, one can achieve high-resolution unambiguous imaging even though the sampling rate is below Nyquist rate. 3) Low-data-rate (Inverse Synthetic Aperture Radar) ISAR imaging, in which compressed sampling is complete by multiplication of echoes with (0, 1) random sequences and integration. This will sufficiently reduce the needed rate for Analog-to-Digital conversion. The 1-norm minimization is utilized to high accuracy image reconstruction.
压缩感知理论提供了一种革命性的解决思路:引入信号的稀疏性,利用少量非相关的压缩采样,通过稀疏优化算法实现信号的高精度重构,避免了盲目追求高分辨率的探测器。本文对压缩感知及其在成像中的应用进行了系统的研究。
理论方面,研究了基于调和变换和数据驱动的信号稀疏表示方法,并利用典型遥感图像库进行实证分析比较。以最小1范数凸优化为主,详细讨论了稀疏重构模型与算法。提出了基于累加互相关性的重构条件,并给出了稀疏重构的误差理论上界。该重构条件综合了已有的相关性和RIP两种准则的优点。同时,利用累加互相关性推广了基于互相关性的测量矩阵优化设计准则。
方法方面,不同于传统从信号匹配滤波的角度解释的成像方法,本文将图像重构视为Fredholm第一类积分方程的求解,利用Hilbert空间的算子理论分析了求解过程病态性的原因,研究通过加入稀疏性约束项将其转为良态的,在此基础上,引入测量矩阵以完成压缩采样,从而建立压缩成像方法。通过系统调制传递函数建模和稀疏重构误差分析,建立了压缩成像性能分析模型,定量地分析了信号稀疏度、测量数据量以及测量噪声等因素对图像重构精度的影响。
应用方面,首先讨论了光学压缩成像的实现模式,包括:1)提出压缩感知量子成像,说明随机热光源符合可重构条件,利用稀疏优化算法显著提高了成像质量。2)研究了焦平面编码的高分辨率成像模式,讨论了基于多路技术和掩膜或DMD编码的压缩采样方式,并提出了变分辨率智能成像模式。3)研究了CMOS低数据率成像模式,通过模拟域向量-矩阵相乘完成投影测量,显著降低了数据率。其次,讨论了压缩成像在雷达系统中的应用,包括1)多测速体制下的高精度雷达目标定位,通过运动轨迹的样条函数表示和节点优化,以及基于稀疏约束的系统误差估计,实现了高精度的弹道解算。2)随机噪声雷达的稀疏重构成像,克服了传统方法背景噪声电平较高的缺陷,在低于Nyquist频率的采样率下,实现无模糊高分辨率成像。3)低数据率ISAR成像,利用随机(0, 1)序列相乘和积分器完成压缩采样,显著降低AD转化的速率,并通过最小1范数方法完成高精度图像重构。
The military applications call for more stringent requirements for the accuracy, timeliness and diversity of information. As a key approach to acquire it, the imaging systems need a substantial performance increase, such as resolution enhancement. In the box of Shannon sampling theorem, raising the resolution means cutting the pixel size of sensor and increasing the number of pixels. This will enlarge the complexity and difficulty of system nonlinearly.
Compressive sensing (CS) has given a revolutionary solution: Based on the sparsity of signal and sub-Nyquist non-coherent compressed sampling, one can recover the original signal by sparse optimization. It avoids the blind pursuit of an excessively high-resolution sensor. This paper investigates the theory of CS and its application in imaging system.
In theoretical aspect, we have studied the Harmonic analysis based and data driven method for sparse representation. A contrast analysis of these methods is given by testing them on a typical database of remote-sensing images. Take the 1 -norm minimization as main factor, the models and algorithms of sparse recovery have been discussed. We have proposed a recovery condition based on cumulative mutual coherence (CMC), and given a theoretical upper bound of the recovery error. This recovery condition combines the advantages the mutual coherence (MC) and restricted isometry property (RIP). Meanwhile, we have extended the optimization criterion of measurement matrices by using CMC.
Unlike the traditional imaging methods which begin with the matched filtering, we take the process of imaging as solving a Fredholm first kind integral equation. The theory of operators Hilbert space in employed to analyze the ill-conditioning of the solution. The constraint of sparsity is introduced to turn it to well-conditioned. After that, the measurement matrix is imported to finish the compressed sampling. Finally, the compressive imaging method is established. We have developed the performance analysis model of compressive imaging via modulation transfer function and error analysis of sparse recovery. The quantitative effects of sparsity of signal, amount of measurements, and noise on the accuracy of image reconstruction are given.
As for practical aspect, firstly, we have considered several realization modes of optical compressive imaging. 1) We have proposed the compressive quantum imaging, and pointed out that the chaotic thermal lights satisfy the recovery condition. So, the quality of imaging can be enhanced thanks to the sparse recovery. 2) We have studied the high-resolution imaging based on focal plan coding. The technology of multiplexing and amplitude mask or (Digital Micro-mirror Device) DMD is used to complete the compressed sampling. A resolution-varied imaging mode is given. 3) We have discussed a low-data-rate imaging mode by using CMOS. The compressed sampling is finished by vector-matrix multiplication in analog domain, and the data rate is significantly reduced. Secondly, we have investigated the application of CS in radar system. 1) The target poisoning in multi-range-rate radar system. By the spline function modeling with knots optimization of the target trajectory, and systematic error estimation via sparse restriction, we have realized a high-precision trajectory solving method. 2) Random noise radar sparse image reconstruction, which can overcome the shortcoming that traditional imaging methods suffer from high background noise floor. Using the new method, one can achieve high-resolution unambiguous imaging even though the sampling rate is below Nyquist rate. 3) Low-data-rate (Inverse Synthetic Aperture Radar) ISAR imaging, in which compressed sampling is complete by multiplication of echoes with (0, 1) random sequences and integration. This will sufficiently reduce the needed rate for Analog-to-Digital conversion. The 1-norm minimization is utilized to high accuracy image reconstruction.
引文
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