The cornea is the front of the eye. Its inner cell layer, called the endothelium, is important because it is closely related to the light transparency of the cornea. An in vivo observation of this layer is performed by using specular microscopy to evaluate the health of the cells: a high spatial density will result in a good transparency. Thus, the main criterion required by ophthalmologists is the cell density of the cornea endothelium, mainly obtained by an image segmentation process. Different methods can perform the image segmentation of these cells, and the three most performing methods are studied here. The question for the ophthalmologists is how to choose the best algorithm and to obtain the best possible results with it. This paper presents a methodology to compare these algorithms together. Moreover, by the way of geometric dissimilarity criteria, the algorithms are tuned up, and the best parameter values are thus proposed to the expert ophthalmologists.
The eye is the first sense organ responsible for human vision. The human eye functions like a camera to refract light and produce a focused image which stimulates neural responses transmitted to the brain vision centers. crystalline lens is made of compacted protein fibers and is anchored in place by muscles attached to the wall of the eyeball. Contraction of these muscles causes the lens to change its shape and curvature, thus improving the focusing power. Refracted light passes through the eye cavity and strikes the inner surface at the back, known as the retina. The retina contains the specialized nerve cells called rods and cones that detect the intensity and the frequency of the incoming light. Light stimulates the rods and cones, which creates neural impulses that are transmitted to the brain through a network of nerve cells bunched together to form the optic nerve that exits from the back of the eyeball and passes to the brain.
The cornea is the transparent, spherical surface covering the front of the eye. It is a powerful refractive surface, providing about 2/3 of the eye’s focusing power. Healthy cornea has no blood vessel, which accounts for its clarity. But it is rich in nerve endings and so it is extremely sensitive to pain. The tears and aqueous humor, a watery fluid circulating in the cavity behind it that contains glucose and several electrolytes, nourish the cornea. The cornea is a highly organized tissue consisting of cells and protein arranged in three main layers: epithelium: this is the outermost layer comprising about 10% of the total thickness. Along with the tear film that bathes the outer surface of the eye, it provides a protective function preventing the entry of foreign material into the eye; stroma: it makes up to 90% of the corneal thickness. It consists primarily of water (78%) and layered collagen fibers (16%) that give the cornea its strength, elasticity, and shape. It also contains cells scattered between the fibers that produce the stromal constituents. The lattice-like arrangement and uniform spacing of the collagen fibers are essential for corneal transparency; endothelium: this is the innermost layer facing the aqueous and consists of a single layer of hexagonal cells. It pumps water out of the cornea and hence plays a vital role in keeping it in a dehydrated state. Without this pumping action, the stroma would accumulate water and become hazy and finally opaque (corneal oedema) leading to loss of vision.
The cornea must remain transparent to refract light properly and the corneal endothelium ensures the integrity and transparency of the cornea.
The corneal endothelium consists of a single layer of closely packed, flat, hexagonally shaped cells covering the back surface of the cornea. In the human cornea at birth, there are more than 4000 cells/mm2. With age, the number of endothelial cells gradually decreases, but because they cannot regenerate, neighboring cells spread out to fill the gap leading to an alteration of cell shape (pleomorphism) and size (polymegathism). The mean endothelial cell density (ECD) in adults is generally between 500 and 3500 cells/mm2. Cell density, as well as variation in size and shape, can be examined by specular microscopy in living human subjects. These methods permit early diagnosis of any damage of the corneal endothelium.
Since the cornea is transparent, cornea cells can easily be observed in vivo with a specular microscope. This technology comes from the early 1980s. Those optical microscopes can acquire an image of the cells on a very little surface (0.08 mm2 compared to the endothelium surface of about 100 mm2; see Figure
It is necessary to evaluate the quality of the human corneal endothelium in several circumstances (for example, after accidents, surgery, or trauma). The main cause is corneal grafting.
The two criteria required for the evaluation are the endothelial cell density (ECD, in cells/mm2): there are several threshold values: for example, an ECD lower than 400 cells/mm2 does not enable maintaining the cornea transparency. An ECD lower than 1000 cells/mm2 is a contraindication for using intraocular lens implants. the morphometry of endothelial cells: their size regularity (called the polymegathism, i.e., the variation of areas of the cells) and their shape regularity (called the pleomorphism, i.e., the percentage of hexagon-like cells) induce a good quality of the cornea.
Different methods exist to perform the segmentation of images of endothelial cells. Among those, three methods give the better results [
The presented algorithms have a common structure. First, they filter the original image. Second, they aim to find some markers of the cells, and then they perform a morphological operation (a watershed; see [
This method has been proposed in [
The algorithm is summarized in Algorithm
(1) (2) ASF (3) (4) (5) (6)
This method is more recent than the previous one [
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
This is the most recent method proposed in [
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
Table
Summary of the control parameters of the three presented image segmentation algorithms.
Method | Parameters |
Description |
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Vincent and Masters Algorithm |
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Angulo and Matou Algorithm |
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Gavet and Pinoli Algorithm |
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The evaluation of a segmentation quality is a common problem encountered when developing a segmentation method. Like the segmentation methods themselves, the image segmentation evaluation criteria can be classified into region-based or contour-based approaches, although they usually can be adapted from one class to the other. The segmentation processes of the corneal endothelium result in the contours of the cells, but the proposed comparison methods are also suitable for segmented regions.
This paper deals with supervised segmentation evaluation, that is, involving a criterion that compares the result of the segmentation process to a ground truth image (usually manually segmented by an expert of the application field). This is usually preferred to unsupervised evaluation (where some kind of intraregion homogeneity is involved), but the bias introduced by the expert does not have to be neglected (see Section
The following notations are first introduced:
This paper will not present an exhaustive view of supervised evaluation of segmentation criteria. The reader can have a look at [
The two detailed criteria have been chosen because they are tolerant towards spatial variations. One could also use other frequently used criteria proposed in the literature [
The figure of merit [
An image segmentation process refers to the action of partitioning the spatial domain of an image into adjacent regions, each of them preserving a certain homogeneity following a given criterion. Thus, a computer program is able to answer the following binary question: is this pixel inside the region of interest or not?
To formalize this mathematically, let
This paper deals with the case where contours are detected and the segmentation result is a binary image; that is,
The
First, let us recall that the symmetric difference set between two segmentations
The Minkowski addition [
If
In [
The main properties of
The usual concept to compare mathematical objects is the metric notion, defined by four axioms (identity, separation, symmetry, and triangle inequality; see [
The problem of the experts reference segmentation is crucial because subject to variations between experts and sometimes also for one expert. To deal with this problem, some articles use an average result, like [
For one original gray-tone image, the experts have manually drawn their segmented image several times, and the
Method for fixing the tolerance parameter. In this example,
Thus, the
The different segmentation algorithms presented in the previous sections require to setup the values of the so-called control parameters. The choice of the control parameter values for a specific application issue is generally not trivial, especially for nonimage analysis experts. This section explains the generic way of selecting the best parameters in average for the considered three image segmentation methods.
Let
Let
Let
In the following, we consider an image database of
Let
Let
This way of finding the best parameter set is also called leave-one-out cross validation.
Some noise may be present in the computed values (mainly because of a too poor image quality). To be more tolerant towards these perturbations, the trimmed mean (sometimes called truncated mean) is also employed: in the addressed application issue, given parts of the sample are discarded at the high end.
If
Notice that the trimmed mean corresponds to the classical mean for
The median of
In order to observe the influence of one control parameter in the segmentation results, it is interesting to fix every control parameter but the considered one, and see if there is an impact on the quality of the segmentation. Let
The
The results will be presented in Tables
For comparison purposes, we also provide the best criterion value that could have been obtained on the test partition, denoted by
This section presents the results for the three aforementioned image segmentation methods.
An image database of
The summary of the optimal control parameters values is presented in Table
Results for Algorithm
Criterion |
Optimal parameters | Trimmed mean ( |
Median |
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0.11 |
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0.08 |
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0.10 | 0.115 |
fom |
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0.52 |
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0.47 |
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0.50 | 0.520 |
It appears that
Projection on the optimal values of the control parameters
The results are presented in Table
Results for Algorithm
Criterion |
Optimal parameters | Trimmed mean ( |
Median |
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0.15 |
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0.08 |
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0.11 | 0.150 |
fom |
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0.54 |
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0.46 |
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0.51 | 0.525 |
Projection on the optimal values of the control parameters for the method of Angulo and Matou. See Table
The summary of the optimal control parameter values is presented in Table
Results for Algorithm
Criterion |
Optimal parameters | Trimmed mean ( |
Median |
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0.10 |
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0.06 |
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0.08 | 0.099 |
fom |
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0.50 |
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0.45 |
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0.49 | 0.506 |
Cross-validation information for Algorithm
Partition number | Optimal parameter values |
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1 |
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0.146 | 0.118 |
2 |
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0.099 | 0.096 |
3 |
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0.110 | 0.109 |
4 |
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0.117 | 0.115 |
5 |
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0.104 | 0.104 |
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Mean |
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Cross-validation information for Algorithm
Partition number | Optimal parameter values |
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1 |
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0.512 | 0.512 |
2 |
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0.524 | 0.510 |
3 |
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0.490 | 0.488 |
4 |
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0.533 | 0.533 |
5 |
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0.540 | 0.540 |
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Mean |
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Cross-validation information for Algorithm
Partition number | Optimal parameter values |
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1 |
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0.162 | 0.148 |
2 |
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0.157 | 0.155 |
3 |
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0.141 | 0.141 |
4 |
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0.109 | 0.109 |
5 |
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0.180 | 0.158 |
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Mean |
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For both
Projection on the optimal values of the control parameters for the method of Gavet and Pinoli. See Table
Table of the 30 reference segmented images of the database. They have been manually drawn by an expert ophthalmologist from a human corneal endothelium image database (see Figure
Table of the 30 specular microscopy images of corneal endotheliums of the database. They are segmented by the proposed method and by an ophthalmologist (see Figure
The control parameter
The control parameter
According to the numerical values, the method of Gavet and Pinoli outperforms the methods from Vincent and Masters, and Angulo and Matou (see the values of
The
Cross-validation information for Algorithm
Partition number | Optimal parameter values |
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1 |
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0.510 | 0.510 |
2 |
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0.565 | 0.558 |
3 |
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0.624 | 0.598 |
4 |
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0.448 | 0.443 |
5 |
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0.479 | 0.479 |
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Mean |
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Cross-validation information for Algorithm
Partition number | Optimal parameter values |
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1 |
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0.078 | 0.078 |
2 |
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0.100 | 0.099 |
3 |
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0.111 | 0.093 |
4 |
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0.109 | 0.101 |
5 |
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0.100 | 0.091 |
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Mean |
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Cross-validation information for Algorithm
Partition number | Optimal parameter values |
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1 |
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0.506 | 0.502 |
2 |
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0.490 | 0.486 |
3 |
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0.486 | 0.480 |
4 |
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0.484 | 0.477 |
5 |
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0.562 | 0.548 |
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Mean |
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Moreover, the optimal parameter values obtained for the different training partition do not vary a lot and are really similar to those proposed in Tables
In this paper, three segmentation methods suitable for binarizing the optical specular microscopy gray-tone images of human corneal endotheliums have been presented. These methods involve different control parameters. This is always a hard problem for the user because he has no time to manually tune up his computer softwares (and especially his image segmentation softwares). Two dissimilarity criteria have been employed (
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank Pr. Gain and Pr. Thuret from the University Hospital of Saint-Etienne for providing them with their images and their expertise in the field of ophthalmology. This work was partially supported by the ANR Project TecSan 2012, CorImMo 3D.