Image Processing

The images were processed by first applying nearest neighbor deconvolution [26] to each three image stack in order to remove out of focus fluorescence from the central image plane. As described in [27], the image produced at the output of the microscope system (o_j for focal plane j) is the sum of all focal planes in the specimen (i_k) convolved with their corresponding plane of the point-spread function (impulse response) of the microscope (s_k-j). The output image is described by

$$o_j = i_j \otimes s_0 + \sum_{k=j-m'}^{j+m}i_k \otimes s_{k-j}$$ (2.1)

where $\otimes$ denotes the convolution operation, m' is the number of focal planes in the specimen below the current plane, m is the number of focal planes above the current plane, and the subscripts indicate the focal plane at which the corresponding function is evaluated (s₀ is the 2D in-focus point spread function, s_-1 is the point spread function evaluated one focal plane below the in-focus plane, etc.)

One major assumption of nearest neighbor deconvolution, and one that greatly simplifies data collection and computation is that the planes immediately above and below the in-focus plane contribute the most to blurring. In this case, Equation 2.1 simplifies to

$$o_j = i_j \otimes s_0 + i_{j-1} \otimes s_{-1} + i_{j+1} \otimes s_{+1}$$ (2.2)

To undo the blurring defined in Equation 2.2, one needs to solve for $i_j \otimes s_0$, the `true' value of the in-focus object plane. Since i_j+1 and i_j-1 are not known, however, they must be approximated by o_j+1 and o_j-1. Making this substitution and solving Equation 2.2 for $i_j \otimes s_0$,

$$i_j \otimes s_0 = [o_j - c(o_{j-1} \otimes s_{-1} + o_{j+1} \otimes s_{1})]$$ (2.3)

where c is an empirically chosen constant indicating how much of a contribution the out-of-focus planes contribute to the in-focus plane. Based on [26], c was chosen to be 0.45 and the stacks of images were deconvolved using Equation 2.3.

After deconvolution, a rectangular region containing a single cell was defined for each image. Only the pixels from the single, deconvolved image that were within this region were subject to further processing steps. The background fluorescence, defined as the most common pixel value in the region, was subtracted from all pixels. Finally, the images were thresholded using a constant multiple of the background fluorescence for that image. This multiple was 4.0 for all probes except Hoechst 33258, for which it was 1.5. These numbers were arrived at empirically by assessing the quality of images thresholded using various values. Pixels at or above this threshold were used in subsequent processing steps, those below the threshold were set to 0. In order to make feature calculations insensitive to changes in overall image brightness, each pixel value in the thresholded image was divided by the total fluorescence in that image.