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对于处在
$({x_0},{y_0})$ 的单个荧光分子或发光粒子所发出的光的强度经过光学系统在相机上成像可以用一个二维的高斯函数近似表示:$$ I\left( {i,j} \right) = {N_0}\frac{{4\ln 2}}{{\pi {w^2}}}\exp \left( - 4\ln 2\frac{{{{\left( {i - {x_0}} \right)}^2} + {{\left( {j - {y_0}} \right)}^2}}}{{w_{}^2}}\right) + b $$ (1) 式中:
${N_0}$ 为荧光分子辐射的光子数,激发光激发的荧光光子数一般会服从对数正态分布[8];$(i,j)$ 表示相机像素所在的位置;$w$ 为荧光强度分布的半高全宽值;$b$ 为背景信号噪声。根据高斯函数的半高全宽与标准差的关系$w = 2\sqrt {2\ln 2} \sigma $ ,$\sigma $ 为高斯函数的标准差,也可以将公式(1)写为:$$ I\left( {i,j} \right) = \frac{{{N_0}}}{{2\pi {\sigma ^2}}}\exp \left( - \frac{{{{\left( {u - {x_0}} \right)}^2} + {{\left( {j - {y_0}} \right)}^2}}}{{2\sigma _{}^2}}\right) + b $$ (2) 如果一共有Q个荧光分子发光,那么探测器上接收到的总光强可用求和公式得到:
$$ I\left( {i,j} \right) = \sum\limits_{k = 1}^Q {\frac{{{N_k}}}{{2\pi {\sigma ^2}}}\exp \left( - \frac{{{{\left( {u - {x_k}} \right)}^2} + {{\left( {j - {y_k}} \right)}^2}}}{{2\sigma _{}^2}}\right) + b} $$ (3) 由于光信号在光电探测器上成像过程中会引入散粒噪声,在像素
$(i,j)$ 处接收到$q$ 个光子的概率可以表示为[12]:$$ {P_f}(i,j) = \frac{{\exp ( - {I_{i,j}})I_{i,j}^q}}{{q!}} $$ (4) -
在STORM的单分子定位中常用的是单测量矢量压缩感知模型(Single Measurement Vector-Compressed Sensing, SMV-CS),在该模型中,通常将采集到的一系列图像进行单张处理,如将一张图像转换为列矢量
${\boldsymbol{b }}$ ,CS模型中的观测矩阵由点扩展函数构建。所以CS模型具体表达式如下[8]:$$ \begin{gathered} {\text{minimize}}\;{\left\| {\boldsymbol{s}} \right\|_1} \hfill \\ \;{\text{subject}}\;{\text{to}}\;{{\boldsymbol{s}}_i} \geqslant 0\;{\text{and}}\;\left\| {{\boldsymbol{As}} - {\boldsymbol{b}}} \right\|_2 \leqslant \varepsilon {\left( {\sum {{{\boldsymbol{b}}_j}} } \right)^{1/2}}\\ \end{gathered} $$ (5) 式中:
${\boldsymbol{s}}$ 为待求的重构的单张分子定位图像,它的值可以反映出分子的位置和重构的分子所发出的光子数多少;观测矩阵${\boldsymbol{A}}$ 由系统的点扩展函数构造,其第$j$ 列为假设分子处在向量${\boldsymbol{s}}$ 的第$j$ 行所形成的图像。约束条件让重构的像素值大于等于零,因为分子发出的光子数不可能小于零,第二项约束是重构的荧光分子生成的图像与采集图像的方差小于等于$\varepsilon $ ,$\varepsilon $ 的值可以控制重构信号的稀疏程度与重构结果和采集图像的匹配程序之间的平衡。为了解决单测量矢量压缩感知处理速度过慢、不利于实时成像的问题,需要对模型进行优化处理,故采用多测量矢量压缩感知模型,采集的所有图像转换成一列,然后组成一个矩阵
${\boldsymbol{B}}$ ,则图像采集可以表示为${\boldsymbol{B}} = {\boldsymbol{AS}} + {\boldsymbol{E}}$ 。所以多测量矢量压缩感知模型变为:$$ \begin{gathered} {\text{minimize}}\;{\left\| {\boldsymbol{S}} \right\|_1} \\ {\text{subject}}\;{\text{to}}\;{\boldsymbol{S}}_i^{} \geqslant 0\;{\text{and}}\;\left\| {{\boldsymbol{AS}} + {\boldsymbol{B}}} \right\| \leqslant {{\boldsymbol{E}}} \\ \end{gathered} $$ (6) 该问题可以通过多重稀疏贝叶斯学习算法进行求解。观测矩阵
${\boldsymbol{A}}$ 与单测量矢量压缩感知模型的观测矩阵相同。其中,重构矩阵${\boldsymbol{S}}$ 中的每一列对应一帧图像的重构,这一列的行索引与荧光分子所处的空间位置信息对应,列向量里面的值对应荧光分子发出的光子数大小,与单测量矢量一次求解结果类似。多测量矢量是一次对所有采用的图像进行重构,每列对应一帧图像的重构结果,然后对所有列求和,最后将列向量转换为超分辨图像进行显示。 -
压缩感知无论在信号处理还是在图像处理方面都取得了很大成功[13-14]。它指出:如果原始信号是稀疏的(信号里大部分信号为0或相对其他值很小)或通过某种变换能够变成稀疏信号,则可以把带有噪声的信号通过压缩感知算法精确地恢复出其原来的信号。公式(6)为多测量矢量(Multiple Measurement Vector, MMV)压缩感知优化问题。在合适的噪声和信号统计下,多重稀疏贝叶斯学习(Multiple Sparse Bayesian Learning,M-SBL)算法原始模型是最小化以下目标函数[15]:
$$ {L}^{\gamma }(\gamma )={\rm{Tr}}({\Sigma }^{-1}{{\boldsymbol{BB}}}^{*})+T\mathrm{log}\left|\Sigma \right| $$ (7) 其中
$$ \Sigma = {\sigma ^2}{\boldsymbol{I}} + {\boldsymbol{A}}{\boldsymbol{\varGamma}} {\boldsymbol{A}}^*\;,\;\;{\boldsymbol{\varGamma}} = {\text{diag}}(\gamma ) $$ (8) 式中:
${\sigma ^2}$ 表示测量噪声方差;I为单位矩阵;${{\boldsymbol{A}}^*}$ 表示矩阵${\boldsymbol{A}}$ 的转置矩阵。最小化目标函数(公式(6))可以等价于正则最小二乘化框架:$$ \mathop {{\text{min}}}\limits_{\boldsymbol{S}} {L^{\boldsymbol{S}}}({\boldsymbol{S}}),\;\;{{{L}}^{\boldsymbol{S}}}({\boldsymbol{S}}) = \left\| {{\boldsymbol{B}} - {\boldsymbol{AS}}} \right\|_F^2 + {\sigma ^2}{{{g}}_{{\text{msbl}}}}({\boldsymbol{S}}) $$ (9) 其中
$$ {{{g}}_{{\text{msbl}}}}({\boldsymbol{S}}) \equiv \mathop {\min }\limits_{\gamma \geqslant 0} \;{\rm{Tr}}({{\boldsymbol{S}}^*}{\boldsymbol{\varGamma}} {\boldsymbol{S}}) + T\log \left| {{\sigma ^2}{\boldsymbol{I}} + {\boldsymbol{A}}{\boldsymbol{\varGamma}} {{\boldsymbol{A}}^*}} \right| $$ (10) M-SBL算法的具体步骤如下:
1) 最小化
${\boldsymbol{S}}$ :在第k次迭代,对于一个给定的估计${\gamma ^{(k)}}$ ,对于公式(9)中的${\boldsymbol{S}}$ 可以找到一个封闭的解:${{\boldsymbol{S}}^{(k)}} = {{\boldsymbol{\varGamma }}^{(k)}}{{\boldsymbol{A}}^*}{({\sigma ^2}{\boldsymbol{I}} +{\boldsymbol{ A}}{{\boldsymbol{\varGamma }}^{(k)}}{{\boldsymbol{A}}^*})^{ - 1}}{\boldsymbol{B}},\;{\boldsymbol{\varGamma}} = {\rm{diag}}({\gamma ^{(k)}})$ 。2)最小化参数
$\gamma $ :对于一个给定的${{\boldsymbol{S}}^{(k)}}$ ,${\gamma ^{(k + 1)}}$ 需要找到更新规则。其更新规则如下:$$ \gamma _i^{k + 1} = \dfrac{{\dfrac{1}{N}{\displaystyle\sum\nolimits_{t = 1}^{{T}}{{\left| {s_{it}^{(k)}} \right|}^2}} }}{{1 - \displaystyle\sum _{tt}^{(k)}/\gamma _i^{(k)}}} $$ (11) 式中:
$\Sigma _{tt}^{(k)}$ 表示$\Sigma _{}^{(k + 1)} = {\sigma ^{2(k)}} + {\boldsymbol{A}}{{\boldsymbol{\varGamma}} ^{(k)}}{{\boldsymbol{A}}^*}$ 的$(t,t)$ 元素。3) 噪声方差
${\sigma ^2}$ 估计:噪声方差的更新规则如下:$$ {\sigma ^{2(k + 1)}} = \frac{{\left\| {{\boldsymbol{B}} - {\boldsymbol{A}}{{\boldsymbol{S}}^{(k + 1)}}} \right\|_F^2}}{{n - m + \displaystyle\sum\nolimits_{t = 1}^{{T}} {\dfrac{{\displaystyle\sum\nolimits _{tt}^{(k + 1)}}}{{\gamma _i^{(k + 1)}}}} }} $$ (12) 式中:
$\Sigma _{tt}^{(k + 1)}$ 表示$\Sigma _{}^{(k + 1)} = {\sigma ^{2(k)}} + {\boldsymbol{A}}{{\boldsymbol{\varGamma}} ^{(k + 1)}}{{\boldsymbol{A}}^*}$ 的$(t,t)$ 元素。将多测量矢量压缩感知模型应用于超分辨荧光显微成像中,其迭代次数一般设置为20次,其解最后收敛。最后通过公式(13)计算得到超分辨成像的解:
$$ \mathop s\limits^ \wedge = \left[ \begin{gathered} \sqrt {\sum\nolimits_{t = 1}^{{T}} {{{\left| {s_{1t}^{(k)}} \right|}^2}/T} } \hfill \\ \sqrt {\displaystyle\sum\nolimits_{t = 1}^{{T}} {{{\left| {s_{2t}^{(k)}} \right|}^2}/T} } \hfill \\ \;\;\;\;\;\;\;\;\; \vdots \hfill \\ \sqrt {\sum\nolimits_{t = 1}^{{T}} {{{\left| {s_{nt}^{(k)}} \right|}^2}/T} } \hfill \\ \end{gathered} \right] $$ (13)
Research on super-resolution fluorescence microscopy imaging based on multiple measurement vector compressed sensing
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摘要: 在超分辨荧光显微成像技术中,单分子定位显微方法是被广泛应用的技术之一。根据荧光显微成像原理构造多测量矢量压缩感知模型(Multiple Measurement Vector-Compressed Sensing, MMV-CS),并采用多重稀疏贝叶斯学习算法进行求解,来实现超分辨荧光图像重建。分析了有效像元大小、荧光分子生成的光子数和背景信号泊松化噪声对重建结果的影响,以及在图像进行分块处理时算法运行时间的分析。模拟和实验计算分析表明,当点扩展函数的标准差在160 nm时,有效像元大小在120、160、200 nm能取得较好的重构效果,而在60 nm时效果较差。探测器收集的光子数越多,重构效果越好,随着背景信号光子数增加时,离得越近的样品结构越不能分辨。在同样的分块处理情况下,MMV-CS比同伦算法(L1-Homotopy, L1-H)和凸优化算法(CVX)分别快一个数量级和三个数量级,因此,在研究三维超分辨荧光显微成像时,MMV-CS算法在运行时间上具有更大的优势。Abstract: In the super-resolution microscopy imaging technology, single molecule localization microscopy is one of the widely used techniques. In this paper, in order to achieve super-resolution fluorescence image reconstruction, a multiple measurement vector Compressed sensing (MMV-CS) model was established based on the principle of fluorescence microscopic imaging, and the multiple sparse Bayesian learning algorithm was applied in problem solving. The effects of the effective pixel size, the number of photons generated by fluorescent molecules and the Poisson noise of fluorescence and background signal on the reconstruction results were analyzed. The running time of the algorithm was analyzed with the image subdivided into smaller patches. The results of simulation and experimental calculation show that when the standard deviation of the point spread function is 160 nm, the effective pixel size at 120 nm, 160 nm and 200 nm can achieve good reconstruction effect, while the pixel size at 60 nm results in poor effect. Better reconstruction image quality is achieved with more photons collected by the detector. As the background signal photons increase, the sample structure becomes indistinguishable when the distance is too close. Under the same subdivided condition, MMV-CS is one order of magnitude faster than the Homotopy (L1-H) algorithm and three orders of magnitude faster than the convex optimization algorithm (CVX), which has greater advantages in terms of running time for the application of MMV-CS in 3D super-resolution fluorescence microscopy.
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图 7 L1-H算法和MMV-CS算法对高密度图像进行重建比较。(a) 1500帧图像累加的宽场荧光图像;(b)取1500帧里的一帧图像显示;(c)采用L1-H算法重构结果;(d)采用MMV-CS算法重构结果
Figure 7. Comparison of reconstructions images of L1-H and MMV-CS algorithms based on high-density images. (a) Wide-images of directly accumulated microtubule images by 1500 frames; (b) A frame of 1500 frames; Reconstructions images using (c) L1-H algorithm and (d) MMV-CS algorithm
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