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压缩感知理论利用信号的稀疏性,用远低于Nyquist采样定理要求的采样次数对信号进行采样或高噪声和有损测量时,也能很好地恢复出原始信号。压缩感知理论公式是:
$ \min \|\boldsymbol{X}\|_{0}, (\boldsymbol{Y}=\boldsymbol{A} \boldsymbol{X}) $
(1) 式中, Y为采样信号,是N×1的列向量,A是N×M的测量矩阵,X是M×1的原始信号。目标信号的稀疏或变换后稀疏是压缩感知的先决条件,在原始信号X中,有$ K \ll M$个元素是非零的,其余大部分信号为零或近似为零,则称信号是K稀疏的。N是采样数,它影响着压缩感知恢复信号的能力[14]。
目前常用的压缩感知重构算法主要分为三大类[15]:(1)贪婪方法。包括匹配追踪(matching pursuit, MP)[16]、正交匹配追踪(orthogonal matching pursuit, OMP)[17]等,其主要思想是通过迭代寻找匹配信号的最优值; (2)松弛方法。它是基于l1范数的最小化,主要有GPSR、基追踪算法(basis pursuit, BP)[18],这一类方法精度高、需要的测量个数少,但计算复杂度高; (3)非凸算法。该方法计算复杂度和精度都是介于以上两种方法之间,典型的有迭代重加权重构算法[19]。此外,还有软、硬迭代阈值等[20]、贝叶斯压缩感知重构算法[21]等也是使用较多的压缩感知重构算法。
本文中采用GPSR算法来重构原始图像,由于(1)式中‖X‖0为非凸函数,无法求解,为获得次优解,将(1)式中对0范数的求解改写为求解次优解,即:
$ \min \|\boldsymbol{X}\|_{1}, \left(\|\boldsymbol{A} \boldsymbol{X}-\boldsymbol{Y}\|_{2} \leqslant \varepsilon\right) $
(2) 式中, ε是与噪声有关的一个参量,根据参考文献[22], 可将(2)式转化为下式:
$ \min \frac{1}{2}\|\boldsymbol{Y}-\boldsymbol{A} \boldsymbol{X}\|_{2}^{2}+\tau\|\boldsymbol{X}\|_{1} $
(3) 式中, τ>0表示可自定义的正则化参量,对于(2)式,任意ε总有τ与其对应,从而保证(2)式、(3)式有共同解。该算法由于ε的参与而具有一定的抗噪能力,因此利用(3)式求解的算法被称作基追踪降噪算法。但是由于宽场显微成像中测量矩阵的数据量大, 导致(3)式不能够快速高效地求解出超分辨图像,因此根据参考文献[23], 将(3)式进行再次优化,从而简化计算过程,降低计算时间。优化算法采用GPSR-BB(Barzilai-Borwein),这种算法计算复杂度低于基追踪降噪的方式,对于有限等距约束的要求低于正交匹配追踪算法。因此,能够在测量矩阵数据大和噪声干扰的情况下快速准确地重构出原始信号。
压缩感知实现快速超分辨荧光显微成像
Fast super-resolution fluorescence microscopy by compressed sensing
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摘要: 为了发展能够同时兼顾超分辨、快速成像和视场的荧光显微镜, 以促进其在活细胞或微观动态过程成像的应用, 将压缩感知应用到超分辨荧光显微镜中, 利用投影梯度稀疏重构算法对单帧荧光宽场图像重构, 并进行了理论分析、仿真和实验验证。结果表明, 该方法能够突破光学衍射极限, 成像分辨率达到180nm, 相比衍射极限提高1.8倍。此结果说明压缩感知能够实现单帧宽场超分辨荧光显微成像, 相比现有的方法在成像速度上有巨大的提升。
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关键词:
- 显微 /
- 超分辨 /
- 压缩感知 /
- 投影梯度稀疏重构算法
Abstract: In order to develop fluorescent microscope that can simultaneously take into account super-resolution, fast imaging and field of view, and promote its application in imaging of living cells or micro-dynamic processes, the compressed sensing was applied to super-resolution fluorescence microscopy. The algorithm of gradient projection for sparse reconstruction was used to reconstruct single fluorescence wide field image. Theoretical analysis, simulation and experimental verification were carried out. The results show that, this method can break through the optical diffraction limit. The imaging resolution is 180nm. Compared with the diffraction limit, it is 1.8 times higher. Compressed sensing can realize single-frame wide-field super-resolution fluorescence microscopy imaging. Compared with the existing methods, the imaging speed has been greatly improved. -
Figure 2. Imaging resolution and influence of sampling numbers on resolution
a—original image of two fluorescence molecules separated by 180nm b—wide-field image (160nm/pixel) of two fluorescence molecules separated by 180nm c—reconstruction results at sampling number 121 d—Gaussian fitting curve of intensity distribution on white line in Fig. 2a~Fig. 2c e—original image of two fluorescence molecules at a distance of 200nm f—wide-field images (160nm/pixel) of two fluorescence molecules at distance of 200nm g—reconstruction results at sampling number of 49 h—Gaussian fitting curve of intensity distribution on white line in Fig. 2e~Fig. 2g
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