物理可实现的压缩成像理论与方法研究
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摘要
压缩传感理论是一种新兴的信号采集与处理的新理论,其突破了传统的Shannon/Nyquist采样定理的限制,主要得益于自然界中绝大多数信号都普遍具有的稀疏先验知识,可以从远少于Nyquist采样率的线性非相干测量中精确重建原信号。显然,从不完全的测量数据中重建完整信号是一个非适定的逆问题。压缩传感应用范围涉及信号处理的诸多领域都有着巨大的应用价值,如信息论、信号处理、光学成像、模式识别、信号采集和通信等领域。
压缩成像是压缩传感理论最重要的研究领域之一,第一个成功应用的压缩成像实例是美国Rice大学基于压缩传感理论设计的单像素相机,只使用一个探测器元素对场景成像。该相机有一个可以数字微控制的镜阵列,由一个伪随机二进制数组驱动,将感兴趣地场景投影到阵列上,投影聚集的强度使用单一的探测器测量,然后从得到的观测值重建原始图像。这种结构的一个主要优点是任何二进制投影矩阵可以很容易地在这个系统中实施,使现有的压缩传感理论可以直接应用到测量。单像素相机非常完美地与压缩传感理论相契合,一次随机调制仅仅获得一个测量值,在太赫兹成像等小部分成像领域成功地获得了实用。但是,单像素相机的最大缺陷之处在于同一时刻一次只能得到一个测量值,对于其它更广泛的高分辨率的应用领域,其获取足够测量的时间太长而不太实用。因此,Stern等人提出一种改进的压缩成像方法,可以在单次曝光条件下获得所有的测量值,在其他更广泛的成像领域中具有更大的应用潜力。
本文借鉴Stern的单次曝光思想,围绕的压缩成像的理论与方法、成像架构、物理可实现的压缩成像系统以及三维压缩成像理论与方法中遇到的一系列问题展开深入研究。
本文的研究工作及创新点主要集中在如下几个方面：
1)探讨了双透镜加确定性光学相位掩膜所构成的一种成像系统,给出压缩成像的基本原理和公式描述,并以托普利兹和循环两种确定性相位掩膜矩阵为例验证确定性相位掩膜成像方法的有效性。与Stern提出的单次曝光随机相位掩膜调制压缩成像方法相比,采用托普利兹和循环等确定性相位掩膜矩阵更利于物理实现和快速计算。此外,本文推导出压缩双透镜与确定性相位掩膜组成的成像系统满足压缩传感理论中对应的测量矩阵的任意两列之间不相关所需满足一些基本的条件,并给出具体的推导分析过程。
2)提出了几种新的随机间距稀疏测量矩阵,包括随机间距稀疏托普利兹／循环和随机三元间距稀疏托普利兹/循环矩阵,并给出这些测量矩阵满足相关限制等距性质的理论证明。与高斯矩阵等常用测量矩阵相比,新提出的矩阵随机独立元个数大大减少,重建速度有了较大的提高；然后将这几种新的随机间距稀疏测量矩阵应用到压缩双透镜成像系统中,验证其作为相位掩膜矩阵在实现对图像信息随机调制的有效性；结果表明所提出的随机间距稀疏相位掩膜矩阵在物理上更容易物理实现,存储量低,易于数据传输；并且在保证重建性能的同时,可以显著地节约了后端压缩传感重建的时间,对于压缩传感理论的实际应用具有重要意义。此外,还提出一种块状确定性矩阵作为相位掩膜矩阵,在也具有易于物理实现的优势的同时,更适宜多通道成像领域的应用,利于压缩成像理论走向实际应用。
3)将压缩传感理论应用于超分辨率成像重建问题,给出了在不同模糊内核作用下从低分辨率到高分辨率的超分辨率图像重建方法,结果显著优于直接插值方法。此外,还提出一种改进的基于4-f傅里叶光学成像架构的频域相位随机编码压缩成像方法,实现了图像信息的有效记录与恢复；针对实际可记录的测量值非负的限制,利用卷积原理分析实现测量值非负的基本原理,并给出改进的频域相位编码成像系统实现非负测量值的记录与成功重建,利于大尺度图像的压缩编码与重建,及其在实际中的应用与推广。为了实际掩膜的物理实现,本文进一步给出了一种改进的二元随机相位掩膜编码,规避了均匀随机相位掩膜的实际制作难度太大的困难。
4)提出一种基于4-f傅里叶光学成像架构的0/1频域振幅编码孔径以及与随机间距光栅或一般的均匀间距的光栅相结合的压缩成像方法。与传统的针孔相机相比,大大增加了光的利用率,提高了对噪声的鲁棒性。与随机相位掩膜编码相比,振幅编码孔径更易于物理实现。在此基础上,设计了一种改进的0/1频域振幅编码孔径压缩成像系统,保证了非负性约束,使得实际的物理测量矩阵与理论计算的测量矩阵之间有更好的一致性。
5)探讨压缩传感理论在三维成像领域中的应用,提出一种从一个二维全息图重建3D层析相干成像的多波长压缩全息成像方法；将散斑非相干压缩成像方法从单波长扩展到多波长和把压缩全息成像理论应用于稀疏孔径合成压缩全息成像等。在稀疏孔径合成压缩全息成像应用中改善了从多个低分辨率的传感器阵列合成高分辨率数字全息图像中遇到的相位不稳定的问题。最后给出物理实验验证了压缩全息成像方法的有效性。
Compressive sensing is a novel emerging theory of signal acquisition and processing, which break through the limitations of the traditional Shannon/Nyquist sampling theorem, the main benefit from that the vast majority of signals in the nature generally has the prior knowledge of sparsity, which can help to reconstruct the original signal accurately from the far less linear incoherent measurements than the Nyquist sampling rate. Obviously, recovering the real signal from incomplete measurement data is an ill-posed inverse problem. Compressive sensing theory has a huge potential value in many areas ranging from most signal processing applications, such as the field of information theory, signal processing, optical imaging, pattern recognition, signal acquisition and communications.
Compressive imaging is one of the most important research areas of compressive sensing theory, the first successful instance of compressive imaging applications is the single pixel camera designed by Rice University based on compressed sensing theory which use only one detector for scene imaging. The camera has a control of digital micro-mirror array, driven by a pseudo-random binary array, and reflects the the scene of interest onto a single detector array which gather the intensity of this projection, and then reconstruct the original image from the observation value. A major advantage of this structure is any binary projection matrix can be easily implemented in this imaging system, so that the compressed sensing theory can be naturally applied to the measurement. Single-pixel camera perfectly agrees with the compressive sensing theory, which using a random modulation only to obtain a measured value, and has successfully gained practical application in a small part of the imaging area such as Terahertz imaging. However, the most severely drawback of the single-pixel camera is that can only obtain a measured value at one time, however, in other more extensive high-resolution applications, the time to obtain sufficient measurements is too long and thus is not very practical. Therefore, Stern proposed an improved compressive imaging method, that all measured values can be obtained with a single exposure conditions, and has greater potential application in other more widely imaging areas.
This thesis draws on Stern's single exposure ideology, surrounded with a series of problems encountered in compressive imaging theories and methods, imaging architecture, physical realizable imaging system and three-dimensional compressive imaging theory for deeply research.
The main research works and contributions of this thesis are outlined as follows.
1) A novel imaging system consisting of double lens and deterministic optical phase mask is proposed, and the basic principles and formulas for compressive imaging are described, and the validity of the phase mask imaging method with Toeplitz and Circulant two deterministic phase mask matrix for instance is verified in the numerical simulations. Compared with the random phase mask modulation compressive imaging method with single-exposure proposed by Stern, using Toeplitz and Circulant and other deterministic phase mask matrix are more conducive to physical implementation and fast calculation. In addition, any two columns of measurement matrix are corresponding to compressive imaging system consisting of double-lens and deterministic phase mask to meet some basic conditions and specific derivation analysis are derived.
2) Several novel random spacing sparse measurement matrices, including the random spacing sparse Toeplitz/Circulant and random ternary spacing sparse Toeplitz/Circulant matrices are proposed, and the proofs for such measurement matrices to satisfiy the restricted isometric property are given. Compared with the Gaussian measurement matrices, the novel proposed matrices, independent component and storage spaces are reduced significantly, and the speed of reconstruction have greatly improved; then these novel types of random spacing sparse measurement matrices are applied into compressive double lens imaging system, the validity of them as a phase mask matrices to achieve image information random modulation is verified; The simulation results of proposed random spacing sparse phase mask matrix demonstrate that proposed masks is superior for its easier physical implementation, low storage capacity, ease of data transfer; and with the same reconstruction performance and the significantly reduced reconstruction time, which has great significance for practical application of compressed sensing theory. In addition, blocked measurement matrices as the phase mask matrices are also proposed, which have the advantage of easy physical implementation, and are more suitable for multi-channel imaging applications, and can help compressive imaging theory to practical application.
3) Compressive sensing was introduced into image super-resolution reconstruction problem, numerical simulation results demonstrate that reconstruction from low resolution to high resolution with different blurring kernel, that are much better than the direct interpolation method. In addition, an improved optical imaging method based on the4-f Fourier frequency domain phase random encoding for compressed imaging is proposed, which can capture image information and recover the image effectivrly;
Due to the measured values recorded in real-world should satisfy non-negative constraints, the convolution principle to analysis the basic principle for how to implementing the measurement values non-negative is analysized, and an improvement frequency domain phase encoding imaging method is proposed to implement non-negative recording of the measurement values and successful reconstruction after then, which is conducive to large-scale image compressed encoding and reconstruction, and its practical application and promotion. For the realization of actual mask, an improved encoding of binary random phase mask is also given in this thesis, to avoid the practical difficulty for producing uniform random phase mask.
4) A Fourier optical imaging architecture based on the4-f0/1frequency domain amplitude mask aperture imaging method with random spacing grating or evenly spaced grating combined are proposed. Compared with the traditional pinhole camera, the utilization of illumination light can be greatly increased in our method to improve the robustness against noise. Compared with the random phase mask encoding, the amplitude coded aperture are much easier for physical implementation. Based on this point, an improved frequency-domain amplitude0/1mask aperture compressive imaging system is designed, non-negative constraints is guaranteed, making a better consistency between the actual physical measurement matrices and the theoretical calculation of the measurement matrices.
5) Compressed sensing theory is applied in three-dimensional imaging, and a multi-wavelength compressive holography method is proposed which can reconstruct a3D coherent tomography imaging from a two-dimensional hologram; compressive imaging method with single wavelength speckle incoherent is extended to multi-wavelength, and compressive holography imaging is also applied into sparse aperture synthetic holography imaging which improves the phase instability problems encountered in sparse aperture synthetic holography imaging for mosaics of multiple low-resolution and low cost2D sensor arrays which can achieve large aperture; and the effectiveness of compressive holography imaging method is verified by physical experiments.
压缩成像是压缩传感理论最重要的研究领域之一,第一个成功应用的压缩成像实例是美国Rice大学基于压缩传感理论设计的单像素相机,只使用一个探测器元素对场景成像。该相机有一个可以数字微控制的镜阵列,由一个伪随机二进制数组驱动,将感兴趣地场景投影到阵列上,投影聚集的强度使用单一的探测器测量,然后从得到的观测值重建原始图像。这种结构的一个主要优点是任何二进制投影矩阵可以很容易地在这个系统中实施,使现有的压缩传感理论可以直接应用到测量。单像素相机非常完美地与压缩传感理论相契合,一次随机调制仅仅获得一个测量值,在太赫兹成像等小部分成像领域成功地获得了实用。但是,单像素相机的最大缺陷之处在于同一时刻一次只能得到一个测量值,对于其它更广泛的高分辨率的应用领域,其获取足够测量的时间太长而不太实用。因此,Stern等人提出一种改进的压缩成像方法,可以在单次曝光条件下获得所有的测量值,在其他更广泛的成像领域中具有更大的应用潜力。
本文借鉴Stern的单次曝光思想,围绕的压缩成像的理论与方法、成像架构、物理可实现的压缩成像系统以及三维压缩成像理论与方法中遇到的一系列问题展开深入研究。
本文的研究工作及创新点主要集中在如下几个方面：
1)探讨了双透镜加确定性光学相位掩膜所构成的一种成像系统,给出压缩成像的基本原理和公式描述,并以托普利兹和循环两种确定性相位掩膜矩阵为例验证确定性相位掩膜成像方法的有效性。与Stern提出的单次曝光随机相位掩膜调制压缩成像方法相比,采用托普利兹和循环等确定性相位掩膜矩阵更利于物理实现和快速计算。此外,本文推导出压缩双透镜与确定性相位掩膜组成的成像系统满足压缩传感理论中对应的测量矩阵的任意两列之间不相关所需满足一些基本的条件,并给出具体的推导分析过程。
2)提出了几种新的随机间距稀疏测量矩阵,包括随机间距稀疏托普利兹／循环和随机三元间距稀疏托普利兹/循环矩阵,并给出这些测量矩阵满足相关限制等距性质的理论证明。与高斯矩阵等常用测量矩阵相比,新提出的矩阵随机独立元个数大大减少,重建速度有了较大的提高；然后将这几种新的随机间距稀疏测量矩阵应用到压缩双透镜成像系统中,验证其作为相位掩膜矩阵在实现对图像信息随机调制的有效性；结果表明所提出的随机间距稀疏相位掩膜矩阵在物理上更容易物理实现,存储量低,易于数据传输；并且在保证重建性能的同时,可以显著地节约了后端压缩传感重建的时间,对于压缩传感理论的实际应用具有重要意义。此外,还提出一种块状确定性矩阵作为相位掩膜矩阵,在也具有易于物理实现的优势的同时,更适宜多通道成像领域的应用,利于压缩成像理论走向实际应用。
3)将压缩传感理论应用于超分辨率成像重建问题,给出了在不同模糊内核作用下从低分辨率到高分辨率的超分辨率图像重建方法,结果显著优于直接插值方法。此外,还提出一种改进的基于4-f傅里叶光学成像架构的频域相位随机编码压缩成像方法,实现了图像信息的有效记录与恢复；针对实际可记录的测量值非负的限制,利用卷积原理分析实现测量值非负的基本原理,并给出改进的频域相位编码成像系统实现非负测量值的记录与成功重建,利于大尺度图像的压缩编码与重建,及其在实际中的应用与推广。为了实际掩膜的物理实现,本文进一步给出了一种改进的二元随机相位掩膜编码,规避了均匀随机相位掩膜的实际制作难度太大的困难。
4)提出一种基于4-f傅里叶光学成像架构的0/1频域振幅编码孔径以及与随机间距光栅或一般的均匀间距的光栅相结合的压缩成像方法。与传统的针孔相机相比,大大增加了光的利用率,提高了对噪声的鲁棒性。与随机相位掩膜编码相比,振幅编码孔径更易于物理实现。在此基础上,设计了一种改进的0/1频域振幅编码孔径压缩成像系统,保证了非负性约束,使得实际的物理测量矩阵与理论计算的测量矩阵之间有更好的一致性。
5)探讨压缩传感理论在三维成像领域中的应用,提出一种从一个二维全息图重建3D层析相干成像的多波长压缩全息成像方法；将散斑非相干压缩成像方法从单波长扩展到多波长和把压缩全息成像理论应用于稀疏孔径合成压缩全息成像等。在稀疏孔径合成压缩全息成像应用中改善了从多个低分辨率的传感器阵列合成高分辨率数字全息图像中遇到的相位不稳定的问题。最后给出物理实验验证了压缩全息成像方法的有效性。
Compressive sensing is a novel emerging theory of signal acquisition and processing, which break through the limitations of the traditional Shannon/Nyquist sampling theorem, the main benefit from that the vast majority of signals in the nature generally has the prior knowledge of sparsity, which can help to reconstruct the original signal accurately from the far less linear incoherent measurements than the Nyquist sampling rate. Obviously, recovering the real signal from incomplete measurement data is an ill-posed inverse problem. Compressive sensing theory has a huge potential value in many areas ranging from most signal processing applications, such as the field of information theory, signal processing, optical imaging, pattern recognition, signal acquisition and communications.
Compressive imaging is one of the most important research areas of compressive sensing theory, the first successful instance of compressive imaging applications is the single pixel camera designed by Rice University based on compressed sensing theory which use only one detector for scene imaging. The camera has a control of digital micro-mirror array, driven by a pseudo-random binary array, and reflects the the scene of interest onto a single detector array which gather the intensity of this projection, and then reconstruct the original image from the observation value. A major advantage of this structure is any binary projection matrix can be easily implemented in this imaging system, so that the compressed sensing theory can be naturally applied to the measurement. Single-pixel camera perfectly agrees with the compressive sensing theory, which using a random modulation only to obtain a measured value, and has successfully gained practical application in a small part of the imaging area such as Terahertz imaging. However, the most severely drawback of the single-pixel camera is that can only obtain a measured value at one time, however, in other more extensive high-resolution applications, the time to obtain sufficient measurements is too long and thus is not very practical. Therefore, Stern proposed an improved compressive imaging method, that all measured values can be obtained with a single exposure conditions, and has greater potential application in other more widely imaging areas.
This thesis draws on Stern's single exposure ideology, surrounded with a series of problems encountered in compressive imaging theories and methods, imaging architecture, physical realizable imaging system and three-dimensional compressive imaging theory for deeply research.
The main research works and contributions of this thesis are outlined as follows.
1) A novel imaging system consisting of double lens and deterministic optical phase mask is proposed, and the basic principles and formulas for compressive imaging are described, and the validity of the phase mask imaging method with Toeplitz and Circulant two deterministic phase mask matrix for instance is verified in the numerical simulations. Compared with the random phase mask modulation compressive imaging method with single-exposure proposed by Stern, using Toeplitz and Circulant and other deterministic phase mask matrix are more conducive to physical implementation and fast calculation. In addition, any two columns of measurement matrix are corresponding to compressive imaging system consisting of double-lens and deterministic phase mask to meet some basic conditions and specific derivation analysis are derived.
2) Several novel random spacing sparse measurement matrices, including the random spacing sparse Toeplitz/Circulant and random ternary spacing sparse Toeplitz/Circulant matrices are proposed, and the proofs for such measurement matrices to satisfiy the restricted isometric property are given. Compared with the Gaussian measurement matrices, the novel proposed matrices, independent component and storage spaces are reduced significantly, and the speed of reconstruction have greatly improved; then these novel types of random spacing sparse measurement matrices are applied into compressive double lens imaging system, the validity of them as a phase mask matrices to achieve image information random modulation is verified; The simulation results of proposed random spacing sparse phase mask matrix demonstrate that proposed masks is superior for its easier physical implementation, low storage capacity, ease of data transfer; and with the same reconstruction performance and the significantly reduced reconstruction time, which has great significance for practical application of compressed sensing theory. In addition, blocked measurement matrices as the phase mask matrices are also proposed, which have the advantage of easy physical implementation, and are more suitable for multi-channel imaging applications, and can help compressive imaging theory to practical application.
3) Compressive sensing was introduced into image super-resolution reconstruction problem, numerical simulation results demonstrate that reconstruction from low resolution to high resolution with different blurring kernel, that are much better than the direct interpolation method. In addition, an improved optical imaging method based on the4-f Fourier frequency domain phase random encoding for compressed imaging is proposed, which can capture image information and recover the image effectivrly;
Due to the measured values recorded in real-world should satisfy non-negative constraints, the convolution principle to analysis the basic principle for how to implementing the measurement values non-negative is analysized, and an improvement frequency domain phase encoding imaging method is proposed to implement non-negative recording of the measurement values and successful reconstruction after then, which is conducive to large-scale image compressed encoding and reconstruction, and its practical application and promotion. For the realization of actual mask, an improved encoding of binary random phase mask is also given in this thesis, to avoid the practical difficulty for producing uniform random phase mask.
4) A Fourier optical imaging architecture based on the4-f0/1frequency domain amplitude mask aperture imaging method with random spacing grating or evenly spaced grating combined are proposed. Compared with the traditional pinhole camera, the utilization of illumination light can be greatly increased in our method to improve the robustness against noise. Compared with the random phase mask encoding, the amplitude coded aperture are much easier for physical implementation. Based on this point, an improved frequency-domain amplitude0/1mask aperture compressive imaging system is designed, non-negative constraints is guaranteed, making a better consistency between the actual physical measurement matrices and the theoretical calculation of the measurement matrices.
5) Compressed sensing theory is applied in three-dimensional imaging, and a multi-wavelength compressive holography method is proposed which can reconstruct a3D coherent tomography imaging from a two-dimensional hologram; compressive imaging method with single wavelength speckle incoherent is extended to multi-wavelength, and compressive holography imaging is also applied into sparse aperture synthetic holography imaging which improves the phase instability problems encountered in sparse aperture synthetic holography imaging for mosaics of multiple low-resolution and low cost2D sensor arrays which can achieve large aperture; and the effectiveness of compressive holography imaging method is verified by physical experiments.
引文
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