Three-dimensional single-molecule localization microscopy imaging based on compressed sensing
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摘要: 本文建立了一种三维压缩感知模型以实现对高密度荧光分子图像的快速三维定位。首先，根据荧光显微的三维点扩展函数成像理论，设计测量矩阵，并建立压缩感知模型。接着，对荧光显微成像过程进行了模拟，并采用凸优化方法（CVX）、正交匹配追踪（Orthogonal Matching Pursuit，OMP）算法和同伦算法对建立的压缩感知模型中模拟生成的图像进行了定位分析，分别从恢复率、定位精度、重构时间几方面进行了对比。最后，采用同伦算法对模拟的生物样品和实验室采集的细胞进行了三维定位，并获得了三维超分辨图像。对比结果表明：在重构密度和定位精度接近的情况下，同伦算法比CVX方法的重构速度快2个数量级。同伦算法较OMP算法的定位精度要高一倍。采用同伦算法来实现三维的超分辨荧光显微成像在节约计算时间、实现实时成像方面具有一定的意义。Abstract: In order to achieve fast three-dimensional localization of high-density fluorescent molecular images, a three-dimensional compressed sensing model was established and studied using the CVX method, the Orthogonal Matching Pursuit(OMP) algorithm and a homotopy algorithm. The models’ measurement matrix was then designed. Firstly, the system’s theory and design were both developed using the three-dimensional point-spread function imaging theory of fluorescence microscopy. Then, the process of fluorescence microscopic imaging was simulated, through which the images generated in the established compressed sensing model were analyzed using the CVX method, OMP algorithm and homotopy algorithm. The recall rate, localization accuracy and reconstruction time were compared. Finally, the simulated biological samples and the collected cells in the laboratory were analyzed using the homotopy algorithm, and thus three-dimensional super-resolution imaging was achieved. It can be seen from the comparative results that the homotopy algorithm is two orders of magnitude faster than the CVX method when the reconstruction density and localization accuracy have little deviation. The localization accuracy of the homotopy algorithm is twice higher than that of the OMP algorithm. The homotopy algorithm is meaningful for 3D super-resolution fluorescence microscopy imaging, which can save computing time and achieve real-time imaging.
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图 1 荧光图像对应样本进行三维网格细分图。 （a）模拟生成荧光分子成像；（b）荧光图像对应的荧光分子三维空间细分；（c）一个像素和整个图像对应网格细分数
Figure 1. 3D mesh subdivision process of fluorescence image corresponding sample. (a) Simulated fluorescence molecular image; (b) the three-dimensional subdivision of sample corresponding to fluorescent image; (c) the number of the subdivision corresponding to a pixel and the whole image
图 2 对整个图像进行分块处理。（a）从一帧图像里取一小块；（b）重合边缘；（c）对取出的小块图像进行边缘扩展；（d）对重构结果保留有效区域
Figure 2. The original image subdivided into smaller patches. (a) A small patch taken from one frame of image； (b) superposition edges; (c) edge expansion of the small block of image; (d) the retaining valid areas of reconstruction results
图 3 模拟生成的荧光分子图像（绿色星号为三维空间随机分布的荧光分子；三维空间的底面图像为荧光分子模拟生成的图像）
Figure 3. Simulated fluorescent molecule image (green asterisks are randomly distributed fluorescent molecules in 3D space; the bottom image in 3D space is the image generated by the simulated fluorescent molecules)
图 4 CVX，OMP和L1-H 三种算法对模拟生成的荧光分子图像的重构结果。（a）恢复率；（b）横向XY用标准差（Stdev）显示定位精度；（c）Z向用标准差显示定位精度；（d）算法运行时间
Figure 4. Reconstructed results of simulated fluorescence molecule image by three kinds of algorithms. (a) Recovery rate; (b) localization accuracy of horizontal XY shown with the standard deviation; (c) localization accuracy of axial Z shown with the standard deviation; (d) algorithm running time
图 5 模拟生物实验的三维重构结果，图中的比例尺为800 nm。 （a）模拟生成随机分布的荧光分子累加结果，横向范围为6.9 μm × 6.9 μm，轴向范围为−200~200 nm；（b）随机生成一帧图像；（c）随机生成200幅图像进行累加的结果；（d）用算法分别对200帧图像进行重构定位出分子整合成的一张三维超分辨图像。
Figure 5. 3D reconstructed results of simulated bioexperiment (scale bar is 800 nm). (a) Accumulation results of randomly distributed fluorescent molecules with a transverse range of 6.9 μm×6.9 μm and an axial range from −200 nm to 200 nm. (b) A randomly generated frame of image. (c) Accumulation results of randomly generated 200 frames of images. (d) A 3D super-resolution image obtained by reconstructing 200 frames of images.
图 6 三维超分辨图像的重构结果。（a）由2 000帧荧光图像累加得到的宽场图像；（b）本文算法重构的三维超分辨图像
Figure 6. Reconstructed results of 3D super-resolution image (scale bar is 2 μm). (a) Wide-field image obtained by accumulating 2 000 frames of fluorescent images. (b) 3D super-resolution image reconstructed by the proposed algorithm.
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