Preza C, Miller M I, Thomas L J, McNally J G
Biomedical Computer Laboratory, Washington University, St. Louis, Missouri 63110.
J Opt Soc Am A. 1992 Feb;9(2):219-28. doi: 10.1364/josaa.9.000219.
The inverse problem involving the determination of a three-dimensional biological structure from images obtained by means of optical-sectioning microscopy is ill posed. Although the linear least-squares solution can be obtained rapidly by inverse filtering, we show here that it is unstable because of the inversion of small eigenvalues of the microscope's point-spread-function operator. We have regularized the problem by application of the linear-precision-gauge formalism of Joyce and Root [J. Opt. Soc. Am. A 1, 149 (1984)]. In our method the solution is regularized by being constrained to lie in a subspace spanned by the eigenvectors corresponding to a selected number of large eigenvalues. The trade-off between the variance and the regularization error determines the number of eigenvalues inverted in the estimation. The resulting linear method is a one-step algorithm that yields, in a few seconds, solutions that are optimal in the mean-square sense when the correct number of eigenvalues are inverted. Results from sensitivity studies show that the proposed method is robust to noise and to underestimation of the width of the point-spread function. The method proposed here is particularly useful for applications in which processing speed is critical, such as studies of living specimens and time-lapse analyses. For these applications existing iterative methods are impractical without expensive and/or specially designed hardware.
通过光学切片显微镜获得的图像来确定三维生物结构的逆问题是不适定的。虽然通过逆滤波可以快速获得线性最小二乘解，但我们在此表明，由于显微镜点扩散函数算子的小特征值求逆，它是不稳定的。我们通过应用乔伊斯和鲁特的线性精度规范形式[《美国光学学会志A》1, 149 (1984)]对该问题进行了正则化。在我们的方法中，通过将解约束在由与选定数量的大特征值相对应的特征向量所张成的子空间中来进行正则化。方差与正则化误差之间的权衡决定了估计中求逆的特征值数量。所得的线性方法是一种单步算法，当求逆正确数量的特征值时，能在几秒钟内产生均方意义上最优的解。灵敏度研究结果表明，所提出的方法对噪声和点扩散函数宽度的低估具有鲁棒性。这里提出的方法对于处理速度至关重要的应用特别有用，例如活体标本研究和延时分析。对于这些应用，现有的迭代方法在没有昂贵和/或专门设计的硬件的情况下是不切实际的。
