Dr François-Xavier Coudert
email: fx.coudert@chimie-paristech.fr

Alain Boutin

Alex Fuchs

Peter Neimark

Ravikovitch

Molecular Mechanism of Swing Effect in Zeolitic Imidazolate Framework ZIF-8: Continuous Deformation Upon Adsorption

Keywords: metal-organic framework, adsorption, flexibility, molecular dynamics, first-principles simulations

Zeolitic Imidazolate Framework ZIF-8 displays flexibility of its framework by rotation of its imidazolate linker. This "swing effect" has been widely described in the literature, both experimental and theoretical, as a bistable system where the linker oscillates between two structures: "open window" and "closed window". By using quantum chemistry calculations and first principles molecular dynamics simulations, it is shown that the deformation upon adsorption is in fact continuous upon pore loading, with thermodynamics of packing effects the reason behind stepped adsorption isotherms experimentally observed. Finally, we study a variant of ZIF-8 with a different linker, highlighting the influence of the linker and the balance of microscopic interactions on the framework's flexibility.

Introduction

Zeolitic imidazolate frameworks (ZIF) are a subclass of metalorganic frameworks (MOF) 1,2,3,4 formed from imidazolate linkers and metal cations, linked together into three-dimensional zeolitic frameworks. 5,6,7,8 Like MOFs, ZIFs feature tuneable nanoporosity (e.g., by changing the size of their linker), structural flexibility (dictated by relatively weak coordination bonds and intermolecular forces), and versatility of their inner surface (through functionalization of the organic ligand). They also feature strengths typical of the zeolites family, in terms of thermal, mechanical and chemical stability. Finally, due to the large diversity of four-connected nets, over 100 different ZIF structures have been reported so far, with many more potential structures existing within a small range of enthalpy of formation. 9,10,11 ZIFs present a great potential for applications such as gas capture (CO2 in particular), 12 sensing, 13 encapsulation and controlled delivery, 14 and fluid separation. 15,16,17,18 ZIF-8, in particular, is commercially available, has been widely studied, and has excellent performance at lab scale for the separation of many strategic mixtures, including C2/C3 hydrocarbons, CO2/CH4, CO2/N2, biobutanol fermentation mixture, phenolic isomers, etc. It is a low density porous framework with the sodalite (sod) topology and chemical formula Zn(mim)2, where mim = 2-methylimidazolate (see Figure 1). It features large spherical cages, called sodalite cages, separated by 6-ring windows of small aperture (~3.4 Å diameter as determined from the crystallographic structure), and 4-rings that connect different cages. It has high thermal, mechanical and chemical stability, 5,19 and can be readily assembled into membranes 20 or form thin films. 21 It undergoes amorphization under mechanical pressure 22 or ball milling. 23 Moreover, studies have highlighted that ZIF-8 exhibits some flexibility of its framework in the form of a torsional motion of its imidazolate linkers, a phenomenon called the "swing effect". Both experimental and theoretical works on ZIF-8 have noted a discrepancy between experimental results of gas adsorption and diffusivity, and the values that could be predicted from molecular simulations in rigid structures or simple considerations comparing the ZIF-8 window size and the adsorbates' kinetic diameter. 24 This ability of molecules larger than the geometric window size to diffuse through the ZIF-8 framework was attributed to flexibility of the structure. 25 The best-documented evidence of this effect was the computational characterization of self and transport diffusion coefficients of methane, ethane, ethylene, propane, propylene, nbutane, and 1-butene, by the Sholl group. 26 By comparing simulation results to experimental transport properties, the authors showed that framework flexibility plays a critical role in accurately describing molecular diffusion.

Moreover, there is ample direct evidence of a structural transition occurring in ZIF-8 upon loading, first reported by Moggach et al. in an high-pressure experiment using a methanol/ethanol mixture as hydrostatic medium. 27 They demonstrated that at a pressure of 1.47 GPa, the intrusion of methanol molecules into the ZIF-8 nanopores triggers a reversible single-crystal to single-crystal structural transition. The "high pressure" (or HP) phase has the same space group (I43m) as ZIF-8, a slightly larger unit cell volume (4974.8 Å 3 vs. 4900.5 Å 3 ), and exhibits a reorientation of the imidazolate linkers, which open the 6-ring windows. In later work, Fairen-Jimenez et al.

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demonstrated by in situ powder XRD measurements that the HP phase of ZIF-8 can also be observed upon adsorption of N2 at 77 K. These authors also demonstrated that the stepped N2 adsorption isotherm (at 77 K) could not be explained by Grand Canonical Monte Carlo (GCMC) simulations in a rigid "ambient pressure" (AP) ZIF-8 structure. They showed, on the contrary, that its lower pressure part matches the GCMC simulations performed in the AP structure, while the saturation uptake corresponded with GCMC simulations in the (rigid) HP structure. Some of the same authors later studied adsorption of CO2, CH4 and longer alkanes in ZIF-8, at temperatures between 120 K and 300 K. 29 While no clear step can be seen on these experimental isotherms, they used GCMC simulations and Density Functional Theory (DFT) calculations to prove that adsorption could not be fully described by the AP structure only, but that two different structural configurations (AP and HP) of ZIF-8 are needed to properly describe the adsorption performance of this material. In particular, they demonstrated that the HP structure presents an additional adsorption site within its 4-ring window.

Finally, another study provided high-resolution adsorption and desorption isotherms of CO, N2, O2 and Ar at 77 and 90 K, which all exhibit multiple steps as well as a hysteresis loop in the higherpressure step. 30 As in the earlier work on N2, Ania et al. compared these experimental isotherms to GCMC calculations performed on the rigid AP and HP structures, and attributed the stepped adsorption behavior to the packing arrangement of the guest molecules, as governed by polarizability, molecular size and shape of the gases.

However, while the thermodynamics and energetics of the ZIF-8 swing effect have been studied from the energetics and thermodynamics of adsorption in the two crystallographic structures AP and HP, little has been said of the nature of this structural transformation, what happens in between, and what is the dynamics of imidazolate linker swing motion in the material in the presence of adsorbate. To our knowledge, no experimental data or molecular simulation on the single-crystal to single-crystal AP→HP transformation has determined whether it is a continuous deformation of the ZIF-8 structure, a phase transition between metastable phases of the material (like the MIL-53 breathing transition, for example), or a "gate opening" phenomenon.

Finally, there is little information in the literature about the swing effect of the imidazolate linkers in other ZIFs of the sod topology, and the possible influence of linker functionalization on the framework dynamics. In the case of ZIF-7, a Zn(benzenimidazolate)2 framework with great separation potential, the existence of a structural transformation was established by Aguado et al. 31 van den Bergh et al. 32 were able to give some microscopic insight into the energetics of adsorption of hydrocarbons in ZIF-7. Zhao et al. 33 determined the crystallographic structure of ZIF-7-II, obtained from ZIF-7 upon solvent evacuation, and showed that a dense phase with the same topology, ZIF-7-III, can also be triggered in the presence of water. Later, Du et al. showed the existence of a high-temperature transition, and elucidated the interplay between temperature effects and CO2 guest adsorption by establishing the phase diagram of ZIF-7 in (T, PCO2) parameter space. 34 Finally, the Hupp group 35 was able to obtain an ZIF of sod topology with unsubstituted imidazolate (im) as linked through solvent-assisted linker exchange: the resulting material, SALEM-2, incorporates as much as 85% of im vs. mim. This change in the linker, while it has no bearing on the framework, appears to have an impact on the N2 adsorption properties. However, no detailed study of the imidazolate swing effect at the microscopic level has been performed.

In this paper, I look at the nature of the ZIF reorientation transition, and in particular whether it is a transition between two metastable phases of ZIF-8, a continuous adsorption-induced deformation, or a type of adsorbate-induced gate opening. 36 In order to bring microscopic insight into the transition, it is studied at the atomistic scale in terms of both energetics as well as thermal motion, with and without adsorbates. Because this effect is expected to be particularly difficult to capture with classical force fields and the strong assumptions they make on interatomic interactions, the issue is addressed through quantum chemistry simulations, using both zero Kelvin energy minimizations and finite temperature molecular dynamics.

Results and Discussion

Quantum chemistry calculations

First, I performed quantum chemistry calculations starting from the experimental AP and HP structures, optimizing each structure's geometry -both atomic positions and unit cell parameters -in the absence of guests. The ZIF-8 AP structure is fairly well reproduced (see Table 1), but the HP structure spontaneously relaxes into AP, with extremely good precision. Thus, HP is not a metastable structure for the bare ZIF-8, i.e. not a local minimum in the energy profile of the material. This is in contrast, for example, with adsorption-induced transitions such as the "breathing" transition in the MIL-53 family. 37 Then, I tried to see if the HP structure could be observed by applying pure (isotropic) mechanical stress to ZIF-8. To mimic the effect of adsorption-induced stress near saturation uptake (i.e. at large loading), where the HP structure is experimentally observed, I studied the effect of negative (outward) pressure on ZIF-8. Starting from the relaxed AP structure, an isotropic stress is applied and the deformation is measured (see Figure S1). A linear elastic domain is found up to P = -1.5 GPa, and the unit cell volume of the HP phase would correspond to an isotropic pressure of P = -0.3 GPa. However, imposing an isotropic stress on ZIF-8 does not lead to imidazolate reorientation, but on the contrary, the unit cell expands homogeneously, without any "swing" of the linker. The AP-HP transition is thus not due merely to an isotropic stress.

First principles MD of guest-free host

In order to look at nature of thermal motion of the imidazolate linker, I ran first principles MD simulations of the empty AP ZIF-8 structure in the (N, V, T) ensemble, at temperatures of 77, 150 and 300 K. The spontaneous "swinging" motion of the imidazolate linkers is observed in all three trajectories. It is then characterized by the dihedral angle φ Zn3-Zn2-Zn1-CH3 of the imidazolate around the Zn1-Zn2 axis (see Figure S4), where the "reference" of 0° is the 6-ring of Zn (i.e. the window connecting the cages). In the crystallographic AP structure, the window is not perfectly planar (φ ~ 7°), while in the crystallographic HP structure, φ ~ 35°. Figure 2 shows histograms of values taken by φ during our dynamics, where the thermally-activated swinging motions are seen to increase in amplitude with temperature: ~15° at 77 K, ~20° at 150 K and ~25° at 300 K. However, the distribution is in all cases centered around φ = 0°, and never reaches the value of 35° observed in the HP structure. On the lower panel of Figure 2, we see the potential of mean force (PMF) as a function of dihedral angle, F(φ) = -kT log P(φ), obtained from the histograms above. The profiles obtained at different temperatures coincide and the free energy is quadratic, which showing that the "thermal swinging motion" seen in the empty host is a soft vibration mode of wide amplitude. The frequency of vibration, as calculated from the MD simulations, is very low at νswing = 60(±10) cm -1 . The nature of this motion is in good agreement the experimentally observed peak at 1.95 THz by terahertz time-domain spectroscopy, 38 and with the lowfrequency terahertz vibrations observed experimentally in guestfree ZIF-8 by inelastic neutron scattering at room temperature, and studied by static DFT calculations in the limit of the harmonic regime. 39 The same modes were also shown to be linked to the know shear-instability of ZIF-8, and its pressure-induced amorphization through shear softening.

40

Moreover, the study of the correlations between the motions of the various methylimidazolate linkers shows that they are totally uncorrelated -as is reflection on the matrix of correlation plots at 77 K shown in Figure S2. We also see from Figure 2 that the energy needed to bring one imidazolate linker from 0° to 35° is ~15 kJ/mol. That value is of the same order of magnitude as adsorption enthalpies, explains why guest adsorption can trigger such a large swing motion of the ZIF-8 imidazolate linker, compared to the amplitude of thermal motions.

Grand canonical Monte Carlo

I then performed Grand canonical Monte Carlo simulations of N2 adsorption in ZIF-8, using the two crystallographic structures AP and HP (Figure S3). We confirm Düren et al's findings 28 that while GCMC isotherms in the AP structure describe reasonably the first part of the isotherm, they fail to reproduce the saturation uptake, which is on the other hand reasonably described by GCMC simulations using the HP structure. However, it is clear from the higher-resolutions isotherms published by Calero et al. 30 that the in the intermediate pressure range, neither the isotherm of the rigid AP or rigid HP structures reproduce the experimental results. In addition, GCMC simulations were also performed at high loading in using two hypothetical structures: one with the AP atomic positions but HP unit cell parameters, and one with HP atomic positions but AP unit cell parameters (red and pink points near P = P0 on Figure S3). The purpose of these is to check whether the adsorption properties of the two phases are dominated by their linker positions or the total unit cell volume. We can see that the effect of unit cell volume change is marginal, confirming that the main effect is that of the swing of the imidazolate linkers.

First principles MD with guest molecules

The behavior of ZIF-8 at intermediate N2 loading was then investigated, by means of first principles molecular dynamics. Starting configurations were taken from GCMC simulations, with 11, 27, 41 and 49 N2 molecules adsorbed per unit cell, at a temperature of 77 K. From the analysis of these trajectories, Figure 3 presents the histograms of the imidazolate swing angle φ, for all four loadings and compared to the empty host. We see that at low loading (11 N2), the adsorbed molecules have limited effect on the imidazolate linker, increasing slightly the amplitude of swing motions (with a half-width going from 8° to 12°) but the distribution remains centered near 0°. At higher loading (27, 41, and 49 N2 molecules), we see an important change in the nature of the distribution of swing angles: its maximum shifts from ~0° to 14°, 20°, and 25° (for 27, 41, and 49 guest molecules respectively). This is the manifestation of the opening of the windows, which occurs as a gradual transformation of the ZIF-8 structure upon adsorption, from the guest-free metastable AP structure to a guest-loaded HP structure with open windows. It contrasts with the previous picture of an abrupt, first-order adsorption-induced transition 28,41 between preexisting phases of the bare ZIF-8 framework. This is consistent with our quantum chemical calculations, which show that the HP structure is not metastable in the absence of guest molecules. It is also in good agreement with recent indirect experimental evidence obtained by 2 H NMR 42 and later by in situ inelastic neutron scattering, 43 showing that the linker swing motion is retained at full loading -although the exact nature of the transition and behaviour at intermediate loading was not detailed.

It should be noted that, at all loadings simulated, a few events of diffusion of N2 through the 6-ring windows of ZIF-8 are observed. From the "rigid" AP structure, N2 diffusion is not expected from geometric considerations, as the window diameter of 3.4 Å is smaller than the N2 kinetic diameter of 3.6 Å. Thus this shows that even a "small" swing amplitude (of the order of 15°) is enough to allow rather "fast" diffusion of N2 in the pores of ZIF-8, which is observed twice in a 18 ps dynamics at 77 K. Furthermore, I analyzed the spatial distribution of N2 molecules adsorbed at loadings of N = 41 and N = 49 molecules per unit cell. In order to get the necessary statistics, GCMC results were used for that analysis. They confirm that the opening of the ZIF-8 window allows for a denser packing of N2, with the possibility of having 2 molecules per window (at N ~ 49 in the HP structure) against only 1 molecule per window (at N ~ 41 in the AP structure). This packing effect can explain the observed steps on the experimental adsorption-desorption isotherms, 27,28,29,30 similarly to what has been demonstrated for other flexible porous materials in both theoretical 44,45 and experimental 46 works.

Linker functionalization

Finally, I studied the influence of the nature of the ZIF-8 linker by comparing the swing motions of ZIF-8 (whose linker is 2methylimidazole) with a ZIF with the same sodalite (sod) topology but built on an unsubstituted imidazolate linker -a structure similar to that of material SALEM-2, obtained from ZIF-8 by postsynthetic exchange of the linkers. 35 Quantum chemical calculations show that in SALEM-2 the most stable structure still resembles the AP structure, with a swing angle close to zero. However, molecular dynamics at 77 K reveals (Figure 4) that the swing motions are of a much larger amplitude for SALEM-2 than for ZIF-8. In fact, at 77 K, the "thermal" swing motion goes all the way up to 35° (instead of 15° for ZIF-8). The free energy required for a rotation of the linker from 0° to 35° is a mere 3.5 kJ/mol for SALEM-2, compared to 15 kJ/mol for ZIF-8. This "extra" stabilization of the "AP" structure of ZIF-8 (compared to its unsubstituted variant) can be ascribed to the dispersive interactions between the methyl groups and the neighbouring windows, which favour the in-plane configuration. As a consequence, I predict that the "swing effects" in SALEM-2 are much less severe, if they are visible at all, than in ZIF-8. This also shows that the "swing effect" of ZIF-8, which were studied here, is not a universal phenomenon for ZIF structures, but linked to the specific nature of the ZIF-8 framework.

Conclusions

In conclusion, combined quantum chemistry calculations and first principles molecular dynamics simulations demonstrate that the "swing effect" or "window opening" in ZIF-8 is in fact a gradual adsorption-induced deformation of the ZIF-8 framework. The "open window" (or "high pressure") structure is not metastable in the absence of adsorbate, and the opening transition is continuous and not of the first order (as was suggested in prior literature). This gradual deformation leads to stepped isotherms in some cases because of packing effects of the adsorbed phase. Experimentally, stepped isotherms have been observed for small molecules at low temperature, which is exactly when such packing effects can be expected to dominate (dense and wellorganized adsorbed phases). Thus, GCMC simulations at fixed volumes, even when considering two (AP and HP) or more phases, are not well-suited to study guest adsorption in ZIF-8. Studies in the osmotic ensemble are required for a complete thermodynamic description of adsorption. Furthermore, it is shown that extent of the swinging motions of the imidazolate linker depend largely on the microscopic nature of the linker, and predict that the "swing effect" would be much less important on a ZIF-8 with unsubstituted imidazole linker. This opens the way to finetuning of the ZIFs' adsorption and permeation properties by chemical tuning, pre-or post-synthesis, of their linkers. First principles molecular dynamics is a useful tool to investigate the microscopic behavior of these soft porous materials and predict the impact of adsorbates on their structure.

Computational Methods

Quantum chemistry calculations. The structures of all MOF materials considered were fully relaxed by optimizing both atomic positions and unit cell parameters simultaneously, starting from the experimental crystallographic structure. First principles calculations were performed in the density functional theory approach with periodic unit cell, full use of the crystal's symmetry elements and localized basis sets as implemented in the CRYSTAL14 code. 47 The B3LYP hybrid exchange-correlation functional 48 was used, with empirical correction for the dispersive interactions following the scheme of Grimme. 49 All electron basis sets were used for all atoms: H (3-1p1G), C (6-31d1G), N (6-31d1G) 50 and Zn (86-411d31G). 51 The accuracy of this methodology is now well established for the calculation of MOF and ZIF structures, their relative energies, 52 and elastic constants. 53 , 54 Unit cell parameters obtained for ZIF-8 was 16.86 Å (vs. 16.99 Å experimentally 5 ), while for SALEM-2 it is 17.01 Å (experimentally 16.83 Å 35 ) -although that latter experimental value was determined not for a pure imidazolatebased SALEM-2 but for a mixed-ligand ZIF of composition Zn(im)1.7(mim)0.3. Calculations were repeated with the PBESOL0 exchange-correlation functional, and the conclusions were identical. After full optimization in the crystal's symmetry group, negative frequencies remained corresponding to the free rotation of the ZIF-8 methyl top.

First principles molecular dynamics. First principles molecular dynamic (FPMD) simulations of the ZIF-8 and SALEM-2 empty frameworks, as well as the ZIF-8 framework with various quantities of adsorbed N2, were performed with Born-Oppenheimer dynamics based on Density Functional Theory for the calculation of atomic forces. They were carried out using the QUICKSTEP/CP2K package, 55 which is based on a hybrid Gaussian plane-wave approach combining a Gaussian basis for the wave functions with an auxiliary plane wave basis set for the representation of the density. 56 The gradient-corrected BLYP functional 57 , 58 was employed with empirical correction for the dispersive interactions using the Grimme "D3" method. 59 The cutoff for the density plane-wave basis set was set to 400 Ry at the finest level, and double-ζ valence polarized (DZVP) basis sets were used for all atoms.

All molecular simulations were performed using periodic boundary conditions on a single unit cell of the material (ZIF-8 or SALEM-2). Hydrogen atoms were deuterated in order to allow for a larger time step (0.5 fs) in the integration of the equations of motion. Simulations were run in the canonical ensemble (N,V,T), and the temperature was controlled by a Nosé-Hoover thermostat chain. 60 For each simulation, an initial equilibration period of 5 ps was used, with a thermostat time constant of 400 fs, before a production run lasting 15 to 25 ps, during which the thermostat time constant is set to 1000 fs. Results were then checked against shorter (N, P, T) runs (10 ps, with 250 fs thermostat time constant and 1 ps barostat time constant) to validate the little influence of the unit cell volume change on the results obtained. For example, for the empty ZIF-8 framework at 77 K, constant-pressure simulations found an average unit cell parameter of 16.94 Å (vs. 16.86 Å for the energy-minimized structure).

From the molecular simulations, the dihedral angle φ was used to characterize the swing of each imidazolate linker (Zn3-Zn2-Zn1-CH3; see Figure S4). The frequency of the swinging motion νswing was then calculated from the trajectories, by Fourier transform of the φ(t) autocorrelation function, and the width of the FT peak gives the uncertainty of ±10 cm -1 .

Grand Canonical Monte Carlo. In order to characterize the thermodynamics of adsorption of N2 in the rigid ZIF-8 AP and HP structures, force field-based Monte Carlo simulations were performed in the grand canonical ensemble. The N2 molecules were described by a rigid two points and three charges TraPPE model. 61 The ZIF-8 structure was described as in Ref. 62. These force field parameters are presented in the Supporting Information. Lorentz-Berthelot rules were used to calculate ZIF-N2 Lennard-Jones parameters. Series of GCMC simulations were performed at various values of N2 fugacity, which was then related to gas pressure by the ideal gas law. Long-range electrostatic interactions were taken into account using the Ewald summation technique. In order to improve the efficiency of the calculations, electrostatic and repulsion-dispersion interaction energies between the rigid ZIF-8 framework and adsorbed N2 molecules were precomputed on a grid (with a grid mesh of 0.1 Å) and stored for use during the simulation.

Figure 1 .

 1 Figure 1. Left: representation of a 2 × 2 supercell of the ZIF-8 structure, highlighting the SOD topology of the frame-work with brown lines drawn between neighboring Zn 2+ ions (in green). Top right: views of the 6-ring windows of ZIF-8's sodalite cages, and the window as viewed from above in the AP structure. Bottom right: lateral views in the AP and HP structures, showing the deformation of the window and reorientation of the 2-methylimidazolate linkers.

Figure 2 .

 2 Figure 2. Upper panel: histogram of the imidazolate swing angle φ (in degrees), as a function of temperature. Lower panel: potential of mean force as a function of the dihedral angle.

Figure 3 .

 3 Figure 3. Histograms of the swing angle of the imidazolate linkers, for ZIF-8 with various quantities of N2 adsorbed at 77 K.

Figure 4 .

 4 Figure 4. Histograms of the swing angle of the imidazolate linkers, for ZIF-8 (black) and SALEM-2 (red) at 77 K.

Table 1 .

 1 Properties (unit cell volume and cell parameters) of the ZIF-8 AP and HP phases, before and after DFT energy minimization in the absence of guest molecules.

	Structure	Experimental	Volume of DFT-	Unit cell
		volume used as	relaxed		parameter a
		starting point	structure
	ZIF-8 AP	4907.12 Å	3	4790.62 Å	3	16.85766 Å
	ZIF-8 HP	5006.60 Å	3	4790.54 Å	3	16.85756 Å

Acknowledgements

Funding from the Agence Nationale de la Recherche under project "SOFT-CRYSTAB" (ANR-2010-BLAN-0822) is acknowledged. This work was performed using HPC resources from GENCI (project A0010807069). I thank Aurélie Ortiz, Anne

Entry for the Table of Contents

The flexibility of zeolitic imidazolate framework ZIF-8 and its "swing" effect are investigated by first-principles molecular dynamics.

François-Xavier Coudert*

Molecular Mechanism of Swing Effect in Zeolitic Imidazolate Framework ZIF-8: Continuous Deformation Upon Adsorption


Maxime Rousselot

Xavier Ducloux

Olivier Le Meur

Rémi Cozot

QUALITY METRIC AGGREGATION FOR HDR/WCG IMAGES

Keywords: Image quality, High Dynamic Range, Machine learning, Image processing

High Dynamic Range (HDR) and Wide Color Gamut (WCG) screens are able to display images with brighter and darker pixels with more vivid colors than ever. Automatically assessing the quality of these HDR/WCG images is of critical importance to evaluate the performances of image compression schemes. In recent years, full-reference metrics, such as HDR-VDP-2, PU-encoding metrics, have been designed for this purpose. However, none of these metrics consider chromatic artifacts. In this paper, we propose our own full-reference quality metric adapted to HDR and WCG content that is sensitive to chromatic distortions. The proposed metric is based on two existing HDR quality metrics and color image features. A support vector machine regression is used to combine the aforementioned features. Experimental results demonstrate the effectiveness of the proposed metric in the context of image compression.

INTRODUCTION

High Dynamic Range (HDR) and Wide Color Gamut (WCG) technologies allow displaying always brighter and darker pixel with more vivid color. An HDR image can represent a range of luminance closer to the perceptible range by the Human Visual System (HVS) while WCG images are able to represent more colors than ever. Current HDR screens can display a luminance that can go below 0.01 cd/m 2 and/or up to 10,000 cd/m 2 . Presently, no monitors are able to faithfully represent WCG normalized by the BT.2020 [START_REF]Parameter values for ultra-high definition television systems for production and international programme exchange[END_REF]. However, highly promising solutions are developed towards this direction. HDR/WCG contents require new methods for assessing their quality. Image Quality Assessment (IQA) is a key tool to evaluate the performances of diverse image/video processing such as compression. Subjective evaluations are reliably assessing Quality. However they are often very time-consuming and not practical in many applications such as the quality monitoring for live video applications. Objective metrics are often more convenient and quite accurate to perform IQA. However, the IQA metrics used for Standard Dynamic Range (SDR) images using the legacy gamut BT.709 [START_REF]Parameter values for the hdtv standard for production and international program exchange[END_REF] are not able to process the refined information brought by the new representations. In recent years, full-reference objective quality metrics adapted to HDR video content have been proposed to tackle this problem. Two main strategies can be found in the literature. The first is to design dedicated metrics for HDR contents by modeling the HVS like HDR-VDP-2 [START_REF] Mantiuk | Hdrvdp-2: A calibrated visual metric for visibility and quality predictions in all luminance conditions[END_REF][START_REF] Narwaria | Hdr-vdp-2.2: a calibrated method for objective quality prediction of high-dynamic range and standard images[END_REF] or HDR-VQM [START_REF] Narwaria | Hdr-vqm: An objective quality measure for high dynamic range video[END_REF]. The second strategy is to adapt the SDR quality metrics, such as PSNR, SSIM [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF], MS-SSIM [START_REF] Wang | Multiscale structural similarity for image quality assessment[END_REF], VIF [START_REF] Hr | An information fidelity criterion for image quality assessment natural scene statistics[END_REF] and FSIM [START_REF] Zhang | Fsim: A feature similarity index for image quality assessment[END_REF], to HDR context. All these metrics assume that pixels values are perceptually linear (or perceptually uniform). This is why Aydn et al. [START_REF] To. Aydın | Extending quality metrics to full luminance range images[END_REF] designed the Perceptually Uniform (PU) function to remap the luminance of HDR images. This function has the same behavior than the classic gamma function for luminance corresponding to the SDR luminance range (below 80cd/m 2 ) and keeps perceptual linearity above this range. However, all above-mentioned metrics have still one limitation: they do not consider chromatic distortions. In a previous work [START_REF] Rousselot | Quality assessment of hdr/wcg images using hdr uniform color spaces[END_REF], we adapted SDR metrics, including color metrics, using a remapped version of perceptually uniform color space created specifically for HDR and WCG. We concluded that existing quality metrics still perform poorly for assessing the HDR chromatic distortions while performing quite well on luminance dimension. In this paper, we propose an HDR quality metric able to assess luminance and chromatic distortions caused by image compression. To obtain such a metric, we combine two full-reference quality metrics and two color image features using a Support Vector Regressor (SVR). Combining several features and quality metrics for improving the overall prediction performance was already proposed for SDR images and video. We can mention CQM [START_REF] Okarma | Combined full-reference image quality metric linearly correlated with subjective assessment[END_REF], CF-MMF and CD-MMF [START_REF] Liu | Image quality assessment using multi-method fusion[END_REF], FVQA [START_REF] Jy | A fusion-based video quality assessment (fvqa) index[END_REF], EVQA [START_REF] Jy Lin | Evqa: An ensemble-learning-based video quality assessment index[END_REF] and the metric developed by Netflix, VMAF [START_REF] Li | Toward a practical perceptual video quality metric[END_REF]. Recently, Choudhury et al. [START_REF] Choudhury | Combining quality metrics for improved hdr image quality assessment[END_REF] proposed HDR-CQM, a combination of full-reference metrics adapted to HDR contents. The main idea is to combine, using support vector machine regression, the scores of a subset of metrics selected from a list of quality metrics. Although performing well, the proposed metric does not tackle the issue of WCG chromatic distortions. The paper is organized as follows. In Section 2, we elaborate on the proposed metric and the features being used. Section 3 describes the training methodology. In Section 4, we evaluate the performance of our metric. Section 5 is the conclusion.

PROPOSED METHOD

The proposed metric relies on the mixture parameters of several quality metrics and image features; the combination is learned by using supervised machine learning. The main idea is to take advantage of each metric / feature strengths while compensating individual weakness. A Support Vector Regression machine with a Gaussian radial basis function kernel is used to map the features to a quality score. Figure 1 presents the overall architecture of the proposed method. The metric binary is available at www-percept.irisa. fr/software/ .

Considered HDR quality metrics

Two HDR metrics were selected for their good performances on several databases [START_REF] Rousselot | Quality assessment of hdr/wcg images using hdr uniform color spaces[END_REF]: HDR-VDP-2 and FSIM Jz az bz .

• HDR-VDP-2 [START_REF] Mantiuk | Hdrvdp-2: A calibrated visual metric for visibility and quality predictions in all luminance conditions[END_REF] (version 2.2.1 [START_REF] Narwaria | Hdr-vdp-2.2: a calibrated method for objective quality prediction of high-dynamic range and standard images[END_REF]): HDR-VDP-2 require numerous parameters such as the angular resolution, the surround luminance and the spectral emission of the screen. To ensure that HDR-VDP-2 score is coherent, we use the same angular resolution (60 pix/deg, the angular resolution of three of the databases), for each studied database (cf. Section 3.1). We have shown in a previous work [START_REF] Rousselot | Impacts of viewing conditions on hdr-vdp2[END_REF] that HDR-VDP-2 quality score shift when the angular resolution change but not in an expected way: HDR-VDP-2 score increases when the angular resolution decreases (or when the observer is closer to the screen).

• FSIM Jz az bz : FSIM [START_REF] Zhang | Fsim: A feature similarity index for image quality assessment[END_REF] is a full-reference metric adapted to SDR images. To adapt it to HDR images we use a remapped version of the Jzazbz [START_REF] Safdar | Perceptually uniform color space for image signals including high dynamic range and wide gamut[END_REF] color space as defined in [START_REF] Rousselot | Quality assessment of hdr/wcg images using hdr uniform color spaces[END_REF]. FSIM is a color-blind metric and only considers luminance distortions.

The two metrics considered above are only considering luminance distortions. To consider the chromatic dimension, we add two chromatic-based features.

Chromatic-based visual features

As we aim to extend the quality metric to color images, we add two features able to measure chromatic distortion. These features are inspired by the MS-SSIM features [START_REF] Wang | Multiscale structural similarity for image quality assessment[END_REF], the multi-scale version of the SSIM index [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF]. SSIM conducts a comparison on three levels: luminance, contrast and structure. Each factor is estimated pixelwise through a sliding window in order to achieve a distortion map. To obtain a global quality score, a mean is applied on this map. In MS-SSIM, contrast and structure are compared on different scales of the images. Our color features are created using the contrast (c) and structure (s) comparison function on the color components of the images. Because the HVS is less sensitive to high frequency variation in color, we use the scale 5 of the images, i.e. the images after 5 downsampling with a ratio of 2. The feature, MCS5 az , is calculated as following:

MCS5(X, Y ) = 1 M N M m=1 N n=1 c(xmn, ymn) • s(xmn, ymn) (1)
where X and Y are the reference and distorted images, respectively. xmn and ymn represent pixel values and M and N the image sizes.

For both original and degraded images, we calculate this feature on the Jzazbz color component of the images: MCS5 az for the az component, and MCS 5 bz for the bz component.

In addition, we also add a last feature to characterize the image complexity. Indeed this characteristic affects our perception of the image distortion.

Image spatial information

The last feature allows us to characterize the complexity of the image. Image complexity affects the image quality perception as it is related to the HVS spatial masking effects. It is measured on the Jz component of the reference images. We selected Spatial Index SI as recommended by the International Telecommunication Union (ITU) [START_REF]Subjective video quality assessment methods for multimedia applications[END_REF]. The spatial information SIJ z corresponds to the standard deviation of the image luminance plane which has been filtered by a Sobel filter: SI = std[Sobel(X)], where X is the reference image.

Mapping the features to the quality scores

To summarize, we compute from the original and distorted images a vector of 5 parameters: 2 parameters deal with the luminance component, 2 concerns considers chroma dimensions and one is used to characterize the image complexity. The relationship between these features and the quality scores is learned thanks to the SVR machine (cf. Figure 1). More details are given in section 3.3.

TRAINING METHODOLOGY

Image quality databases

To train the proposed model, a large collection of annotated HDR images is required. Unfortunately, there is, to the best of our knowledge, only a small number of HDR image databases annotated with subjective quality scores. In addition those databases are composed of a rather small amount of images. To obtain a suitable database for our experiment, we considered the five databases presented in Table 1: Narwaria et al. [START_REF] Narwaria | Tone mapping-based high-dynamic-range image compression: study of optimization criterion and perceptual quality[END_REF], Korshunov et al. [START_REF] Korshunov | Subjective quality assessment database of hdr images compressed with jpeg xt[END_REF], Zerman et al. [START_REF] Zerman | An extensive performance evaluation of full-reference hdr image quality metrics[END_REF], 4Kdtb [START_REF] Rousselot | Quality assessment of hdr/wcg images using hdr uniform color spaces[END_REF] and HDdtb [START_REF] Rousselot | Impacts of viewing conditions on hdr-vdp2[END_REF]. The first four databases were used for the training phase and HDdtb was considered as an independent test database used to validate our proposed metric. Three of them, Narwaria et al., Korshunov et al. and Zerman et al. includes images rendered on the same display (the SIM2 HDR47ES4MB), which has a measured dynamic range going from 0.03 to 4250 cd/m 2 and is only able to display a BT709 gamut. The HDdtb and the 4Kdtb were created using the SONY BVM-X300 monitor which is able to handle the WCG BT2020 and has a peak brightness at 1000 cd/m 2 . The luminance of a black pixel was too low to be measured by the equipment (<0.02 cd/m 2 ). We assume a luminance for the black pixel at 0.001 cd/m 2 . All the databases focus on compression artifacts with the exception of the HDdtb which also evaluates two different distortions: Gamut mismatch and Gaussian noise. Also, it must be emphasised that the 4Kdtb and HDdtb databases were designed to highlight metric sensitivity to chromatic distortions. Images of the 4Kdtb and the HDdtb database were compressed using three different methods:

• HEVC compression following the recommendation ITU-T H Suppl.15 [START_REF]Conversion and coding practices for hdr/wcg y'cbcr 4:2:0 video with pq transfer characteristics[END_REF]. This recommendation proposes different processes for increasing the quality of the compression like a chroma Qp adaptation. This adaptation is of special interest for this study because it corrects errors due to the compression of the chrominances.

• HEVC compression without the chroma Qp adaptation.

• HEVC compression with 8 bits quantization for the chrominances instead of 10 during the compression (4Kdtb only). As presented in Table 1, images of the 4Kdtb have a higher resolution than the other databases with an angular resolution around 120 pix/degree. To be as homogeneous as possible, we choose to downsample them to obtain a 60 pix/degree resolution before using the different quality metrics or train our model. Indeed, most quality metrics are not adapted to high resolution images.

Before the training, we limited the dynamic range of images accordingly to the used monitor available luminance during the conducted subjective test. We assume that the screen saturates if the image luminance is below or above its ability. The main characteristics of these databases can be found in table 1. A more detailed description of each database can be found in [START_REF] Rousselot | Quality assessment of hdr/wcg images using hdr uniform color spaces[END_REF].

Combining several databases

One problem to solve is the combination of multiple databases. Indeed, each subjective test is different, using a different protocol or using a different range of image quality. Thus, the subjective score or Mean Opinion Score

Support Vector Regression

To train the SVR, we first conduct a grid-search using all of the training databases to tune the hyper-parameters to obtain optimal results. Once the best hyper-parameters are found, we split our training database into two sets of images, one for training the SVR (80% of images) and one for testing (20% of images). This split is done randomly with the constraint that the same content can not be present in the training set and testing set. We then train the SVR using the function fitrsvm of Matlab. To obtain a more robust estimation of the proposed model performance and to eliminate selection bias, we repeat this split 1000 times. We evaluate the performance of our proposed metric using the me- As recommended in [START_REF] Am | Video quality experts group: Current results and future directions[END_REF], before computing these indicators, a non-linear regression is applied to the objective quality scores thanks to a logistic function:

Qi = a + b 1 + e -(Q i -c) d ( 2
)
where Qi is the score of the quality metrics on the image i and Qi the mapped quality score. Parameters a, b, c and d are determined by the regression conducted by the lsqcurvefit function of Matlab.

RESULTS

Proposed metric performance

In this section, we present the performance of the proposed metric on the 1000 test sets. The median performances across 1000 Train-Test Combinations can be found in Table 2. We compare the score of the proposed metric to five other metrics: HDR-VDP-2 and four SDR metrics adapted to HDR using two uniform color space Jzazbz and ICtCp [START_REF]Image parameter values for high dynamic range television for use in production and international programme exchange[END_REF] using the methodology presented in [START_REF] Rousselot | Quality assessment of hdr/wcg images using hdr uniform color spaces[END_REF]: MS-SSIM Jz az bz , MS-SSIMICtCp, FSIM Jz az bz , FSIMICtCp. All of those metrics are color-blind and only consider luminance distortions. To ensure that each feature is meaningful, we also report four

with all features except HDR-VDP2, metric (3) with all features except FSIM Jz az bz and metric (4) with all features except SIJ z . Table 2 shows that the proposed model provides superior performances than any other metric. To ensure that the good performances of the proposed model are homogeneous across all databases, we also report the median SROCC of the 1000 trained metrics, database by database in Table 3. Because the number of images in the test set coming from one database was rather small, we use the complete databases to calculate the performance indexes and not only the images used in the test set. We observe that achieving high accuracy on one database does not prevent the model performances on another database. Our proposed model reaches higher performances when compared to other metrics and especially to HDR-VDP-2 and FSIM Jz az bz , the two metrics that are also forming the basis of our model.

Validation on an independent database

In this section, we provide further evidence that the proposed metric can be generalized to other databases. More specifically, we evaluate its performance on the HDdtb database. In Table 4, we report results obtained on the complete HDdtb database and the results on the HDdtb images with compression artifacts only. It is worth mentioning that such artifacts are of same nature that those we trained our model for. The proposed metric outperforms five state-of-the-art metrics for all performance indexes when considering the complete database. As it was expected, the gain is higher when considering only the images with compression artifacts.

Sensitivity to color distortions

One of the metric goals is to assess not only the luminance distortions but also chromatic distortions. To illustrate this behavior, we use the images of the 4Kdtb database. Indeed, the difference between the compression strategies is chrominances management. As illustrated on Figure 3 (left), observers are sensitive to the different modes of compression. Scores from HDR-VDP-2 and from the proposed metric are given on the right hand side. As expected, HDR VDP2 is not sensitive to chromatic distortions while the proposed metric is.

CONCLUSION

In this paper, we proposed a new full-reference quality metric adapted to HDR/WCG content. This metric is a combination of two full-reference quality metrics as well as color image features and the image spatial information. Those features are fused into one metric with an SVR method. Experimental results show that the proposed metric can accurately assess the quality of images distorted by compression. Moreover, contrary to most HDR metrics, the metric is sensitive to chromatic distortions as well as luminance distortions. However, due to the characteristics of the database used to train and test the metric, it is not possible to assess its performance for distortion beyond image compression. Databases with more various kind of distortions are needed to improve the proposed metric.

Fig. 1 :

 1 Fig. 1: Architecture of the proposed full-reference HDR quality metric.

  (MOS) given to an image is not necessarily the absolute quality score of this image. In order to make our training database, it is required to combine the scores of the four databases, namely Narwaria et al., Korshunov et al., Zerman et al. and 4Kdtb. For this purpose, we align the MOS of the different subjective tests into a common quality scale thanks to the Iterated Nested Least Square Algorithm (INLSA) proposed in [25]. INLSA allows to determine a common scale. This algorithm is based on the assumption that objective quality metrics are linearly correlated with the subjective scores. Four objective metrics were selected: HDR-VDP-2, VIF Jz az bz , MS-SSIM Jz az bz and FSIM Jz az bz . The figure 2 illustrates the impact of the INLSA algorithm. It represents the HDR-VDP2 score in function of the MOS before the INLSA algorithm (MOS are re-scale between 0 and 1) and after the INLSA algorithm.

Fig. 2 :

 2 Fig. 2: HDR-VDP2 score in function of MOS (a) before the INLSA algorithm (b) after the INLSA algorithm

Table 3 :

 3 Median SROCC across 1000 Train-Test Combinations database by database: (a) 4Kdtb (b) khorshunov et al. (c) Narwaria et al. (d) Zerman et al. MS-SSIM Jz az bz 0.8306 0.9648 0.9088 0.8109 MS-SSIMICtCp 0.8447 0.9529 0.8714 0.9260 FSIM Jz az bz 0.8849 0.9663 0.8466 0.9031 FSIMICtCp 0.9049 0.9477 0.8645 0.8863 HDR-VDP-2 0.8678 0.9516 0.8909 0.9289 Proposed metric 0.9095 0.9704 0.9240 0.9474

Fig. 3 :

 3 Fig. 3: Subjective and objective scores in function of HEVC Qp for the image Bike 30s.

Table 1 :

 1 Review of 5 HDR image quality databases in terms of number of observers, number of images, subjective test protocol, kind of distortion, used display, used gamut and image size.

	Name	#Obs #Img	Protocol	Distortion	Display	Gamut	Size
	Narwaria et al. [21]	27	140	ACR-HR	JPEG	SIM2 HDR47ES4MB	BT.709	1920 × 1080
	Korshunov et al. [22]	24	240	DSIS (side by side)	JPEG-XT	SIM2 HDR47ES4MB BT.709	944 × 1080
	Zerman et al. [23]	15	100	DSIS	JPEG, JPEG-XT JPEG2000	SIM2 HDR47ES4MB	BT.709	1920 × 1080
	HDdtb [18]	15	96	DSIS (side by side) noise, Gamut mismatch HEVC, Gaussian	Sony BVM-X300	BT.2020	944 × 1080
	4Kdtb [11]	13	96	DSIS (side by side)	HEVC, Quantization	Sony BVM-X300	BT.2020 1890 × 2160

Table 2 :

 2 Median performances across 1000 Train-Test Combinations on the test set.

		SROCC KRCC	PCC	RMSE
	MS-SSIM Jz az bz	0.8096	0.6246 0.7963	16.6
	MS-SSIMICtCp	0.9197	0.7507 0.9237	10.74
	FSIM Jz az bz	0.8955	0.7267 0.8953	12.2
	FSIMICtCp	0.8807	0.7104 0.8757	13.52
	HDR-VDP-2	0.9186	0.7496 0.9179	10.9
	metric (1)	0.9234	0.7597 0.9244	10.46
	metric (2)	0.9165	0.7563 0.9235	10.51
	metric (3)	0.9113	0.7413 0.9101	11.36
	metric (4)	0.9294	0.7692 0.9271	10.28
	Proposed metric	0.9421	0.7899 0.9376	9.5
	other trained metrics using only a subsection of all features. Met-
	ric (1) is trained with only HDR-VDP-2 and FSIM Jz az bz , metric

Table 4 :

 4 Performances of several metrics for (a) the complete database HDdtb (b) Only the HDdtb images with compression artifacts.

	(a)	SROCC KRCC	PCC	RMSE
	MS-SSIM Jz az bz	0.8557	0.6646 0.8603	14.11
	MS-SSIMICtCp	0.8464	0.6462 0.8763	13.33
	FSIM Jz az bz	0.9069	0.7278 0.9187	10.94
	FSIMICtCp	0.8099	0.6080 0.8234	15.70
	HDR-VDP-2	0.8685	0.6721 0.8715	12.55
	Proposed metric	0.9110	0.7357 0.9250	9.05
	(b)	SROCC KRCC	PCC	RMSE
	MS-SSIM Jz az bz	0.8752	0.7165 0.8860	13.13
	MS-SSIMICtCp	0.8651	0.6749 0.8923	12.83
	FSIM Jz az bz	0.9173	0.7619 0.9297	10.47
	FSIMICtCp	0.8271	0.6437 0.8289	15.90
	HDR-VDP-2	0.8739	0.6866 0.9018	12.28
	Proposed metric	0.9422	0.7931 0.9510	8.53