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Frequency analysis Due to the fairly regular formation of the corneal endothelium Figure 1 , frequency analysis is commonly used to obtain an average cell size. Some years later, this relation was further explored through the analysis of synthetic images [ 8 , 17 , 18 ].
The procedure of the most recent approaches to estimate endothelial cell density [ 6 , 7 , 9 ] is to project the two-dimensional frequency spectrum onto a one-dimensional function.
One salient spatial frequency on this function is then found, often a peak.
The stochastic watershed algorithm For more complex analyses of the endothelial layer, a full segmentation of the cells is needed. In confocal images of the endothelial layer, the cells are visualized as bright areas separated by dark lines Figure 1.
An excellent method to segment such images is the watershed algorithm [ 19 , 20 ]. This algorithm segments the image into regions with one local minimum each.
The segmentation lines run along the brightest line in between each pair of local minima. Thus, the confocal images of the endothelium must be inverted first, such that the cell boundaries are bright lines. Unfortunately, due to the noise in the image, each cell typically has many local minima, and therefore will be segmented into many regions by the watershed algorithm.
One solution is to apply the H-minima transform before the watershed algorithm [ 21 ]. This transform suppresses all local minima that have a depth less than the parameter h.
However, it is often impossible to find a value h such that each cell has exactly one local minimum. An alternative to avoid oversegmentation is to apply a seeded watershed [ 22 ].
One seed point needs to be placed in each cell; a minimum imposition algorithm will transform the image such that these seeds will be the only local minima in the image. Consequently, the watershed algorithm will generate a single region for each seed.
The logic required to automatically place one seed point in each cell is far from trivial, and thus this method is often used with some manual interaction.
In the stochastic watershed approach, the seeded watershed is repeatedly applied to the image with randomly placed seeds [ 23 ]. The segmentation results are added together to build a map that shows the likelihood of each pixel belonging to a boundary. This map is referred to as probability density function, PDF. More salient boundaries will have a larger value in the PDF. A segmentation is obtained from this map by applying a smoothing, the H-minima transform, and finally the classical watershed algorithm.
Bernander et al. Methods Materials We used a set of 52 confocal images of corneal endothelium. All patients were examined in the first year after Descemet stripping automated endothelial keratoplasty. Before the examination, the eyes were anesthetized with one drop of 0.
A coupling gel Vidisic; Dr. Mann Pharma, Berlin, Germany was applied between the objective lens and the corneal surface. The cropped images had a mean area of px2 range [,]. According to the manufacturer, the lateral sampling density was 0. He placed a point in the middle of each cell and drew an outline around the set of cells polygonal frame method [ 25 ]. Thus, in the outlined region evaluation area Aeval is a known number of complete cells neval.
We consider the morphometric quantities that were then determined by the NAVIS software as ground truth. We also used the fully automatic segmentation mode of the NAVIS software to obtain a second set of measurements for comparison. Data was collected in accordance with the tenets of the Declaration of Helsinki and informed consent was obtained from all participants.
Frequency analysis Below, we review published methods to derive the characteristic frequency from the frequency spectrum of the image, and propose a new one.
Next, we determine the frequency for each sample in the frequency spectrum, and round the value to the nearest frequency in the list above.
Finally, we compute the mean of all values assigned to the same frequency. The position of the local maximum with the highest value is the characteristic frequency.
We refer to this method as modeRM. Bucht et al. We refer to this method as 1stRM. Radial maximum of the frequency spectrum Ruggeri et al. We compute the radial maximum in an analogous way to the radial mean, and determine the precise location of the peak by fitting a parabola. We refer to this method as 2ndRMAX. We compute the radial projection of the power spectrum in an analogous way to the radial projections above. However, Foracchia and Ruggeri [ 6 ] computed the cumulative radial mean by integrating the frequency spectrum over increasingly large circles, then computed the derivative of this cumulative distribution.
Depending on the method used to determine the derivative, this results in either the same or a very similar function to that obtained by our implementation. The size parameter of this smoothing operator was not given in their paper, but presumably has a very strong effect on the result, especially in the case of the mode. We chose to not apply any smoothing. The methods meanPS and medianPS require the application of a high-pass filter to the image. We indicate this parameter by appending it to the method name: meanPS 3, medianPS 5, etc.
Enhancing the ring prior to computing the radial mean This paragraph proposes a new method, based on modeRM. We therefore developed a simple procedure to completely remove the central peak from the frequency spectrum before computing the radial mean.
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