Imaging models for differential-interference-contrast (DIC) microscopy are presented. Two- and three-dimensional models for DIC imaging under partially coherent illumination were derived and tested by using phantom specimens viewed with several conventional DIC microscopes and quasi-monochromatic light. DIC images recorded with a CCD camera were compared with model predictions that were generated by using theoretical point-spread functions, computer-generated phantoms, and estimated imaging parameters such as bias and shear. Results show quantitative and qualitative agreement between model and data for several imaging conditions.

We derive an algorithm for maximum-likelihood image estimation on the basis of the expectation-maximization (EM) formalism by using a new approximate model for depth-varying image formation for optical sectioning microscopy. This new strata-based model incorporates spherical aberration that worsens as the microscope is focused deeper under the cover slip and is the result of the refractive-index mismatch between the immersion medium and the mounting medium of the specimen. Images of a specimen with known geometry and refractive index show that the model captures the main features of the image. We analyze the performance of the depth-variant EM algorithm with simulations, which show that the algorithm can compensate for image degradation changing with depth.

We tested the most complete optical model available for computational optical-sectioning microscopy and obtained four main results. First, we observed good agreement between experimental and theoretical point-spread functions (PSF's) under a variety of imaging conditions. Second, using these PSF's, we found that a linear restoration method yielded reconstructed images of a well-defined phantom object (a 10-microns-diameter fluorescent bead) that closely resembled the theoretically determined, best-possible linear reconstruction of the object. Third, this best linear reconstruction suffered from a (to our knowledge) previously undescribed artifactual axial elongation whose principal cause was not increased axial blur but rather the conical shape of the null space intrinsic to nonconfocal three-dimensional (3D) microscopy. Fourth, when 10-microns phantom beads were embedded at different depths in a transparent medium, reconstructed bead images were progressively degraded with depth unless they were reconstructed with use of a PSF determined at the bead's depth. We conclude that (1) the optical model for optical sectioning is reasonably accurate; (2) if PSF shift variance cannot be avoided by adjustment of the optics, then reconstruction methods must be modified to account for this effect; and (3) alternative microscopical or nonlinear algorithmic approaches are required for overcoming artifacts imposed by the missing cone of frequencies that is intrinsic to nonconfocal 3D microscopy.

An iterative phase-estimation method for the calculation of a specimen's phase function or optical-path-length (OPL) distribution from differential-interference-contrast (DIC) microscopy images is presented. The method minimizes the least-squares discrepancy measure by use of the conjugate-gradient technique to estimate the phase function from multiple DIC images acquired at different specimen rotations. The estimate is regularized with a quadratic smoothness penalty. Results from testing the method with simulations and measured DIC images show improvement in the estimated phase when at least two rotationally diverse DIC images instead of a single DIC image are used for the estimation. The OPL of a cell that is estimated from two DIC images was found to be much more reliable than the OPL computed from single DIC images (which had a coefficient of variation equal to 15.8%).

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 timelapse analyses. For these applications existing iterative methods are impractical without expensive and/or specially designed hardware. INTRODUCTIONLight microscopy is a powerful tool for the noninvasive examination of biological specimens. Specimens are frequently labeled with fluorescent probes that are specific for certain cells or for defined molecular components within cells. Visualization of these fluorescent probes in three dimensions is critically important for understanding the three-dimensional (3-D) architectures of cells and cellular components.Computational optical-sectioning microscopy is one method for reconstructing 3-D images of living biological structures from data acquired by using light microscopy. These data are obtained by first labeling the specimen with a dye that fluoresces when exposed to light. Then a series of two-dimensional (2-D) images is collected while the specimen is stepped through focus. Since photons are detected from anywhere in the specimen, each 2-D image contains information from both the current in-focus plane and the out-of-focus planes. The effect of the photons from out-of-focus regions can be described by modeling the microscope's optics' and the detection process. In this paper we propose a linear, regularized algorithm that uses such a model to reconstruct 3-D images of the specimen from a sequence of corrupted 2-D images.A number of image-processing algorithms have been applied with various degrees of success in computational optical sectioning.2 6 Results from these algorithms indicate that processing of optical-section images can lead to improvements. The simplest and fastest processing method considers only the out-of-focus contributions from nearest-neighbor focal planes i...