Younes Makoudi

Frank Palmino

Madjid Arab

Eric Duverger

Frédéric Chérioux

Complete Supramolecular Self-Assembled Adlayer on a Silicon Surface at Room Temperature

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The controlled positioning and assembling of polyfunctional molecules on surfaces are attracting considerable interest. 1 For example, in the view to create new electronic devices, physicists have had to rely increasingly on the originality of chemists in designing molecular systems possessing useful properties. 2 A rich variety of molecular nanostructures such as clusters, wires, and extended 2D networks have been selectively assembled on surfaces and directly characterized by scanning tunneling microscopy (STM). In this approach, the selective molecular assembly has been controlled by tuning directional intermolecular attractions such as hydrogen bonds or Van der Waals interactions. [START_REF] Lehn | Supramolecular Chemistry; Concepts and PerspectiVes[END_REF] Many remarkable examples of such complex architectures have been obtained by supramolecular self-assembly on metallic 4 or HOPG 5 substrates. Nevertheless, the use of semiconductive interfaces remains one of the best choices for the development of operational post-CMOS devices. Despite numerous attempts, the observation of complete organic molecules adlayer on semiconductive substrates has not been reported in the literature yet. This failure could be explained by strong molecule-substrate interactions (for example the formation of Si-C σ bonds), which can disrupt the growth of the supramolecular edifice. 6 Recently, several strategies have emerged to prevent the formation of covalent bonds with silicon surfaces: (i) passivation of the surface by inserting doping elements (for example B or C atoms) 7 (ii) overriding of the adsorbate-substrate interactions by the formation of strong supramolecular dimers, 8 or (iii) shielding the π-skeleton by the insertion of charged atoms. 9 In this paper, we are detailing the engineering of a complete adlayer of organic nanolines by supramolecular self-assembly on Si(111)-B 3x 3R30°s urface at room temperature. This breakthrough has been achieved thanks to the combination of boron atoms used as doping elements to passivate the silicon-based surface, with the formation of strong hydrogen bonds between pairs of dipolar molecules.

To circumvent the problem of silicon surface reactivity with π-conjugated molecules, the Si(111)-B 3x 3R30°surface has been used as substrate. This surface possesses the unique particularity to show depopulated dangling bonds due to the presence of boron atoms underneath the top silicon layer, leading to a weak interaction π-conjugated organic molecules/surface. 7b Therefore, the growth of the supramolecular edifice should be only controlled by the molecule-molecule interactions as reported in most work using metallic substrates.

All STM experiments were performed with an Omicron microscope and were carried out in an ultrahigh vacuum chamber with a pressure lower than 2 × 10 -10 mbar. STM images were acquired in constant-current mode at room temperature. 4-Aminobenzonitrile were purchased from Aldrich and purified by column chromatography (silica gel, petroleum ether as eluent) and then, by recrystallization (ethanol) before being used. 4-Aminobenzonitrile are push-pull molecules where the amino function is the donor group and CN is the electron withdrawing group. The length of the molecule is 0.68 nm, and the permanent dipole moment is close to 4.9 D (Figure 1a).

Figure 1b is a large-scale STM image acquired on the Si(111)-B 3x 3R30°surface at room temperature after the deposition of 4-aminobenzontrile. A well-ordered adlayer of molecules can be clearly observed. The formation of a parallel nanowire network is revealed by the high-resolution STM images, with an average distance measured between two nanolines equal to 0.5 ( 0.1 nm (Figure 1b). The elemental building blocks of the nanolines are constituted by two spots. Each spot (0.6 ( 0.1 nm) is assigned to one molecule (0.68 nm). The length of a pair of protrusions is 1.7 ( 0.1 nm, which corresponds to two molecules separated by 0.6 ( 0.1 nm (Figure 1c). A close examination of the large-scale STM image (Figure 1b) shows the presence of two types of nanolines. This is highlighted in Figure 2a.

In one case, the nanowire is straight (Figure 2a, S) whereas in the other case, it takes a saw-toothed shape (Figure 2a, S-T). In the case of straight nanolines, the elemental cell is formed by one pair of molecules. The strong permanent dipole moment of 4-aminobenzonitrile (4.9 D) induces large dipole-dipole interactions between molecules which can takes the well-known antiferroelectric configuration, with the cyano groups pointing at one toward the other (Figure 2c) despite the electrostatic repulsion. 1

The size of the elemental (i.e. a pair of molecules) cell, about 0.8 × 1.7 nm 2 measured by STM, is in agreement with the distances obtained with the model.

In the case of saw-toothed nanolines, the elemental cell takes a parallelogram shape with a size being about 1.7 × 1.7 nm 2 and is constituted by two pairs of molecules (Figure 2a). Each pair preserves the antiferroelectric conformation of two molecules in order to keep the stability of the edifice. This reorganization can be explained by the formation of two hydrogen bonds between the nitrogen atom of the cyano group and the hydrogen atom on the 2-position of the aromatic ring (see Figure 3). These interactions are strongly supported by the electronic skeleton of 4-aminobenzonitrile. 11 Consequently, each pair of molecules can slip toward its neighbor to promote the interactions between the molecules and to align the three involved atoms, C-H •••N (Figure 3a). As all experiments were performed at room temperature, the diffusion of molecules on the substrate is possible and can lead to the reorganization of nanolines.

After this rearrangement, the nanolines are still preserved but they take a symmetric saw-toothed structure instead of a straight structure. This new conformation is more stable than the well-known antiferroelectric distribution because two hydrogen bonds per pair of molecules reinforce the interactions between the molecules in order to stabilize the complete adlayer. This hypothesis is justified by the presence, in the large scale STM images, of 12% of nanolines in the straight conformation and 88% in the symmetric saw-toothed conformation (see Figure 1b and4a).

In a very large-scale STM image (Figure 4a), a modulation of the intensity appears as previously reported by K. Kern et al. in the case of deposition of (Z) 4- (2-pyrid-4-yl-vinyl))benzoic acid on Ag(111). 12 According to these authors, the origin of this modulation is presumably explained by a vertical displacement of molecules to reduce strains. In our study, a vertical displacement of molecules is also visible (about 0.30 nm in Figure 4b). In accordance to Kern et al., the modulation of the intensity can be explained by the space limitations in molecular adlayer domains.

To sum up, for the first time, the formation of a complete adlayer of organic molecules has been achieved on semiconductive substrate at room temperature. This remarkable result is explained by the use of passivate Si(111)-B 3x 3R30°surface and the formation of nanolines constituted by dipolar molecules. These molecules are paired to adopt the antiferroelectric arrangement. They optimize their dipole-dipole interactions and they are linked by two hydrogen bonds between two consecutive neighboring pairs of molecules. The high stability of the adlayer seems to be justified by the addition of these two types of interactions. This new method provides a versatile and easy approach to extend the supramolecular organization developed on metallic substrate to semiconductive substrates at room temperature.

  Palmino, Madjid Arab, Eric Duverger, and Fre ´de ´ric Che ´rioux* Institut FEMTO-ST, CNRS, UniVersite ´de Franche Comte ´, 32 AVenue de l'ObserVatoire, F-25044 Besanc ¸on cedex, France

10

 10

Figure 1 .

 1 Figure 1. (a) Chemical structure of 4-aminobenzonitrile; (b) large scale STM image of the nanolines network constituted by the adsorption 4-aminobenzonitrile on Si(111)-B 3x 3R30°surface at room temperature (40 × 32 nm 2 , Vs )-1.7 V, and I ) 0.6 nA); (c) z-profile along the dark line.

Figure 2 .

 2 Figure 2. (a) High-resolution STM image (16 × 15 nm 2 ) of the two types of nanolines of the 4-aminobenzonitrile adlayer with (S) straight nanowire and (S-T) saw-toothed nanowire; (b) superimposed ball model of 4-aminobenzontrile on straight nanolines on an STM image (7.5 × 8.8 nm 2 ) and (c) ball model of 4-aminobenzontrile in antiferroelectric conformation. The imaging conditions were Vs )-1.7 V and I ) 0.6 nA.

Figure 3 .

 3 Figure 3. (a) Superposed ball model of 4-aminobenzontrile on saw-toothed nanolines on STM image (8 × 10 nm 2 , Vs )-1.7 V and I ) 0.6 nA); (b) hydrogen bonds between neighbor pairs of molecules.

Figure 4 .

 4 Figure 4. (a) Large scale STM image (60 × 25 nm 2 , Vs )-1.7 V and I ) 0.6 nA) of the nanolines network on Si(111)-B 3x 3R30°surface at room temperature and (b) z-profile along the dark line.

Acknowledgment. The authors thank Dr. C. Joachim (CEMES, Toulouse, France) for fruitful discussions. This work is supported by the Communaute ´d'Agglome ´ration du Pays de Montbe ´liard.


Dr Younes Makoudi

Dr Madjid Arab

Dr Frank Palmino

Dr Eric Duverger

Christophe Ramseyer

Fabien Picaud

Frédéric Cherioux
email: frederic.cherioux@femto-st.fr

F Chørioux

A Stable Room-Temperature Molecular Assembly of Zwitterionic Organic Dipoles Guided by a Si(111)-7x7 Template Effect

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A Stable Room-Temperature Molecular Assembly of Zwitterionic Organic Dipoles Guided by a Si(111)-7 7 Template Effect** Younes Makoudi, Madjid Arab, Frank Palmino, Eric Duverger, Christophe Ramseyer, Fabien Picaud, and FrØdØric ChØrioux*

The adsorption of functional molecules on surfaces plays a vital role in the emerging field of nanoelectronics. [1] In this context, molecular and supramolecular ordering, which are key steps in the development of complexes architectures, are controlled by a balance between intermolecular forces and molecule-substrate interactions. [2] This restructuring is often driven by cooperative molecule-substrate interactions involving many molecules and is not directly related to the shape of the individual molecules.

The deposition of organic molecules and preservation of their entire skeleton, especially on semiconductor surfaces, is still a challenge, although much progress has been made in the development of metallic surface-based devices with organic molecules at low temperature, where the molecule-surface interactions are weak and diffusion remains low. [3] However, the use of metallic substrates is less attractive for potential applications as semiconductors, especially in the field of nanoelectronics.

The deposition of molecules on a semiconductor surface at room temperature with complete control of the adsorption site and without altering their aromatic behavior is still a challenge in view of the development of complex architectures. [4] Polanyi and co-workers, for example, have shown that the adsorption of chlorododecane on Si(111)-7 7 at room temperature leads to the formation of a bistable dimeric corral of self-assembled molecules and have demonstrated on/off electronic switching of a single silicon adatom using molecular field effects. [5] These authors have also described a new strategy for preventing the dissociation of haloalkanes on Si(111)-7 7 up to 373 K. This stability is ensured thanks to the formation of dimeric circular corrals that override the molecule-substrate interactions. [5] Herein, we propose a new concept for the room-temperature deposition of p-conjugated organic molecules, without any modification of their elec-tronic structure, at specific adsorption sites on the surface of the semiconductor Si(111)-7 7 by adsorbing p-conjugated zwitterionic molecules on this semiconductor surface. The presence of a negative charge on the target molecules offsets the electrophilic character of the Si(111)-7 7 adatoms and preserves the p skeleton of the organic molecules after deposition. The half-cells of Si(111)-7 7 act as a template that guides the molecular assembly of achiral molecules, as shown by the induction of chirality in the modified areas. Experimental STM images as well as theoretical calculations are in perfect agreement with the proposed concept.

The electronic properties of molecules are often modified during their adsorption on a Si(111)-7 7 surface. Indeed, Si À C s bonds are usually formed by reaction of the electrondeficient silicon adatoms of Si(111)-7 7 with the electronrich carbon atoms of the organic molecules. [6] We propose an original concept to prevent these cycloadditions by using the negative site of a zwitterionic molecule as a shield (by electrostatic interactions) for the molecule on the electrophilic adatoms of Si(111)-7 7. Zwitterionic molecules are ideal candidates since they are neutral but carry formal positive and negative charges on different atoms. In principle, the surface should offset its electron deficit with the negative charge of the molecules instead of its extended electron-rich p-conjugated system. To prove our concept, we synthesized 4methoxy-4'-(3-sulfonatopropyl)stilbazolium (MSPS), which is terminated by a negatively charged sulfonato (SO 3 À ) group (Figure 1 a), as a model zwitterionic organic molecule. All experiments were carried out in an ultra-high vacuum chamber with a pressure lower than 2 10 À10 mbar. The molecules were sublimed from a Knudsen cell at 390 K onto the Si(111)-7 7 surface at room temperature. STM images were acquired in the usual constant-current mode at room temperature.

Figure 1 b,c shows STM images taken with two opposite polarities, with atomic resolution, of both the Si(111)-7 7 reconstruction and the adsorbed molecules. One half-cell exhibits a novel threefold star at the center of the image for both filled and empty states. Each arm of the star consists of two matching protrusions, one of which is more intense than the other. The distance between the centers of the two paired protrusions (0.67 nm) is much shorter than the length of a free MSPS molecule (1.3 nm). This difference can be explained by a conformational change caused by MSPS-substrate interactions (see below). Three protrusions are located exactly on top of corner adatoms and the other three are found between a rest-atom and an adjacent center adatom. In the filled states (Figure 1 b), the brighter protrusions are situated exactly on top of corner adatoms, whereas in the empty states (Fig-ure 1 c) the brighter protrusions appear between a rest-atom and a center adatom. The sulfonato moiety in MSPS possesses a negative charge, and the methoxy group is electron-poor owing to conjugation with the pyridinium ring. Therefore, the intense protrusions observed in the filled states on top of corner adatoms can be attributed to the sulfonato groups (Figure 1 b) and the intense protrusions observed in the empty states between a rest-atom and an adjacent center adatom to the methoxy groups (Figure 1 c).

The position of the MSPS molecules in this assembly can be determined unequivocally thanks to the use of two different polarities when recording the STM images. The star shape in each 7 7 half-cell can be attributed to the assembly of three MSPS molecules by molecule-substrate interactions, which change the conformation of the three molecules from trans to cis. The proposed structural model of the 7 7 assembly is superimposed on the STM images in Figure 1 d.

At very low molecule coverage this adsorption and formation of star shapes occurs preferentially on the faulted half-cell, which is known to be more reactive (Figure 1 b). [7] The occupied half-cells only contain three MSPS molecules, as described above, which means that this assembly process implies high molecular diffusion onto the surface at room temperature.

The assembly of three molecules into this particular star shape can be explained by the template effect of the surface, which possesses threefold symmetry. This assembly process is probably guided by the template effect of the substrate. Moreover, the intensities of the protrusions at each polarity are not particularly affected by a change of the bias voltage, which means that they are not likely to be due to the wellknown formation of C À Si s bonds between adsorbed molecules and a substrate. [6,8] The nature of MSPS adsorption on a Si(111)-7 7 surface has been investigated by ab initio calculations at the extended density functional theory (DFT) level using the Vienna Ab Initio Simulation Package (VASP). [9] To simulate the entire system (MSPS + Si(111)-7 7) we first optimized the geometry of one free MSPS molecule in vacuo in the cis conformation using Gaussian 03 at the Hartree-Fock 6-31G(d) level of theory. [10] Three of these optimized molecules were then positioned in the cis conformation thanks to the help of the high-resolution STM images discussed above. The adsorption energy was then varied at room temperature for different assemblies of molecules and surface distances (d), which correspond to the separation between the Si adatoms and sulfur atom of the sulfonato group. Finally, the local densities of states (LDOS) were determined at each point for a constant current and at bias voltages of À2 and + 2 V, respectively, with the Tersoff-Hamann scheme. [START_REF] Tersoff | [END_REF] The adsorption energy for the entire system is depicted in Figure 2 a as a function of the distance d. The equilibrium distance, which corresponds to the energy minimum of À2.66 eV (À0.887 eV per molecule) is close to 0.26 nm, in agreement with a physisorbed state. The LDOS images calculated at the equilibrium distance compare well with the experimental STM images (Figures 2 b and2 c). Figure 2 d shows the DOS of the whole stable system at d = 0.26 nm. The two weak interactions observed correspond to a very small electronic transfer from the sulfonato groups to the corner adatoms and from methoxy moieties to Si atoms, respectively, which means that there is no bond formation between any single oxygen atom and a silicon adatom. An interaction between the molecular assembly and the substrate nevertheless exists, but it depends only on the distance, in agreement with a weak electrostatic interaction between the oxygen atoms of the sulfonato moieties and the electrophilic Si adatoms. These results confirm that we have developed a new strategy for stabilizing zwitterionic p-conjugated molecules on a Si(111)-7 7 surface by taking advantage of the electrostatic interactions between silicon adatoms and the anionic groups of these molecules. These interactions are strong enough to stabilize the molecular assembly at room temperature and produce a template effect whereby the Si(111)-7 7 surface imposes a cis conformation on the MSPS molecules in the Si(111)-7 7 half-cells.

Figure 3 shows two half-cells, each of which contains an assembly of three MSPS molecules that is a mirror image of the other reflected in a mirror plane. This means that the threefold stars are chiral even though free MSPS is achiral. The chirality of these molecular assemblies can be explained by the template effect of the Si(111)-7 7 surface. The sulfonato group can swivel upon the corner adatom, whereas the surface forces the methoxy moieties to be fixed between a rest-atom and a center adatom (Figure 3). The two possible enantiomers in this case, which are formed by a clockwise or anticlockwise folding, are termed the (P)-enantiomer and the (M)-enantiomer, respectively. Most previous studies regarding chiral structures have been concerned with metal surfaces, [12] and relatively little work has been reported for semiconductor surfaces. [13] However, the chirality induced in these previous studies on semiconductors is due to the formation of at least one stereogenic center after the chemisorption, for example by cycloaddition, of molecules on silicon or germanium atoms. The results reported here show, to the best of our knowledge, the first example of chiral assembly of achiral molecules on Si(111)-7 7.

In conclusion, the results reported herein are remarkable from several different points of view, especially the molecular assembly of organic dipoles at room temperature, the chirality of the assembly, and the stability and conservation of the pconjugated skeleton on the semiconductor surface at room temperature. This zwitterionic strategy, where the sulfonato group acts as an electrostatic shield that protects the p skeleton of organic molecules against the dangling bonds of the semiconductor surface, may become an important method for the deposition of p-conjugated molecules.

Experimental Section

The Si(111) substrate was heated under ultra-high vacuum by direct current. Clean Si(111)-7 7 surface reconstruction was obtained by repeated cycles of heating at 1200 8C and slow cooling to room temperature. Deposition of the MSPS molecules from an Mo crucible onto the sample at room temperature was performed at 120 8C and a pressure lower than 10 À10 mbar.

Electron-ion interactions were described using the projectoraugmented wave (PAW) method, which was expanded within a planewave basis-set up to a cut-off energy of 400 eV. [14] Electron exchange and correlation effects were described by the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE GGA) exchange-correlation functional. [15] The Brillouin zone was sampled using the gamma point only. Simulations were performed using a volume box (21.67 nm 3 ) containing one 7 7 unit cell (DAS model with four silicon layers). [16] The adsorption energy was calculated from the difference between the total energy of the entire system (substrate + molecules) when interacting and the sum of the total energies of each free component.

All synthetic procedures and calculations are fully described in the Supporting Information.

Keywords: chirality • scanning probe microscopy • semiconductors • surface chemistry • zwitterions

Figure 1 .

 1 Figure 1. a) Molecular structure of MSPS. b, c) High-resolution, roomtemperature STM images (7 7 nm 2 ) of MSPS deposited on Si(111)-7 7 in the filled state (sample negative bias voltage: V s = À1.6 V) and in the empty state (sample positive bias voltage: V s =+1.9 V), respectively. d) Both the above STM images with superimposed ball and structural models showing the assembly of three MSPS molecules. The structural model of Si(111)-7 7 (right) with a superimposed ball model of the self-assembly of three MSPS molecules summarizes the STM images in the two polarities.

Figure 2 .

 2 Figure 2. a) Adsorption energy as a function of distance. b, c) Integrated LDOS images of three MSPS molecules on Si(111)-7 7 (left) and with a superimposed ball model (right) with a negative bias voltage (À2 V, b) and a positive bias voltage (+ 2 V, c). d) A side view of the density of states of MSPS adsorbed on Si(111)-7 7 showing weak molecule-substrate interactions.

Figure 3 .

 3 Figure 3. A room-temperature STM image (V s = À2.1 V, 7 9 nm 2 ) showing the two enantiomers of the chiral supramolecular selfassembly of MSPS deposited on Si(111)-7 7 (left) and a schematic model of these two enantiomers on Si(111)-7 7 (right).

Prof. Dr. C. Ramseyer, Dr. F. Picaud Laboratoire de Physique MolØculaire, UMR CNRS 6624 16 Route de Gray, 25030 Besancon cedex (France) [**] The authors thank Dr. C. Joachim (CEMES, Toulouse, France) for fruitful discussions. This work was supported financially by the CommunautØ d'AgglomØration du Pays de MontbØliard.


Abderezak Salmi

Kamal Hammouche

Ludovic Macaire
email: ludovic.macaire@univ-lille.fr

Constrained feature selection

Keywords: Color texture segmentation, pairwise constraints, constrained feature selection, constrained spectral clustering

come

Introduction

Color-texture image segmentation is one of the fundamental low-level problems in computer vision. [START_REF] Jain | Color image segmentation techniques: A survey[END_REF] Specifically, it aims to segment a color-texture image into disjointed homogeneous regions in terms of colors and textures. Although a wide variety techniques have been developed, color-texture image segmentation remains an open and challenging problem in computer vision due to the high variability of color textures. Color-texture segmentation by pixel classification is divided into two main steps. First, the color and texture features that characterize pixels must be selected in order to build a discriminating feature space.

Second, a classification procedure analyzes pixel distribution in the feature space in order to assign each pixel to a class. Depending on the availability of the labeled pixels, color-texture image segmentation can be performed in a supervised, unsupervised, or semi-supervised learning context. Deep learning approaches such as convolutional neural networks (CNNs) have shown good performance for supervised color-texture segmentation. [START_REF] Huang | Empirical curvelet based fully convolutional network for supervised texture image segmentation[END_REF] However, these algorithms require a large labeled pixel subset whose construction is time-consuming. Deep CNNs have also been recently developed for unsupervised color-texture image segmentation. [START_REF] Huang | Empirical curvelet based fully convolutional network for supervised texture image segmentation[END_REF] But, the training step of these CNNs is supervised and requires a dictionary of labeled textures. In contrast, the unsupervised segmentation methods based on clustering models such as slope difference distribution clustering, k-means, mean-shift, and spectral clustering can classify pixels without any prior knowledge about the classes. [START_REF] Wang | Image segmentation by combining the global and local properties[END_REF][START_REF] Tao | Color image segmentation based on mean shift and normalized cuts[END_REF][START_REF] Shi | Normalized cuts and image segmentation[END_REF] Besides the afore mentioned methods, several techniques have been applied for unsupervised color-texture segmentation. [START_REF] Ilea | Image segmentation based on the integration of colour-texture descriptors a review[END_REF][START_REF] Kiechle | Model-based learning of local image features for unsupervised texture segmentation[END_REF][START_REF] Haindl | A competition in unsupervised color image segmentation[END_REF] However, the parameters of unsupervised algorithms must be carefully tuned in order to yield satisfactory accuracy.

To overcome problems caused by supervised and unsupervised algorithms, weakly supervised or semi-supervised segmentation algorithms use limited amounts of supervised information. The prior knowledge is then expressed by a few labeled pixels or by a few links between pixels. A must-link constraint specifies that two pixels must belong to the same class, whereas a cannot-link constraint specifies that two pixels must belong to different classes. These pairwise constraints can be obtained from the available labeled pixels. Conversely, the labels of pixels cannot be deduced from the pairwise constraints. The labeled pixels or the pairwise constraints can be interactively given by a user. Semi-supervised algorithms aim to classify pixels by propagating the labels [START_REF] Quan | Weakly-supervised sparse coding with geometric prior for interactive texture segmentation[END_REF] or by incorporating the pairwise constraints into popular clustering algorithms such as k-means, estimation-maximization, hierarchical clustering 10 and spectral clustering. [START_REF] Luxburg | A tutorial on spectral clustering[END_REF] Spectral clustering, which is based on graph theory, computes the Laplacian matrix from a similarity matrix between pixels and analyzes the distribution of pixels that are projected into a low-dimensional subspace composed of the eigenvectors of the Laplacian matrix. A trivial way to integrate pairwise constraints in spectral clustering is to enforce the similarities between pixels to 1 and 0 for must-link and cannot-link constraints, respectively. [START_REF] Kamvar | Spectral learning[END_REF][START_REF] Ahn | Face and hair region labeling using semi-supervised spectral clustering-based multiple segmentations[END_REF] As the number of given pairwise constraints is low, they can be propagated across the similarity matrix. [START_REF] Chen | Spectral clustering: a semi-supervised approach[END_REF][START_REF] Zhu | Constrained clustering with imperfect oracles[END_REF][START_REF] Xia | Oriented grouping-constrained spectral clustering for medical imaging segmentation[END_REF] Propagation rules use Gaussian processes [START_REF] Sourati | Accelerated learning-based interactive image segmentation using pairwise constraints[END_REF][START_REF] Lu | Constrained spectral clustering through affinity propagation[END_REF] or the similarities between constrained and unconstrained pixels. [START_REF] He | Constrained clustering with local constraint propagation[END_REF] Han et al. [START_REF] Han | Segmentation with selectively propagated constraints[END_REF] proposed the propagation of pairwise constraints only on a randomly selected subset of pixels. The selectively propagated constraints were then used to adjust the weights of the similarity matrix. Constrained spectral clustering can be also formulated as a constrained optimization problem, where only linear equality constraints (analogous to must-link constraints) are exploited for image segmentation. [START_REF] Yu | Segmentation given partial grouping constraints[END_REF] Another formulation of constrained spectral clustering under general convex constraints has been developed for gray level image segmentation. [START_REF] Ghanem | Dinkelbach ncut: An efficient framework for solving normalized cuts problems with priors and convex constraints[END_REF] These constrained spectral clustering methods are based on a pixel similarity that is computed using Euclidean distance in the original feature space. Nevertheless, using all the features may bias the pairwise similarity computation and thus degrade the clustering performance. Indeed, they would provide irrelevant, redundant, and even contradictory information. To compute relevant similarity, the Mahalanobis distance is modified so that it minimizes the distance between the must-link pixels and maximizes the distance between the cannot-link pixels. [START_REF] Zou | A new constrained spectral clustering for sar image segmentation[END_REF][START_REF] Jia | Learning distance metric for semi-supervised image segmenta-tion[END_REF] Furthermore, the constraints can be propagated to the nearest neighborhood pixels that are found in the discriminating feature sub-spaces, rather than those that are identified in the original feature space. [START_REF] Zhu | Constrained clustering with imperfect oracles[END_REF] However, the random forest procedure used for this purpose, requires a time consuming training step.

In this paper, we propose an original approach for color-texture segmentation based on constrained spectral clustering. Specifically, it consists of propagating the pairwise constraints in order to select the most relevant features. Rather than computing a score for each color-texture feature, we propose a new constraint score that evaluates the relevance of a subset of color-texture features. Thus, the similarity matrix that is computed in this selected subspace is truer to the data structure and respects as much as possible the given must-link and cannot-link constraints.

We develop an efficient color-texture image segmentation that combines constrained feature selection and spectral clustering (CFS-SC). Its main contributions are as follows:

-Pairwise must-link and cannot-link constraints are involved in feature selection to improve the color-texture image segmentation performance using spectral clustering.

-A new similarity-based constraint score is proposed to select the optimal number of relevant features.

The rest of this paper is organized as follows. Section 2 introduces the related work, including a brief overview of the constrained spectral clustering and primary state-of-theart semi-supervised constraint scores. Our proposed color-texture image segmentation is detailed in Sec. 3. Section 4 provides experimental segmentation results achieved with four benchmark datasets. Finally, our conclusions are given in Sec. 5.

Related work

In this section, we first introduce spectral clustering and detail how to include the constraints for spectral clustering and feature selection.

Spectral clustering

Let X be the set of n sample pixels represented by the matrix X = [x 1 , x 2 , . . . , x n ] ∈ R n×d , where x i = [x i1 , x i2 , . . . , x ir , . . . , x id ] T ∈ R d represents the i-th pixel of X, and x ir ,(r = 1, . . . , d), the r-th feature value of the i-th pixel. Let

F d = {f 1 , f 2 , . . . , f r , . . . , f d } be the set of d features that is represented by X T = [f 1 , f 2 , . . . , f r , . . . , f d ], where f r = [x 1r , x 2r , . . . , x nr ] ∈ R n
is the r-th feature vector.

Spectral clustering represents dataset X by an un-directed weighted similarity graph G = (V, E, W) in which each sample pixel corresponds to a node. V is a non-empty set that contains all nodes, and E is the set of edges between any two nodes in V . Each edge in E is weighted by a similarity value w ij (i, j = 1, 2, . . . , n) between two nodes. The similarity matrix W that gathers similarities between all pairs of nodes is positive semi-definite and symmetric. Generally, the similarity w ij between two pixels x i and x j is computed by the following Gaussian kernel function 11

w ij = exp ( - δ 2 (x i , x j ) 2σ 2 ) i, j = 1, 2, . . . , n (1)
where σ is a scaling parameter and δ(x i , x j ) is the Euclidean distance between the two pixels

x i and x j .
Spectral clustering is an unsupervised classification method that computes the eigen spectrum of the Laplacian matrix, which is deduced from the similarity matrix W, to separate the dataset into clusters. The most commonly used un-normalized Laplacian matrix is

L = D -W while the most used normalized Laplacian matrices are L Sym = D -1/2 LD -1/2
and L Asym = D -1 L. [START_REF] Shi | Normalized cuts and image segmentation[END_REF][START_REF] Luxburg | A tutorial on spectral clustering[END_REF] D ∈ R n×n is the diagonal degree matrix whose elements are d ii =

∑ n j=1 w ij . To identify k clusters, the data are mapped onto a low-dimensional space that is based on the k-dominant eigenvectors corresponding to the smallest eigenvalues of the Laplacian matrix. Finally, an unsupervised clustering algorithm such as k-means identifies the k clusters of so mapped data. 5, 11

Constraint spectral clustering

In the semi-supervised learning context, the prior knowledge is expressed by a small number of labeled pixels or pairwise constraints. In this paper, we consider that few labeled pixels (prototypes) characterize the k classes ω l , l = 1, . . . , k of the input image I. These prototypes can be interactively selected by the user or randomly selected from the ground truth. Let X l (X l ⊂ X) be the subset of p prototypes associated with the class ω l . From the set of prototypes denoted as X P

( X P = ∪ l=1,...,k X l ) , we can build the set M of (k • p • (p -1))
must-link pairs that are composed of two prototypes belonging to the same class:

M = { (x i , x j ) ∈ X 2 | ∃ l = 1, . . . , k so that x i ∈ X l and x j ∈ X l } . ( 2
)
We can also build the set C of (k • (k -1) • p 2 ) cannot-link pairs that are composed of two prototypes belonging to different classes:

C = { (x i , x j ) ∈ X 2 | ∃ (l, m); l ̸ = m; so that x i ∈ X l and x j ∈ X m } . ( 3
)
Of all possible pixel pairs that can be extracted from X, those belonging to M or C are called constrained pairs while the remaining pairs are called unconstrained pairs. Furthermore, the pixels that are not prototypes are called unlabeled sample pixels, and are gathered in subset X U = X/X P (with

|X U | = n -(k • p)).
A straightforward way to integrate must-link and cannot-link pairs in spectral clustering is computing the similarity matrix W M,C . [START_REF] Basu | Constrained clustering: Advances in algorithms, theory, and applications[END_REF][START_REF] Kamvar | Spectral learning[END_REF] For a given must-link pair (x i , x j ) ∈ M , w M,C ij is set to 1; for a given cannot-link pair (x i , x j ) ∈ C, w M,C ij is set to 0. Thus, the modified matrix W M,C is defined as follows:

w M,C ij =                1 if (x i , x j ) ∈ M 0 if (x i , x j ) ∈ C exp ( - δ 2 (x i ,x j ) 2σ 2 ) otherwise. i, j = 1, 2, . . . , n (4)
The next steps of constrained spectral clustering are not specific and are summarized in

t ij = u ij √ ∑ k j=1 u 2 ij 5.
Consider each row of T M,C as a point, and cluster all points into k clusters by k-means algorithm. Output: k clusters of dataset X.

The performance achieved by constrained spectral clustering strongly depends on the pairwise similarities between the unconstrained pixels, which are based on the Euclidean distance δ in the original d-dimensional feature space (see Eq. ( 4)). Nevertheless, using all of these features may bias the pairwise similarities. To deal with this problem, we propose to select the relevant subset of features among those available. In the semi-supervised context, many feature selection methods have been proposed to make full use of prior knowledge. [START_REF] Sheikhpour | A survey on semisupervised feature selection methods[END_REF] In the next section, we focus on the constraint scores that are used by semi-supervised feature selection.

State-of-the-art about semi-supervised constraint scores

Feature selection based on semi-supervised constraint scores involves the analysis of both pairwise constraints and unlabeled sample pixels. Specifically, it considers both the discriminating power of the pairwise constraints and the local properties of the unlabeled sample pixels. In the context of spectral graph theory, the dataset X can be represented by the nearest neighbor graph G KNN whose node v i is connected to v j when x j is one of the Knearest neighbors (KN N s) of x i , the distance being computed in the original d-dimensional feature space. The similarity matrix W KNN ∈ R n×n is defined as follows:

w KNN ij =        w ij if x i ∈ KNN(x j ) or x j ∈ KNN(x i ) 0 otherwise (5)
The two graphs G M and G C can also be built from the sets of must-link constraints M and cannot-link constraints C. The corresponding similarity matrices W M ∈ R n×n and W C ∈ R n×n are defined as follows:

w M ij =        1 if (x i , x j ) ∈ M 0 otherwise (6) w C ij =        1 if (x i , x j ) ∈ C 0 otherwise. (7)
Zhao et al. [START_REF] Zhao | Locality sensitive semi-supervised feature selection[END_REF] introduced the semi-supervised constraint score C 1 r , which combines the similarity matrix W C constructed from the cannot-link constraints (Eq. ( 7)) and the similarity matrix W KNN1 ∈ R n×n , which is built from the set of must-link constraints and unlabeled sample pixels as follows:

w KNN1 ij =                γ if (x i , x j ) ∈ M 1 if (x i ∈ X U or x j ∈ X U ) and ( x i ∈ KNN(x j ) or x j ∈ KNN(x i ) ) 0 otherwise ( 8
)
where γ is a constant parameter. In our experiments, γ is set to 100 and K is set to 5 as in Ref. 26.

The constraint score C 1 r is defined as

C 1 r = f T r L KNN1 f r f T r L C f r ( 9
)
where L KNN1 = D KNN1 -W KNN1 is the un-normalized Laplacian matrix of W KNN1 , and D KNN1

is the degree matrix computed from W KNN1 .

Kalakech et al. [START_REF] Kalakech | Constraint scores for semi-supervised feature selection: A comparative study[END_REF] proposed the semi-supervised constraint score C 2 r that is less sensitive to the constraint sets (10) where fr =

C 2 r = fT r L fr fT r D fr • f T r L M f r f T r L C f r
∑ n i=1 x ir d ii ∑ n i=1 d ii = f T r D1
1 T D1 , fr = f r -fr , and 1 = [1, . . . , 1] T . L and D are deduced from the similarity matrix W (Eq. ( 1)

). L M = D M -W M and D M = ∑ n j=1 w M ij .
Benabdeslem and Hindawi proposed another constraint score C 3 r , which combines the similarity matrices W C and W KNN2 . [START_REF] Benabdeslem | Constrained laplacian score for semi-supervised feature selection[END_REF] The similarity matrix W KNN2 is expressed as

w KNN2 ij =        w ij if ( (x i , x j ) ∈ M ) or ( x i ∈ KNN(x j ) or x j ∈ KNN(x i ) ) 0 otherwise (11)
The score C 3 r is defined as follows: 28

C 3 r = f T r L KNN2 f r f T r L C D KNN2 f r (12)
A second semi-supervised constrained Laplacian score, referred to C 4 r , has been proposed by Benabdeslem and Hindawi in Ref. 29 as follows: (13) where the diagonal matrix D KNN is deduced from the similarity matrix W KNN (see Eq. ( 5)).

C 4 r = f T r L KNN3 f r f T r L C D KNN f r
The Laplacian matrix L KNN3 is computed from the similarity matrix W KNN3 , which is defined as follows:

w KNN3 ij =                        w 2 ij + w ij if ( (x i , x j ) ∈ C ) and ( x i ∈ KNN(x j ) or x j ∈ KNN(x i ) ) w 2 ij if ( (x i , x j ) ∈ M ) and ( x i / ∈ KNN(x j ) and x j / ∈ KNN(x i ) ) w ij if ( x i ∈ X U or x j ∈ X U ) and ( x i ∈ KNN(x j ) or x j ∈ KNN(x i ) ) 0 otherwise (14)
Recently, Yang et al. [START_REF] Yang | Semi-supervised feature selection for audio classification based on constraint compensated laplacian score[END_REF][START_REF] Yang | Semi-supervised minimum redundancy maximum relevance feature selection for audio classification[END_REF] introduced the new semi-supervised constraint score C 5 r , which takes advantage of the local geometrical structure of unlabeled data samples as well as the constraint information deduced from prototypes as follows:

C 5 r = 2(f r ) T L KNN4 f r ( fr ) T D KNN4 fr + 2( fP r ) T L P fP r -( fP r ) T D P fP r ( 15
)
The similarity matrix W KNN4 is expressed as

w KNN4 ij =                                1 if ( (x i , x j ) ∈ M ) and ( x i ∈ KNN(x j ) or x j ∈ KNN(x i ) ) λ if ( (x i , x j ) ∈ M ) and ( x i / ∈ KNN(x j ) and x j / ∈ KNN(x i ) ) (1 -γ)w ij if ( (x i , x j ) ∈ C ) and ( x i ∈ KNN(x j ) or x j ∈ KNN(x i ) ) w ij if ( x i ∈ X U or x j ∈ X U ) and ( x i ∈ KNN(x j ) or x j ∈ KNN(x i ) ) 0 otherwise ( 16
)
where γ and λ are the parameters set to the empirical values of 0.9 and 0.5, respectively. [START_REF] Yang | Semi-supervised feature selection for audio classification based on constraint compensated laplacian score[END_REF][START_REF] Yang | Semi-supervised minimum redundancy maximum relevance feature selection for audio classification[END_REF] The cells of W P are set to 1/|X l | when two data samples are prototypes that belong to the same class and to 0, otherwise

w P ij =        1/|X l | if x i ∈ X l and x j ∈ X l 0 otherwise (17)
Note that fP r = f P r -fP r . L P = D P -W P is the un-normalized Laplacian matrix of W P , and D P is the degree matrix computed from W P .

Because feature scores are based on the similarity matrices that are computed in the original d-dimensional feature space, they may be corrupted by irrelevant features. Furthermore, as features are ranked according to their individual scores, the selected features can be correlated and may lead to low pixel classification performance.

3 Proposed color-texture image segmentation This section details our CFS-SC method for semi-supervised color-texture image segmentation. The prior knowledge provided by a user is expressed by a small number of labeled pixels that represent the prototypes of each color-texture class. The binary similarities between the prototypes are integrated into a matrix that quantifies the pairwise similarities of the pixels, and pixel spectral clustering that analyzes similarity matrix is used to perform semi-supervised segmentation.

Because some color and texture features may be irrelevant or redundant, the computation of pairwise similarities, which is based on the Euclidean distance in the full feature space, is biased. Thus, the segmentation performance from spectral pixel clustering can be degraded.

To circumvent this problem, a new constrained feature selection is included in the color- texture image segmentation that is outlined in Fig. 1. The proposed constrained feature selection and spectral clustering is first applied to a limited number of sampled pixels.

Pixel sampling procedure

Spectral clustering is based on Laplacian matrix eigen-decomposition, whose computational complexity is O(n 3 ), where n is the number of pixels that are sampled from the color image I, whose size is N pixels. From I, we build the subset X = {x 1 , x 2 , . . . , x n } of n sample pixels. This set includes the subset X P of the (k • p) prototypes and the subset X U of the (n -(k • p)) unlabeled sample pixels that are picked up thanks to a regular hexagonal grid. Once the subset X has been classified by constrained spectral clustering, a specific

Color and texture features

To segment the color-texture image, the pixels are characterized by both Haralick features (spatial features) [START_REF] Haralick | Textural features for image classification[END_REF] and Gabor features (spectral features). [START_REF] Jain | Unsupervised texture segmentation using gabor filters[END_REF][START_REF] Liu | From bow to cnn: Two decades of texture representation for texture classification[END_REF] Haralick texture features describe correlation between neighboring pixels and are computed from co-occurrence matrices. In a color image I, each pixel x is represented by three color components R, G, and

B. Six co-occurrence matrices (three between-component matrices M R,R [x] , M G,G [x],
and

M B,B [x], and three within-component matrices M R,G [x], M R,B [x], and M G,B [x]
) are locally computed for each pixel x by considering all co-occurrences in a window with a size of (2ω + 1) × (2ω + 1) centered on pixel x (see Fig. 2) (ω is set to 11 for our experiments).

Overall, 84 Haralick features are extracted from six co-occurrence matrices. [START_REF] Haralick | Textural features for image classification[END_REF] Gabor features are powerful texture descriptors generated from the convolution of Gabor filters. The impulse response of a Gabor filter with specific parameters (scale γ and orientation θ) depends on spatial coordinates (u, v) and is expressed as

g(u, v; γ, θ) = exp -1 2 ( u ′2 σu 2 + v ′2 σv 2
)

exp j2πγu ′ (18)
with u ′ = u cos θ+v sin θ and v ′ = -u sin θ+v sin θ. γ is the central frequency of the filter and σ u and σ v are the space constants of the Gaussian envelope along the horizontal and vertical axes, respectively. A bank of Gabor filters, tuned with several scales (frequencies) and orientations, are designed to characterize different textures at different scales and orientations in the input image. [START_REF] Jain | Unsupervised texture segmentation using gabor filters[END_REF] Seven scales (γ

⋆ = 2 √ 2, 4 √ 2, 8 √ 2, 16 √ 2, 32 √ 2, 64 √ 2,

Proposed constrained feature selection

Among extracted features, some of them are irrelevant for the faithful computation of pairwise similarities. Therefore, analyzing all of these features could degrade the clustering performance. To avoid this problem, we propose applying a constrained feature selection using constraint feature scores to select the most relevant features among the d available features. [START_REF] Sheikhpour | A survey on semisupervised feature selection methods[END_REF] As mentioned in Section 2, the state-of-the-art feature scores based on the similarity matrices evaluate the features one by one and ignore any correlation between features.

Thus, clustering algorithms that operate in a subspace of individually relevant features do not necessarily provide favorable results. [START_REF] Benabdeslem | Efficient semi-supervised feature selection: constraint, relevance, and redundancy[END_REF] In this section, we propose a new constraint score that estimates the relevance of a subset of features, making it possible to identify an optimal number of relevant features. The relevance of F m is evaluated by means of the distance between a target similarity matrix ŴSS , which is defined from the given constraints, and the similarity matrix W(F m ) computed with the subset F m of features. The score ε SS (F m ), which should be as low as possible, is expressed as follows:

ε SS (F m ) = ∥W(F m ) - ŴSS ∥ 2 (19)
where ∥ • ∥ 2 is the Euclidean norm. Thus, ε SS (F m ) can be rewritten as

ε SS (F m ) = n ∑ i=1 n ∑ j=1 ( w ij (F m ) -ŵSS ij ) 2 (20)
W(F m ) ∈ R n×n is the similarity matrix computed on the sample pixel set X that is characterized by the subset of features

F m w ij (F m ) = exp ( - δ 2 ( x (m) i , x (m) j ) 2σ 2
)

i, j = 1, 2, . . . , n (21)
where δ(x

(m) i , x (m) j )
is the Euclidean distance between two sample pixels x i and x j characterized by the subset of m features F m .

ŴSS ∈ R n×n is the target matrix whose cells correspond to must-link pairs and are set to 1, while cells corresponding to the cannot-link pairs are set to 0. ŴSS is then a binary matrix defined as follows:

ŵSS ij =        1 if (x i , x j ) ∈ M SS 0 otherwise. ( 22
)
The cell ŵSS ij is set to 1 when (x i , x j ) belongs to M SS (0 otherwise), which is a new set of must-link pairs that results from the propagation of prototype subsets X l , l = 1, . . . , k to unlabeled sample pixels

M SS = { (x i , x j ) ∈ X 2 | ∃ l = 1, . . . , k so that N P (x i ) ∈ X l and N P (x j ) ∈ X l } (23)
The nearest prototype N P (x i ) is the prototype whose distance in the original d-dimensional feature space from a sample pixel x i ∈ X is the smallest

N P (x i ) = arg min z ∈ ∪ l=1,...,k X l ( δ 2 (x i , z) ) . ( 24
)
Thus, the pair (x i , x j ) belongs to M SS when the nearest prototypes of x i and x j both belong to the subset X l of prototypes that is associated to the class ω l . Because N P (x i ) is x i

when x i belongs to prototype subset X l , set M is included in M SS (M ⊂ M SS ). Thus, M SS
increases the contribution of the pairwise constraints M for semi-supervised feature selection.

Finally, ε SS (F m ) makes it possible to assess the ability of the feature subset to preserve the pairwise constraint sets M and C, and the extended pairwise constraints provided by M SS .

These extended pairwise constraints should represent the geometric structure of the classes well.

Feature selection procedure

Because the score ε SS evaluates the relevance of a subset of features simultaneously, we use a simple sequential forward feature selection technique. [START_REF] Siedlecki | On automatic feature selection[END_REF] To evaluate the relative relevance of d features, we first consider each feature one by one (m = 1). The feature f r ,which minimizes

ε SS (F 1 ) with F 1 = {f r },
f + r = arg min fr ∈ F d \F m-1 ( ε SS (F m-1 ∪ {f r }) ) . b. Update F m = F m-1 ∪ {f + r }. 3. Select the number m of features such that m = arg min m=1,2,...,d ( ε SS (F m )
) .

Output: Subset of m relevant features F m.

Constrained spectral clustering

Once the subset F m of relevant features has been selected, the set of n sample pixels X is classified by the constrained spectral clustering algorithm (see Algorithm 1). For this purpose, we propose to replace the k-means algorithm with the nearest neighbor algorithm, which uses the available k prototype subsets X l , l = 1, . . . , k. Moreover, the matrix W M,C of Eq. ( 4) is computed in the m-dimensional feature space and becomes

w M,C ij (F m) =                1 if ( x i , x j ) ∈ M 0 if ( x i , x j ) ∈ C exp ( - δ 2 (x ( m) i ,x ( m) j ) 2σ 2 ) otherwise. i, j = 1, 2, . . . , n (25)

Out-of-sample classification

The last step of the proposed image segmentation is to assign the subset Y of (N -n) out-ofsample pixels to clusters that have been previously determined by the constrained spectral clustering. Generally, this procedure is performed by assigning each out-of-sample pixel y i to the class of its nearest sample pixels x i ∈ X in the m-dimensional feature space. [START_REF] Zou | A new constrained spectral clustering for sar image segmentation[END_REF] In this paper, we propose to classify the out-of-sample pixels in this low-dimensional space, where the k clusters of sample pixels have been determined.

For this purpose, we propose to embed out-of-sample pixels y i onto the k-dimensional subspace that is formed by the k-dominant eigenvectors u l , which correspond to the k smallest eigenvalues λ l of the Laplacian matrix L M,C Sym computed on the set X of sample pixels. [START_REF] Bengio | Out-of-sample extensions for lle, isomap, mds, eigenmaps, and spectral clustering[END_REF] The projection of out-of-sample pixel y i on the l th k-dominant eigenvector is defined as follows:

z l (y i ) = 1 (1 -λ l ) n ∑ j=1 u lj w( m) (y i , x j ) l = 1, . . . , k (26)
where w( m) (y i , x j ) is the equivalent normalized similarity between the out-of-sample y i and the sample pixel x j .

w( m) (y i , x j ) = 1 n w ( m) (y i , x j ) √ E [ w ( m) (y i , x) ] E [ w ( m) (x j , x ′ )] ) (27)
E[•] represents the average operator such that E [ w ( m) (y i , x) ] = 1 n ∑ n j w ( m) (y i , x j ).
The similarity w ( m) (y i , x j ) between two pixels y i and x j is defined as follows:

w ( m) (y i , x j ) = exp ( - δ 2 ( y ( m) i , x ( m) j ) 2σ 2 ) ( 28
)
The embedded out-of-sample pixel y i onto the k-dimensional subspace is normalized to have the unit length

zl (y i ) = z l (y i ) √ ∑ k j=1 z j (y i ) 2 l = 1, . . . , k. ( 29
)
Following this, each out-of-sample pixel y i is assigned to the cluster of the nearest sample pixel of z(y i ) in the k-dimensional subspace.

As in Ref.

2, a refinement procedure can be applied to further improve the segmentation accuracy. This procedure consists of keeping all regions larger than 0.5% of N while all other regions are relabeled with the class index corresponding to the largest adjacent region.

Finally, all steps of the proposed CFS-SC method for color-texture image segmentation are summarized in Algorithm 3.

Experimental results

In this section, we evaluate our CFS-SC method for color-texture image segmentation. We first evaluate the performance of the proposed semi-supervised feature score ε SS . For this purpose, we consider two color-texture images from the Prague texture segmentation benchmark. [START_REF] Haindl | Texture segmentation benchmark[END_REF] Next, we present segmentation results obtained by the proposed algorithm and we compare them with those obtained by several state-of-the-art algorithms on four benchmark datasets of color and grayscale textures.

Evaluation of the semi-supervised constrained feature selection

To evaluate the performance of the proposed constraint score ε SS , we compare it with the state-of-the-art constraint scores C 1 , C

M,C = [u 1 , . . . , u k ] ∈ R n×k corresponding to k smallest eigenvalues λ M,C = [λ 1 , . . . , λ k ] ∈ R k . 8. Construct the matrix T M,C ∈ R n×k from U M,C by normalizing each row of U M,C to have unit length t ij = u ij √ ∑ k j=1 u 2 ij 9.
Handle each row of T M,C as a point, and regroup them into k clusters using the nearest neighbor algorithm with the p prototypes of each class. -Compute m most relevant features.

-Compute the normalized kernel W defined by Eq. ( 28).

-Compute the embedded point z(y i ) defined by Eq. ( 26).

-Compute the normalized embedded point z(y i ) using Eq. ( 29).

-Assign y i to the cluster of the nearest sample point, among T M,C , of z(y i ).

Refinement procedure (optional).

Output: Segmented image.

selection procedures are performed on the sample pixel set X, and repeated over 100 runs. At are tested to study the influence on the quality of feature selection. Furthermore, two images with 6 and 9 textures from the Prague dataset are used to illustrate our experiments regarding parameter adjustment (see Fig. 3).

Constraint scores with respect to number of features

The performance obtained by the proposed constraint score ε SS is compared with that reached by the constraint scores C c (c = 1, . . . , 5). increase with respect to the number m of features. Similar results are observed for other numbers of prototypes and sample pixels (these results are omitted from this paper to avoid excessive length). We can conclude that only ε SS presents a minimum value that can be considered as the optimal number of features.

Accuracy of unlabeled pixels

The performance of the constraint score ε SS is evaluated according to the classification accuracy of the set X U of unlabeled sample pixels, which is obtained by the constrained spectral clustering described in Sec.

Optimal number of selected features

As indicated in Sec. 4.1.1, the curve of constraint score ε SS versus the number of features presents a minimum that can be considered as the optimal number of features. To validate this assertion, we simultaneously display in Fig. 7 the curves of ε SS and the accuracy versus the number of features achieved when n is set to 300 and p is set to 5. We can see that the accuracy T c ( m) obtained with the subset of features F m that corresponds to the minimum of ε SS coincides with or is close to the maximum value of the accuracy T c (m * ). In order to compare the performance of our constraint score with the state-of-the-art scores according to the number of pairwise constraints, the number of selected features for each score is set to half of all features (i.e. m = 90). Figure 9 shows the average accuracy obtained over 100 runs by the constraint scores versus number p of prototypes when n is set to 300. The accuracy obtained with ε SS is higher than those obtained with C c (c = 1, . . . , 5)

for any value of p. multi-phase segmentation framework (PCA-MS) [START_REF] Mevenkamp | Variational multi-phase segmentation using highdimensional local features[END_REF] that uses a filter bank for feature extraction, 3) the model-based learning of local image features (MLLIF) 7 that learns features from images without ground truth segmentation, 4) the factorization-based texture segmentation (FSEG) algorithm 41 that characterizes pixels by local LoG/Gabor spectrum features, 5) the unsupervised fully convolutional network for texture (FCNTunsup) algorithm, and 6) the dictionary learning based sparse representation (DLSRC) [START_REF] Yang | Unsupervised images segmentation via incremental dictionary learning based sparse representation[END_REF] that extracts the features from wavelet transform and co-occurrence matrices.

Four semi-supervised algorithms: 1) the weakly-supervised sparse coding geometric priori (WSSCGP) [START_REF] Quan | Weakly-supervised sparse coding with geometric prior for interactive texture segmentation[END_REF] that requires a small amount of prototypes to perform segmentation. WSS-CGP adopts the same local LoG/Gabor spectrum features than FSEG, 2) FSEG + , which is a semi-supervised version of FSEG in which the same prototypes used in WSSCGP are integrated in FSEG, [START_REF] Quan | Weakly-supervised sparse coding with geometric prior for interactive texture segmentation[END_REF] 3) our proposed semi-supervised method using only the constrained spectral clustering (CSC), without feature selection procedure, i.e., with 180 features, and 4) our proposed semi-supervised method CFS-SC.

Eight supervised algorithms: 1) the MRF algorithm based on a Markov random field pixel classification model, [START_REF] Kato | Color image segmentation and parameter estimation in a markovian framework[END_REF] 2) the COF algorithm that uses the co-occurrence features and the nearest neighbor classifier, [START_REF]Prague texture segmentation data generator and benchmark[END_REF] 3) Con-Col algorithm, 39 4) the supervised fully convolutional network for texture (FCNTsup) algorithm, 2 5) the empirical wavelet transform based fully convolutional network for texture (EWT-FCNT) 2 that combines the empirical wavelet transform with FCNTsup, 6) U-Net, [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF] 7) Deep Visual Model (DA), [START_REF] Wang | Deep visual attention prediction[END_REF] and 8) the pyramid scene parsing network (PSP-Net). [START_REF] Zhao | Pyramid scene parsing network[END_REF] The results of supervised algorithms (Table 2) were obtained using the normal Prague dataset, which contains 20 test images. [START_REF]Prague texture segmentation data generator and benchmark[END_REF] The results of semi-supervised and unsupervised algorithms (Table 1) were obtained over the large Prague dataset (80 test images), which includes the 20 test images of the normal Prague dataset. [START_REF]Prague texture segmentation data generator and benchmark[END_REF] The segmentation results of FCNT, texNcut, FSEG, MRF, COF, and Con-Col were reported on the Prague benchmark website. [START_REF]Prague texture segmentation data generator and benchmark[END_REF] Those of DLSRC, WSSCGP, and FSEG + were described in Ref. 9 and in Ref. and WSSCG used 81 prototypes per class. These prototypes are included in square regions of (9 × 9) whose centers are marked by the user inside each texture region.

Next, we examine tables 1 and 2 according to three general comparisons.

Comparison with unsupervised algorithms

In Table 1 we can see that all semi-supervised algorithms, except FSEG + , outperformed the unsupervised algorithms. Moreover, our proposed methods with (CFS-SC) and without refinement (CFS-SC-nr) exhibited the best results for all the criteria except for GCE and LCE.

Comparison with semi-supervised algorithms

To assess the relevance of the proposed semi-supervised feature selection on the segmentation performance, we compare the segmentation accuracy benchmark results obtained by our method with feature selection (CFS-SC) and without feature selection (CSC). The results in Table 1 clearly show the benefit provided by the proposed feature selection on the segmentation. When we compare the semi-supervised methods, we observe that the performance of the proposed semi-supervised method (CFS-SC), even without refinement, is superior to that of FSEG + and WSSCGP.

Segmentation results on Histology dataset

The Histology dataset [START_REF] Ozolek | Teratomas derived from embryonic stem cells as models 50 for embryonic development, disease, and tumorigenesis[END_REF][START_REF] Mccann | Images as occlusions of textures: A framework for segmentation[END_REF] contains 36 color images of size 128 × 128 pixels that represent two types of tissue. The ground truth images were determined by professional pathologists through visual inspection. We use the same setup as for the Prague texture experiment except for the Haralick parameter ω that we set to 5. To assess the performance of our methods CFS-SC and CSC without and with refinement, we compare their results with several methods including, FSEG + , WSSCGP, DLSRC, and FSEG as well as an another unsupervised method, called occlusion of random textures segmentation method (ORTSEG). [START_REF] Mccann | Images as occlusions of textures: A framework for segmentation[END_REF] The quantitative evaluation is based on the same measures used for the Prague dataset. Table 3 shows that the proposed algorithm, even without refinement, largely outperforms the other methods. Fig. 11 displays the segmentation results achieved on some images of Histology dataset. To highlight the accuracy of segmentation results, the boundaries of detected regions are embedded in the original images. We clearly observe that the boundaries between the two tissues are better detected by our method and close to the ground truth.

Segmentation results on Outex dataset

Our proposed method CFS-SC is designed to segment color-texture images, but it is also suitable for grayscale texture segmentation. To assess its performance for grayscale textures, we apply it on the popular texture dataset Outex (US_00000 test suite) [START_REF] Ojala | Outex-new framework for empirical evaluation of texture analysis algorithms[END_REF] error (BCE). [START_REF] Pont-Tuset | Measures and meta-measures for the supervised evaluation of image segmentation[END_REF] All these measures provide a number between 0% and 100%, the latter corresponding to perfect segmentation. Fig. 12 shows some segmentation results obtained on the Outex dataset. A visual inspection illustrates the efficacy of the proposed method (CFS-SC) compared with the other segmentation methods. Table 4 gives quantitative measurements for the Outex dataset based on the previously mentioned mesures. We clearly observe that the proposed algorithm (with and without refinement) largely outperforms the two unsupervised algorithms (PCA-MS, FSEG), the semi-supervised algorithm (WSSCGP) and especially the three supervised methods (U-Net, DA, PSP-Net). However, CFS-SC provides lower results than EWT-FCNT whose expensive learning reaches impressive performance.

Segmentation results on Berkeley dataset

To study the efficiency of our proposed method, we applied CFS-SC on some natural images selected from the Berkeley database. [START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF] To evaluate segmentation performances, three quantitative measures, namely, F-measure, [START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF] probabilistic rand index (PRI), [START_REF] Pont-Tuset | Measures and meta-measures for the supervised evaluation of image segmentation[END_REF] and recall (CO), are computed. The values of F-measure and PRI fall in [0,1], and the larger, the better. These images, displayed on Fig. 13, are often considered because they include some uncertain and complex color-texture patterns. We compare our method CFS-SC with robust self-sparse fuzzy clustering algorithm (RSSFCA) that is a very recent image segmentation method. [START_REF] Jia | Robust self-sparse fuzzy clustering for image segmentation[END_REF] Fig. 13 shows that the obtained segmentation results from a qualitative and quantitative point of view are better than those of RSSFCA and close to ground truth.

Algorithm 1 .Algorithm 1

 11 Constrained spectral clustering algorithm. Input: dataset X represented by X = [x 1 , x 2 , . . . , x n ] ∈ R n×d , number k of desired clusters, set M of must-link constraints, and set C of cannot-link constraints. 1. Compute the similarity matrix W M,C defined by Eq. (4). 2. Compute the Laplacian matrix L M,C Sym associated to W M,C . 3. Compute the spectrum of L M,C Sym , and extract k dominant eigenvectors U M,C = [u 1 , . . . , u k ] ∈ R n×k . 4. Construct the matrix T M,C ∈ R n×k from U M,C by normalizing each row of U M,C to have unit length

Fig 1

 1 Fig 1 Flowchart of the proposed color-texture image segmentation (CFS-SC).

Fig 2

 2 Fig 2 Illustration of local Haralick texture feature computing.

  and 128 √ 2), and four orientations (θ ⋆ = 0°, °45°, 90°, and 135°) are considered as in Ref. 33 for each one of three color components R, G, and B (with ⋆ = R, G, or B). As magnitude response of each filter corresponds to a feature, 84 Gabor features are computed at each pixel. Besides texture features, color features are represented by color components. Two devicedepend color spaces, namely (R, G, B) and (H, S, V ), and two device-independent color spaces, namely (X, Y, Z) and (L * , a * , b * ) are considered in this paper. The conversion from standard (R, G, B) to (H, S, V ), (X, Y, Z) and (L * , a * , b * ) color spaces is performed using CIE standard illuminant D65 as reference white. 35 Finally each pixel is characterized by a set of d = 180 features (84 Haralick texture features, 84 Gabor texture features, and 12 color features) F m = {f 1 , . . . , f d }. Each of the d = 180 pixel features is normalized between 0 and 1 to ensure that the scales of all features are equal.

3. 3 . 1

 31 Proposed constraint scoreThe proposed constraint score, denoted ε SS (F m ), uses the similarity matrices to evaluate the relevance of a subset of m features F m = {f 1 , . . . , f m } (m = 1, 2, . . . , d) and exploits both the pairwise constraint sets and the distribution of unlabeled sample pixels in the F m feature space.

3. 3

 3 Segmentation by classification of the out-of-sample pixel set Y 10. For each out-of-sample pixel y i :

1Fig 3

 3 Fig 3 Two test color-texture images from the Prague dataset (first row) and their corresponding ground truth images (bottom row): (a) image 7_1_1 and (b) image 17_1_1.

Figure 4

 4 illustrates the variation of the constraint scores ε SS , C 1 , C 2 , C 3 , C 4 , and C 5 with respect to the number m of features when the number p of prototypes is set to 5 and the number n of sample pixels is set to 300.The curves of ε SS are quasi convex, whereas those of scores C c (c = 1, . . . , 5) monotonically

Fig 4

 4 Fig 4 Constraint scores versus number m of selected features for the two test images: (a) image 7_1_1 and (b) image 17_1_1.

3 . 4 .

 34 First, we study the influence of the number p of prototypes by class and the number n of sample pixels on the accuracy achieved with ε SS . Next, we compare the accuracy obtained using our constraint score with those obtained using state-of-the-art constraint scores.Accuracy versus number p of prototypes. The number p of prototypes by class is related to the number of pairwise constraints and may modify the classification result provided by the constrained spectral clustering. To stay in the semi-supervised learning context, we have chosen small values for p ranging from 2 to 5. Figure5shows the variation of the average accuracy of X U classification obtained with ε SS over 100 runs according to the number p of prototypes and the number n of sample pixels. As expected, the curves of this figure show that, for a given number n of sample pixels, the accuracy achieved with our constraint score increases with respect to the number p of prototypes by class.Accuracy versus number n of sample pixels. The number n of sample pixels is related to the spatial sampling of the image. On the one hand, n must be large enough to reach a satisfying spatial definition of the detected regions. On other hand, n must be small enough to reduce the computational complexity of the constrained spectral clustering of the unlabeled sample pixels. To satisfy a trade-off between the spatial definition and complexity, the n range is [200 -400], which corresponds on average to 0.10% of the number N of pixels. To show

Fig 5

 5 Fig 5 Accuracy versus number m of features obtained using the proposed semi-supervised constraint scores on test images for different numbers of n and p. Left column: image 7_1_1 and right column: image 17_1_1. (a)-(b) n = 200, (c)-(d) n = 300, and (e)-(f) n = 400.

Fig 6

 6 Fig 6 Accuracy versus number m of features obtained using the proposed semi-supervised constraint score on the two test images for different numbers n of sample pixels: (a) image 7_1_1 and (b) image 17_1_1.

Fig 7

 7 Fig 7 Accuracy-score ε SS versus number m of selected features for the test images: (a) image 7_1_1 and (b) image 17_1_1.

Fig 8 Fig 9

 89 Fig 8 Accuracy versus number m of features for the two test images: (a) image 7_1_1 and (b) image 17_1_1.

  7 for MLLIF. The results of EWT-FCNT, U-Net, DA, and PSP-Net were taken from Ref. 2, and those of PCA-MS from Ref. 40. The majority of these methods involved a refinement step to improve their performances, except for texNcut, MRF, COF, and Con-Col. It is important to note that our methods CSC and CFS-SC were run with n = 300 sample pixels and p = 5 prototypes for each texture class, while FSEG +

Fig 10

 10 Fig 10 Exemplary segmentation results on the Prague color benchmark. From top to bottom: Original images, ground truth, FCNTsup, EWT-FCNT, FCNTunsup, texNcut, WSSCGP, CFS-SC-nr, and CFS-SC.

Fig 11

 11 Fig 11 Exemplary segmentation results on the Histology dataset. From top to bottom: Original images, ground truth, FSEG, ORTSEG, WSSCGP, and CSC-FS.

  and compare its results with those of state-of-the art algorithms: ORTSEG, PCA-MS, FSEG, EWT-FCNT, U-Net, DA, PSP-Net, and WSSCGP. This dataset contains 100 composite texture images of size 512 × 512 pixels, each one being a composite of five training textures. They are generated by mixing twelve different texture images with different rotations according to the regions depicted by the ground truth. 49 For convenience, we analyze only the texture features that are computed from the gray level images. The size of neighborhood ω used for computing the Haralick features is set to 5. To quantitatively assess segmentation performance, we use the same measures as in Ref. 2 : recall (CO), normalized variation of information (NVOI), swapped directional hamming distance (SDHD), Van Dongen distance (VD), swapped segmentation covering (SSC), bipartite graph matching (BGM) and bidirectional consistency

Fig 12

 12 Fig 12 Exemplary segmentation results on the Outex dataset. From top to bottom: Original images, Ground truth, PSP-Net, U-Net, EWT-FCNT, PCA-MS, FSEG, WSSCGP, and CFS-SC.

Fig 13

 13 Fig 13 Segmentation results on some images of the Berkeley database. The values of CO, F-measure, and PRI for each result are presented in parentheses.

  is selected and combined with each of the remaining d -1 features to form (d -1) feature subsets, denoted as F 2 . The corresponding d -1 scores ε SS (F 2 ) are then computed, and the pair of features that minimizes ε SS (F 2 ) is retained. When m features among the d features have been selected, the (m + 1)-th feature that minimizes ε SS (F m+1 ) when combined with the m previously chosen features, is selected. This sub-optimal procedure is iterated until d features have been ordered. Finally, the subset F m that corresponds to the minimum of ε SS (F m ) is selected (see Algorithm 2). Algorithm 2 Feature selection procedure. Input: Set of d feature F d = {f 1 , . . . , f r , . . . , f d }. 1. Create empty set of features F 0 = {∅}. 2. For m = 1 to d a. Select the most relevant feature f +

	r

  Color-texture image segmentation by CFS-SC method. Input: -Image parameters: Image I, number k of pixel classes; -Feature extraction parameters: Haralick parameter (ω), Gabor parameters (θ, γ); -Constrained feature selection and spectral clustering parameters : number n of sample pixels, number p of prototypes by class. 3.1 Sampling and pairwise constraint set generation 1. Extract n sample pixels X from I and regroup remaining out-of-sample pixels in the subset Y . 2. Select p pixel prototypes by class and generate the sets of pairwise constraints M and C. 3. Compute d color texture features for each sample pixel of X. 3.2 Constrained feature selection and spectral clustering of sample pixel set X 4. Select m most relevant features F m using our constrained feature selection (see Algorithm 2). 5. Construct the similarity matrix W M,C (F m) ∈ R n×n from X using Eq. (25). 6. Compute the normalized Laplacian matrix L M,C Sym associated to W M,C (F m). 7. Compute the spectrum of L M,C sym , and find k dominant eigenvectors U

[START_REF] Huang | Empirical curvelet based fully convolutional network for supervised texture image segmentation[END_REF]

, C 3 , C 4 , and C 5 (see section 2.3). The scaling parameter σ used to compute similarity matrices is set to 1 for all experiments. Feature Algorithm 3

Table 1

 1 Results of unsupervised and semi-supervised methods on the large Prague dataset (80 test images). The arrows ↑ | ↓ denote the required criterion direction. Here, 'nr' means no segmentation refinement.

				Unsupervised					Semi-supervised
	Method	FCNT DLS unsup RC	tex Ncut	PCA MLL FSEG FSEG + WSS CSC CSC MS IF CGP nr	CFS SC-nr	CFS SC
	↑ CS	79.34 77.46 72.54 72.27 77.73 69.18	75.97	84.18 79.73 82.97 93.82 94.74
	↓ OS	13.67 28.40 10.92 18.33 15.92 14.69	3.38	10.95 0.14	0.14	0.00	0.00
	↓ US	6.25	0.00	9.61	9.41	6.31	13.64	5.53	0.00	0.00	0.00	0.00	0.00
	↓ ME	3.80	7.13 10.25 4.19	3.93	5.13	11.82	6.90 14.64 11.92 1.98	1.41
	↓ NE	3.80	7.39	9.83	3.92	3.92	4.63	11.49	7.04 15.48 12.34 2.59	1.85
	↓ O	6.47	8.58	7.33	7.25	7.68	9.25	9.12	6.84	8.59	7.32	4.35	3.70
	↓ C	22.88 29.48 8.17	6.44 24.24 12.55	9.34	7.09	7.33	6.02	3.87	3.39
	↑ CA	84.17 83.41 80.58 81.13 82.80 78.22	80.26	87.05 84.33 86.01 91.89 92.62
	↑ CO	87.97 87.36 86.89 85.96 86.89 84.44	88.09	91.58 90.50 91.68 95.51 95.98
	↑ CC	94.15 95.16 88.28 91.24 93.65 87.38	88.19	94.85 91.92 92.92 95.94 96.31
	↓ I.	12.03 12.64 13.11 14.04 13.11 15.56	11.91	8.42	9.50	8.32	4.49	4.02
	↓ II.	1.42	1.19	2.36	1.59	1.50	2.53	2.47	1.28	1.79	1.64	0.75	0.71
	↑ EA	88.97 89.70 86.39 87.08 88.03 84.24	87.40	92.53 90.69 91.76 95.61 96.04
	↑ MS	85.23 84.74 80.33 81.84 83.93 78.81	82.46	88.63 85.75 87.51 93.27 93.97
	↓ RM	3.12	2.42	3.69	5.11	3.27	4.74	2.99	1.89	1.99	1.91	0.81	0.77
	↑ CI	89.91 90.44 86.97 87.81 89.03 85.03	87.76	92.86 90.94 92.02 95.67 96.09
	↓ GCE	6.46	9.56 11.92 8.35	7.40	9.35	15.08	10.21 14.62 12.86 8.00	7.26
	↓ LCE	4.75	7.17	6.85	5.61	5.62	6.08	11.71	7.71 11.52 9.79	6.28	5.59
	↓ dD	7.79	9.08	9.18	9.06	8.57	10.01	10.34	7.21	9.28	8.12	4.48	4.02
	↓ dM	4.88	5.40	6.03	5.89	5.30	7.01	6.59	4.40	5.57	4.95	2.66	2.41
	↓ dVI	14.75 15.18 14.19 14.54 14.88 14.33	14.32	14.52 14.43 14.32 14.18 14.13

Table 2

 2 Results of our method CFS-SC and supervised methods on the normal Prague dataset (20 test images). The arrows ↑ | ↓ denote the required criterion direction. Here, 'nr' means no segmentation refinement.Although the prior knowledge used by the supervised methods is more important than that used by our semi-supervised method, we can observe in Table2that our proposed method, both with and without refinement, largely outperformed the classical supervised methods (MFR, COF, and Con-Col). The methods based on the deep learning (FCNTsup, DA, U-Net, PSP-Net, and EWT-FCNT) outperform CFS-SC for most criteria. However, our method CFS-SC is competitive with deep learning methods except EWT-FCNT that provides exceptional results due to expensive learning step. For example, the CFS-SC does not provide under-segmentation nor over-segmentation (US and OS measures are equal to 0) compared to deep learning methods. It is also noticeable that our method CFS-SC without refinement outperformed the FCNTsup without refinement.

					Supervised					Semi-supervised
	Method	MRF	COF	Con FCNT FCNT EWT U-Net DA Col sup-nr sup FCNT	PSP Net SC-nr CFS	CFS SC
	↑ CS	46.11	52.48 84.57 87.52	96.01	98.45 96.71 94.18 96.45 94.03	94.97
	↓ OS	0.81	0.00	0.00	0.00	1.56	0.00	1.71	0.00 0.17	0.00	0.00
	↓ US	4.18	1.94	1.70	0.00	1.20	0.00	0.00	1.18 0.41	0.00	0.00
	↓ ME	44.82	41.55	9.50	6.70	0.78	0.37	0.68	3.42 1.23	1.63	1.09
	↓ NE	45.29	40.97 10.22	6.90	0.89	0.46	0.48	3.24 1.12	2.32	1.58
	↓ O	14.52	20.74	7.00	7.46	2.72	0.93	0.72	3.13 2.75	4.42	3.83
	↓ C	16.77	22.10	5.34	6.16	2.29	1.04	0.70	1.32 2.39	4.45	3.80
	↑ CA	65.42	67.01 86.21 87.08	93.95	97.67 95.86 94.53 93.89 91.85	92.61
	↑ CO	76.19	77.86 92.02 92.61	96.73	98.78 96.91 96.23 96.06 95.49	95.97
	↑ CC	80.30	78.34 92.68 93.26	97.02	98.81 97.38 97.01 96.41 95.91	96.29
	↓ I.	23.81	22.14	7.98	7.39	3.27	1.22	3.09	3.77 3.94	4.51	4.03
	↓ II.	4.82	4.40	1.70	1.49	0.68	0.25	0.41	0.58 0.69	0.79	0.76
	↑ EA	75.40	76.21 91.72 92.68	96.68	98.77 97.01 96.24 96.08 95.59	96.04
	↑ MS	64.29	66.79 88.03 88.92	95.10	98.17 95.37 94.35 94.08 93.24	93.96
	↓ RM	6.43	4.47	2.08	1.38	0.86	0.24	0.61	1.07 0.70	0.79	0.70
	↑ CI	76.69	77.05 92.02 92.81	96.77	98.78 97.08 96.41 96.15 95.65	96.08
	↓ GCE	25.79	23.94 11.76 12.54	5.55	2.33	2.13	3.50 4.67	8.07	7.30
	↓ LCE	20.68	19.69	8.61	9.94	3.75	1.68	1.46	2.47 3.52	6.40	5.75
	↓ dD	20.35	17.86	7.50	-	3.06	1.21	1.45	2.41 2.59	4.50	4.02
	↓ dM	13.25	10.62	4.69	-	1.96	0.74	0.77	1.35 1.56	2.76	2.48
	↓ dVI	14.51	14.22 13.99	-	13.80	13.68 13.68 13.71 13.77 14.01	13.96
	4.2.3 Comparison with supervised algorithms

For a visual comparison, Fig.

10

shows some segmentation results obtained by our CFS-SC method, WSSCGP, and by four top-performing methods whose segmented images were available in Ref.

39

. For clarity, the black lines highlight the boundaries of segmented texture regions. Overall, CFS-SC, FCNTsup, and EWT-FCNT approaches provide satisfactory visual segmentation results that are close to the ground truth.

Table 3

 3 Results on the Histology dataset. The arrows ↑ | ↓ denote the required criterion direction and the best score is marked in boldface. Here, 'nr' means no segmentation refinement.

		Unsupervised		Semi-supervised
	Method	DLS ORT FSEG FSEG + WSS CSC RC SEG CGP	CFS CFS SC-nr SC
	↑ CS	82.98 72.53 38.98	82.06 86.90 82.54 96.93 97.07
	↓ OS	3.34 1.71 35.95	1.78	3.29 0.00	0.00	0.00
	↓ US	2.82 2.78	0.00	10.25	1.94 0.00	0.00	0.00
	↓ ME	9.69 20.09 19.96	7.67	5.95 11.33 0.00	0.00
	↓ NE	11.49 20.31 21.05	8.25	7.02 11.99 0.00	0.00
	↓ O	7.64 10.17 24.77	13.71	6.70 8.69	3.26	3.10
	↓ C	6.96 10.57 24.23	7.42	6.21 9.62	3.47	3.31
	↑ CA	86.80 81.81 67.17	84.90 87.69 84.71 94.12 94.37
	↑ CO	91.44 88.84 72.38	90.70 92.07 91.20 96.93 97.07
	↑ CC	94.96 91.27 92.27	92.02 95.28 91.99 97.00 97.14
	↓ I.	8.56 11.16 27.62	9.30	7.93 8.80	3.07	2.93
	↓ II.	5.08 8.98	6.99	11.40	5.07 8.57	3.45	3.27
	↑ EA	92.60 88.71 78.78	90.66 93.15 91.21 96.94 97.08
	↑ MS	87.17 83.43 67.63	86.05 88.11 86.8 95.40 95.61
	↓ RM	6.55 9.93 17.29	7.42	5.82 5.22	1.06	1.13
	↑ CI	92.89 89.37 80.49	91.00 93.40 91.4 96.95 97.09
	↓ GCE 10.42 12.36 14.10	9.64	9.73 13.65 5.77	5.52
	↓ LCE	6.77 7.23 11.24	6.12	6.48 9.68	4.82	4.51
	↓ dD	7.54 9.73 18.55	7.65	6.90 8.66	3.07	2.93
	↓ dM	12.88 17.19 26.18	14.00 11.99 14.96 5.90	5.65
	↓ dVI	5.88 5.64	7.23	5.61	5.85 5.68	5.43	5.42

Table 4

 4 Results on the Outex dataset. Each row corresponds to a segmentation quality measure and the best score is marked in boldface. Here, 'nr' means no segmentation refinement. .65 83.86 96.52 72.83 70.70 86.00 85.41 87.88 88.70 SSC 70.88 62.23 84.32 98.27 64.25 60.65 84.22 89.11 92.42 93.01 SDHD 80.64 65.08 89.67 99.13 71.40 67.77 88.00 94.71 96.02 96.34 BGM 76.42 64.97 89.24 99.13 70.85 67.27 87.83 93.59 96.02 96.34 VD 83.87 80.60 91.39 99.13 80.47 78.75 91.64 94.15 96.02 96.34 BCE 69.50 61.08 82.45 98.02 61.89 58.58 82.63 86.64 91.19 91.77 CO 76.42 64.97 89.24 98.36 70.28 67.28 86.72 93.59 96.02 96.34

		Unsupervised	Supervised	Semi-supervised
	Method	ORT FSEG SEG	PCA EWT U-Net DA MS FCNT	PSP WSS CFS CFS Net CGP SC-nr SC
	NVOI	77.99 73

Table 5

 5 The average computing time (second) per image of FSEG, WSSCGP, and CFS-SC on the tree datasets. (FE: Feature extraction, OS: Other steps.) .74 26.89 18.21 45.10 67.87 10.69 78.56 constrained spectral clustering. A new constraint score is developed in order to select a subset of features at one time. As the proposed score evaluates the similarity between the sample pixels in the examined feature subspace, the selected features are better analyzed by constrained spectral clustering. Experiments on four benchmark datasets showed that the proposed constraint score outperforms the main state-of-the-art constraint scores in the semi-supervised learning context. Moreover, the result demonstrated that the proposed constrained feature selection and the constrained spectral clustering algorithm is very effective for color-texture image segmentation.Finally, classical color-texture features and a simple sampling technique were used to conduct the experiments presented in this paper. It may to be advantageous to integrate other powerful texture descriptors, such as LBP, wavelets, or pre-trained CNNs, with a more elaborate sampling technique to further improve the segmentation accuracy. Moreover, we have considered that the prior knowledge is represented by class prototypes from which the pairwise constraints are deduced. All available constraints contribute to efficient feature selection and spectral clustering. In future work, we intend to generalize our method to prior knowledge that is only formalized by constraints.

		Prague (512 × 512) Histology (128 × 128) Outex (512 × 512)
	Methods		color image		color image	grayscale image
		FE	OS	Total	FE	OS	Total	FE	OS Total
	FSEG	0.41 0.67	1.08	0.02 0.04	0.06	0.14 0.41	0.55
	WSSCGP 0.90 31.92 32.82	0.06 0.61	1.21	0.72 26.63 27.35
	CFS-SC	486 31.74 517

Segmentation results on Prague dataset

Here, we present the results obtained by our proposed semi-supervised segmentation method with (CFS-SC) and without refinement (CFS-SC-nr; see Algorithm 3) on the Prague dataset and compare them with those of several state-of-the-art segmentation methods. The Prague dataset, known to be difficult to segment, contains 80 texture mosaics synthetically generated from 114 color-texture images in 10 categories (see ground truth in Ref. 39). The texture class number in each of the images, which are sized at 512 × 512 pixels, ranges from 3 to 12.

The assessment of segmentation performance is based on the conventional measures provided by the Prague texture segmentation website, [START_REF]Prague texture segmentation data generator and benchmark[END_REF] The best score is displayed as bold in each table.

The methods involved in comparison include:

Six unsupervised algorithms: 1) the texNcut algorithm that is based on spectral clustering algorithm [START_REF] Shi | Normalized cuts and image segmentation[END_REF] and which uses texture features computed on super-pixels, 2) the variational

Processing time

To provide an overview on the computational requirements of the proposed CFS-SC algorithm, we estimated its processing time on an Intel Core i7 3.60GHz computer with 8GB RAM. Table 5 displaysthe average computing time of FSEG, WSSCGP, and CFS-SC for the tree datasets, i.e. according to size and type (color or grayscale) of image. The processing time of these methods can be divided into two parts: the processing times consumed by feature extraction step and by the other steps (feature selection and pixel classification steps).

From this table, we can see that the processing time required by the CFS-SC algorithm is the highest. For example, it was about 518 seconds for a (512 × 512) image. This relatively long processing time led us revisit each step of our algorithm. The majority of this time was consumed by the feature extraction step (486 seconds) and more precisely by the Haralick feature extraction (403 seconds). The remaining time (34 seconds) was mainly shared by the three basic procedures of our CFS-SC algorithm; namely, the constrained feature selection (20.7 seconds), the constrained spectral clustering (0.033 seconds), and the classification of out-of-sample pixels (11.01 seconds). Apart from the feature extraction step, the algorithm runs relatively quickly. It should be emphasized that it is always possible to reduce the computational time for feature extraction step and that the computational time cannot be considered as a crucial drawback,given the appreciable segmentation results achieved by the CFS-SC algorithm on all the tested datasets.

Conclusion

In this paper, we proposed an efficient semi-supervised approach for color-texture image segmentation. The proposed method combines the constrained feature selection and the

List of Figures