Let I, a P-component image (P ≥ 2). The principle is declined in several steps (see the flowchart in Figure 1).

a) We first construct the compact histogram related to the image I in order to solve the problem of coding and manipulation of nD histograms. Indeed, the compact histogram stores only the E cells actually occupied in clear it removes the empty cells. For an image with P components, a table of size E × P for storing the tuples in the lexicographic morphological order and a table of size E × 1 corresponding to the number of associated pixels (the number of tuples).

Let Hc be the nD compact histogram of the image I containing only the tuples

and i a fixed row of Hc.

Let j describing the number of planes of the image varying from 1 to P. A tuple in the nD compact histogram can be described by the triplet ( i , H c i , E i ) where E i = c a r d ( H c i ) represents the cardinal or the number of a tuple at the row i in H c .

Let I = [ I 1 , ⋅ ⋅ ⋅ , I j , ⋅ ⋅ ⋅ , I P ] and H c j i the j^th multimodal component of the tuple located at row i of Hc, with I_j the j^th component of the image.

b) Then we segment each component (I_j) of the image I by multi-thresholding into a fixed number of classes K (K ≥ 2).

Let K j = K max j be the number of classes obtained for the component I_j of the image I. Let be H_j the marginal monodimensionnal compact histogram of the j^th component of the image I(I_j) and Ni the number of pixels of a gray level of H_j at the row i.

To generalize, let us call H j i the gray level at the row i of the j^th 1D compact histogram of the image I(I_j) and by N j i the number of pixels associated.

The multi-thresholding used here is inspired by the Otsu method applied to the compact marginal 1D histograms H_j. Thus, for a given component j, it produces K j − 1 thresholds denoted [ t 1 j , t 2 j , … … . , t K m a x − 1 j ] = T j ordered in ascending order for a partition of the image into K classes [ C 1 j , C 2 j , ⋅ ⋅ ⋅ , C K j ] where K = K max = K j . They take their values in H_j. Here, we denotes T j the vector of the segmentation thresholds of the component I_j.

The thresholds are obtained by the optimization of Equation (1) in which σ_Intra defines the standard deviation intra-classes whose expression is explicitly given by Equation (2) and other parameters defined by Equations ((3) and (4)).

{ t 1 j , t 2 j , ⋅ ⋅ ⋅ , t K − 1 j } = arg { max t ∈ H j σ I n t r a 2 { t 1 j , t 2 j , ⋅ ⋅ ⋅ , t K − 1 j } } (1)

σ I n t r a 2 = ∑ n = 1 K w n ( μ n − μ T ) (2)

w n = ∑ m = t n − 1 j t n j P m (3)

μ n = ∑ m = t n − 1 j t n j m . P m w n (4)

where P_m and µ_T are respectively the probability of a gray level pixel m and the total average of the gray levels of the j^th component of the image I.

c) On the basis of the thresholds calculated on each component of the image, if K 1 = K 2 = ⋅ ⋅ ⋅ = K j = ⋅ ⋅ ⋅ . = K − 1 then we construct K − 1 threshold vectors S_k with k varying from 1 to K − 1, such that the threshold vector of rank k (S_k) has the scalar thresholds of rank k ( t k j ) from each component of the image I, with S k = ( t k 1 , t k 2 , ⋅ ⋅ ⋅ t k P ) .

In the contrary case, a cross product combination of the partitions of the components of the image is realized.

d) In the first case where the number of classes is fixed and equal to K, the tuples of the nD compact histogram (Hc) are first ordered with respect to the vector thresholds S_k according to a vectorial order of choice and then the K partitions are created. In the case of a complete order, all tuples are immediately assigned to the different classes, otherwise the remaining pixels are assigned to the nearest class. In the second case where the number of classes is variable by class, an over-segmentation is performed and then an interclass fusion is applied to obtain at a maximum the K desired classes.

Note that in this paper, we used the marginal vectorial order, the lexicographic vectorial order and the hybrid vectorial order.

[Iseg]=Fonction FonctSegMorphologiqSemiVectorial (I, K)

// I : Multicomponent image

// K : Number of classes

// Iseg : Segmented image into K classes

P = FonctNberComponents (I) // Return the number of components of I

Hc_nD = FonctHistoCompactnD (I) // Return the nD compact histogram of I

For j =1 to P

H(j).Histo_1D = HistoCompactnD (Ij) // return the 1D compact histogram of components of I

T(j).T = FonctSegMultiThresold (H(j).Histo_1D) // Multi-thresholding of Histo_1D

K(j).K = FonctNberComponents (T(j).T) // Return the number of threshold of Hj

End

If ( K(1).K = K(2).K =….= K − 1 ) then

For k =1 to K − 1

S(k).S = FonctVectorsThresold(T(j).T) // Construction of threshold vectors (K-uplets)

End

H = [Hc_nD ; S(1 à K − 1).S] // concatenation of Hc_nD tuples and threshold vectors

H_ord =FonctOrdonnement (H, ChoiceOrder) // Order tuples according to ChoiceOrder

Classes.C=FonctConstructionClasses (H_ord, S(1 à K − 1).S ) // Classification into K classes

Iseg=FonctImageSegmentée (I, Classes.C) // Image segmentée

Else

ClassesSurSegmente.C=FonctCombinatorial (H_ord, S(1 à K − 1).S ) // over-segmentation

Classes.C=FonctFusionInterClasses (ClassesSurSegmente.C) // Fusion interclasses into K classes

Iseg=FonctImageSegmentée (I, Classes.C) // Segmented image

End

In order to highlight the relevance of the proposed approach, segmentations have been performed on natural images using marginal order, lexicographic order and hybrid order with absolute referent. Some original images and their segmentation results by the proposed method and the K-means method are presented in Figures 2-4 by varying the number of classes from 2 to 4. Moreover, a series of images of forsythia of which we have the ground truths (reference

Figure 2. Segmentation into 2 classes by the proposed method and K-means.

Figure 3. Segmentation into 3 classes by the proposed method and K-means.

Figure 4. Segmentation into 4 classes by the proposed method and K-means.

Figure 5. Segmentation of Forsythia images into 2 classes by the proposed method and K-means.

expert segmentations from [18] ) have been segmented into two classes as shown in Figure 5.

These segmentation results clearly show the visual relevance of our approach through its various variants. However, a quantitative evaluation is required to validate our method, hence the following section.

Different assessment methods exist in the literature and can be grouped into two classes: supervised methods and unsupervised methods. In this work, on the one hand for the supervised evaluation, we used Vinet’s criterion [14] which calculates the rate of conformity errors of the segmentation maps by an optimal matching; this criterion has been applied to the series of images of Forsythia, of which we have ground truths. On the other hand, for the unsupervised evaluation, we have retained the Levine-Nazif Intra-region criterion and the Zeboudj criterion [14] because they are commonly used and also because of their relevance.

The Levine-Nazif Intra-region criterion favors segmentation with homogeneous classes and relies on intra-class energy. However, Zeboudj’s criterion takes into account intra-class energy and inter-class energy, thus favoring segmentations with homogeneous and well-separated classes.

The latter two criteria give the best score to the best segmentation. The evaluation results of our segmentation approach and those of K-means are presented in the tables below with an accuracy of 10⁻⁴.

In Table 1, we present some results of supervised evaluation by the criterion of Vinet, in green the best segmentation and in red the second best segmentation. The values recorded in the table correspond to the error rate between the segmented image and the ground truth. It appears that the proposed method presents better segmentations globally than K-means. On the three variants of the proposed method, vectorial morphological segmentation in hybrid order is the best, then comes the lexicographic order.

Table 2 presents the unsupervised evaluation of natural images by the Levine-Nazif criterion. This table shows that for k varying from 2 to 3 the vectorial morphological segmentation in marginal order and K-means have the same performances and are the best on all the images processed. When the number of classes k ≥ 4, the three variants converge to the same interclass fusion giving identical segmentations and a performance identical to K-means.

Table 3 illustrates the non-supervised evaluation by the Zeboudj criterion. It is found that for this criterion, the proposed approach gives better results and inversely, it is the case of the images House and Lena.

Consequently, it could be said that the relevance of the segmentation methods is related to the content of the image and the criterion used.

In order to demonstrate the robustness to the noise of our approach, we first added Gaussian noise of different average powers Pb to the synthetic image Savoise [15] . An example of adding noise is illustrated in Figure 6. Then, the noisy images have been segmented into different number of classes k varying

Figure 6. Original Savoise image, noisy Savoise images and their segmentations into 3 classes by the proposed method and K-means.

from 2 to 5, an illustration of which is given in Figure 6.

Indeed, an evaluation of the relevance of the segmented images is presented in Table 4 where the best segmentation is the one whose criterion value is the smallest and indicated in green. Moreover, when the value of the criterion is less than 5%, the segmentation is qualified as good. On the whole of the images processed, it can be seen through this table, for some, that the proposed method is robust to the noise K-means and inversely. This could be related to the content of the image.