Jan Brugues and Daniel J. Needleman, Nonequilibrium Fluctuations in Metaphase Spindles: Polarized Light Microscopy, Image Registration, and Correlation Functions Proc. SPIE , 2010, 7618.
Wiki Entry by Robin Kirkpatrick, AP 225, Fall 2011
The authors use polarized light microscopy to study the dynamics of the metaphase spindle. The internal spindle fluctuation are measured by computing a spatiotemproal correlation function. This is done after correcting for the net spindle motion (rotation and translation) using an image registration algorithm. The observed birefringence is due to microtubules, thus the authors use polarized light microscopy to measure the spindle dynamics.
The authors use polarized light microscopy to measure the dynamics of the mitotic spindle. Briefly, nuclei formation of CSF-arrested egg extracts was induced by addition of demembrenated sperm and calcium. These were further driven to metaphase by addition of metaphase-arrested extract. Time lapse images were taken after imaging with an LC-Polscope. Briefly, light of a certain ellipticity is generated and passes through a sample volume. The LC-Polscope measures the optical retardance and the orientation of the slow optical axis. The optical axis is in the direction of microtubule alignment, thus measurement of the dynamics of the optical axis is of interest.
The author's goal is to measure the dynamics of the spindle using a correlation based approach. In order to achieve accurate correlation sequences, one must accurately account for the translational and rotational motion of the entire spindle. The authors use the retardance (rather that the orientation) image to determine the rotation and displacement of the object from the prior frame. However, the spindle changes shape from frame to frame, thus the method used must be insensitive to fluctuations in boundary. The authors use image thresholding to determine a mask. The mask is determined after contrast enhancement using a gamma adjustment, thresholding, and erosion and dilation to remove erroneous objects. A correlation of the normalized intensity of over the masked region is done with the prior frame for different rotation angles and translation distances. The location (angle and translational distance) of the maximum correlation is the guessed moved distance. For a variety of angles, the correlation is computed using an FFT based approach. The correlation is maximized with respect to translational distance, and the angle is found by fitting correlation vs angle plots to a cubic spline. The distances that maximize the correlation correspond to the distance the object has moved from the prior frame and the angle refers to the degree of rotation of the object from the prior frame. An illustration of this technique is shown in Figure 3.
The authors test their algorithm for a variety of simulated noise levels (Gaussian noise). The results are shown below in Figure 4.
To infer properties regarding internal dynamics of the spindle, one may be interested in computing the correlation function for a spatio-temporal process. For a process A(x, y, t), we define the correlation function as c(η, ξ, τ) = δA(x, y, t)δA(x + η, y + ξ, t + τ ) (3) where δA(x, y, t) is a fluctuation around the mean at position (x, y) and time t. However, time and space are discrete,s o the correlation function must be estimated. There are two common estimators of the correlation function, namely the biased and unbiased estimators. The authors argue in favor of the biased estimator for several reasons, one of which is that the bias is reduced as a as lag increases, which is often seen physically. Furthermore, the fourier techniques can be used to compute the biased, but not the unbiased estimator. The also notes that biased estimators are valid autocovariance sequences, while unbiased estimators are not ( I think, not going to prove it here). The results of computing the biased and unbiased estimators is shown below in Figure 6. It is noted that the biased estimator decays nicely to zero for large lags (which is expected physically).
The authors also note that they can compute the correlation function of the tensorial order parameter Q by knowing the direction of orientation theta via the following relations after subtracting the temporal mean at each point.
The authors present their methods for computing the correlation function from time lapse images, which involved a carefully constructed registration algorithm to remove errors from translational and rotational motion of the spindle.