Graziella Goglio
email: graziella.goglio@u-bordeaux.fr

Arnaud Ndayishimiye

Alain Largeteau

Catherine Elissalde

View point on hydrothermal sintering: Main features, today's recent advances and tomorrow's promises

Keywords: Hydrothermal Sintering, Low-Temperature Water-Assisted Densification, Ceramics, Nanostructuration, Low-Temperature Sintering Process

The development of new high performance materials faces the challenge of implementing low temperature densification processes to overcome current technological limitations. In this context, the hydrothermal sintering, inspired by the natural processes of geological and biological mineralization, has recently emerged as a major opportunity to develop new and/or optimized materials that respond to today's scientific, technological and related socio-economic issues. The purpose of this viewpoint paper is to present opinions and propose future outlook for hydrothermal sintering based on the most recent achievements.

New materials have always accompanied scientific and technical progress. Currently, research is directed towards materials with new properties, improved performance to cost ratio, limited environmental impact over their entire life cycle. More particularly, ceramics and high performance assemblies have a major impact in automotive, electronic applications, energy and environment, mechanical engineering, medical technology... These materials, due to their exceptional properties, are able to meet the specific requirements of different applications, especially when a high level of functionality and reliability is required. These ceramic-type materials are oxides, sulfates, carbonates, phosphates, silicates, non-oxides and can be crystallized or amorphous.

The development of new high-performance ceramics, but also the design of complex assemblies, requires the mastery of the shaping processes, including suitable sintering processes. Currently most of advanced ceramic materials are fabricated from powders and sintered above 1000°C to reach at least 95% of their theoretical densities. The decrease in surface free energy, which is a driving force for sintering, might be promoted either by applying pressure or by enhancing diffusional processes using fast heating routes. Thus, intense international research has led to the development of numerous sintering techniques to enhance the performances of conventional sintering (CS) such as two-step (TSS), flash (FS), high pressure (HPS), spark plasma (SPS), rate controlled (RCS), microwave (MWS) sintering... [1][2][3][4][5][6] If it is well admitted that pressure is beneficial for densification via particle sliding and rearrangement, plastic deformation, and pore shrinkage, the high temperatures required by these processes create several technological barriers: (i) advanced materials need to be produced by energyand cost-efficient processes to ensure the feasibility of industrial scalability, (ii) materials that are metastable or that decompose at low temperature are difficult to sinter with such processes and (iii) co-sintering of multimaterials is hindered by differences in thermal stability, the rate and the onset temperature of shrinkage, and the physical and/or chemical compatibilities between the components [7][8].

To decrease these processing temperatures, the use of nanopowders (with particle size in the range 10 -100 nm [9]) as starting materials appears as a key solution since the high surface-to-volume ratio of nanoparticles provides a strong driving force to promote the diffusion process, especially at elevated temperatures [START_REF] Khalil | Ceramic materials-Progress in modern ceramics[END_REF]. Another advantage of nanostructured ceramics is that, unlike the conventional technical ceramics, the pore volume fraction should be limited below a few tens of ppm [START_REF] Chaim | [END_REF]. It was shown that oxide nanoparticles with equiaxed morphology and narrow size distribution are a prerequisite to form green pellets with homogeneous density and low pore-to-particle size ratio, and thus to optimize the final relative density of the nanoceramics while preserving the initial nanograin size at full densification [START_REF] Chaim | [END_REF]. Finally one should consider that the use of nanoparticles is a route to design increasingly performing ceramics, providing a deep knowledge of the impact of the nanoscale structure on bulk properties. The recent advances in processing high quality nanocrystalline powders have favored the design of well controlled functional nanoceramics [12].

However, the decrease of the sintering temperature associated with the use of nanopowders is limited when full densification is targeted and high temperatures are still required to achieve densification.

Consequently, one should consider that the competition between densification, coarsening processes and agglomeration often yields microstructures with overly coarse grains, which is detrimental to densification [START_REF] Rahman | Sintering of Ceramics[END_REF].

Then it appears clearly that an indisputable breakthrough in the sintering approaches would consist in combining pressure (below 350MPa) and low temperature (at least below 500°C). In this way, hydrothermal sintering appears as a powerful technology that surmounts the aforementioned barriers.

First, similarly to advanced sintering processes such as SPS and hot pressing, this process uses an applied force to generate an internal pressure to improve particle rearrangement and densification of materials. It combines also the benefit of low temperatures that allow a better control of defects, prevent from solid state diffusion and grain growth and can be suitable for materials exhibiting low transition or decomposition temperatures. In addition, it overcomes the weakness of low temperature regime with respect to reactivity. Indeed, the main feature of hydrothermal sintering is to take advantage of hydrothermal conditions that promote diffusion of chemical species and optimize reactivity considered as an efficient driving force. The nature of solvent, the pH and the addition of mineralizers are then important levers to operate to optimize densification.

1-A brief history on this geologically inspired process

Dense ceramic materials can form in the nature under mild temperatures in the presence of water. As an example, besides biomineralization, the formation of ceramic-dense materials such as limestone also occurs in nature through large scale geological processes. Here the basic physicochemical mechanism may involve dissolution-precipitation process, referred to pressure solution creep, which relies on the mass transport from the contact zone between touching particles to the surrounding liquid phase and eventually to nearby non-contacting surfaces [START_REF] Gratier | [END_REF][15][16][17][18]. Consequently, the distance between the particle centres decreases, enabling global shrinkage and densification. The motion of ions during pressure solution creep is driven by the high-stress concentration at the contact point when the particle compact is subjected to an externally applied mechanical load. In response to such a stress concentration, ions at the contact point dissolve from the solid particle into the interfacial liquid film and eventually diffuse along the grain boundary before reaching the continuous liquid phase and precipitate on a lower-stressed particle surface. In addition, when the applied pressure is sufficiently high, plastic deformation through dislocation motion and/or viscous flow of the solid phase may participate to the densification process. It was shown that, for a given temperature, the densification rate is increased by reducing the particle size and increasing the applied pressure.

The hydrothermal sintering process is totally inspired from this geological process. It was first developed by N. Yamasaki et al. [19] and was called Hydrothermal Hot Pressing (Fig. 1.a). Here, a powder with water is externally and mechanically compressed in an autoclave, under hydrothermal conditions (100°C<T<350°C; 22.5MPa<P<220MPa) over short periods of time (from a few minutes to a few hours). As expected, the main driving force of such a process is the stress gradient within grains induced by external compression leading to the activation of the dissolution/precipitation phenomena at the solid/liquid interface. Here, water is expelled from the sample during densification and recovered in specific spaces for water retreat. Water both acts as a solvent, a pressure transmitting medium and a mass transport medium. Thus, hydrothermal conditions promote the reactivity at the solid/liquid interfaces, as diffusion processes in the solid phase are hindered at such low temperatures. This process was set up to densify metastable materials (anatase) [20][21], materials with mild temperature decomposition (Ca/SrCO3 [22][23], hydroxyapatite [24][25]), porous ceramics (porous hydroxyapatite [26], zeolites [START_REF] Nakahira | AIP Conference Proceedings[END_REF]), nanomaterials without coarsening (Sn1,24Ti1,9403,66(OH)1,50F1,42 [START_REF] Xie | [END_REF]), amorphous nanomaterials (SiO2 [29][30]), tailored thermoelectric materials (Ca3Co4O9 [31][32],

NaxCo2O4 [33]) or to bond different materials (hydroxyapatite bonded to titanium [34] or to magnesium alloy [35]).

We have recently developed a new apparatus inspired by the hydrothermal hot pressing tool described by Yamasaki and Yanasigawa [19], but with improved performances due to the specific design of both the pistons and the autoclave (Fig. 1.b) [36]. Here, the glass-filled teflon gaskets are maintained below 100°C, i.e. far from their maximum operating conditions (300°C), thanks to a double system (cooling fins and water cooling circuit), so they can ensure the tightness of the system even when samples are submitted to higher temperatures. As a result, temperature and pressure can reach 500°C and 350MPa, respectively [36]. Thanks to this optimization of the device, the range of experimental parameters is wide enough to tune the conditions to get either subcritical or supercritical solvent, which may have an influence on the elementary steps of the sintering. Depending on the nature of the solvent that can be either aqueous or non-aqueous, the process is called hydrothermal sintering or solvothermal sintering, respectively. These terms will be preferred to hydrothermal (or solvothermal) hot pressing because the word "hot" is contradictory with the experimental conditions applied and related induced mechanisms.

Fig. 1. Hydrothermal sintering apparatus designed by a) Yamasaki et al. [19] and b) Goglio et al. [36]

In b), the design of pistons and specific cooling systems favor improved performances.

2-Recent advances on hydrothermal sintering. Illustration with the use of silica as a common thread

To explore all the potentialities of hydrothermal sintering and evaluate its adaptability towards the elaboration of a wide range of materials, we have focused on amorphous 50-70 nm SiO2 nanoparticles. They have been selected as starting powders because (i) they are considered as a suitable platform for tuning surface chemistry, (ii) they exhibit a spontaneous tendency for self-organization, (iii) they offer the possibility to study sintering with or without structural transition and (iv) the dissolution of silica (according to the equation SiO2 + 2 H2O H4SiO4 [37]) is congruent, which guarantees to exclude selective precipitation phenomena and makes the system easier to control.

2.1-A synergy between mechanical-chemical effects and chemical ones driven by water

The amount of water is a crucial parameter in hydrothermal sintering as it strongly influences the dissolution and mass transport steps. In this way, silica is a suitable for hydrothermal densification as the particles raw from synthesis are naturally hydrophilic due to their surface hydroxylation, which enhances the physisorption of water. Both physisorbed and chemisorbed water plays a crucial role during hydrothermal sintering [38]:

-the physisorbed water in the starting particles acts as the solvent and promotes dissolutionprecipitation. The amount of this weakly bonded water can be tuned by a previous calcination step of silica nanoparticles and strongly influences the spread of densification. If physisorbed water is removed from the silica sample before sintering experiment, the absence of inter particle neck growth is noticed after treatment at 300 °C and 190 MPa during 90 minutes, in full agreement with a high value of specific area close to that of starting powder (54 m².g -1 and 60 m².g -1 , respectively). However, when the physisorbed water increases up to around 3 wt% and when same pressure and temperature sintering conditions are carried out, the mechanical-chemical process is strongly enhanced leading to a very dense microstructure with a reduced specific area to 14 m².g -1 and residual mesopores.

-the chemisorbed water related to surface hydroxylation favors chemical mechanisms enhanced under temperature and/or pressure and which act in synergy with the mechanical-chemical effects.

The silanol groups both internal and at the surface of the particles and of the pores polycondensate hence leading to chemisorbed water departure and to the formation of siloxane bridges in particular between touching particles. These Si-O-Si inter particle bonds promote the neck formation, thus favoring the densification of the material. In the presence of a water excess that may be trapped into the macropores, the hydrophobic siloxane surface turns to hydrophilic due to a rehydration of the surface. A summary of the complementarity and synergy of the mechanical-chemical and chemical effects is schematically depicted in Fig. 2.

Obviously, this mechanical-chemical induced densification is more efficient when the nanoparticles are initially close packed (the green material being then mesoporous), in order to generate higher stress gradients. As a consequence, any stacking faults in the green material will generate residual macropores in the ceramic. The relative density of the green compact has then to be optimized to further promote the densification. In the case of silica, nanoparticles naturally self-organize during the compaction step. However, if water is added in the system, as-induced local turbulences may disturb this arrangement, favoring the formation of macroporosities in the ceramic [38].

Fig. 2. Synergy between mechanical-chemical and chemical effects [38] 2.2-A process governed by thermodynamical and chemical levers

One main advantage of the process is related to the hydrothermal conditions which strongly promote the reactivity and the diffusion of species in the solvent. Obviously, classical thermodynamic parameters such as pressure and temperature act as important levers to tune these specific conditions.

-

In the case of naturally hydrated nanoparticles with or without added water, a higher relative density is observed when the hydrothermal sintering temperature is 300°C (Fig. 3). On the one hand, below this temperature, the densification is limited by the dissolution due to a low solubility of silica in water; the observed residual mesopores result from the uncomplete filling of the interstices of the initial close packed assemblies. It has been observed that the densification begins around 200°C. On the other hand, microstructural control deteriorates significantly beyond 300°C because of the released water departures increasingly important induced by the spread of densification (departure of additional water, water and chemisorbed water released by polycondensation). The microstructure consequently accommodates these water releases via the formation of macropores in a material which is less dense as the temperature rises above 300°C. Hence, because water remains liquid (or supercritical) throughout the overall process, it appears necessary to control each elementary step in order to find the best compromise between activated dissolution-precipitation mechanisms and the regulation of the water release. Besides these thermodynamic parameters, hydrothermal sintering offers the main advantage to control the densification by chemical parameters.

-The use of a co-solvent makes possible to regulate the dissolution and precipitation steps. For example, the less polar nature of ethanol relative to water makes the dissolution weaker, thus favoring the precipitation in the vicinity of the zone of dissolution and leading to a slower propagation of the densification and to a slower release of the water. Moreover ethanol is more compressible than water which favors its transport into the matrix [START_REF] Jana | Particulate Technology for Delivery of Therapeutics[END_REF]. Consequently, the use of ethanol as a co-solvent favors a fine control of the microstructure through a change from macroporous matrix in the case of pure water to mesoporous one with a pore size all the more lower as the proportion of ethanol is high.

When the sintering is performed at 300°C under 350 MPa during 90 minutes with 20 wt% of solvent, the relative density reaches 88% with mesoporosity in the case of a mix made of 75 vol% ethanol-25 vol% water, while it equals to 85% with macroporosity in the case of pure water (Fig. 5). -A mineralizer can also be used to improve the dissolution step. In the case of silica, it is well known that the dissolution kinetics is strongly improved by a pH increase [START_REF] Iler | The chemistry of silica: solubility, polymerization, colloids and surface properties and biochemistry of silica[END_REF]. As a consequence, lowest NaOH concentrations favor the densification of amorphous silica, reaching 86% of relative density for 0.2M NaOH (300°C, 350 MPa, 20 wt% of solvent, Fig. 6.a) [START_REF] Ndayishimiye | [END_REF]. However, highest sodium hydroxide concentrations favor the stabilization of metastable -quartz. When the solvent is too basic, the dissolution is too fast in comparison to precipitation and diffusive mass transport is enhanced, leading to a highly densified heterogeneous microstructure (Fig. 6.c). However, quasi-complete densification of well-faceted -quartz grains with homogeneous nanometric size has been achieved for 2M NaOH concentration, leading to 98% of relative density (Fig. 6.b) [START_REF] Ndayishimiye | [END_REF]. The epitaxial growth at the grain boundaries is remarkable and characteristic for weak constraints (Fig. 6.d). Finally, a parallel can be made between hydrothermal crystal growth [42] and hydrothermal sintering of -quartz [START_REF] Ndayishimiye | [END_REF] where the driving force is the temperature gradient and the local pressure gradient, respectively. In both cases, the alkaline medium strongly favors the dissolution step by increasing the rate of dissolution.

2.3-Design of complex multimaterials

From a global point of view, one challenge is to perform the sintering of complex assemblies of materials that exhibit differences in their chemical and physical characteristics. In this way, in addition to any problems to solve for a single constituent, one should consider the specific reactivity at interfaces. For example, the high temperatures processed in conventional sintering routes can induce the formation of undesirable interphases between components leading to a detrimental influence on the multimaterial assembly properties. We have then developed a manufacturing strategy based, on the one hand, on grain boundary engineering to design 0-3 type nanocomposites starting from core@shell nanoparticles and, on the other hand, on the implementation of hydrothermal sintering (Fig. 7.a) [43].

We have focused on the (La,Sr)MnO3 (LSMO)-silica system because the introduction of the insulating silica phase at the grain boundaries of polycrystalline metallically conducting manganite promotes the spin-polarized tunneling between neighboring perovskite grains [44][45][46][47][48][49][50][51]. The macroscopic transport is consequently strongly dependent on external magnetic field, which makes such composites promising for spintronic devices. Hydrothermal sintering is suitable to fabricate 0-3 type composites with a relative density of approximately 80-85% where LSMO cores are uniformly and discretely embedded in the insulating matrix of three-dimensional-connectivity that originates from the silica shell densification [43]. Here silica also ensures the confinement of defects at the interfaces and prevents the core growth. Neither interphase formation nor evolution of core chemistry (manganese valency)

was observed as the sintering mechanisms occur at the liquid/solid interfaces, i.e. the water/silica interfaces. Moreover, the thicker the silica shell, the more core@shell nanoparticles behave like silica nanoparticles with respect to hydrothermal sintering (Fig. 7.b,c). This tendency is attributed to the evolution of the compressibility of the core@shell nanoparticles as it strongly influences the intra grain stress gradient induced by external uniaxial compression, hence the dissolution/precipitation phenomena. In addition, one can also observe that depending on their compressibility, the materials will accommodate differently the released water transport, leading to an evolution of the porosity nature (mesoporosity or macroporosity). Obviously, the bottom up strategy is intended to be extended to other systems. For instance, it should be specifically suitable for thermoelectric, multiferroic or piezoelectric complex multimaterials [52][53]. Studart, from ETH Zurich, claim the densification of vaterite at room temperature using CSP [64].

3-Hydrothermal

First of all, comparing energy merit in the different sintering techniques, the development of advanced sintering processes (Flash sintering, microwave sintering…) has led to a significant reduction in energy consumption compared to conventional sintering. CSP has been developed in this specific context and can be considered up to now as one of the most energy efficient technique [65]. Because hydrothermal sintering experiments are carried out in the same pressure and temperature ranges and with equivalent durations, this process also naturally appears as a promising solution to decrease the energy costs with, in particular, a single step.

The mechanisms involved in CSP and hydrothermal sintering are the same but both processes differ from each other due to the balance between thermodynamics and kinetics that strongly influences the predominance of the involved elementary steps and may induce significant differences concerning solubility, dissolution and precipitation kinetics, pore shrinkage, water release and achievable densification.

-In the CSP process, the system is open and water becomes gaseous during the sintering. The temperature both accelerates the dissolution and quickly saturates the solution with the evaporation of the solvent driving the precipitation. The overall process is analogous to the densification of liquid phase sintering but with an enabling transient liquid phase that evaporates at low temperatures. The loss of water induces departure from thermodynamical equilibrium conditions, bringing a kinetic component to the CSP sintering process.

-In the case of hydrothermal sintering, the system is closed. As the applied pressure exceeds the saturation vapor pressure, the solvent remains liquid under subcritical conditions (or becomes a supercritical fluid above its critical point) and the densification is governed by thermodynamics. Here the main difficulty is related to the competition between the propagation of the densification and the solvent release: if the densification is too fast (enhanced by temperature), the microstructure cannot accommodate the induced fast water release and macroporosities are formed. Pressure then appears as the lever to control the water transport. On contrary, slower densification favors the elimination of water from the matrix and impedes the formation of macroporosities.

Consequently, an in-depth knowledge and the mastery of the balance between thermodynamics and kinetics will provide a powerful solution to optimize the densification, and control the grain size, the size distribution and the crystallinity of the precipitating phase.

4-Future outlook and impact

Hydrothermal sintering appears as an enabling technology as it will provide the means to increase the performance and capabilities of materials and sintering processes. In this way, the impact of this process can be expected multifaceted.

-Scientific and technological impact: our better understanding of the densification mechanisms during hydrothermal sintering will be a key in developing a clear roadmap to guide future studies of ultralow temperature sintering and offer widespread application prospects for a large variety of materials. In addition to a technological transfer to other materials such as complex oxides or non-oxide materials, we can foresee the development of a new technology to perform co-sintering of multiple materials to reach specific functionalities. One should notice that the process is obviously not restricted to nanomaterials. Moreover, given the compressibility of the material and the water release controlled by both pressure and temperature, new ceramic materials with controlled porosity can be designed.

Finally, due to the enhancement of the reactivity at the interfaces in hydrothermal (or solvothermal) conditions, the possibility to bond different materials such as a metal with ceramic that has been densified in situ is of outstanding interest both from a scientific, technological and economical point of view. In this way industrial multistep processes could be replaced by a single hydrothermal sintering step.

-Economic impact: the economic impact of hydrothermal sintering is significant in the sense that it appears as a feasible route for cost-effective and sustainable energy ceramic manufacturing processing and integration that enables a broad range of applications, e.g. for the production of electroceramics.

More precisely, short experiments at low temperature ensure lower energetic costs while the use of water as a solvent is associated with cheap consumables (packing seals). Obviously, hydrothermal sintering may have highly advantageous industrial fallouts, especially in the microelectronic and ceramic industries. In this way, this process unambiguously appears as an appropriate solution for the industrial production of small size pieces (up to a few cm 3 ).

-Environmental impact: hydrothermal process does not generate any waste. Moreover, solvents define a major part of the environmental performance of processes in chemical industry and impact cost, safety and health. The use of water as a cheap, green and safe solvent makes hydrothermal sintering an environmentally friendly process.

As a consequence, hydrothermal sintering is a springboard for the design and fabrication of ceramics and complex multimaterials, integrating nanotechnology. It ensures a social and environmental responsibility as it is a low energy consuming process. Both the process and the related knowledge have been considerably improved these last years however better performances can be expected. For example, fast heating and cooling rates will be an asset to control the balance between thermodynamics and kinetics; the specific design of heating routes is therefore of great interest to master the grain morphology, the porosity and the nature of the precipitated phase. Moreover the development of in situ dilatometry measurements would lead to carry out the identification of the rate-controlling densification mechanisms. They will provide important experimental optimizations as they will allow visualizing more precisely the onset of shrinkage. The short-term challenge is then to transfer all the knowledge on hydrothermal and solvothermal processes acquired as part of the synthesis or crystal growth of size and/or morphology controlled materials. These specific conditions which contribute to an optimization of the reactivity at the solid/liquid interfaces are expected to make possible, on the one hand, to control the grain size from nanometric to micrometric and, on the other hand, to stabilize new phases during the precipitation step as evidenced in the case of -quartz. In conclusion, hydrothermal sintering should be considered as a chemically controlled process which opens up a particularly rich range of perspectives.

Fig. 3 .

 3 Fig. 3. Influence of the temperature (a) 150°C, b) 200 °C, c) 300°C, d) 350°C, e) 420°C ) on the densification of silica nanoparticles. Zooms on the microstructures a) and b) are proposed on a') and b'), respectively

Fig. 4 .

 4 Fig. 4. Influence of the pressure on the densification of silica nanoparticles. The microstructures (with zoom on the inserts a') and b')) observed for 100 MPa and 350 MPa are presented in a) and b), respectively.

Fig. 5 .

 5 Fig. 5. Influence of the solvent on the densification with a) pure water and b) ethanol-water 75 vol%-25 vol%

Fig. 6 .

 6 Fig. 6. HR-SEM of ceramics obtained after hydrothermal sintering of silica with 20wt% of a) 0.2 M NaOH, b) 2M NaOH and c) 5M NaOH. d) HR-TEM image on sample b)[START_REF] Ndayishimiye | [END_REF]

Fig. 7 .

 7 Fig. 7. a) Bottom up strategy for the elaboration of 0-3 type composites implementing hydrothermal sintering. Evolution of the microstructure of the composite after hydrothermal sintering (300°C, 350MPa, 20 wt% of 0.2M NaOH, 90min) according to the starting silica thickness: b)5nm and c)10nm[43]

  Sintering and Cold Sintering Processes : close cousins for complementary performances In June 2016, C. Randall et al. from Pennsylvania State University have reported impressive sintering results [54-63] (95% of compactness reached on a large panel of ceramics and composites then evidencing device fabrication) with a process called the "Cold Sintering Process" (CSP) based on hydrothermal sintering. The explored temperature and pressure ranges are respectively 25-350°C and 50-500 MPa, most of the experiments being set up below 200°C and 350MPa. In 2017, Bouville and

Acknowledgments

Authors gratefully thank Pr. Clive Randall and Dr. Jing Guo from Penn State University for fruitful discussions.


Edward Cayllahua-Cahuina

Jean Cousty

Silvio Jamil

F Guimarães

Yukiko Kenmochi

Guillermo Cámara-Chávez

Arnaldo De

Albuquerque Araújo

Hierarchical segmentation from a non-increasing edge observation attribute

Hierarchical image segmentation provides region-oriented scale-spaces: sets of image segmentations at different detail levels in which the segmentations at finer levels are nested with respect to those at coarser levels. Guimarães et al. proposed a hierarchical graph-based image segmentation (HGB) method based on the Felzenszwalb-Huttenlocher dissimilarity. It computes, for each edge of a graph, the minimum scale in a hierarchy at which two regions linked by this edge should be merged according to the dissimilarity. We provide an explicit definition of the (edge-) observation attribute and Boolean criterion which are at the basis of this method and show that they are not increasing. Then, we propose an algorithm to compute all the scales for which the criterion holds true. Finally, we propose new methods to regularize the observation attribute and criterion and to set up the observation scale value of each edge of a graph, following the current trend in mathematical morphology to study criteria which are not increasing on a hierarchy. Assessments on Pascal VOC 2010 and 2012 show that these strategies lead to better segmentation results than the ones obtained with the original HGB method.

Introduction

Hierarchical image segmentation gives a multi-scale representation for image analysis. Hierarchical image analysis was initially explored by [START_REF] Pavlidis | Structural Pattern Recognition[END_REF] and has gained a lot of attention, as confirmed by the popularity of [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF]. Mathematical morphology has been used in hierarchical image analysis with, e.g., hierarchical watersheds [START_REF] Beucher | Watershed, hierarchical segmentation and waterfall algorithm[END_REF][START_REF] Najman | Geodesic saliency of watershed contours and hierarchical segmentation[END_REF][START_REF] Cousty | Incremental algorithm for hierarchical minimum spanning forests and saliency of watershed cuts[END_REF][START_REF] Perret | Evaluation of hierarchical watersheds[END_REF], binary partition trees [START_REF] Salembier | Binary partition tree as an efficient representation for image processing, segmentation, and information retrieval[END_REF][START_REF] Valero | Hyperspectral image representation and processing with binary partition trees[END_REF][START_REF] Randrianasoa | Binary partition tree construction from multiple features for image segmentation[END_REF], quasiflat zone hierarchies [START_REF] Meyer | Morphological scale-space representation with levelings[END_REF][START_REF] Soille | Constrained connectivity for hierarchical image decomposition and simplification[END_REF], and tree-based shape spaces [START_REF] Xu | Connected filtering on tree-based shapespaces[END_REF]. Other methods for hierarchical image analysis consider regular and irregular pyramids [START_REF] Haxhimusa | Irregular pyramid segmentations with stochastic graph decimation strategies[END_REF], scale-set theory [START_REF] Guigues | Scale-sets image analysis[END_REF], multiscale combinatorial grouping [START_REF] Pont-Tuset | Multiscale combinatorial grouping for image segmentation and object proposal generation[END_REF], series of optimization problems [START_REF] Syu | Hierarchical image segmentation based on iterative contraction and merging[END_REF], and generic image classification based on convolutional neural networks [START_REF] Maninis | Convolutional oriented boundaries: From image segmentation to high-level tasks[END_REF].

A hierarchical image segmentation is a series of image segmentations at different detail levels such that the segmentation at a given level can be produced by merging regions of the segmentation at the previous level. As a result, in such hierarchical representation, any region that belongs to a finer detail level is included (or nested) within a region that belongs to a coarser level. The level of a segmentation in a hierarchy is also called an observation scale. In [START_REF] Guimarães | Hierarchizing graph-based image segmentation algorithms relying on region dissimilarity -the case of the Felzenszwalb-Huttenlocher method[END_REF], Guimarães et al. proposed a hierarchical graph-based image segmentation (HGB) method inspired by the Felzenszwalb-Huttenlocher dissimilarity measure [START_REF] Felzenszwalb | Efficient graph-based image segmentation[END_REF]. The HGB method computes, for each edge of a graph, the minimum observation scale in a hierarchy at which two regions linked by this edge should merge according to the dissimilarity.

In this article, we give an explicit definition of the edge observation attribute and Boolean criterion which are at the basis of the HGB method and which make possible to select an observation scale for each edge of the graph. We show that this attribute and the related criterion are not increasing with respect to scales. We furthermore notice that the HGB method handles this non-increasing behavior with a simple rule, namely the min-decision rule, known in mathematical morphology since the work of Salembier et al. [START_REF] Salembier | Antiextensive connected operators for image and sequence processing[END_REF]. Recently, more elaborated strategies have been proposed to deal with attributes which are not increasing with respect to scales [START_REF] Xu | Connected filtering on tree-based shapespaces[END_REF][START_REF] Salembier | Ship detection in sar images based on maxtree representation and graph signal processing[END_REF]. These strategies are very successful in applications. Following these trends, we propose several new strategies to handle the non-increasing behavior of the observation attribute and criterion behind the HGB method. These strategies allow us to select an observation scale for each edge of the graph. Intuitively, they amount to filtering the set of all the scales for which the Boolean observation criterion holds true, before selecting one scale, with a simple strategy (such as the min-decision rule), among the scales remaining after the filtering. Such filtering is also done with rank and connected filters [START_REF] Salembier | Connected operators[END_REF]. Using the segmentation evaluation framework proposed in [START_REF] Perret | Evaluation of hierarchical watersheds[END_REF], we show that the newly observation scale selection strategies significantly outperform the original HGB method on Pascal VOC 2010 and 2012 datasets [START_REF] Everingham | The pascal visual object classes challenge[END_REF][START_REF] Everingham | The PASCAL Visual Object Classes Challenge[END_REF] (see an illustration in Fig. 1). Compared to a preliminary version of this work presented in a conference [START_REF] Cayllahua-Cahuina | A study of observation scales based on felzenswalb-huttenlocher dissimilarity measure for hierarchical segmentation[END_REF], the present article furthermore introduces the observation attribute behind the HGB methods, proposes several new strategies to select edge observation scales, and includes a deeper assessment considering notably a larger image dataset.

Section 2 briefly reminds the main idea of the HGB method. Section 3 introduces the edge observation attribute and criterion behind the HGB method and shows that they are not increasing with respect to scales. Section 4 presents an algorithm to compute all scales for which the observation criterion holds true making possible to consider new observation scale selection strategies as proposed in Section 5. Finally, Section 6 presents the assessment of the newly proposed method.

Hierarchical graph-based image segmentation

This section aims at explaining the method of hierarchical graph-based image segmentation (HGB) [START_REF] Guimarães | Hierarchizing graph-based image segmentation algorithms relying on region dissimilarity -the case of the Felzenszwalb-Huttenlocher method[END_REF]. We first give a series of necessary notions such as quasi-flat zone hierarchies [START_REF] Meyer | Morphological scale-space representation with levelings[END_REF], and then describe the HGB method.

Hierarchies

Given a finite set V, a partition of V is a set P of nonempty disjoint subsets of V whose union is V. Any element of P is called a region of P. Given two partitions P and P of V, P is said to be a refinement of P, denoted by P P, if any region of P is included in a region of P. A hierarchy on V is a sequence H = (P 0 , . . . , P ) of partitions of V, such that P i-1 P i , for any i ∈ {1, . . . , }.

Graph and connected-component partition

A graph is a pair G = (V, E) where V is a finite set and E is a subset of {{x, y} ⊆ V | x y}. Each element of V is called a vertex of G, and each element of E is called an edge of G. A subgraph of G is a graph (V , E ) such that V ⊆ V and E ⊆ E.
If X is a graph, its vertex and edge sets are denoted by V(X) and E(X), respectively.

If two vertices of a graph G are joined by an edge, we say that they are adjacent. From the reflexive-transitive closure of this adjacency relation on a finite set V(G), we derive the connectivity relation on V(X). It is an equivalence relation, whose equivalence classes are called connected components of G. We denote by C(G) the set of all connected components of G. Note that C(G) is a partition of V(G), called the connected-component partition induced by G.

Quasi-flat zone hierarchies

Given a graph G = (V, E), let w be a map from E in the set R of real numbers. For any edge u of G, the value w(u) is called the weight of u (for w), and the pair (G, w) is called an edgeweighted graph. Any edge-weighted graph can be transformed into a series of connected-component partitions, which constitutes a hierarchy, called the quasi-flat zone hierarchy of (G, w) (see a formal definition below). This transform is a bijection between the hierarchies and a subset of the edge weighted graphs called the saliency maps [START_REF] Cousty | Hierarchical segmentations with graphs: Quasi-flat zones, minimum spanning trees, and saliency maps[END_REF]. Hence, any edge-weighted graph induces a quasi-flat zone hierarchy and any hierarchy H can be represented by an edge-weighted graph whose quasi-flat zone hierarchy is precisely H [START_REF] Cousty | Hierarchical segmentations with graphs: Quasi-flat zones, minimum spanning trees, and saliency maps[END_REF]. This bijection allows us to handle any hierarchy with an edge-weighted graph.

Given an edge-weighted graph (G, w), let X be a subgraph of G and let λ be a value of R. The λ-level edge set of X for w is defined by w λ (X) = {u ∈ E(X) | w(u) < λ}, and the λ-level graph of X for w is defined as the subgraph w

V λ (X) of X such that w V λ (X) = (V(X), w λ (X)). Then, the connected-component partition C(w V λ (X)) induced by w V λ (X) is called the λ-level par- tition of X for w.
As we consider only finite graphs and hierarchies, the set of considered levels is reduced to a finite subset of R denoted by E in the sequel of this article. In order to browse values in this set and to round real values to values of E, we define, for any λ ∈ R,

p E (λ) = max{µ ∈ E∪{-∞} | µ < λ}, n E (λ) = min{µ ∈ E∪{∞} | µ > λ}, and nE (λ) = min{µ ∈ E ∪ {∞} | µ ≥ λ}.
In our experiments, the set E is a finite integer interval.

Let (G, w) be an edge-weighted graph and let X be a subgraph of G. The sequence of all λ-level partitions of X for w, ordered by increasing value of λ, is a hierarchy, defined by QF Z(X, w) = (C(w V λ (X)) | λ ∈ E ∪ {∞}), and called the quasi-flat zone hierarchy of X for w. Let H be the quasi-flat zone hierarchy of G for w. Given a vertex x of G and a value λ in E, the region that contains x in the λ-level partition of the graph G is denoted by H λ

x . Let us consider a minimum spanning tree T of (G, w). It has been shown in [START_REF] Cousty | Hierarchical segmentations with graphs: Quasi-flat zones, minimum spanning trees, and saliency maps[END_REF] that QF Z(T, w) of T for w is the same as QF Z(G, w) of G for w. This indicates that the quasi-flat zone hierarchy for G can be handled by its minimum spanning tree.

Hierarchical graph-based segmentation method

The input of the HGB method is the edge-weighted graph (G, w) representing an image, whose pixels correspond to the vertices of G and whose edges link adjacent pixels. The weight w(u) of an edge linking two pixels x and y represents the dissimilarity between the pixels x and y. As the HGB method is grounded in a general graph framework, it can be implemented with any adjacency relation and dissimilarity measure. For the experiments in Section 6, the 4-adacency relation and the L 2 norm in the RGB color space are used.

Before presenting the HGB method, we describe the observation scale dissimilarity [START_REF] Guimarães | Hierarchizing graph-based image segmentation algorithms relying on region dissimilarity -the case of the Felzenszwalb-Huttenlocher method[END_REF] used by the method and which originates from the region merging criterion proposed in [START_REF] Felzenszwalb | Efficient graph-based image segmentation[END_REF].

Observation scale dissimilarity

Let R 1 and R 2 be two adjacent regions. The dissimilarity measure between R 1 and R 2 consists of the so-called

Method 1: HGB method

Input : A minimum spanning tree T of an edge-weighted graph (G, w) Output: A hierarchy H = QF Z(T, f )

1 for each u ∈ E(T ) do f (u) := max{λ ∈ E} ; 2 for each u ∈ E(T ) in non-decreasing order for w do 3 H := QF Z(T, f ) ; 4 f (u) := p E λ H (u) ; 5 end 6 H := QF Z(T, f ) ;
inter-component and within-component differences [START_REF] Felzenszwalb | Efficient graph-based image segmentation[END_REF]. The inter-component difference between R 1 and R 2 is defined by

∆ inter (R 1 , R 2 ) = min{w ({x, y}) |x ∈ R 1 , y ∈ R 2 , {x, y} ∈ E(T )},
while the within-component difference of a region R is defined by ∆ within (R) = max{w ({x, y}) |x, y ∈ R, {x, y} ∈ E(T )}. They lead to the observation scale of R 1 relative to R 2 , defined by

S R 2 (R 1 ) = (∆ inter (R 1 , R 2 ) -∆ within (R 1 )) |R 1 |, where |R 1 | is the cardinality of R 1 .
Then, a symmetric metric between R 1 and R 2 , called the observation scale dissimilarity between R 1 and R 2 , is defined by

D(R 1 , R 2 ) = max{S R 2 (R 1 ), S R 1 (R 2 )}. ( 1
)
This dissimilarity is used to determine if two regions should be merged or not at a certain observation scale in the following.

HGB Method

The HGB method [START_REF] Guimarães | Hierarchizing graph-based image segmentation algorithms relying on region dissimilarity -the case of the Felzenszwalb-Huttenlocher method[END_REF] is presented in Method 1. The input is a graph G representing an image with its associated weight function w, where the minimum spanning tree T of G is taken. Given (T, w), the HGB method computes a new weight function f which leads to a new hierarchy H = QF Z(T, f ). The resulting hierarchy H is considered as the hierarchical image segmentations of the initial image. Thus, the core of the method is the generation of the weight function f for T .

After initializing all values of f to infinity (see Line 1), we compute an observation scale value f (u) for each edge u ∈ E(T ) in non-decreasing order with respect to the original weight w (see Line 2). Hence, at each iteration, the value f (u) is modified. This means that the weight map f is modified and we need to update H (Line 3) such that we still have H = QF Z(T, f ).

An efficient algorithm for the hierarchy update can be found in [START_REF] Cayllahua-Cahuina | Efficient algorithms for hierarchical graph-based segmentation relying on the Felzenszwalb-Huttenlocher dissimilarity[END_REF]. Once H is updated, the value λ H (u) of a finite subset

E of R, with u = {x, y}, is obtained by λ H ({x, y}) = min λ ∈ E | D H λ x , H λ y ≤ λ . (2)
We first consider the regions H λ x and H λ y at a level λ. Using the dissimilarity measure D, we check if D H λ

x , H λ y ≤ λ. Equation (2) states that the observation scale λ H ({x, y}) is the minimum value λ for which this assertion holds.

Non-increasing observation attribute and criterion

In this section, we provide a formal definition of the observation attribute and criterion involved in Equation ( 2). Then, we discuss their non-increasing behavior opening the doors towards new strategies to select observation scale values based on Felzenszwalb-Huttenlocher dissimilarity measure as used in Method 1.

In the remaining part of this section, we consider that H is any hierarchy and that u = {x, y} is any edge of T .

The edge observation attribute (of u = {x, y} in H) is the map A from E into R defined by:

A(λ) = λ -D(H λ x , H λ y ), (3)
for any λ in E. The edge observation criterion (of u in H) is the map C from E in the set {true, f alse} such that C(λ) = true if A(λ) is greater than or equal to 0; otherwise C(λ) = f alse. Let λ be any element in E, we say that λ is a positive observation scale (of u in H) whenever C(λ) holds true. On the contrary, if λ is not a positive observation scale, then we say that λ is a negative observation scale (of u in H).

Observe that the value λ H (x, y) defined in Equation ( 2) is simply the lowest element of E such that C(λ) is true. In other words, λ H (x, y) is the lowest positive observation scale of u in H. Additionally, we denote by λ H (x, y), the largest negative observation scale of u in H.

Intuitively, a positive observation scale corresponds to a level of the hierarchy H for which the two regions linked by u should be merged according to the observation criterion C. On the other hand, a negative observation scale corresponds to a level of the hierarchy for which the two associated regions should remain disjoint. Furthermore, for a positive observation scale λ, the observation attribute A(λ) can be intuitively interpreted as a measure of the strength at which the two associated regions should be merged whereas, for a negative observation scale, the observation attribute can be seen as the opposite of the strength at which the two regions should remain disjoint: when the observation attribute is highly positive, the two regions should be merged with a high confidence whereas when the observation attribute is negative and low, the two regions should remain disjoint with a high confidence.

A desirable property would be that the observation criterion C be increasing with respect to scales, a Boolean criterion C being increasing whenever, for any scale value λ ∈ E, C(λ) holds true implies that C(λ ) holds true for any scale λ greater than λ. Indeed, in such desirable case, any level in E greater than λ H (x, y) would be a positive observation scale, whereas any level not greater than λ H (x, y) would be a negative scale. In other words, we would have λ H (x, y) = n E λ H (x, y) . Hence, it would be easily argued that the observation scale of the edge u must be set to λ H (x, y). However, in general, the observation attribute A and thus the observation criterion C are not increasing (see a counterexample in Fig. 2) and we have λ H (x, y) < n E λ H (x, y) . Therefore, it is interesting to investigate strategies which can be adopted to select a significant observation scale between λ H (x, y) and λ H (x, y). A graphical illustration of different situations which may occur for a given edge u and hierarchy H is presented in Fig. 2. In other words, it is interesting to investigate strategies to transform the observation attribute A and criterion C into an increasing attribute A and an increasing criterion C .

In the framework of mathematical morphology, nonincreasing regional attributes/criteria are known to be useful but difficult to handle. Several rules or strategies to handle nonincreasing criteria have been considered in the context of connected filters. Among them, one may cite the min-and maxrules [START_REF] Salembier | Antiextensive connected operators for image and sequence processing[END_REF], the Vitterbi [START_REF] Salembier | Antiextensive connected operators for image and sequence processing[END_REF] and shape-space filtering strategies [START_REF] Xu | Connected filtering on tree-based shapespaces[END_REF], and the maxTree pruning strategies based on graph signal processing [START_REF] Salembier | Ship detection in sar images based on maxtree representation and graph signal processing[END_REF]. Note that the strategy adopted in Equation ( 2) corresponds to the min-rule and that the strategy consisting of selecting λ H (x, y) corresponds to the max-rule. Our main goal in this article is to investigate other strategies to efficiently

(G, w) (G, f ) H = QF Z(G, f )
(λ) = λ g(λ) = D(H λ x , H λ y ) observation scales dissimilarity values λ3 λ2 λ1 f (λ) = λ observation scales dissimilarity values g(λ) = D(H λ x , H λ y ) λ4 λ5 λ6 λ A(λ) λ A(λ)
Fig. 3: Illustration of the observation attribute and criterion and of possible observation scale selection strategies. The positive observation intervals (the intervals in which the observation criterion holds true) are represented in gray.

On the top, the dissimilarity measure D(H λ x , H λ y ) is depicted in green as a function of λ, whereas, on the bottom, the observation attribute A(λ) is depicted in green as a functions of λ. On the left, the min-, the lower-length and the lower p-rank selection strategies select the scales λ 1 , λ 2 and λ 3 , respectively (for a length threshold which is a little larger than the leftmost gray interval and for p = 0.3), whereas, on the right, the max-, the upper-length and the upper prank selection strategies select the scales λ 6 , λ 4 and λ 5 , respectively. Note that, for the sake of readability, the values of the functions are depicted for all scales taken in R, whereas the definitions are given only for a finite subset E of R.

handle the non-increasing observation criterion C in the context of hierarchical segmentation based on the Felzenszwalb-Huttenlocher region dissimilarity measure. These strategies are based on the analysis of all positive and negative observation scales and their associated attribute values. Therefore, before presenting these strategies, it is necessary to provide a method to obtain all positive and negative observation scales.

Observation intervals: definition and algorithm

As established in the previous section, the observation attribute and observation criterion (obtained by thresholding the observation attribute at value 0) at the basis of the HGB method are not increasing. Hence, in order to find the sets of all scales for which the criterion holds true (i.e., the scales for which the regions linked by an edge u should be merged), it is not enough to find the minimum of the positive observation scales. In [START_REF] Cayllahua-Cahuina | Efficient algorithms for hierarchical graph-based segmentation relying on the Felzenszwalb-Huttenlocher dissimilarity[END_REF], an algorithm to find the minimum observation scales such that the observation criterion holds true is given. Instead, in this section, an algorithm to find all scales at which the observation criterion holds true is provided. The proposed algorithm searches for the bounds of all (maximal) scale intervals for which the criterion holds true. Before presenting the algorithm that computes the bounds of all these intervals, we start with the definition of such intervals, called positive observation intervals.

Let λ 1 and λ 2 be any two real numbers in E ∪ {-∞} such that λ 1 < λ 2 . We denote by λ 1 , λ 2 E the subset of E that contains every element of E that is both greater than λ 1 and not greater than

λ 2 : λ 1 , λ 2 E = {λ ∈ E | λ 1 < λ ≤ λ 2 }.
We say that a subset I of E is an open-closed interval of E, or simply an interval, if there exist two real values λ 1 and λ 2 in E such that I is equal to λ 1 , λ 2 E . Definition 1 (observation interval). Let H be any hierarchy, let u be any edge in E(T ), and let I be an interval. We say that I is a positive observation interval (resp. a negative observation interval) for (H, u) if the two following statements hold true:

1. any element in I is a positive (resp. negative) observation scale for (H, u); and 2. I is maximal among all intervals for which statement [START_REF] Pavlidis | Structural Pattern Recognition[END_REF] holds true, i.e., any interval which is a proper superset of I contains a negative (resp. positive) observation scale for (H, u).

The set of all positive (resp. negative) observation intervals is denoted by Λ H (u) (resp. by Λ H (u)) .

In order to compute Λ H ({x, y}), we follow the strategy presented in [START_REF] Cayllahua-Cahuina | Efficient algorithms for hierarchical graph-based segmentation relying on the Felzenszwalb-Huttenlocher dissimilarity[END_REF], which relies on the component tree of the hierarchy H. The component tree of H is the pair T H = (N, parent) such that N is the set of all regions of H and such that a region R 1 in N is a parent of a region R 2 in N whenever R 1 is a minimal (for inclusion relation) proper superset of R 2 . Note that every region in N has exactly one parent, except the region V which has no parent and is called the root of the component tree of H. Any region which is not the parent of another one is called a leaf of the tree. It can be observed that any singleton of V is a leaf of T H and that conversely any leaf of T H is a singleton of V. The level of a region R in H is the highest index of a partition that contains R in H. Then, the proposed algorithm, whose precise description is given in Algorithm 1, browses in increasing order the levels of the regions containing x and y until finding a value λ such that D(H λ

x , H λ y ) ≤ λ, or, in other words, such that A(λ) > 0 and hence, C = true. This value is then λ H (x, y) defined by Equation ( 2). This value is also the lower bound of the first positive observation interval. If we keep browsing the levels of the regions containing x and y in this tree, as long as D(H λ

x , H λ y ) ≤ λ, we can identify the upper bound of this first positive observation interval. We can further continue to browse the levels of the regions containing x and y in the tree in order to identify all positive observation intervals. Therefore, at the end of the execution, we can return the set Λ H ({x, y}) of all positive observation intervals. From the set Λ H ({x, y}), we can obtain by duality the set Λ H ({x, y}) of all negative observation intervals.

The time complexity of Algorithm 1 depends linearly on the number of regions in the branches of the component tree of H containing x and y since it consists of browsing all these regions from the leaves to the root. In the worst case, at every level of the hierarchy the region containing x is merged with a singleton region. Hence, as there are |V| vertices in G, in this case, the branch of x contains |V| regions. Thus, the worstcase time complexity of Algorithm 1 is O(|V|). However, in many practical cases, the component tree of H tends to be balanced and each region of H results from the merging of two regions of (approximately) the same size. Then, if the tree is balanced, the branch of x contains O(log 2 (|V|)) nodes and the time-complexity of Algorithm 1 reduces to O(log 2 (|V|)).

Selecting observation scales

In this section, we investigate selection strategies to choose an observation scale from the sets of positive (resp. negative) observation intervals, i.e., the set of scales at which the observation criterion holds true (resp. false). In Section 5.1, we present the min-, max-, and rank-rules to select observation scales from either the set of positive or negative observation scales. Note that the min-and max-rules correspond to simple strategies known in mathematical morphology to prune the branch of a tree (aka a hierarchy) based on non-increasing regional attributes and criteria whereas the rank rules extend them to take a more robust decision. Then, in Section 5.2, we propose to improve the results obtained with these rules by pre-filtering out positive or negative observation intervals which might be considered as irrelevant with respect to a given attribute, before applying the min-, max-or rank-rules. Whereas, the selection rules can be seen as a way to transform a non-increasing criterion into an increasing one, the filtering strategies can be seen as regularization strategy for the non-increasing attribute on which the (observation) criterion is based. Overall, the proposed strategy allows us to transform a non-increasing attribute-based criterion into an increasing one based on a regularized version of the observation attribute.

Min-, max-and rank-selection rules

In this section, the symbol K stands for a finite set of observation scales (either negative or positive) and we define several useful rules to pick one observation scale in such set. Let K be any subset of E. We propose the two following selection rules from K to set the value of f (u) in Method 1: min-rule: f (u) := min{k ∈ K}; and max-rule: f (u) := max{k ∈ K}.

Let us now provide a precise definition of the rank rules. The intuitive idea of these rules is to remove from the set K the scales higher or lower than a given quantile, which are then considered as non-significant, before applying the min-and maxrules.

Let n be the number of elements in the set K of possible observation scales, i.e., n = |K|. Let k be any positive integer less than n. We denote by rank k/n (K) the element e of K such that there are exactly k distinct elements in K which are less than e. Let p be any real value between 0 and 1, we set rank p (K) = rank p.n /n (K), where p.n is the largest integer which is not greater than the real value p.n.

Let p be any real value between 0 and 1. We consider the two following additional selection rules from K to set the value of f (u) in Method 1: lower p-rank-rule: f (u) := rank p (K); and upper p-rank-rule: f (u) := rank 1-p (K).

Using the above rules, we now define several strategies which are tested in Section 6 to set the value f (u) in Method 1.

When K is equal to the set of all positive observation scales (resp. the set of all positive observation scales not greater than λ H (x, y)) for given edge u and hierarchy H, the result obtained with the min-rule (resp. lower p-rank rule) is called the min strategy (resp. lower p-rank strategy). With the min-strategy, the value f (u) in Method 1 is set to the minimum scale λ for which the observation criterion holds true. Therefore, the results obtained with this strategy correspond exactly to the results obtained with the method presented in [START_REF] Guimarães | Hierarchizing graph-based image segmentation algorithms relying on region dissimilarity -the case of the Felzenszwalb-Huttenlocher method[END_REF][START_REF] Cayllahua-Cahuina | Efficient algorithms for hierarchical graph-based segmentation relying on the Felzenszwalb-Huttenlocher dissimilarity[END_REF], as described by Equation (2). The lower p-rank strategy is a new proposal to take more robust decisions, this assertion being practically evaluated in Section 6. Intuitively, it can be seen that the lower p-rank strategy considers the lower p percentile of the observation scales between λ H (x, y) and λ H (x, y). Hence, it can be easily seen that, when p is equal to 0, the lower p-rank strategy is exactly the min-strategy.

In this article, we also consider the max-strategy (resp. upper p-rank strategy), that is the result obtained with the max-rule (resp. upper p-rank rule) when K is the set of the negative observation scales (resp. the set of the negative observation scales not less than λ H (x, y)). It can be observed that when the observation criterion is increasing, the min-, max-, lower-p-rank, and upper-p rank strategies all take the same decision. However, as illustrated in Fig. 3, when the observation criterion is not increasing, these rules provide different alternatives to set up the observation scale f (u) in Method 1.

Filtering the observation scales with connected operators

In the previous section, we proposed several strategies to select an observation scale from the positive and negative obser-vation scales. In particular, in order to provide a more robust alternative to the classical min-and max-rules, the p-rank rule is presented. In this section, following the same goal of robustifying the observation scale selection process, we propose to filter the set of positive and negative observation scales with a family of filters, called connected operators, coming from the field of mathematical morphology. The basic idea of the connected operators is to act at the level of the connected components of a set. Basically, each connected component of the input set is either completely preserved or completely discarded but cannot be partially included in the result of the operator. The choice of the connected components which are kept is based on an attribute designed to measure the importance of the connected components. Such filters are very powerful and able to solve many practical issues in image analysis and signal processing as reviewed in [START_REF] Salembier | Connected operators[END_REF]. In this section, we propose to filter the set of positive and negative observation scales with connected operators before selecting an observation scale with the strategies presented in the previous section. We start this section by defining the notion of a component of a set of scales, and then give a formal definition of a connected operator in this framework. We also describe several measures to assess the importance of a component of scales at the end.

Let K be any subset of E. Any interval I = λ 1 , λ 2 that is included in K and maximal for this property is called a component of K. The set of all components of K is denoted by CC(K). For instance, if K is the set of all positive (resp. negative) observation scales for a given pair (H, u), then CC(K) is precisely the set Λ H (u) of all positive observation intervals (resp. the set Λ H (u) of negative observation intervals). Any map from the set of all intervals of E into the set of real numbers is called an (interval) attribute. Let µ be any interval attribute, let t be any real number, and let K be a subset of E. We set

γ µ t (K) = ∪{I ∈ CC(K) | µ(I) > t}.
When the attribute µ is increasing with respect to its parameter (which can be any interval), the operator γ µ t is called the µ-connected opening of parameter t and is indeed an algebraic opening (i.e,, an increasing, anti-extensive and idempotent operator). Hence, using an interval measure or attribute µ, we are able to analyze and decide if a certain observation interval is relevant or not. If an observation interval is not relevant, then, it is simply discarded. Afterward, an observation scale is selected among the remaining scales using one of the strategies presented in the previous section. To decide if an observation interval I is relevant, we consider the following attributes:

We remind from Equation (3) that A(λ) = λ -D(H λ x , H λ y ). An illustration of these attributes is provided in Fig. 4. The attributes of every positive and negative observation interval can 0000 0000 0000 1111 1111 1111 000 000 000 111 111 111

interval I 2 interval I 3 interval I 1 µ 1 (I 1 ) µ 2 (I 2 ) µ 3 (I 3 ) λ A(λ)
Fig. 4: Illustration of the attributes used to filter the observation intervals. The observation attribute A is depicted in green as a function of the scales. The length µ 1 (I 1 ) of the interval I 1 is represented in red, the depth µ 2 (I 2 ) of I 2 is represented in pink and the area µ 3 (I 3 ) of I 3 is shown in blue. Note that, for the sake of readability, the values of the observation attribute A are depicted for all scales taken in R, whereas the definition is given only for a finite subset E of R.

be computed on the fly during the execution of Algorithm 1. In our experiments, the positive observation scales are filtered according to the length (Equation ( 4)), depth (Equation ( 5)) and area (Equation ( 7)) measures, whereas the negative observation scales are filtered according to the length (Equation ( 4)), Ndepth (Equation ( 6)) and Narea (Equation ( 8)) measures. Besides the min-, lower-p-rank, max-and upper-p-rank strategies, which are presented in Section 5.1, we also assess in the next section the results of the min-rule on the result of the µconnected openings applied to the positive observation scales when the measure µ is either the length, the depth or the area (i.e., either µ 1 , µ 2 , or µ 4 ). These three strategies are called the lower-length, lower-depth and lower-area selection strategies, respectively. Similarly, we also consider the results of the maxrule on the result the µ-connected openings applied to the negative observation scales, when the measure µ is either the length, the Ndepth or the Narea (i.e., either µ 1 , µ 3 , or µ 5 ). These three last strategies are referred to as the upper-length, upper-Ndepth and upper-Narea selection strategies, respectively. Overall, in the next section, we assess 10 different strategies to set up the observation scale f (u) at Line 4 of Method 1.

Experiments

This section aims to compare the results of the original HGB method against the results of the new strategies described in this article. Assessing a hierarchical segmentation, per-se, is a difficult task, a hierarchical segmentation being generally not an end-goal but an intermediary representation from which the information that is of practical interest for a given task is extracted. In order to tackle this problem, we use the framework described in [START_REF] Perret | Evaluation of hierarchical watersheds[END_REF]. It considers images and ground-truth from Pascal VOC 2010 dataset [START_REF] Everingham | The pascal visual object classes challenge[END_REF] (2498 images) and Pascal VOC 2012 dataset [START_REF] Everingham | The PASCAL Visual Object Classes Challenge[END_REF] (3427 objects segmented from the 1449 images). One of the key propositions of [START_REF] Perret | Evaluation of hierarchical watersheds[END_REF] is to assess a hierarchy from different view-points corresponding to different use cases of hierarchies. In particular, two aspects are assessed:

• the quality of the "best" cuts or partitions in K regions appearing at the same scale (FHC measure) or at different scales (FOC measure) of the hierarchy. A cut is assessed with respect to a human-provided ground truth and to a segmentation quality measure (Bidirectional Consistency Error). The measure is averaged for every K between 2 and twice the number of regions in the ground truth.

• the ease of finding a set of regions in the hierarchy corresponding to a semantic object when different levels of information are given on the position of the object: an object is considered easy to find when it can be well retrieved with few human-like-provided markers. Here, the retrieved object is a union of regions which, in general, does correspond to any cut of the hierarchy.

The assessment of the first aspect on Pascal VOC 2010 dataset is summarized with a score called FOC+FHC, whereas the evaluation of the second aspect on Pascal VOC 2012 dataset is summarized with the ODM score. More details on the computation of these scores are given in [START_REF] Perret | Evaluation of hierarchical watersheds[END_REF]. Furthermore, as recommended in [START_REF] Perret | Evaluation of hierarchical watersheds[END_REF], to get assessable results, we perform an areasimplification post-processing [START_REF] Perret | Removing non-signicant regions in hierarchical clustering and segmentation[END_REF] of the hierarchies that removes regions smaller than 0.04% of the image size. On a typical hierarchy produced during our experiment, this postprocessing allows the number of non-singleton regions to be reduced from 180000 to about 800. For the strategies which depend on a parameter, a search was performed to find the optimal parameter. The tested parameters were: • all values from 0.01 to 0.4 with an increment step of 0.01 for the lower p-rank and upper p-rank strategies.

The scores obtained with the 10 strategies described in Section 5 are presented in Tables 1 and2. The associated source code and the saliency maps of our results on numerous images are given at https://cayllahe.github.io/hgbcode. A visual inspection of the resulting saliency maps shows that there is significant variation between the considered strategies. From the numerical results, we can observe that filtering the observation intervals leads to a significant improvement against the raw min-and max-strategies, hence over the original HGB method. More specifically, we observe that:

1. for the best cuts assessment:

• among the strategies based on min-rule, lower p-rank obtains the highest score (0.9138);

• among the strategies based on max-rule, upper prank obtains the highest score (0.9357);

• max-rule based strategies systematically perform better than their min-rule counterpart;

2. for the supervised object retrieval assessment:

• among the strategies based on min-rule, lower-area and lower-depth obtain the highest score (0.877);

• among the strategies based on max-rule, upper-Ndepth obtains the highest score (0.870);

• min-rule based strategies systematically perform better than their max-rule counterpart. In order to go one step further, we also tested some strategies with different parameters for the area-simplification postprocessing leading to the following results:

1. for the best cuts assessment, upper-length is the best strategy (FOC+FHC score: 0.9369); and 2. for the supervised object retrieval assessment, lowerlength is the best strategy (ODM score: 0.911).

Overall, we see that there is a clear benefit of filtering the observations scales with the proposed strategies. However, the strategy which leads to the best result depends on the targeted assessment and end-use of the hierarchies.

Conclusions

This article revisits the HGB method to propose new observation scale selection strategies leading to better segmentation results. We introduce the notion of edge observation attribute and criterion, and establish their non-increasing behavior. Then, we propose an algorithm to compute all scales for which the criterion is true. By analyzing these scales, we propose several strategies to select different observation scales among these values. We assess the performance of our strategies on Pascal VOC 2010 and 2012 datasets showing that these strategies outperform the original HGB method. As future work, we will study the impact of the chosen region dissimilarity measure. The inter-and the within-component differences can be computed based on the properties of each region using, for instance, a statistical approach [START_REF] Guimarães | Hierarchical image segmentation relying on a likelihood ratio test[END_REF]. Another promising direction consists of replacing the region size term of the Felzenswalb-Huttenlocher dissimilarity measure by a different measure of the importance of a region, such as compactness or visual saliency [START_REF] Li | Visual saliency based on multiscale deep features[END_REF].

Fig. 1 :

 1 Fig. 1: (Left) original image extracted from the Pascal VOC 2010 dataset, and the saliency maps (an equivalent representation of hierarchical segmentation) resulting from the HGB method using the original observation scale (middle) and from one of our proposed observation scale selection strategies (right). A video version of this illustration showing the successive levels of the hierarchy is available at www.tobeinsertedByPRLwhenAvailable.com.

Fig. 2 :

 2 Fig. 2: Counterexample for the increasing property of the observation attribute A and criterion C: for the edge u = {x, y} with x = f , y = g, we have D(H 1 x , H 1 y ) = 17, D(H 23 x , H 23 y ) = 17, and D(H 25 x , H 25 y ) = 28. Hence, we have A(1) = 1 -17 = -16, A(23) = 23 -17 = 6, and A(25) = 25 -28 = -3, so that C(1) = f alse, C(23) = true, and C(25) = f alse, which shows that A and C are not increasing.

  f

Algorithm 1 : 5 λ 6 λ 9 end 10 λ 17 λ 18 L

 1569101718 Positive observation intervalsInput : The component tree T = (N, parent) of a hierarchy H, an edge u = {x, y} of T , an array level that stores the level of every region of H Output: A set L containing all elements of Λ H ({x, y})1 C x := {x}; C y := {y}; Λ H ({x, y}) = {}; 2 λ := min(level[C x ], level[C y ]); λ prev := -∞; 3 do 4 while D C x , C y > λ do prev := λ ; := min(level[parent[C x ]], level[parent[C y ]]); 7 if level[parent[C x ]] = λ then C x := parent[C x ]; 8 if level[parent[C y ]] = λ then C y := parent[C y ]; lower := max(n E λ prev , nE D(H λx , H λ y ) ); 11 while D C x , C y ≤ λ and (parent[C x ] root and parent[C y ] root do 12 λ prev := λ; 13 λ := min(level[parent[C x ]], level[parent[C y ]]); 14 if level[parent[C x ]] = λ then C x := parent[C x ]; 15 if level[parent[C y ]] = λ then C y := parent[C y ]; 16 end upper := max(n E λ prev , nE λ prev ) ; .add( λ lower , λ upper ); 19 while parent[C x ] root and parent[C y ] root;

length: µ 1 (

 1 I) = |I| ; (4) depth: µ 2 (I) = max {A(λ) | λ ∈ I} ; (5) Ndepth: µ 3 (I) = max {-A(λ) | λ ∈ I} ; (6) area: µ 4 (I) = {A(λ) | λ ∈ I} ; and (7) Narea: µ 5 (I) = {-A(λ) | λ ∈ I} .

• 3 ,

 3 [START_REF] Meyer | Morphological scale-space representation with levelings[END_REF][START_REF] Pont-Tuset | Multiscale combinatorial grouping for image segmentation and object proposal generation[END_REF][START_REF] Salembier | Antiextensive connected operators for image and sequence processing[END_REF][START_REF] Cayllahua-Cahuina | A study of observation scales based on felzenswalb-huttenlocher dissimilarity measure for hierarchical segmentation[END_REF][START_REF] Li | Visual saliency based on multiscale deep features[END_REF] 35, 40, 45, 50, 60, 70, 800, 900, 1000, and 1050, for the lower-area and upper-narea strategies; • 3, 5, 7, 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, and 70, for the lower-depth and upper-Ndepth strategies; • 3, 5, 7, 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 75, and 80, for the lower-length and upper-length strategies; and

Table 1 :

 1 Best cuts assessment results (Pascal VOC 2010 dataset).

	Strategy Param.	FOC	FHC FOC+FHC
	Min Lower-length Lower-area Lower-depth Lower p-rank	-0.4917 0.4010 3 0.4904 0.3993 5 0.4901 0.3988 3 0.4897 0.3986 0.17 0.5020 0.4117	0.8927 0.8897 0.8889 0.8883 0.9138
	Max Upper-length Upper-Narea Upper-Ndepth Upper p-rank	-0.5184 0.4092 35 0.5189 0.4099 15 0.5194 0.4096 20 0.5189 0.4100 0.36 0.5193 0.4164	0.9277 0.9288 0.9290 0.9289 0.9357

Table 2 :

 2 Supervised object retrieval assessment (Pascal VOC 2012 dataset).

	Strategy (min-rule) ODM Strategy ( max-rule) ODM
	Min Lower-length Lower-area Lower-depth Lower p-rank	0.868 0.871 0.877 0.877 0.870	Max-rule Upper-length Upper-Narea Upper-Ndepth Upper p-rank	0.865 0.866 0.867 0.870 0.866