Yann Gavet
email: gavet@emse.fr

Mathieu Fernandes

Johan Debayle

Jean-Charles Pinoli

Dissimilarity criteria and their comparison for quantitative evaluation of image segmentation. Application to human retina vessels

Keywords: @Article{Gavet2014, {Image segmentation, Quantitative evaluation, Segmentation evaluation, Dissimilarity criteria, Retinal blood vessels}, Language = {English}, Publisher= {Springer Berlin

Dissimilarity criteria and their comparison for quantitative evaluation of image segmentation:

application to human retina vessels

Introduction

Image segmentation is one of the most important and hard-to-address steps in image analysis. An image segmentation process consists in partitioning the spatial support of an image into adjacent parts that are pairwise disjoint. When dealing with a 2-D image, that is to say with a 2-D support, such parts are either regions (two dimensional sets), contours (one dimensional sets) or isolated points (zero dimensional sets). Only Euclidean sets will thus be considered. Moreover, only binary images will be considered without the loss of generality for the purpose of this paper. The general segmentation case (segmentation into n dierent sets) will not be discussed as it implies dierent processes, like for example pairing the sets together (see also [START_REF] Pont-Tuset | Measures and meta-measures for the supervised evaluation of image segmentation[END_REF]).

The relevance of an image segmentation is rstly and mainly that of the resulting binary image. Comparing such resulting binary images coming from various segmentation methods (automatically or manually/visually performed by a computer and a human expert, respectively) is therefore required. Since these methods have been selected as performing in front of a particular imaging application issue, they often output similar binary images, which makes their comparative study more tedious. The reader could refer to [START_REF] Mcguinness | A comparative evaluation of interactive segmentation algorithms[END_REF][START_REF] Villegas | Perceptually-weighted evaluation criteria for segmentation masks in video sequences[END_REF][START_REF] Zhang | A survey on evaluation methods for image segmentation[END_REF][START_REF] Philipp-Foliguet | Évaluation de la segmentation d'images: état de l'art, nouveaux indices et comparaison[END_REF] to have a view on some methods for segmentation evaluation.

The quantitative comparative evaluation either deals with the region-based paradigm or with the contour-based paradigm [START_REF] Marr | Vision: A Computational Investigation into the Human Representation and Processing of Visual Information[END_REF]. It can be performed by resorting to a reference method or not (see [START_REF] Chalana | A methodology for evaluation of boundary detection algorithms on medical images[END_REF]). To perform this comparison, numerous criteria have been reported in the specialized literature. In this article, eleven numerical criteria are employed. They are supervised criteria, as they compare a binary image with a ground truth (usually performed manually/visually by an expert). In fact, the quantitative evaluation by means of comparative criteria is the so-called rst-order comparative problem, since it is criterion-dependent.

Consequently, this article also addresses the second-order comparative problem consisting in the quantitative comparison of the eleven criteria themselves. This article focuses on the so-called summary measures [START_REF] Pont-Tuset | Measures and meta-measures for the supervised evaluation of image segmentation[END_REF]. Although ROC curves are a lot richer, their comparison is not straightforward. It is not evident for example to decide whether it is preferable to have false negatives instead of false positives.

The eld of ophthalmology with the segmentation of the blood vessels of retinal images is addressed here. Five image segmentation methods reported in the literature as particularly adapted to such an application issue have been arbitrarily selected. They are not state of the art methods, but they will provide dierent results for performing a comparison of the criteria. The article is organised in six sections. First, the image segmentation is described from a geometrical viewpoint. In section 2, the mathematical notions of metric and dissimilarity are presented and discussed, showing the relevance of the second one. Section 3 focuses on the eleven dissimilarity criteria used in this article for supervised segmentation evaluation. In section 4, ve selected image segmentation methods are presented. Quantitative comparisons are detailed in section 5. Finally, a concluding discussion and perspectives end the article.

Image segmentation

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 with respect to a given criterion. This article deals with the case where the image segmentation result is a binary image which partitions the spatial support S (that will be considered as a bounded set in R 2 ) such that S = i∈I R i where I is nite and the regions (R i ) i∈I are pairwise disjoint (typically open) sets.

Metrics and dissimilarities

In many real applications, it is useful to compare segmentations together. This can be done for evaluating the eciency of segmentation methods together or comparing them to a reference method (see [START_REF] Chalana | A methodology for evaluation of boundary detection algorithms on medical images[END_REF]). Many approaches exist, mostly involving region criteria [START_REF] Zhang | A survey on evaluation methods for image segmentation[END_REF][START_REF] Philipp-Foliguet | Évaluation de la segmentation d'images: état de l'art, nouveaux indices et comparaison[END_REF]. Usually, these methods are mathematically based on distance functions, e.g., metrics.

Distance functions are adapted to perform comparisons of (mathematical) objects belonging to an object collection ξ (for example, ξ is the family of the segmentations on a given image, e.g., with a given denition domain S). A metric is a particular distance function d that is dened on ξ 2 and valued in R + (set of all positive real numbers) , satisfying the four following axioms (see [START_REF] Deza | Dictionary of distances[END_REF]):

(identity) ∀x ∈ ξ, d(x, x) = 0 (separation) ∀x, y ∈ ξ, d(x, y) = 0 ⇒ x = y (symmetry) ∀x, y ∈ ξ, d(x, y) = d(y, x) (triangle inequality) ∀x, y, z ∈ ξ, d(x, y) ≤ d(x, z) + d(z, y)
Nevertheless, metrics are not necessarily the adapted notion. Indeed, it has been shown [START_REF] Tversky | Features of similarity[END_REF][START_REF] Tversky | Similarity, separability and the triangle inequality[END_REF][START_REF] Gavet | Perception visuelle humaine, complétion des mosaïques et application à la reconstruction d'images de l'endothélium cornéen humain en microscopie optique spéculaire[END_REF] that the metric notion is not consistent with the human visual perception. To be more precise, the three last axioms of separation, symmetry and triangle inequality do not hold for the human visual system:

• two objects of the same kind (for example, a computer-drawn square and a manually-drawn square) are considered as visually similar although dierent from a set-theoretic point of view.

• the order of observation of two dierent objects is important, thus, the symmetry is not veried by the human visual system.

• the triangle inequality is not veried. The Fig. 1 illustrates this assertion. Dissimilarity is the relevant notion, which denes a positive real valued function of two variables: a zero value means that two objects are similar, and a large value implies a high dissimilarity.

As the human expertise plays an important role in image segmentation evaluation, the reader should notice that there are two categories of segmentation evaluation: supervised and unsupervised [START_REF] Zhang | A survey on evaluation methods for image segmentation[END_REF]. The unsupervised evaluation involves an absolute criterion that characterizes the accuracy of the segmentation. The supervised evaluation is based on a distance between a reference segmentation result (called gold-standard in several applications elds; it can be the result given by an automatic reference method or by a human expert) and another segmentation result (generally computed).

3 Eleven dissimilarity criteria for supervised segmentation evaluation

The following notations are rst introduced: M (as Manual) is the binary result of a reference segmentation method and X is the result of the evaluated segmentation method.

This section introduces some classical and recent dissimilarity criteria that will be considered for binary segmentation evaluation. This list is not exhaustive, interesting ideas and criteria can be found for example in [START_REF] Cardoso | Toward a generic evaluation of image segmentation[END_REF][START_REF] Jiang | Distance measures for image segmentation evaluation[END_REF][START_REF] Unnikrishnan | Toward objective evaluation of image segmentation algorithms[END_REF], for general segmentation evaluation.

Figure of merit

One of the mostly used criterion for evaluating segmentation is the so-called gure of merit (fom) ( [START_REF] Abdou | Qualitative design and evaluation of enhancement/thresholding edge detector[END_REF][START_REF] Strasters | Three-dimensional image segmentation using a split, merge and group approach[END_REF]) dened in Eq. 1.

Be aware that the original denition does not involve a dierence to 1. It has been introduced in this paper so that the fom criterion becomes a dissimilarity.

f om M (X) = 1 - 1 max{#(M ), #(X)} p∈X 1 1 + d 2 (p, M ) (1)
where d(p, M ) is the Euclidean distance between the pixel p ∈ X to the closest pixel of M and # designates the cardinality, namely the number of pixels of the considered sets M or X.

Dice coecient

Another widely used criterion is the Dice coecient [START_REF] Dice | Measures of the Amount of Ecologic Association Between Species[END_REF], related to the Jaccard index [START_REF] Jaccard | Étude comparative de la distribution orale dans une portion des alpes et des jura[END_REF], and dened by Eq. 2:

dice M (X) = 1 - 2#(M ∩ X) #(M ) + #(X) (2)
As for the gure of merit (fom), the original denition of the Dice coecient does not involve the dierence to 1, but this has been also introduced to become a dissimilarity.

Five criteria with pixel classication

Performance results can also be compared with the use of pixels classication (see Table 1) into four categories:

• True Positive (T P ),

• False Positive (F P ),

• True Negative (T N ),

• False Negative (F N ).

Positive or Negative refer to the detection, true or false refer to the reference from the ground truth. Then, several criteria are dened in Table 2, by evaluating the number of pixels in each category:

• sensitivity (measures the proportion of actual positives which are correctly identied as such, e.g. the percentage of pixels present in the ground truth which are correctly segmented , also known as recall rate ),

• specicity (measures the proportion of negatives which are correctly identied, e.g. the percentage of pixels absent of the ground truth which are present in the segmentation),

• positive predictive value (the proportion of detected pixels that are true positives, also known as precision ),

• negative predictive value (the proportion of negative pixels that will not be present in the segmentation result),

• accuracy (measures the closeness of measurements of a quantity to that quantity's actual (true) value).

Notice that the Dice coecient can also be expressed in terms of pixel classication as dice =

2T P 2T P +F N +F P . 1 ).

h h h h h h h h h h h h h h h

The dissimilarity criterion

The dissimilarity criterion is based on the symmetric dierence of sets, but involves a tolerance with the help of the Minkowski addition [START_REF] Minkowski | Volumen und Oberäche[END_REF]. It is an extension of the Jaccard index [START_REF] Jaccard | Étude comparative de la distribution orale dans une portion des alpes et des jura[END_REF] treated in [START_REF] Gavet | A geometric dissimilarity criterion between jordan spatial mosaics. Theoretical aspects and application to segmentation evaluation[END_REF], where a detailed study of its properties (tolerance to under-or over-segmentation, translation or distortion, smoothing and visual tolerance as well as mathematical properties) has been reported.

The dissimilarity criterion with the tolerance ρ applied to segmented images is dened by the following equation (Eq. 3, see also [START_REF] Gavet | A geometric dissimilarity criterion between jordan spatial mosaics. Theoretical aspects and application to segmentation evaluation[END_REF]):

ρ M (X) = #{(X \ M ⊕ ρN ) ∪ (M \ X ⊕ ρN )} #{M ⊕ ρN } (3)
with N being the disk of radius 1 (as structuring element) and # designating the number of pixels within the set. The Minkowski addition symbol, denoted ⊕, is equivalent to the morphological dilation, N being symmetric, where N is called a structuring element (for example a disk). Practically, ρ is the radius of the ball used to dilate the binary images.

Fuzzy Jaccard index

In the same objective of being perceptually relevant and visually tolerant, [START_REF] Mcguinness | A comparative evaluation of interactive segmentation algorithms[END_REF] proposes a so-called fuzzy Jaccard index. X and M are the segmented sets and the reference (manual) segmentation, respectively, X c and M c are the complements of these sets.

Let N x be the 8-neighborhood of a point x ∈ Z 2 .

B X = {x : x ∈ X ∧ {N x ∩ X c = ∅}} B M = {x : x ∈ M ∧ {N x ∩ M c = ∅}}
A membership function, called fuzzy border pixel maps in [START_REF] Mcguinness | A comparative evaluation of interactive segmentation algorithms[END_REF], is dened as follows, for K = X or M :

BK (x) = exp - x -x 2 2σ 2 x = arg min y∈K x -y
The parameter σ is the fuzzy parameter, used to introduce a tolerance, and similar to ρ for the parameter. Then, the fuzzy Jaccard index is dened as:

F J = 1 - x min BX (x), BM (x)
x max BX (x), BM (x)

Notice that the values are between 0 and 1. This denition is slightly modied from the original denition from [START_REF] Mcguinness | A comparative evaluation of interactive segmentation algorithms[END_REF] to get 0 for an exact match.

Perceptually-weighted evaluation criterion

An evaluation criterion that tries to be perceptually signicant, adapted to object present in videos, is proposed in [START_REF] Villegas | Perceptually-weighted evaluation criteria for segmentation masks in video sequences[END_REF]. For the purpose of binary image segmentation, we will restrict the denition to the static part of this criterion qms, which is dened for a xed frame as follows:

QMs = QMs + + QMs - where QMs + = D + max d=1 w + (d) • # M + d ∩ X QMs -= D - max d=1 w + (d) • # M - d ∩ X c
and with the denitions (see [START_REF] Villegas | Perceptually-weighted evaluation criteria for segmentation masks in video sequences[END_REF]):

• d: distance to the reference mask border;

• D + max and D - max : biggest distance d for, respectively, false positives and false negatives;

• w + (d): weighting function for false positives according to distance d, expressed as w

+ (d) = B 1 + B2 d+B3 ;
• w -(d): weighting function for false negatives, expressed as w -(d) = F S • d, with F S a constant factor;

• M + d and M - d : sets of pixels situated at distance d from the reference mask border, outside or inside the mask, respectively (dist is the distance from pixel x to a set):

M + d = {x|x ∈ M c , dist(x, M ) = d} M - d = {x|x ∈ M, dist(x, M c ) = d} • X is the segmentation result.
Finally, qms = QM S #M

. Practically, the histogram of the distances to the reference mask of all pixels is used to compute this evaluation criterion.

Presentation of the ve selected human retina vessels image segmentation methods

Five image segmentation methods (see Fig. 2) that specically address the human retina vessels extraction have been selected

for the dierent tests on the dissimilarity criteria. This section presents them quickly.

Chaudhuri et al. method (Method 1)

This method [START_REF] Chaudhuri | Detection of blood vessels in retinal images using two-dimensional matched lters[END_REF] is based on the assumption that the retina vessels' intensity distributions may be approximated by the following equation (Eq. 4, where f is the intensity prole in the image along a line perpendicular to a vessel):

f (x, y) = A 1 -ke -d 2 /2σ 2 (4)
where d is the perpendicular distance between the point (x, y) and the straight line passing through the center of the blood vessel in the direction along its length, σ denotes the thickness of the vessel, L designates the length of the vessels and is used in Method 1, and A and k are real number constants reecting the gray level intensity of the local background and the reectance of the vessel. A matched lter (a correlation) is applied with these constraints (σ = 2 and L = 10 are the parameters of the algorithm) and the result is nally binarized.

The Chaudhuri et al method is summarized in Method 1.

Data : Input ← Grayscale image of retina.

σ ← Thickness of detected vessels (pixels). L ← Length of detected vessels (pixels).

Direction← Dierent orientations.

Result : S: Segmentation of the retina image.

Chanwimaluang et al. method (Method 2)

This second segmentation method ( [START_REF] Chanwimaluang | Hybrid retinal image registration[END_REF]) is based on the previous one, beginning by the same matched lter. The thresholding method changes (see Method 2): the connected components with a too small length are removed since it is based on a local entropy measure (see [START_REF] Chanwimaluang | Hybrid retinal image registration[END_REF] for precisions).

Marin et al. method (Method 5)

The fth segmentation method presented in this article has appeared recently ( [START_REF] Marin | A new supervised method for blood vessel segmentation in retinal images by using gray-level and moment invariants-based features[END_REF]). The method is summarized in Method 5.

Data

Quantitative comparisons

The DRIVE database [START_REF] Staal | Ridge-based vessel segmentation in color images of the retina[END_REF] is constituted of retina images and their segmentations. It contains two families of retina images (Fig. 2) : the test family and the training family. They contain images and their manual segmentations either performed by two experts (test set) or by only one (training set). The STARE database [START_REF] Hoover | Locating blood vessels in retinal images by piecewise threshold probing of a matched lter response[END_REF] also contains 20 images of segmented vessels of retina, with the manual reference given by two experts, denoted ah and vk.

This section presents the results of the evaluation of the dierent segmentations on both databases. The rst subsection will begin a discussion on the tolerance parameter of the dissimilarity criterion. Then, all segmentation methods have been applied

to the 40 images of the test set of the DRIVE database and of the STARE database. One manual segmentation is employed as a reference (ground truth) and the second manual segmentation is presented as a control method. The results will be presented in dierent tables for the dierent dissimilarity criteria. The degrees of consistency and discriminancy (see Defs. 5.1 and 5.2 in Sect. 5.3) of the dissimilarity criteria together will be computed and presented to emphasize the good behaviour of the dissimilarity criterion.

Choice of the tolerance parameter of

The dissimilarity criterion satises the identity and separation properties of a metric, but neither obeys the symmetry nor the triangle inequality axioms. For the symmetry property, it is justied by the fact that the reference segmentation already introduces a dissymmetry in the evaluation. Indeed, the dissimilarity criterion can almost be interpreted as a percentage of misdetected pixels. The triangle inequality property is more complicated to interpret in the case of segmentations. It has been deeply discussed in [START_REF] Gavet | A geometric dissimilarity criterion between jordan spatial mosaics. Theoretical aspects and application to segmentation evaluation[END_REF].

The monotonicity is the main property. Indeed, the dissimilarity criterion is decreasing in regard to the tolerance parameter

ρ (Eq. 5). ∀(ρ 1 , ρ 2 ) ∈ R 2 + , ρ 1 > ρ 2 ⇒ ρ1 M (X) ≤ ρ2 M (X) (5)
As in every spatial analysis method where a scale factor is employed, the choice of the tolerance parameter is crucial and depends on the application issue. This section proposes a way to choose it. The test set of the DRIVE database is used to make the choice of the tolerance parameter value (two persons, trained by an expert and thus also considered as experts, have drawn the contours of the same retina image). The dissimilarity criterion has been used to compare every manual segmentation to the other. The mean value of the dissimilarity criterion is represented in the Fig. 3.

The reader can consider that two experts should always draw the contours at the same location, but within a certain spatial tolerance, depending on the image size and the precision of the drawing tool. The choice of the tolerance parameter then consists in determining the criterion value under which two segmentations are considered as identical. the results is shown in the graph. In this example, ρ is in pixels, and there is a strong gap between no tolerance (ρ = 0) and a tolerance of one pixel (ρ = 1).

For this (DRIVE) database and the considered size of images (565 × 583 pixels), if we admit that an dissimilarity value such that < 0.02 means identical segmentations, we should then take a tolerance of ρ = 4 (see Fig. 3). The same value will be taken for image from the STARE database, as the scale of observation of the vessels is similar to those in the DRIVE database.

Comparison of the ve segmentation methods by means of the eleven dissimilarity criteria

The 40 images from the databases have been segmented using the ve methods, and their results have been compared to the reference (which is the rst manual segmentation). The second manual segmentation is also included and compared to the reference. As the two manual segmentations were performed in the same conditions, they should give similar results for their evaluations by the considered criteria. The summary of the comparison is presented in the dierent subtables of Tables 3 and2. The absolute values comparison between the criterions are not sucient to evaluate a segmentation result. However, the relative comparison of the values obtained by the dierent segmentation methods is meaningful. The Figs. 4 and 5 illustrate in a visual way the results of Tables 3 and2. The values are normalized for each criterion with the maximal value obtained among the dierent methods for the 40 images of the DRIVE and STARE database. This graph (Fig. 4) shows that Sp and P pv criteria do not classify the manual segmentation as the best segmentation (i.e., with the lower dissimilarity value). Moreover, these two criteria do not sort the image segmentation methods in the same order as the others. Fig. 4 also shows that the dissimilarity criterion is shown to be more discriminating than the others. This is not true in the Fig. 5. It also highlights that it is very dicult to make a (visual) dierence between the methods from Soares and Mendonça. The next subsection 5.

Comparison of the eleven dissimilarity criteria

The quantitative comparison of dissimilarity criteria is not an easy task. This article makes use the denition of consistency and discriminancy [START_REF] Fix | Discriminatory analysis. nonparametric discrimination: Consistency properties[END_REF], more precisely the degree of consistency and the degree of discriminancy, as proposed in [START_REF] Huang | Using auc and accuracy in evaluating learning algorithms[END_REF].

Intuitively, the degree of consistency evaluates the agreement between the dierent criteria, and the degree of discriminancy evaluates their capacity to distinguish the dierent methods. We have chosen not to used nonparametric tests (Friedman, Quade... see [START_REF] García | Advanced nonparametric tests for multiple comparisons in the design of experiments in computational intelligence and data mining: Experimental analysis of power[END_REF]) because the approach are similar (based on ranking methods together) and it would complicate the task of the reader. 3. Mean results of the eleven criteria for the ve segmentation methods (and the second manual segmentation, which is a segmentation from another expert), over all images of DRIVE and STARE databases. Notice that median and standard deviation values do not provide more information than the mean value does. The second manual segmentation is expected to be the best segmentation, i.e. to get the lowest value.

segmentations on a given image, X 1 and X 2 are two segmentations belonging to ξ), let the sets R and S be dened by:

R = {(X 1 , X 2 )|X 1 , X 2 ∈ ξ, f (X 1 ) > f (X 2 ) and g(X 1 ) > g(X 2 )} S = {(X 1 , X 2 )|X 1 , X 2 ∈ ξ, f (X 1 ) > f (X 2 ) and g(X 1 ) < g(X 2 )}
The degree of consistency of f and g is d

• C (0 ≤ d • C ≤ 1)
, where

d • C = #R #R + #S
(# stands for the number of elements of the sets). [START_REF] Huang | Using auc and accuracy in evaluating learning algorithms[END_REF] precises that the degree of consistency

d • C is symmetric, i.e. d • C (f, g) = d • C (g, f ).
Denition 5.2 (Degree of Discriminancy). For two dissimilarity criteria f and g dened on the domain ξ, let The degree of discriminancy for f over g is

P = {(X 1 , X 2 )|X 1 , X 2 ∈ ξ, f (X 1 ) > f (X 2 ) and g(X 1 ) = g(X 2 )} Q = {(X 1 , X 2 )|X 1 , X 2 ∈ ψ, g(X 1 ) > g(X 2 ) and f (X 1 ) = f (X 2 )}
d • D = #P #Q
Notice that the degree d • D is not symmetric.

It is stated in [START_REF] Huang | Using auc and accuracy in evaluating learning algorithms[END_REF] that a criterion can be considered as better as another one if d • C > 0.5 (consistency) and d

• D > 1 (discriminancy).
In the present paper, these two degrees are calculated on the 40 segmented images of DRIVE and STARE databases, evaluated for each dissimilarity criterion with the rst manual segmentation taken as the ground truth.

The degrees of consistency are presented in Tables 3 and4, the criteria can be separated into two groups (by taking values above 0.5): the rst group is constituted of , f om, dice, Se, N pv and Acc, whereas the second group is composed of Sp and P pv.

To compute the degree of discriminancy, as the values are oat numerical values, the equality condition is subject to a small tolerance. The choice has been made to consider that two values are equal if their dierence is lower than 5% (with the normalized values). The degrees of discriminancy are presented in Tables 5 and6.

The Table 7 is the summary of this analysis. It shows that the dissimilarity criterion is a good choice. It has indeed a good consistency with F J and qms. One should notice that Sp is considered as dierent, because not consistent with the other criteria. As expected, because they try to have a perceptual justication, F J, qms, f om and give good results. The Acc criterion does not seem to be a good criterion.

As a discussion, the choice of one segmentation as a reference is arbitrary. It has no inuence in the DRIVE database, because both experts propose almost similar segmentations, but references for the DRIVE database are really dissimilar (vk segmentations present a lot more details than ah segmentations). Thus, the global results show a dierent classication of the segmentation methods (see Figs. 4 and5). The dierence between the F J and the dissimilarity are less evident. The computed values of the degree of consistency are presented in Tables 3 and4. fom dice Se Sp Ppv Npv Acc F1 FJ qms Table 7: As stated in [START_REF] Huang | Using auc and accuracy in evaluating learning algorithms[END_REF], the criterion f is statistically consistent and more discriminating than g if and only if d • C > 0.5 and d • D > 1. In this case, we assert that f is a better dissimilarity criterion than g, which is represented with a star/green cell.

6 Concluding discussion and perspectives

Concluding discussion

The purpose of this article was to address the quantitative evaluation of image segmentation methods by means of comparative criteria (rst-order comparative problem) and the quantitative comparison of the criteria themselves (second-order comparative problem). The application issue of retina vessels' image segmentation has been addressed with ve specic segmentation methods and eleven dissimilarity criteria reported in the specialized literature. This articles emphasizes the problem using expert's manual reference, in the case of the STARE database the two experts give really dierent segmentations. To enhance this study, it would be interesting to have a relative rank of all the methods given by one or several experts, so that the consistency and discriminancy values could reect the experts analysis.

It has been shown that the dissimilarity criterion is consistent with other perceptually based criterions. The practical results on the DRIVE and STARE databases of retina images highlight its properties, particularly its tolerance to small variations in the binary images, and its robustness to various small perturbations in the initial images. The dissimilarity criterion ranks the manual segmentation with the lowest value, which means that it is in agreement with the visual perception.

Moreover, a pathway is now open to select the more appropriate image segmentation method for a particular application issue (e.g., the retina vessels extraction in the present article), in addition to choose the best parameter values for optimally tuning the algorithms of the segmentation methods.

Perspectives

In future studies, the dissimilarity criterion will be used to compare the results of some segmentation methods dedicated to corneal endothelium images, as well as the tuning of the parameter values.

Figure 1 :

 1 Figure 1: Illustration of the triangle inequality axiom from a visual point of view. The human visual system does not verify the triangle inequality, because A and B are partly close, B and C are also partly close, but A and C are really dierent Thus, if d is the visual comparative criterion, yields d(A, C) > d(A, B) + d(B, C).

Table 1 :

 1 T P ) False Positive (F P ) Pixel value = 0 False Negative (F N ) True Negative (T N ) Pixels classication. Each category set (T P , F P , F N or T N ) represents a number of pixels. F N + T N + F P

5 Take

 5 emphasize linear segments of length L and thickness σ; 4 end the minimum result for all directions;

6

  Threshold and remove spurs to get a clean binary image; 7 Apply a mask of the region of interest, if necessary; 8 end Method 1: Retina image segmentation from [5]. For the tests, the color images are converted into grayscale by only taking the Green channel. The values of σ and L are experimentally xed at 2 and 10.

  (a) Original image of retina. (b) Manual segmentation of (a). (c) Segmentation of (a) by the algorithm of Chaudhuri et al. [5] (Method 1). (d) Segmentation of (a) by the algorithm of Chanwimaluang et al. [4] (Method 2).(e) Segmentation of (a) by the algorithm of Soares et al.[START_REF] Soares | Retinal vessel segmentation using the 2-d gabor wavelet and supervised classication[END_REF] (Method 3).(f ) Segmentation of (a) by the algorithm of Mendonça et al.[19] (Method 4).(g) Segmentation of (a) by the algorithm of Marin et al.[START_REF] Marin | A new supervised method for blood vessel segmentation in retinal images by using gray-level and moment invariants-based features[END_REF] (Method 5).

Figure 2 :

 2 Figure 2: Retina image (eye fundus) and its segmentations (from DRIVE database [24] and STARE database [12], by using the ve presented methods of Chaudhuri et al. [5], Chanwimaluang et al. [4]), Soares et al. [23], Mendonça et al. [19] and Marin et al. [16].

4. 3

 3 Soares et al. method (Method 3) Soares et al.[START_REF] Soares | Retinal vessel segmentation using the 2-d gabor wavelet and supervised classication[END_REF] proposed a method applied on the (inverted) green channel and based on wavelet decomposition and pixel classication. The method is summarized in Method 3. Data : Input ← Inverted green channel of image of retina. Result : S: Segmentation of the retina image. 1 begin 2 Preprocessing: extending the Region of Interest (ROI); 3 Wavelet decomposition (using 2D-Gabor wavelet); 4 For each scale, keep the maximum value for all directions; 5 Feature normalization (avoid dimensionality problems); 6 Supervised classication (with manual segmentation on a training set); 7 end Method 3: Retina image segmentation from [23].4.4 Mendonça et al. method (Method 4)A fourth segmentation method has been proposed by[START_REF] Mendonça | Segmentation of retinal blood vessels by combining the detection of centerlines and morphological reconstruction[END_REF]. The main stages are presented in Method 4. Data : Input ← image of retina. Result : S: Segmentation of the retina image. 1 begin 2 Preprocessing: background normalization and thin vessel enhancement; 3 Vessel centerline detection phase (Dierence of Oset Gaussians lters); 4 Vessel segmentation phase (based on mathematical morphology lters); 5 end Method 4: Retina image segmentation from [19].

: 5 :

 5 Input ← image of retina. Result : S: Segmentation of the retina image. Retina image segmentation from[START_REF] Marin | A new supervised method for blood vessel segmentation in retinal images by using gray-level and moment invariants-based features[END_REF].

Figure 3 :

 3 Figure 3: Method for xing the tolerance parameter value. Two experts have manually segmented the 20 training images of the DRIVE database, and the dissimilarity criterion has been applied between the segmentation results. The average value for all

Figure 4 :

 4 Figure 4: Quantitative comparison of the six image segmentation methods evaluated with the eleven dissimilarity criteria, for the images of the DRIVE database. The results have been normalized (i.e. the value 1 is given to the highest value for each dissimilarity criterion). The mean values (on all images) have been represented. The choice of representing the values with lines has been done because it emphasizes the fact that methods are consistent or not. In this graph, the comparison of numerical values has no sense. Instead, the rankings (see Def. 5.1) of the segmentation methods can be compared for the dierent criterions, as well as their discriminancy (see Def. 5.2). Continues with images of the STARE database on Fig. 5.

Figure 5 :

 5 Figure 5: Continued from Fig. 4. The rankings are really dierence, depending on the choosen expert. The manual segmentation is not always given the lowest value, because both references from experts 'ah' and 'vk' are really dierent (one contains detailed vessels, whereas the other is less detailed).

  Results on images of the DRIVE database only.

Table 2 :

 2 Some evaluation criteria based on pixel classication (Table

Table 3 :

 3 3 will quantify these visual facts. Mean results of the eleven criteria for the ve segmentation methods (and the second manual segmentation, which is a segmentation from another expert), over all images of DRIVE and STARE databases. Continues on Table2on page 11.

	mean std. dev. median 0.18 0.05 0.17 Chanwimaluang et al 0.12 Chaudhuri et al 0.07 0.09 Marin et al 0.08 0.03 0.08 Soares et al 0.06 0.05 0.05 Mendonca et al 0.07 0.07 0.05 Manual segmentation 0.04 0.02 0.03	mean std. dev. median 0.60 0.12 0.57 0.10 0.29 Chanwimaluang et al 0.32 Chaudhuri et al 0.35 0.11 0.35 Marin et al 0.31 0.13 0.28 Soares et al 0.32 0.15 0.28 Mendonca et al Manual segmentation 0.25 0.10 0.22
	(a)	criterion, with ρ = 4	(b) f om criterion
	mean std. dev. median 0.50 0.12 0.48 Chanwimaluang et al 0.35 Chaudhuri et al 0.07 0.33 Marin et al 0.29 0.07 0.28 Soares et al 0.27 0.07 0.24 Mendonca et al 0.30 0.12 0.27 Manual segmentation 0.24 0.04 0.23	mean std. dev. median 0.62 0.12 0.60 0.09 0.35 Chanwimaluang et al 0.35 Chaudhuri et al 0.39 0.10 0.39 Marin et al 0.35 0.10 0.33 Soares et al 0.37 0.14 0.32 Mendonca et al Manual segmentation 0.29 0.09 0.28
		(c) dice criterion	(d) Se criterion
	mean std. dev. median 0.01 0.01 0.01 Chanwimaluang et al 0.03 Chaudhuri et al 0.02 0.03 Marin et al 0.01 0.01 0.01 Soares et al 0.01 0.01 0.01 Mendonca et al 0.01 0.01 0.01 Manual segmentation 0.01 0.01 0.01	mean std. dev. median 0.18 0.13 0.14 0.13 0.27 Chanwimaluang et al 0.31 Chaudhuri et al 0.13 0.07 0.13 Marin et al 0.15 0.11 0.15 Soares et al 0.18 0.15 0.17 Mendonca et al Manual segmentation 0.15 0.09 0.16
		(e) Sp criterion	(f ) P pv criterion

Table 2 :

 2 Continued from Table

	mean std. dev. median 0.06 0.02 0.06 Chanwimaluang et al 0.04 Chaudhuri et al 0.01 0.04 Marin et al 0.04 0.02 0.03 Soares et al 0.04 0.02 0.03 Mendonca et al 0.04 0.02 0.03 Manual segmentation 0.03 0.02 0.03	mean std. dev. median 0.07 0.02 0.07 0.02 0.06 Chanwimaluang et al 0.07 Chaudhuri et al 0.05 0.02 0.04 Marin et al 0.05 0.02 0.04 Soares et al 0.05 0.02 0.04 Mendonca et al Manual segmentation 0.04 0.01 0.04
	(g) N pv criterion	(h) Acc criterion
	mean std. dev. median 0.44 0.14 0.40 Chanwimaluang et al 0.21 Chaudhuri et al 0.10 0.19 Marin et al 0.20 0.09 0.19 Soares et al 0.18 0.10 0.15 Mendonca et al 0.19 0.13 0.13 Manual segmentation 0.13 0.08 0.12	mean std. dev. median 0.66 0.10 0.65 0.06 0.47 Chanwimaluang et al 0.46 Chaudhuri et al 0.44 0.10 0.45 Marin et al 0.36 0.10 0.36 Soares et al 0.35 0.11 0.34 Mendonca et al Manual segmentation 0.29 0.09 0.28
	(i) F 1 criterion	(j) F J criterion
	mean std. dev. median 32.91 21.05 26.40 6.22 9.98 Chanwimaluang et al 11.64 Chaudhuri et al 8.78 4.77 7.43 Marin et al 5.26 2.87 4.34 Soares et al 5.48 3.91 4.22 Mendonca et al Manual segmentation 3.58 2.28 2.70
	(k) qms criterion

Denition 5.1 (Degree of Consistency). For two dissimilarity criteria f and g dened on the domain ξ (ξ is the family of the

Table 3 :

 3 Degree of consistency d • C between the dierent dissimilarity criteria, for the DRIVE database. If d • C > 0.5, the two dissimilarity criteria are consistent. This evaluation is symmetric.

	1.00	0.85	0.93	0.86	0.45	0.57	0.87	0.87	0.90	0.97	0.97
	fom 0.85 1.00 0.86 0.86 0.34 0.46 0.86 0.79 0.92 0.86 0.84 dice 0.93 0.86 1.00 0.85 0.47 0.59 0.86 0.92 0.91 0.93 0.92 0.86 0.86 0.85 1.00 0.33 0.44 0.99 0.78 0.87 0.86 0.87 Se 0.45 0.34 0.47 0.33 1.00 0.89 0.34 0.55 0.41 0.45 0.45 Sp Ppv 0.57 0.46 0.59 0.44 0.89 1.00 0.45 0.66 0.52 0.57 0.56 Npv 0.87 0.86 0.86 0.99 0.34 0.45 1.00 0.79 0.88 0.87 0.88 Acc 0.87 0.79 0.92 0.78 0.55 0.66 0.79 1.00 0.84 0.88 0.87 0.90 0.92 0.91 0.87 0.41 0.52 0.88 0.84 1.00 0.91 0.90 F1 0.97 0.86 0.93 0.86 0.45 0.57 0.87 0.88 0.91 1.00 0.98 FJ qms 0.97 0.84 0.92 0.87 0.45 0.56 0.88 0.87 0.90 0.98 1.00
		fom dice Se Sp Ppv Npv Acc F1 FJ qms
	1.00	0.82	0.87	0.61	0.64	0.68	0.62	0.87	0.79	0.90	0.87
	fom 0.82 1.00 0.84 0.68 0.55 0.59 0.68 0.76 0.86 0.85 0.81 dice 0.87 0.84 1.00 0.68 0.59 0.64 0.69 0.85 0.83 0.85 0.88 0.61 0.68 0.68 1.00 0.27 0.32 0.99 0.53 0.58 0.63 0.73 Se 0.64 0.55 0.59 0.27 1.00 0.95 0.28 0.74 0.65 0.63 0.54 Sp Ppv 0.68 0.59 0.64 0.32 0.95 1.00 0.33 0.79 0.67 0.66 0.58 Npv 0.62 0.68 0.69 0.99 0.28 0.33 1.00 0.54 0.58 0.64 0.74 Acc 0.87 0.76 0.85 0.53 0.74 0.79 0.54 1.00 0.82 0.84 0.79 0.79 0.86 0.83 0.58 0.65 0.67 0.58 0.82 1.00 0.82 0.76 F1 0.90 0.85 0.85 0.63 0.63 0.66 0.64 0.84 0.82 1.00 0.87 FJ qms 0.87 0.81 0.88 0.73 0.54 0.58 0.74 0.79 0.76 0.87 1.00
	(a) Degree of consistency d • C for the STARE database, with expert 'ah'.
		fom dice Se Sp Ppv Npv Acc F1 FJ qms
	1.00	0.77	0.91	0.76	0.61	0.68	0.78	0.87	0.76	0.92	0.93
	fom 0.77 1.00 0.83 0.91 0.39 0.46 0.91 0.73 0.95 0.79 0.77 dice 0.91 0.83 1.00 0.83 0.55 0.62 0.85 0.89 0.82 0.88 0.87 0.76 0.91 0.83 1.00 0.39 0.45 0.98 0.73 0.87 0.79 0.78 Se 0.61 0.39 0.55 0.39 1.00 0.93 0.41 0.66 0.40 0.55 0.58 Sp Ppv 0.68 0.46 0.62 0.45 0.93 1.00 0.47 0.73 0.46 0.62 0.64 Npv 0.78 0.91 0.85 0.98 0.41 0.47 1.00 0.75 0.86 0.81 0.79 Acc 0.87 0.73 0.89 0.73 0.66 0.73 0.75 1.00 0.72 0.80 0.83 0.76 0.95 0.82 0.87 0.40 0.46 0.86 0.72 1.00 0.77 0.74 F1 0.92 0.79 0.88 0.79 0.55 0.62 0.81 0.80 0.77 1.00 0.92 FJ qms 0.93 0.77 0.87 0.78 0.58 0.64 0.79 0.83 0.74 0.92 1.00

(b) Degree of consistency d • C for the STARE database, with expert 'vk'.

Table 4 :

 4 Degree of consistency d • C between the dierent dissimilarity criteria, for the STARE database. If d • C > 0.5, the two dissimilarity criteria are consistent. This evaluation is symmetric.

		fom dice	Se	Sp Ppv Npv Acc F1	FJ qms
	NaN	3.00	19.00	4.67	7.33	3.06	5.00	14.60	3.80	0.07	Inf
	fom 0.33 NaN 2.26 2.08 2.92 1.16 2.38 2.65 0.90 0.17 2.50 dice 0.05 0.44 NaN 0.71 1.81 0.51 0.81 2.40 0.37 0.03 1.20 0.21 0.48 1.40 NaN 2.07 0.76 5.00 1.83 0.50 0.10 1.50 Se 0.14 0.34 0.55 0.48 NaN 0.23 0.53 0.84 0.32 0.06 0.64 Sp Ppv 0.33 0.86 1.95 1.31 4.42 NaN 1.46 2.57 0.81 0.17 1.92 Npv 0.20 0.42 1.23 0.20 1.90 0.68 NaN 1.67 0.43 0.09 1.35 Acc 0.07 0.38 0.42 0.55 1.19 0.39 0.60 NaN 0.31 0.05 0.68 0.26 1.11 2.73 2.00 3.09 1.24 2.33 3.22 NaN 0.13 3.23 F1 14.00 5.78 34.50 10.50 15.83 5.80 11.17 21.25 7.83 NaN Inf FJ qms 0.00 0.40 0.83 0.67 1.56 0.52 0.74 1.48 0.31 0.00 NaN

Table 5 :

 5 Degree of discriminancy d • D between the dierent dissimilarity criteria, for the DRIVE database. If d • D > 1, the criterion presented in column is more discriminant than the criterion presented in row (NaN stands for Not a Number. Inf stands forinnity; it is obtained when Q = 0, which means that there is no value where g(X 1 ) > g(X 2 ) and f (X 1 ) = f (X 2 )), for f and g two criterions, X 1 and X 2 two segmentation results.

		fom dice Se	Sp Ppv Npv Acc	F1	FJ qms
	NaN	0.63	0.54	0.13	4.95	0.40	8.60	15.86	0.50	0.26	0.00
	fom 1.58 NaN 0.90 0.25 8.23 0.67 18.86 15.38 0.76 0.43 0.00 dice 1.87 1.11 NaN 0.28 8.38 0.74 16.88 24.40 0.83 0.44 0.00 7.50 4.00 3.60 NaN 55.50 2.33 Inf 66.00 2.67 1.50 0.00 Se 0.20 0.12 0.12 0.02 NaN 0.00 1.76 1.58 0.05 0.05 0.00 Sp Ppv 2.50 1.50 1.36 0.43 Inf NaN 17.50 41.67 1.17 0.64 0.00 Npv 0.12 0.05 0.06 0.00 0.57 0.06 NaN 0.85 0.04 0.02 0.00 Acc 0.06 0.07 0.04 0.02 0.63 0.02 1.17 NaN 0.01 0.02 0.00 2.00 1.31 1.20 0.38 20.80 0.86 22.67 121.00 NaN 0.53 0.00 F1 3.88 2.33 2.25 0.67 22.20 1.56 46.67 43.33 1.88 NaN 0.00 FJ qms Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf NaN
		(a) Degree of discriminancy for the STARE database, with expert 'ah'.
		fom dice Se	Sp Ppv Npv Acc F1	FJ qms
	NaN	0.65	1.07	0.84	2.28	1.60	1.32	2.20	0.89	0.40	19.25
	fom 1.55 NaN 2.05 2.11 3.96 2.54 7.20 3.21 3.17 0.93 6.69 dice 0.94 0.49 NaN 0.74 2.09 1.50 1.41 2.22 0.81 0.56 6.46 1.20 0.47 1.36 NaN 2.66 1.85 Inf 2.14 1.14 0.72 6.06 Se 0.44 0.25 0.48 0.38 NaN 0.19 0.58 0.66 0.38 0.28 1.41 Sp Ppv 0.63 0.39 0.67 0.54 5.17 NaN 0.83 0.98 0.56 0.39 2.26 Npv 0.75 0.14 0.71 0.00 1.71 1.21 NaN 1.27 0.47 0.45 3.50 Acc 0.45 0.31 0.45 0.47 1.51 1.02 0.78 NaN 0.50 0.31 3.13 1.12 0.32 1.23 0.88 2.67 1.79 2.12 2.00 NaN 0.67 5.59 F1 2.50 1.08 1.79 1.38 3.54 2.59 2.21 3.25 1.48 NaN 19.80 FJ qms 0.05 0.15 0.15 0.16 0.71 0.44 0.29 0.32 0.18 0.05 NaN

(b) Degree of discriminancy for the STARE database, with expert 'vk'.

Table 6 :

 6 Degree of discriminancy d • D between the dierent dissimilarity criteria, for the STARE database, using two dierent experts as the reference.


Yann Gavet
email: gavet@emse.fr

Jean-Charles Pinoli

{yann Gavet

Jean-Charles Pinoli}

Comparison and supervised learning of segmentation methods dedicated to specular microscope images of corneal endothelium

Comparison and supervised learning of segmentation methods dedicated to specular microscope images of corneal endothelium

Yann Gavet, Jean-Charles

1 Introduction

Human eye and cornea

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 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 human cornea

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 is extremely pain sensitive. 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, hence plays a vital role to keep it in a dehydrated state. Without this pumping action, the stroma would accumulate water, become hazy, and finally opaque (corneal oedema) leading to loss of vision.

Physiology of the human corneal endothelium

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/mm 2 . 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 500-3500 cells/mm 2 . 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.

Principles of specular microscopy

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 (0.08mm 2 compared to the endothelium surface of about 100mm 2 , see Fig. 5). The image is then analyzed by a computer software (embedded in the microscope) to provide both cell density and morphometry.

The physical principle of this type of microscope is the specular reflection (i.e. the same reflection as for a mirror).

As the light goes through the several different layers of the cornea, it is reflected at each interface ( [START_REF] Bourne | Some optical principles of the clinical specular microscope[END_REF][START_REF] Laing | Clinical specular microscopy[END_REF]). The deeper the layer is in the cornea, the darker it appears on the image. This explains the intensity variations within the images, which physically corresponds to the superposition of several different layers.

Quality criteria of the corneal endothelium

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/mm 2 ). There are several threshold values: for example, an ECD lower than 400 cells/mm 2 does not enable to maintain the cornea transparency. An ECD lower than 1000 cells/mm 2 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.

Three image segmentation methods

Different methods exist to perform the segmentation of images of endothelial cells. Among those, three methods give the better results ( [START_REF] Gavet | Visual perception based automatic recognition of cell mosaics in human corneal endothelium microscopy images[END_REF]). The present article first recalls their related algorithms and will then compare their results with regard to several criteria.

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 [START_REF] Beucher | Use of watersheds in contour detection[END_REF]) to get closed contours for each cell. Notice that these three algorithms make important use of mathematical morphology operators (see, e.g., [START_REF] Soille | Morphological Image Analysis: Principles and Applications[END_REF]).

Vincent and Master's method

This method has been proposed in [START_REF] Vincent | Morphological image processing and network analysis of cornea endothelial cell images[END_REF]. It is based on the fact that cell borders' intensities are lower than the cell interiors' intensities, and represent somehow a local maximum of intensity that is retrieved by a morphological operation called a h-maxima ( [START_REF] Soille | Morphological Image Analysis: Principles and Applications[END_REF]). To avoid the problem of noise, a first filtering process is performed by the way of a morphological alternate sequential filter. It involves two parameters:

• h is a value for the h-maxima operation (an intensity), that gives the final markers of the cells.,

• o is the order of the morphological alternate sequential filter.

The algorithm is summarized in algorithm 1.

Data: Input ← Greyscale image of cornea endothelial cells.

o ← Order of alternate sequential filter.

h ← Value for the h-maxima.

Result: S: Segmentation of the cornea endothelial cells.

Angulo and Matou's method

This method is more recent than the previous one ( [START_REF] Angulo | Automatic quantification of in vitro endothelial cell networks using mathematical morphology[END_REF]). The cell markers are detected by the mean of a distance map after some filtering process (performed by a morphological opening and a morphological top-hat). The difference with Vincent and Masters's algorithm is that a first segmentation of the contours is performed and then is improved by the use of the watershed [START_REF] Beucher | Use of watersheds in contour detection[END_REF]. The algorithm 2 details the algorithm. Five parameters are required.

Th three first ones, s 1 , s 2 and g, are used for the filtering process. The last two ones, t and h are used to get the markers that will be used by the watershed.

Gavet and Pinoli's method

This is the most recent method proposed in [START_REF] Gavet | Visual perception based automatic recognition of cell mosaics in human corneal endothelium microscopy images[END_REF]. The improvements to the other methods comes from the fact that the first segmentation of the borders of the cells better performs due to the elimination of non-linear parts of the skeleton (see details of algorithm 2 and algorithm 3). The Gavet and Pinoli's method requires five parameters,

Data: Input ← Greyscale image of cornea endothelial cells. s 1 , s 2 ← Structuring element sizes.
g ← Size of gaussian filter.

t ← Treshold value.

h ← Value for the h-maxima.

Result: S: Segmentation of the cornea endothelial cells. and its algorithm is summarized in algorithm 3. After a filtering process (parameter o), the contours are first roughly detected by the use of the openings by segments (parameters t s and then o,c,e). This operators intend to detect linear cell contours. Then, these contours are corrected by the mean of a distance map and a constrained watershed (parameter s), as proposed by the two previous methods.

Summary of the control parameters

The Table 1 summaries the control parameters used by the three presented image segmentation methods. The main problem here is: how to select the values of these parameters? The following section will try to answer to this problem by using two comparison criteria (namely the dissimilarity criteria ǫ and f om), and by using them on an image database to find the best parameter values.

Image segmentation evaluation

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 of segmented regions.

Data: Input ← Greyscale image of corneal endothelial cells.

o ← Order of alternate sequential filter.

t s ← Length of segments.
c ← Structuring element size for a closing operation.

s ← Structuring element size for an opening operation.

Result: Segmentation of the cornea endothelial cells.

begin This article deals with supervised segmentation evaluation, i.e. 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 intra-region homogeneity is involved), but the bias introduced by the expert do not have to be neglected (see Sect. 3.4).

ASF ← Alternate Sequential Filter of size o of Input; R ← (Input -ASF ) < 0 (residu computation); U ← Supremum (union) of Openings of R

Basic notations

The following notations are first introduced: I R and I X represent two binary images that correspond to a reference segmentation method R and the evaluated segmentation method X, respectively. Both I R and I X are considered as sets of contours. In this article, R and X may be employed for I R and I X in order to alleviate the notations and more deeply for emphasizing the geometrical problems. A point p ∈ I X or p ∈ I R means a point present in the related segmented binary image.

Classical dissimilarity criteria

This article will not present an exhaustive view of supervised evaluation of segmentation criteria. The reader can have a look at [START_REF] Gavet | A geometric dissimilarity criterion between jordan spatial mosaics. theoretical aspects and application to segmentation evaluation[END_REF] for a more complete presentation and a comparison.

The two detailed criteria have been chosen because they are tolerant to spatial variations. One could also use other frequently used criteria proposed in the literature [START_REF] Ghose | A supervised learning framework of statistical shape and probability priors for automatic prostate segmentation in ultrasound images[END_REF][START_REF] Schaap | Standardized evaluation methodology and reference database for evaluating coronary artery centerline extraction algorithms[END_REF][START_REF] Kirişli | Standardized evaluation framework for evaluating coronary artery stenosis detection, stenosis quantification and lumen segmentation algorithms in computed tomography angiography[END_REF], like the Hausdorff distance, the Dice coefficient (also known as the Jaccard index), or classification criteria (specificity, sensibility). The main drawback of these criteria is that a small noise (like a misdetected point) implies a high comparison value. This is why people introduced a distance ponderation, like the p-th point in the Hausdorff distance, or the following figure of merit.

The figure of merit ( [START_REF] Abdou | Qualitative design and evaluation of enhancement/thresholding edge detector[END_REF][START_REF] Strasters | Three-dimensional image segmentation using a split, merge and group approach[END_REF]), is defined by:

f om R (X) = 1 - 1 max{#(M ), #(X)} p∈X 1 1 + d 2 (p, R)
where d(p, R) is the Euclidean distance between the pixel p ∈ X to the closest pixel of R, and # the number of pixels of the considered segmentation result R or X (which are non-empty images, at least for R).

Partitioning

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 to the following binary question: is this pixel inside the region of interest or not?

To formalize this mathematically, let S be a binary image resulting from a segmentation process, defined by the number of regions (number of labels L, K = [1; L]) that partitions the spatial domain D and by the set R of adjacent regions R i that fulfill D:

∀(i, j) ∈ K, i = j, R i ∩ R j = ∅ (1)
and

R = R i , i ∈ K (2)
This article deals with the case where contours are detected and the segmentation result is a binary image, i.e., L = 2, K = [1; 2] (label 1 stands for the background, label 2 for the detected contours).

The ǫ dissimilarity criterion

The ǫ dissimilarity criterion is based on the symmetric difference ∆ of sets, but this latter lacks some tolerance, which is introduced by the Minkowski addition.

Symmetric difference set

First, let us recall that the symmetric difference set between two segmentations R and X (R ⊂ D and X ⊂ D), denoted ∆(R, X), is defined by Eq. 3:

∆(R, X) = (R ∪ X) \ (R ∩ X) = (R \ X) ∪ (X \ R) (3)

Minkowski addition

The Minkowski addition ( [START_REF] Minkowski | Volumen und Oberfläche[END_REF]) defines an algebraic operation between sets in the n-dimensional Euclidean space R n . It will be used to spatially "enlarge" the segmentations X or R to be compared, in order to be less sensitive to small spatial variations or noises.

If X is a set (segmentation result) and B is another set (generally the unit ball), the Minkowski sum of X and B, denoted X ⊕ B, is then defined by:

X ⊕ B = {x + b|x ∈ X, b ∈ B} X ⊕ B = b∈B {x + b|x ∈ X} ( 4
)
where ⊕ is the Minkowski addition symbol. In the field of Mathematical Morphology ( [START_REF] Soille | Morphological Image Analysis: Principles and Applications[END_REF]), it is equivalent to the dilation, and B is called a structuring element (for example the unit ball).

Definition

In [START_REF] Moretti | Phantom-based performance evaluation: Application to brain segmentation from magnetic resonance images[END_REF], each pixel in the segmentation result is attributed a distance value to the reference segmentation, and an histogram of these distances is thus computed. Then, a statistical analysis of this histogram is performed. In the same spirit, we propose a dissimilarity criterion that is tolerant to small spatial variations. The ǫ dissimilarity criterion with the tolerance ρ applied to segmented images is defined in the case of discrete images (R is the reference segmentation result and X is the considered segmented image) by the Eq. 5 ( [START_REF] Gavet | A geometric dissimilarity criterion between jordan spatial mosaics. theoretical aspects and application to segmentation evaluation[END_REF]):

ǫ ρ R (X) = #{(X \ (R ⊕ ρN )) ∪ (R \ (X ⊕ ρN ))} #{R ⊕ ρN } (5)
with N being the structuring element of radius 1 (typically the unit ball) and # designating the number of pixels in the set (# is the cardinal operator, counting the number of non zero valued pixels in the sets X or R). Practically, ρ is the radius of the ball used to dilate the binary images, thus forming a tolerance tube around the original set X or R. This article will propose a way of selecting the right value for ρ.

The main properties of ǫ are:

• ǫ ρ R (R) = 0, which means that when R is compared to itself, the numerical result is 0,

• ǫ ρ R (R) -→ ρ→∞ 0, which means that if the tolerance increases, the numerical value tends to 0.

• ρ is the tolerance value, thus ǫ is tolerant to small spatial variations (like translations, rotations, over-and under-segmentations).

Discussion about the notion of metric

The usual concept to compare mathematical objects is the metric notion, defined by four axioms (identity, separation, symmetry and triangle inequality, see [START_REF] Deza | Dictionary of distances[END_REF]). If a metric has important mathematical properties, it has been proved that the human visual system does not follow them ( [START_REF] Tversky | Features of similarity[END_REF][START_REF] Tversky | Similarity, separability and the triangle inequality[END_REF]). For example, the human visual system does not always consider two distinct objects as different (the separation property is thus not verified). This is also true for the triangle inequality and the symmetry property ( [START_REF] Gavet | A geometric dissimilarity criterion between jordan spatial mosaics. theoretical aspects and application to segmentation evaluation[END_REF]). This is important to notice that ǫ is not a metric: separation, symmetry and triangle inequality are not verified. This is why it is called a dissimilarity criterion.

Bias in experts manual segmentation: choice of the tolerance value

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 [START_REF] Hameeteman | Evaluation framework for carotid bifurcation lumen segmentation and stenosis grading[END_REF]. Some others do not take these into account and use only one reference segmentation as an absolute truth. The proposed ǫ dissimilarity criterion deals with this problem by the choice of the tolerance parameter ρ. The ρ value will in fact reflect the precision of the manual segmentation of the expert.

For one original gray-tone image, the experts have several times manually drawn their segmented image, and the ǫ dissimilarity criterion have been used to compare every manually segmented image to the others. The mean value of the ǫ dissimilarity criterion is represented in the Fig. 1. The reader can consider that an expert will always (try to) draw the contours at the same location within a certain spatial tolerance (i.e., within the tolerance tube), depending on the image size and the precision of the drawing tool. From Fig. 1, we conclude that a reasonable tolerance value is typically ρ = 2 is the present application.

Thus, the ǫ dissimilarity criterion is able to deal with the bias in the experts reference segmentation as well with the noises present in the segmentation results themselves. The next section will focus on cornea endothelium images and their segmentations.

Image segmentation method tuning

The different segmentation algorithms presented in the previous sections require to setup the values of the socalled control parameters. The choice of the control parameter values for a specific application issue is generally not trivial, especially for non image analysis experts. This section explains the generic way of selecting the best parameters in average for the considered three image segmentation methods. Figure 1: Method for fixing the tolerance parameter. In this example, ρ is in pixels, and there is a strong gap between no tolerance (ρ = 0) and a tolerance of one pixel (ρ = 1). The scale is logarithmic.

Method

Definitions

Let p be the control parameters set of a given algorithm A. For example, for algorithm 1, p = {s, h}. Then, A p (I)

is the result of the segmentation process by algorithm A with the parameter set p on the input image I.

Let C denote the criterion used to compare the segmentation results with the reference. In this article, C will be either the dissimilarity criterion ǫ or f om.

Let Q be an evaluation function of the quality of the segmentation, depending on the considered criterion C, defined as follows:

Q C (A p , I) = C(A p (I), R)
where R is the reference segmentation of the image I. To simplify the notations, R will be used instead of R(I).

Best parameter set

In the following, we consider an image database of N gray-tone images, each being associated with a reference segmented image. What we are looking for is the best parameter set, i.e., the parameter set that will result in the best segmented images considering the reference R and a specific comparison criterion (among the two considered criteria, f om and ǫ).

Let p be the best parameter set regarding the mean of all quality values on the N gray-tone images of the database, yielding to:

p(A, C) = arg min p 1 N I Q C (A p , I) = arg min p mean I Q C (A p , I)
Let Q be the minimal mean value of Q on the N images of the database, yields:

Q(A, C) = min p 1 N I Q C (A p , I) = mean I Q C (A p, I)
This way of finding the best parameter set is also called leave-one-out cross validation.

Trimmed mean

Some noise may be present in the computed values (mainly because of a too poor image quality). To be more tolerant to 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 k ∈ [0; 0.5] is the percentage of discarded values, then:

pk (A, C) = arg min p Tmean I,k Q C (A p , I) Qk (A, C) = min p Tmean I,k Q C (A p , I)
Notice that the trimmed mean corresponds to the classical mean for k = 0, namely:

p0 = p Q0 = Q 4.1.

Median

The median of Q, denoted med Q, is also a classical way to avoid noise perturbations in such measurements, yielding to:

p(A, C) = arg min p med I Q(A p , I) Q(A, C) = min p med I Q(A p , I)

Projection

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 P be a parameter of the set p. Let QP k be the (trimmed) mean evolution of Q when the parameter P is varying and the other parameters are fixed at values of pk . The parameters are chosen among those defined in Table 1.

K-fold cross validation

The K-fold cross validation consists in validating the learning process by splitting the database into K folds, using K -1 folds as the learning database and the last one as the test database [START_REF] Hastie | The elements of statistical learning[END_REF]. This is repeated K times such that each fold will be used as the test database. The result is a discrepancy value that reflects the pertinence of the learning. It will be noticed CV in the different result tables, which is the mean result value over each test partition.

The results will be presented in tables in Sect. 4.3 in details. For each method and for each partition, the learned parameter values are presented as well as the corresponding result of the criterion value for the test partition. Let Q CV i be the value of the criterion for the partition i, and

Q CV = mean i {Q CV i }.
For comparison purposes, we also provide the best criterion value that could have been obtained on the test partition, denoted QCV i (for partition i), and QCV for the mean on all partitions. The value of QCV should be only a little smaller than Q CV .

Quantitative comparison results

This section presents the results for the three aforementioned image segmentation methods.

An image database of N = 30 gray-tone images of the human corneal endothelium acquired with a specular microscope is employed to evaluate the segmentation processes realized by the different algorithms. This image database (see Fig. 5) contains gray-tone images and also the related experts' segmented images (manually performed).

Vincent and Masters' method

The summary of the optimal control parameters values is presented in Table 2.

It appears that ǫ and f om do not provide the same results for the optimal value of parameter h. One shall notice that the value of Qh 0.5 does not vary a lot for both ǫ and f om criteria (see Figs. 2(b) and 2(c)). This means that the choice of h appears as not crucial.

Criterion

Optimal parameters Trimmed mean (k = 0. See Fig. 2 for illustrations. Masters. See Table 2 for numerical values.

Angulo and Matou's method

The results are presented in Table 3. Both ǫ and f om give the same results. In addition, the projections (Figs. 3(e) and 3(c)) show that s 2 and h are useless. Thus, the top-hat transform can be avoided, and the h-maxima operation can be replaced by the computation of the maxima of the distance map.

Gavet and Pinoli's method

The summary of the optimal control parameter values is presented in For both f om and ǫ criteria, the optimal control parameter values are identical. The filtering parameter o used in the alternate sequential filter must be higher than 4. This parameter is linked to the length of an edge of a cell, and thus, some corneal endothelium with big cells would get a better result with a higher value of o (this explains the almost constant values that can be observed in Fig. 4(c)).

The control parameter s used for filtering markers must be chosen with a low value (less than 7). In this case, the effect is not really noticeable.

The control parameter t s is strictly related to the length of the cells edges (its purpose is to retrieve the linear parts of the detected borders and thus eliminate the noise). See Fig. 4 for illustrations.

Comments

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 Q CV , Q, Q0.5 or Q). Clearly, the superiority of this method comes from the extraction of the linear parts of the cell borders.

The K-fold cross-validation (see Tables 5,[START_REF] Gavet | Visual perception based automatic recognition of cell mosaics in human corneal endothelium microscopy images[END_REF][START_REF] Gavet | A geometric dissimilarity criterion between jordan spatial mosaics. theoretical aspects and application to segmentation evaluation[END_REF][START_REF] Ghose | A supervised learning framework of statistical shape and probability priors for automatic prostate segmentation in ultrasound images[END_REF][START_REF] Hameeteman | Evaluation framework for carotid bifurcation lumen segmentation and stenosis grading[END_REF][START_REF] Hastie | The elements of statistical learning[END_REF] shows that the learning is not far from the optimal value, in terms of ǫ or fom, as well as in terms of optimal parameter values. This can be concluded from comparing the mean ǫ or f om value from the test partitions (Q CV ) to the best possible value ( QCV ): there is only a small difference between these values.

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 2, 3 and4

Conclusion and perspectives

In this article, 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 (ǫ dissimilarity criterion [START_REF] Gavet | A geometric dissimilarity criterion between jordan spatial mosaics. theoretical aspects and application to segmentation evaluation[END_REF] and Pratt's figure of merit f om [START_REF] Abdou | Qualitative design and evaluation of enhancement/thresholding edge detector[END_REF]) to tune up the segmentation algorithms in regard to the expert manual segmentation. As a result, this article proposes the optimal control parameter values to use for these images. It also proposes to avoid some operations since their parameters do not really influence the segmentation results. More generally, this article highlights the relevance of the ǫ dissimilarity criterion with a spatial tolerance, suitable to handle the problem of the bias in the reference segmentation. This ǫ dissimilarity criterion is adapted to compare binary contour images as well as binary sets, for 2D or even 3D images. In a near future, we expect to report such a criterion to compare gray-tone images.

Figure 5: 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 Fig. 6). These images come from [START_REF] Gavet | A geometric dissimilarity criterion between jordan spatial mosaics. theoretical aspects and application to segmentation evaluation[END_REF].

Figure 6: 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 Fig. 5). These images come from [START_REF] Gavet | A geometric dissimilarity criterion between jordan spatial mosaics. theoretical aspects and application to segmentation evaluation[END_REF].

1 begin 2 ASF← 3 M 4 W 5 S 6 end Algorithm 1 :

 1234561 Alternate Sequential Filter of order o of Input; ← h-maxima of ASF and binarization (threshold 0) ; ← Watershed constrained by M on Input image; ← Pruning of W to eliminate the lines touching the borders of the image; Vincent and Masters's algorithm for detecting the human corneal endothelium cells ([START_REF] Vincent | Morphological image processing and network analysis of cornea endothelial cell images[END_REF]).

1 begin 2 G ← Gaussian filter of Input; 3 O 4 T 5 B 6 S ← Skeletonization of T ; 7 DM← 8 M 9 W 10 S 2 :

 123456789102 ← Opening of G with s 1 ; ← Top-Hat of O with s 2 ; ← Binarization of T with threshold t; Distance Map from skel; ← h-maxima of DM (markers); ← Watershed constrained by the markers M in the inverse of DM ; ← Pruning of W to eliminate the lines touching the borders of the image; 11 end Algorithm Angulo and Matou's algorithm for detecting the cells ([2]).

ρ

  Errors made by an expert ophtalmologist in logscale ǫ ρ

Figure 2 :

 2 Figure 2: Projection on the optimal values of the control parameters o and h for the method of Vincent and

Figure 3 :

 3 Figure 3: Projection on the optimal values of the control parameters for the method of Angulo and Matou. See

Figure 4 :

 4 Figure 4: Projection on the optimal values of the control parameters for the method of Gavet and Pinoli. See

Table 2 :

 2 Results for algorithm 1, method of Vincent and Masters, where the control parameters are p = {o, h}.

	5)	Median	Q CV

Table 4 .

 4

	Criterion	Optimal parameters Trimmed mean (k = 0.5) p Q p0.5 Q0.5	Median p	Q	Q CV
	ǫ	{4, 4, 1, 28, 1} 0.15 {3, 4, 1, 22, 1}	0.08	{4, 3, 1, 16, 1} 0.11 0.150
	fom	{3, 4, 1, 21, 1} 0.54 {3, 4, 1, 21, 1}	0.46	{3, 4, 1, 8, 1} 0.51 0.525

Table 3 :

 3 Results for algorithm 2, method of Angulo and Matou, where the control parameters are p = {s 1 , g, s 2 , t, h}. See Fig.3for illustrations.

Table 3

 3

for numerical values.

Table 4 :

 4 Results for algorithm 3, method of Gavet and Pinoli, where the control parameters are p = {c, t s , o, s}.

	Criterion	Optimal parameters Trimmed mean (k = 0.5) p Q p0.5 Q0.5	Median p	Q	Q CV
	ǫ	{2, 7, 4, 6}	0.10	{2, 7, 4, 6}	0.06	{2, 7, 4, 6} 0.08 0.099
	fom	{2, 7, 4, 7}	0.50	{2, 7, 4, 7}	0.45	{2, 7, 4, 8} 0.49 0.506

Table 4

 4

for numerical values.

Table 5 :

 5 . Cross validation informations for algorithm 1, method of Vincent and Masters, with ǫ criterion.

	Partition number Optimal parameter values	Q CV i	QCV i
	1	{2, 8}	0.146	0.118
	2	{2, 12}	0.099	0.096
	3	{2, 9}	0.110	0.109
	4	{2, 12}	0.117	0.115
	5	{2, 12}	0.104	0.104
	mean		Q CV = 0.115 QCV = 0.108
	Partition number Optimal parameter values	Q CV i	QCV i
	1	{2, 19}	0.512	0.512
	2	{2, 18}	0.524	0.510
	3	{2, 19}	0.490	0.488
	4	{2, 19}	0.533	0.533
	5	{2, 19}	0.540	0.540
	mean		Q CV = 0.520 QCV = 0.516

Table 6 :

 6 Cross validation informations for algorithm 1, method of Vincent and Masters, with f om criterion.

	Partition number Optimal parameter values	Q CV i	QCV i
	1	{4, 5, 1, 25, 1}	0.162	0.148
	2	{4, 4, 1, 28, 1}	0.157	0.155
	3	{4, 4, 1, 28, 1}	0.141	0.141
	4	{4, 4, 1, 28, 1}	0.109	0.109
	5	{4, 4, 1, 30, 1}	0.180	0.158
	mean		Q CV = 0.150 QCV = 0.142

Table 7 :

 7 Cross validation informations for algorithm 2, method of Angulo and Matou, with ǫ criterion.

Acknowledgment

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.

Partition number Optimal parameter values

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.