Eric Darrigrand
email: eric.darrigrand-lacarrieu@univ-rennes1.fr

Nabil Gmati
email: nabil.gmati@enit.rnu.tn

Rania Rais
email: rania.rais@lamsin.rnu.tn

Schwarz method justification of a coupling between finite elements and integral representation for Maxwell exterior problem

come

Schwarz method justification of a coupling between finite elements and integral representation for Maxwell exterior problem

non borné (problèmes (5) et (6)). Cette réinterprétation de la méthode CEFRI a été initialement présentée dans [START_REF] Ben Belgacem | On the Schwarz algorithms for the elliptic exterior boundary value problems[END_REF]. Dans le cas d'une géométrie sphérique, elle nous permet d'établir le taux de convergence de la méthode itérative inhérente au problème (2) : la méthode de Schwarz avec recouvrement total converge si le bord Γ de l'obstacle est assez éloigné de la frontière artificielle Σ. Ce résultat est confirmé par des tests numériques réalisés avec des épaisseurs différentes entre Γ et Σ (voir Fig. 2). Cette relecture de la méthode CEFRI offre également une technique de préconditionnement.

Pour des raisons pratiques de mise en oeuvre numérique, la prise en compte de la condition essentielle imposée sur le bord Γ, se fait par une stratégie de pénalisation qui consiste à remplacer la condition au bord E × n γ = 0 du problème (2) par ε p (n γ × curlE) + E × n γ = 0 où ε p > 0 est choisi petit. Après discrétisation, cette méthode nous amène à résoudre un système linéaire impliquant une matrice éléments finis classique A ainsi qu'un bloc plein C dû à la représentation intégrale. La matrice du système linéaire à résoudre A + C est mal conditionnée. En appliquant le préconditionneur suggéré par la méthode de Schwarz sur l'équation induite de la discrétisation du problème de diffraction (3), le système devient alors

(I + A -1 C)E = A -1 F, (1)

Introduction

We are interested in the resolution of the 3D exterior time-harmonic Maxwell equations. To this aim, we focus on a combination of finite elements and integral representation ( [START_REF] Hazard | On the solution of time-harmonic scattering problems for Maxwell's equations[END_REF]). This strategy leads to an equivalent problem on a reduced bounded domain delimited by the surface of the scatterer and an artificial boundary with exact artificial boundary condition. No a priori condition is required on the distance between the scatterer and the artificial boundary but a difficult issue consists in the elaboration of a resolution strategy. A relevant idea was suggested in [START_REF] Jin | A novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering and radiation Problems[END_REF]. We propose the interpretation of this idea as an application of the Schwarz method, following the work done in [START_REF] Ben Belgacem | On the Schwarz algorithms for the elliptic exterior boundary value problems[END_REF] for Helmholtz equation. Hence, the theory on the Schwarz method justifies the use of Krylov solvers and the choice of a preconditioner.

In the next section, we derive the formulation of the system to be solved. Section 2 is devoted to the Schwarz interpretation of the resolution strategy suggested in [START_REF] Jin | A novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering and radiation Problems[END_REF]. In Section 3, the speed of convergence is estimated in the case of a spherical scatterer. Finally, some numerical results illustrate the theoretical properties. boundary Σ (see Fig. 1), the exterior problem reduces to a problem on a bounded domain Ω delimited by Γ and Σ (see [START_REF] Hazard | On the solution of time-harmonic scattering problems for Maxwell's equations[END_REF]): Find E such that

         curl curlE -t -1 ∇(divE) -k 2 s E = 0 in Ω, E × n γ = 0, divE = 0 on Γ, T ν1 (E) = T ν1 (E inc -I Γ (E)) and N ν2 (E) = N ν2 (E inc -I Γ (E)) on Σ, (2)
where n γ is the exterior unit normal of the domain Ω i on Γ; the regularization term t -1 ∇(divE) allows the use of a Galerkin finite-elements method (see [START_REF] Hazard | On the solution of time-harmonic scattering problems for Maxwell's equations[END_REF]) and the regularization parameter t -1 depends on the permittivity and the permeability of the air; k s is the wave number; ν 1 and ν 2 are complex numbers which have a negative imaginary part. The two operators T ν1 and N ν2 are defined by

T ν1 E = curlE × n σ + ν 1 n σ × (E × n σ ) and N ν2 E = divE + ν 2 E
• n σ with n σ the exterior unit normal of the domain Ω on Σ. The boundary conditions on Σ are derived from the integral representations satisfied by the scattered field and identified by the following expression ( [START_REF] Hazard | On the solution of time-harmonic scattering problems for Maxwell's equations[END_REF]): for x ∈ Ω e ,

I Γ (E)(x) = -k 2 s Ω RG t (x, .)E + Ω curlRG t (x, .)curlE +t -1 Ω divRG t (x, .) T divE -t -1 Γ divG t (x, .) T (E •n γ )dγ,
where

G t = G ks I + 1 k 2 s
Hess(G ks -G kp ) is the outgoing Green tensor associated with the differential operator curl curl -t -1 ∇(div) -k 2 s I of the regularized Maxwell equation; I is the identity matrix in R 3 ; Hess stands for Hessian operator; k p = √ tk s and G k is the fundamental solution of Helmholtz equation; R is a linear operator that maps every regular function ϕ defined on Γ into a regular function Rϕ defined on Ω which satisfies Rϕ = ϕ on Γ and Rϕ = 0 on Σ. The consideration of the Hilbert space

H t = E ∈ H(curl, Ω) / divE ∈ L 2 (Ω), E × n γ = 0 on Γ, E × n σ ∈ L 2 (Σ) 3 , E • n σ ∈ L 2 (Σ) ,
enables one to write a variational formulation of the problem (2): Find E ∈ H t such that

(A t + C t )E = F t , (3)
where the operators A t and C t : H t → H t are defined as follows

(A t E, E ′ ) t = Ω (curlE•curlE ′ +t -1 divEdivE ′ -k 2 s E•E ′ )+ Σ (ν 1 (n σ ∧E)•(n σ ∧E ′ )+t -1 ν 2 (n σ •E)(n σ •E ′ ))dσ, (C t E, E ′ ) t = Σ T ν1 (I Γ (E)) • E ′ dσ + t -1 Σ N ν2 (I Γ (E))(n σ • E ′ )dσ,
and

F t is given by (F t , E ′ ) t = Σ (T ν1 (E inc ) • E ′ + t -1 N ν2 (E inc )(n σ • E ′ ))dσ, where (•, •) t is the scalar product on H t .
The problem (3) is well posed as explained in [START_REF] Hazard | On the solution of time-harmonic scattering problems for Maxwell's equations[END_REF] and the operator A t is invertible.

Schwarz method interpretation

In order to solve the system (3), Jin and Liu [START_REF] Jin | A novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering and radiation Problems[END_REF] suggested to consider C t in the right hand side. An application of the fixed point algorithm leads to finding E n+1 such that

         curl curlE n+1 -t -1 ∇(divE n+1 ) -k 2 s E n+1 = 0 in Ω, E n+1 × n γ = 0, divE n+1 = 0 on Γ, T ν1 (E n+1 ) = T ν1 (E inc -I Γ (E n )) and N ν2 (E n+1 ) = N ν2 (E inc -I Γ (E n )) on Σ. (4)
In this paper, we focus on an original mathematical justification of convergence of the algorithm expressed by Jin and Liu. We interprete the algorithm defined by (4) as a Schwarz method. This interpretation has been initially proposed for the case of Helmholtz equation in [START_REF] Ben Belgacem | On the Schwarz algorithms for the elliptic exterior boundary value problems[END_REF]. The strategy is designed by the Total Overlapping Schwarz Method. Indeed the overlapping area is the total domain Ω. We hereby extend their work to the case of Maxwell equations: it consists in replacing equivalently the problem (4) by the two following subproblems. The first one is a transmission problem:

                         curl curlE 2n+1 -t -1 ∇(divE 2n+1 ) -k 2 s E 2n+1 = 0 in Ω i ∪ Ω e , n γ × [E 2n+1 ] = 0, n γ × [curlE 2n+1 ] = -n γ × curlE 2n on Γ, [divE 2n+1 ] = 0, n γ • [E 2n+1 ] = -n γ • E 2n on Γ, lim ρ→∞ ||x||=ρ ||curlE s 2n+1 × n γ -ik s n γ × (E s 2n+1 × n γ )|| 2 dγ = 0, lim ρ→∞ ||x||=ρ | √ t -1 divE s 2n+1 -ik s E s 2n+1 • n γ | 2 dγ = 0. ( 5
)
The second one consists in finding E 2n+2 such that

         curl curlE 2n+2 -t -1 ∇(divE 2n+2 ) -k 2 s E 2n+2 = 0 in Ω, E 2n+2 × n γ = 0, divE 2n+2 = 0 on Γ, T ν1 (E 2n+2 ) = T ν1 (E 2n+1 ) and N ν2 (E 2n+2 ) = N ν2 (E 2n+1 ) on Σ. (6)
The solution E 2n+1 of (5) has an explicit solution given by an integral representation. By inserting this representation in the second condition of (6) we effectively obtain the solution of (4). At the iteration n, the Schwarz algorithm is defined by A t E n+1 = -C t E n + F t . Numerically, we use the scheme suggested by Jin and Liu and do not use the subproblems ( 5) and (6). The intermediate problems ( 5) and (6) are used for theoretical justifications. This enables one to derive convergence estimations that cannot be obtained directly from the system (4). In Section 3, we investigate an analytical calculation of the rate of convergence of the Schwarz method in a spherical configuration.

Analytical estimation of the convergence for a spherical scatterer

In the case where Ω i is a perfectly conducting ball, we investigate the rate of convergence of the Total Overlapping Schwarz method. Let us consider the scatterer to be a ball of radius R * . We suppose that the artificial boundary Σ is a sphere concentric to Γ with radius R > R * . We first introduce some notations: We denote by j l the spherical Bessel function of degree l, by h l the spherical Hankel function of the first kind of degree l and H l (r) = h l (r) + rh ′ l (r), J l (r) = j l (r) + rj ′ l (r). We introduce the tangential vector spherical harmonics on the unit sphere S 1 , U lm = (1/ l(l + 1))∇Y m l and V lm = n γ × U lm , where Y m l , l > 0, m = -l, ..., l are the orthonormal scalar spherical harmonics (complete basis of L 2 (S 1 )). The sets of U lm and V lm form a complete orthonormal basis for T 2 (S 1 ) := a :

S 1 → C 3 / a ∈ (L 2 (S 1 )) 3 , a • n γ = 0 .
We define the error (w n ) n on the field E at each iteration by:

w 2n+1 = E 2n+1 -E in Ω e , w 2n+1 = E 2n+1 in Ω i , w 2n+2 = E 2n+2 -E in Ω.
For all n, we define Λ n = T ν1 (w 2n ) and δ n = N ν2 (w 2n ). In order to prove that the error (w n ) n converges to zero, we first show that Λ n+1 = KΛ n and δ n+1 = Lδ n with K : T 2 (S 1 ) → T 2 (S 1 ) and L : L 2 (S 1 ) → L 2 (S 1 ) two linear maps. K (resp. L) has a diagonal representation in the basis (U lm , V lm ) lm of T 2 (S 1 ) (resp. Y m l of L 2 (S 1 )). Let us denote by τ 1,lm , τ 2,lm (resp. τ 3,lm ) the eigenvalues of K (resp. L). These eigenvalues define the rate of convergence of the Total Overlapping Schwarz method. Taking into account the boundary and transmission conditions, we obtain:

τ 1,lm = 1 - H l (k s R * ) J l (k s R * ) J l (k s R) -ik s Rj l (k s R) H l (k s R) -ik s Rh l (k s R) -1 , τ 2,lm = 1 - h l (k s R * ) j l (k s R * ) J l (k s R) -ik s Rj l (k s R) H l (k s R) -ik s Rh l (k s R) -1 , τ 3,lm = 1 - H l (k s R * ) J l (k s R * ) j l (k s R) h l (k s R) -1
.

The convergence of the Total Overlapping Schwarz method is ensured if |τ i,lm | < 1, ∀i = 1, ..., 3, ∀l. The reader can see that these eigenvalues are independent of the parameter m. For R * = 1, the asymptotic behavior of the spherical Bessel functions for large l leads to the asymptotic estimation τ i,lm ∼ (1-R 2l ) -1 , i = 1, ..., 3. As a consequence, for small values of R, there exists a finite number of coefficients τ i,lm outside of the unit disk, and for sufficiently large values of R, all the coefficients τ i,lm are in the interior to the unit disk. We conclude that the Schwarz method has a linear convergence for R large enough. The numerical tests illustrate this theoretical result. In Fig. 2, we consider R * = 1 and R = R * + e with different values of e: λ/100, λ/10 or λ/5. The cases e = λ/100 and e = λ/10 exhibit some coefficients larger than 1 while the maximum value of |τ 2,lm | is strictly lower than 1 at the considered wavenumbers for the thickness e = λ/5 but the results are strongly dependent on the wavenumber. Similar asymptotic observations can be done on |τ 1,lm | and |τ 3,lm |. As a consequence, a Krylov method is a relevant alternative to the algorithm defined by (4): due to the properties of Krylov solvers demonstrated in [START_REF] Gmati | Comments on the GMRES Convergence for Preconditioned Systems[END_REF], the convergence of a Krylov method is ensured for the resolution of the problem (3) using A t as a preconditioner.

Preconditioner for a Krylov solver

The previous work suggests the use of A t as a preconditioner to solve the problem (3) using a Krylov solver. We hereby consider the resolution of problem (3) using the biconjugate gradient stabilized method. After a finite-elements discretization, the linear system is written under the form (A + C)E = F and the preconditioned system becomes (I + A -1 C)E = A -1 F where the matrix C involves the integral operators and the matrix A related to the differential operators involves a term resulting from the essential condition considered by a penalization strategy: ε p (n γ × curlE) + E × n γ = 0 on Γ, with ε p > 0.

The numerical implementation were done using and developing new integrands in the library Mélina++ [5]. In this section, we validate the efficiency of the preconditioning strategy by the consideration of an intermediate problem the solution of which is known:

                   curl curlE -∇(divE) -k 2 s E = 0 in Ω e , E × n γ = G 1 1 × n γ , divE = divG 1 1 on Γ, lim ρ→∞ ||x||=ρ ||curlE s × n γ -ik s n γ × (E s × n γ )|| 2 dγ = 0, lim ρ→∞ ||x||=ρ |divE s -ik s E s • n γ | 2 dγ = 0, (7)
where G 1 1 is the first vector component of G 1 . The scatterer is the unit sphere and the artificial boundary Σ is the sphere concentric to Γ with radius R = 1.5. Fig. 3-left shows the convergence of the relative error with respect to the finite-elements discretization: the relative error is plotted with respect to the average size of the mesh elements, for different values of the penalization parameter ε p , with k s = 3. Fig. 3-right illustrates a superlinear convergence of the biconjugate gradient stabilized solver applied to the preconditioned system for different values of the wavenumber with ε p = 10 -4 .

Figure 1 .

 1 Figure 1. Left: exterior unbounded domain Ωe, right: bounded domain Ω.

Figure 2 .

 2 Figure 2. Modulus of τ 2,lm for thickness e = λ/100 (left), λ/10 (center) and λ/5 (right). Cases ks = 1, 10 or 30.

Figure 3 .

 3 Figure 3. Relative l 2 -error with respect to the discretization (left), behavior of the residuals, case ksh = 2π/10 (right)

Scattering by a perfect conductorLet us consider Ω i a bounded scatterer in R 3 with a regular boundary Γ and Ω e its unbounded complementary. We are concerned with the scattering of a time-harmonic electromagnetic wave by the perfect conductor Ω i . Our purpose is to determine the total field E = E s + E inc where E inc is the incident wave and E s is the scattered field, solution to the regularized Maxwell equations with essential boundary


Manish Narwaria

Matthieu Perreira

Da Silva

Patrick Le Callet

Romuald Pepion

IMPACT OF TONE MAPPING IN HIGH DYNAMIC RANGE IMAGE COMPRESSION

Tone mapping or range reduction is often used in High Dynamic Range (HDR) visual signal compression to take advantage of the existing image/video coding architectures. Thus, it is important to study the impact of tone mapping on the visual quality of decompressed HDR visual signals. To our knowledge, most of the existing studies focus only on the quality loss in the resultant low dynamic range (LDR) signal (obtained via tone mapping) and typically employ LDR displays for subjective viewing experiments. As a result, it is not clear how tone mapping affects the perceptual quality of the decompressed HDR signal produced by inverse tone mapping. To address that, we present the results and analysis of a comprehensive subjective study in which the decompressed HDR images were displayed on an HDR display. Our study reveals that, indeed, the perceptual quality of the decompressed HDR signal is dependent on the tone mapping method employed.

INTRODUCTION

Conventional imaging systems (both for capturing and displaying) cannot handle the vast range of luminance typically present in a real scene due to inherent hardware limitations (for example, the camera sensor might have a limited dynamic range as a result of which intensity values in the scene which are above a certain threshold get saturated). Thus, visual signals are typically captured such that they represent colors which should appear on a display device and not necessarily proportional to scene intensities. Such images and videos have low dynamic range (LDR) and are device-referred or output-referred. Obviously, LDR signals cannot typically reproduce the contrast (or dynamic range) of the real scene. That is, the finer visual details are lost thereby reducing the visual quality. Therefore, HDR imaging has attracted attention in both industry and academia [START_REF] Banterle | Advanced High Dynamic Range Imaging: Theory and Practice[END_REF] towards improving the visual experience of the end users by producing compelling pictures that have much larger contrast in comparison to the traditional LDR pictures.

With respect to HDR capture, the requirement of powerful camera sensors can be bypassed by software processing, thanks to exposure fusion [START_REF] Banterle | Advanced High Dynamic Range Imaging: Theory and Practice[END_REF]. However, software processing for displaying HDR (known as tone mapping in which the dynamic range of the HDR image/video is reduced so that it matches with that of the display) invariably leads to loss of contrast and other finer details. As a result, it reduces the visual quality of the HDR scene and so this method of displaying HDR cannot be considered to be accurate. Thus, the more precise way of displaying HDR is the use of specialized displays with more powerful backlight structure resulting in a much larger dynamic range [START_REF] Seetzen | High Dynamic Range Display Systems[END_REF]. Such displays (often referred to as HDR displays) are now starting to become available in the consumer market (for instance, the SIM2 HDR display [4]).

Given that HDR images and videos capture more visual details as compared to the conventional LDR ones, they require a lot more storage space. Due to this, HDR image/video compression is an active research area. It is worth noting at this point that the existing coding architectures (for LDR images and videos) have become widely adopted standards supported by almost all software and hardware equipment dealing with digital imaging. As a result, it is of great interest to design HDR compression schemes that are compatible with existing coding architectures (for example [START_REF] Ward | JPEG-HDR: A Backwards-Compatible High Dynamic Range Extension to JPEG[END_REF], [START_REF] Sugiyama | HDR Compression Using Optimized Tone Mapping Model[END_REF]). In this, dynamic range reduction (or tone mapping) is usually the first step. The resultant LDR image/video is then compressed using existing codecs. Next, inverse tone mapping is employed to convert the decompressed LDR signal to HDR.

As already mentioned, TMOs are not necessarily transparent and as a result, the decompressed signal is not only affected by coding/compression error but also by the TMO. Thus, it is of interest to examine how some of the existing TMOs perform when it comes to HDR image compression. In Fig. 1, we have indicated the two stages at which visual quality loss occurs. It can be seen that stage 1 (i.e. HDR to LDR conversion via tone mapping) pertains to LDR quality assessment using the conventional LDR display and has been widely studied [START_REF] Narwaria | Effect of Tone Mapping on Visual Attention Deployment[END_REF]. On the other hand, stage 2 demands HDR quality assessment i.e. the decompressed HDR image has to be displayed on an HDR display and subjectively rated. However, a survey of literature reveals that this has not been comprehensively investigated. Our study, therefore, aims to fill that gap and analyze how different tone mapping methods affect the visual quality of decompressed HDR images.

TONE MAPPING BASED HDR COMPRESSION METHOD

In this section, we provide a brief description of the tone mapping based HDR image compression method [START_REF] Sugiyama | HDR Compression Using Optimized Tone Mapping Model[END_REF] used in this study and whose block diagram is shown in Fig. 1.

In this method, the first step is the tone mapping of the HDR image to obtain the LDR version. Next, a non-linear function similar to the µ-Law encoding is defined as

  ) 1 ln( ) / ( 1 ln ) (       x s s x LawTMO (1)
where s is a scaling parameter and µ controls the depth of the logarithm function. These parameters are determined by non-linear least squares optimization as

     i i i s LDR HDR LawTMO E 2 , ) ( min   (2)
where HDR i and LDR i respectively denote the i th pixel of the HDR and the corresponding tone mapped LDR image. Once the two parameters are determined, the HDR image is transformed by the function defined in (1) and quantized to 8 bits to obtain an image that has lower dynamic range as compared to HDR. As a result, this image can be compressed by an existing codec (we used JPEG 2000 in this paper) and the encoded bit stream along with the two function parameters are sent to the decoder. Along with the usual JPEG 2000 decoding, the decoder also applies the inverse function 1 

LawTMO



in order to transform the decoded image to HDR. The inverse function is defined as [START_REF] Seetzen | High Dynamic Range Display Systems[END_REF] It can be seen that the range reduction function in Eq.

  1 ) ( ) 1 ln( ). / ( 1        s x e s x LawTMO
(1) and its inverse in Eq. ( 3) are both completely determined by s and µ. Further, as indicated by Eq. ( 2), these two control parameters are derived based on the premise that the transformed image is as close to the tone mapped LDR image as possible. This obviously implies that the TMO used has a direct impact on the visual quality of the final decompressed HDR image. In the next section, we investigate this aspect in more details.

DETAILS OF SUBJECTIVE VIEWING TESTS

This section gives the details of the test setup used for the subjective viewing experiments.

Test scenes

To include a wide range of still image content and varying illumination conditions, we used 6 HDR still image scenes whose tone mapped versions are shown in Fig. 2. These include outdoor only scenes ('Carpark_IVC', 'Forest_path' and 'Lake') and both indoor and outdoor scenes ('Bausch_lot', 'Office_IVC' and 'Moto'). The resolution of the test scenes was HD (1920 × 1080). Note that in HDR, the luminance values are expressed in cd/m² and Fig. 4 (a) shows the luminance range (order of magnitude) of the test scenes used in this paper.

Tone mapping operators (TMOs)

The existing TMOs can be broadly classified into two categories namely local operators and global operators. As the name implies, local operators employ a spatially varying mapping which depends on the local image content. As opposed to this, global operators use the same mapping function for the whole image. Local TMOs are usually better at preserving the local details but are less effective in reproducing the overall brightness and contrast. This happens because of their design in which more emphasis is placed on preserving details in small neighborhoods and there may be no explicit global operation towards range reduction. On the other hand, global TMOs preserve the overall contrast better at the expense of local details. Global TMOs are generally computationally more efficient than the local ones. So while local and global TMOs have their own advantages and disadvantages, they inevitably degrade the visual quality. In this paper, we selected 5 TMOs: 3 local and 2 global. The local TMOs include the ones proposed by Ashikmin [START_REF] Ashikhmin | A tone mapping algorithm for high contrast images[END_REF], Reinhard [START_REF] Reinhard | Photographic tone reproduction for digital images[END_REF] and Durand [START_REF] Durand | Fast bilateral filtering for the display of high-dynamic range images[END_REF]. These are denoted as AL, RL and Dur respectively.

Note that the former two TMOs operate locally but also have the option for global processing. For global TMOs, we chose the logarithmic TMO and the global version of the TMO proposed by Reinhard [START_REF] Reinhard | Photographic tone reproduction for digital images[END_REF]. For the algorithmic details of the TMOs used, the reader is referred to their respective references. We used the HDR Toolbox available in [START_REF] Banterle | Advanced High Dynamic Range Imaging: Theory and Practice[END_REF] for the TMO implementations.

Test material preparation

All the TMOs used in this paper require one or more parameter value(s) to produce an acceptable quality tone mapped image. Moreover, these parameter(s) are highly content dependent. So these were selected on the basis of a pilot study for the six HDR still image contents [START_REF] Narwaria | Effect of Tone Mapping on Visual Attention Deployment[END_REF]. We used the JPEG2000 compression scheme for compressing the transformed HDR images. Seven bit rates were chosen such that the resulting visual quality covers the entire range i.e. from excellent (rating 5) to bad (rating 1). As a result, we obtained a total of 210 decompressed HDR images (6 reference scenes × 5 TMOs × 7 bit rates).With the inclusion of the 6 reference scenes, we have a total of 216 still HDR images, i.e. 216 conditions = 6 reference scenes × 36 conditions per reference image, to be evaluated by subjects.

Apparatus and Rating Methodology

Observers were seated in a standardized room conforming to the International Telecommunication Union Recommendation (ITU-R) BT500-11 recommendations [START_REF]Methodology for the subjective assessment of the quality of television pictures[END_REF]. For displaying the HDR images, SIM2 HDR47E S 4K display [4] was used. The HDR47E S 4K is a 47-inch, 1080p LCD TV with maximum displayable luminance of 4000 cd/m². The viewing distance was set to three times the height of the screen (active part), that is approximately 178 cm and the room illumination was set to 130cd/m².

For rating the compressed HDR images, we adopted the absolute category rating with hidden reference (ACR-HR) which is one of the rating methods recommended by the International Telecommunication Union (ITU) in Rec. ITU-T P.910 [START_REF]Subjective video quality assessment methods for multimedia applications[END_REF]. For rating overall quality, a five-level scale is used: 5 (Excellent), 4 (Good), 3 (Fair), 2 (Poor) and 1 (Bad). A total of 29 observers (14 males and 15 females) were used to subjectively evaluate the visual quality of the compressed HDR images. All were naive (not expert in image or video processing) for the purpose of this study. We also employed post-experiment screening of the subjects in order to reject any outliers in accordance with the Video Quality Experts Group (VQEG) multimedia test plan [START_REF] Hands | Video Quality Experts Group (VQEG) Multimedia Group Test Plan[END_REF]. Analysis per processed image and per source (i.e. reference) image was performed and in our case, none of the observers was rejected. The mean opinion score (MOS) for each stimuli was obtained by averaging the scores for that stimuli from all the observers.

QUALITATIVE ANALYSIS OF RESULTS

We obtained a total of 6264 subjective ratings (216 conditions each rated by 29 observers). This represents substantial ground truth to draw reasonably reliable conclusions.

The bit-rate vs the MOS (higher indicates better quality) plots for the 6 scenes are shown in Fig. 3. We have also indicated the associated 95% confidence intervals in these plots. We can make the following observation from these plots:

(1) The small confidence intervals indicate a high degree of agreement between the observers and are evidence in favor of the reliability of the obtained subjective ratings. (2) Recall that we used ACR-HR methodology in which the reference image (i.e. uncompressed HDR image in our case) is also shown to the observers without informing them. One can observe that the reference image was rated the highest for all the six. (3) The reader will notice from these plots that nearly the entire range of quality levels (from bad to excellent) have been covered for all the six scenes. As a result, the data obtained is not biased towards any specific content, TMO or bit rate. (4) In general, the TMO curves are distinguishable i.e. each TMO resulted in different quality HDR images. This is an important observation since it clearly implies that the visual quality of the decompressed HDR image is dependent on the type of TMO employed. (5) In general, the local TMO, AL overall performs the best as it leads to better visual quality at nearly all bit rates.

Interestingly, for the scene 'Forest_path', the MOSs for different TMOs are much closer with significantly overlapping confidence intervals. A plausible explanation for this is related to how TMOs operate. In general, TMOs attempt a trade-off between preserving details in very dark and very bright scene regions. As a result, a scene with very dark (for instance, shadows) and very bright (for instance, sunlight) regions is the quite challenging for TMOs. Obviously some TMOs achieve a better said tradeoff than others leading to different quality tone mapped scenes. With this in mind, the reader will notice that the other five scenes used in this paper have several darker regions which typically correspond to shadows, lower level of room illumination etc. In contrast, the scene 'Forest_path' has mainly bright regions (confirmed by the order of magnitude of the luminance ranges shown in Fig. 4 (a)). Consequently, most TMOs can preserve details leading to more similar visual quality and hence closer subjective ratings.

STATISTICAL ANALYSIS OF RESULTS

From the previous section, we find that indeed, different TMOs lead to quite different visual quality with one TMO performing better than others in different conditions i.e. depending on the scene and compression rate. Therefore, it will be interesting to further examine if the subjective ratings for scenes processed by one TMO are statistically different to the ones processed by another TMO. This would give insights into if a particular TMO leads to statistically better (or worse) quality as compared to another TMO.

Parametric Statistical Testing

We conducted a two-tailed Student's t-test [START_REF] Snedecor | Statistical Methods[END_REF] to compare the subjective score distribution (or population) from each of the five TMOs. However, t-test requires that each of the population being compared follows a Gaussian distribution. We verified this via Jarque Bera (JB) test [START_REF] Snedecor | Statistical Methods[END_REF] which determines normality by matching the skewness and kurtosis of the data with those of a normal distribution. The JB test statistic is defined as

         2 2 ) 3 ( 4 1 6 K S n JB (4)
Here, n is the number of observations, S and K denote skewness and kurtosis respectively. Samples from a normal distribution have S = 0 and K = 3. The results of the JB test revealed that none of the score distributions were strictly Gaussian. It may be pointed out however, that the JB test is strict in the sense that even small deviations from normality may result in the rejection of the assumption that the data follows normal distribution. Therefore, we also considered a 'relaxed' criteria to determine normality. For that we computed the higher order statistics (skewness and kurtosis) of each distribution. Since the Gaussian distribution has K value of 3, commonly, K values between 2 -4 can be deemed Gaussian approximately [START_REF] Montgomery | Applied Statistics and Probability for Engineers[END_REF]. Further given that S = 0 for normal distribution, one can assume approximate normality if S values are close to 0. The S and K values are reported in Table 1. From these, we can assume that the distributions are approximately Gaussian.

The null hypothesis under test is that the scores obtained from the five TMOs (for all the scenes and observers) are independent random samples from normal distributions with equal means (but unknown variances) against the alternative hypothesis that means are not equal:

Non-Parametric Statistical Testing

For the t-test, we assumed that score distributions are Gaussian (at least approximately). To further verify if such assumption is valid, we carried out the Wilcoxon rank-sum statistical test [START_REF] Snedecor | Statistical Methods[END_REF]. It is a non-parametric test and uses population medians (instead of the mean as in the case of t-test). The null hypothesis that the two distributions have the same median against the alternative hypothesis. As an important advantage, Wilcoxon rank-sum test (non-parametric) does not require the assumption of Gaussian distribution. Therefore, it is expected to be more accurate for scenarios in which the distribution is non-Gaussian.

Discussion of the Results of Statistical Tests

We carried out the two statistical tests (t-test and Wilcoxon rank-sum test) by using the subjective scores corresponding to the decompressed HDR images obtained from a TMO across all the test conditions, from all the subjects and for all the 6 source (i.e. reference) scenes.

That is, we compared 1218 subjective scores (29 observers × 6 reference scenes × 7 compression rates) obtained for one TMO with those for another TMO. Therefore, for five TMOs we have the 'blue colored squares' denote that the TMO mentioned in the row is statistically the same (i.e. leads to similar visual quality) as that in the column. The 'green colored squares' in Fig. 4 (b) imply that the TMO indicated in the row is statistically better than the one in the column and 'red color squares' represent the opposite case. One can see that out of ten comparisons between the different TMOs only in one case (corresponding to Log and RG TMOs), the results are statistically indistinguishable. In the remaining cases, one TMO was found to be statistically better (or worse) than another. This therefore confirms the point made in the previous section that the visual quality of the decompressed HDR image is dependent on the type of TMO. Interestingly, we found that the results from the Wilcoxon rank-sum statistical test the same as those from the t-tests. That is, the two statistical tests lead to the same conclusions. This probably justifies why the assumption of approximate Gaussianity is valid, at least in this case. Of course, there was no evidence before the start of the tests to believe that the two tests would lead to the same statistical conclusions.

A closer look at Fig. 4 (b) shows that AL TMO statistically outperforms the remaining TMOs and is followed by RL TMO (recall that both are local TMOs). On the other hand, Dur TMO is statistically the worst. Log and RG TMOs (both are global TMOs) lead to very similar visual quality scores and are statistically indistinguishable. This suggests that local TMOs might lead to better visual quality in the decompressed HDR images due to preserving more details. Dur TMO (local) however does not follow this trend at least for the scenes used in this paper. Therefore, while the experimental results indicate that the local TMOs might be better, it might be unfair to generalize that local TMOs always perform better than the global ones.

CONCLUSIONS

Most of the existing studies have focused only on the quality aspects of the tone mapped LDR content and typically use LDR displays for subjective viewing. Consequently, how different TMOs affect visual quality of the decompressed HDR images when viewed on an HDR screen has not been studied. To provide answer to this question, we analyzed the impact of TMOs on the visual quality using a total of 6264 ratings obtained from the HDR subjective viewing tests. The results indicate that different TMOs indeed lead to quite different perceptual quality in the decompressed HDR images. Further, we carried out both parametric and non-parametric statistical analysis and found that statistical differences exist between the visual quality ratings of the images processed by different TMOs.

Figure 1 .Figure 2 .

 12 Figure 1. Tone mapping based HDR image compression method.

Figure 3 .

 3 Figure 3. Bit rate Vs Mean Opinion Score (MOS) plots for the six scenes used in this paper. The respective scene names have been indicated in each plot. The error bars denote 95% confidence intervals. The legend for all the plots is the same as in the first plot. (Figure best viewed in color)

  results for the t-test are shown in Fig. 4 (b) in which

Table 1 .

 1 Skewness and kurtosis values of the score distributions

		AL	Dur	Log	RG	RL
	Skewness (S)	0.0638	0.8027	0.5844	0.6096	0.4442
	Kurtosis (K)	1.7668	2.7625	2.4469	2.1642	2.1621


Manish Narwaria

Matthieu Perreira

Da Silva

Patrick Le Callet

Romuald Pepion

On Improving the Pooling in HDR-VDP-2 towards Better HDR Perceptual Quality Assessment

Keywords: High Dynamic Range (HDR), perceptual quality, HDR-VDP-2

High Dynamic Range (HDR) signals capture much higher contrasts as compared to the traditional 8-bit low dynamic range (LDR) signals. This is achieved by representing the visual signal via values that are related to the real-world luminance, instead of gamma encoded pixel values which is the case with LDR. Therefore, HDR signals cover a larger luminance range and tend to have more visual appeal. However, due to the higher luminance conditions, the existing methods cannot be directly employed for objective quality assessment of HDR signals. For that reason, the HDR Visual Difference Predictor (HDR-VDP-2) has been proposed. HDR-VDP-2 is primarily a visibility prediction metric i.e. whether the signal distortion is visible to the eye and to what extent. Nevertheless, it also employs a pooling function to compute an overall quality score. This paper focuses on the pooling aspect in HDR-VDP-2 and employs a comprehensive database of HDR images (with their corresponding subjective ratings) to improve the prediction accuracy of HDR-VDP-2. We also discuss and evaluate the existing objective methods and provide a perspective towards better HDR quality assessment.

INTRODUCTION

High Dynamic Range (HDR) has been gaining popularity in academia and industry in recent times [START_REF] Banterle | Advanced High Dynamic Range Imaging: Theory and Practice[END_REF] . The reason is that with HDR we can represent the real physical luminance of a natural scene. As opposed to this, traditional low dynamic range (LDR) content allows limited range due to the limitations of capture and display devices. Therefore, LDR usually defines a white point or the maximum reachable luminance. For example, typical 8-bit representation assumes 255 as the maximum level. This has the consequence of shrinking the actual scene intensities within the defined limits obviously leading to loss of visual details and in turn the perceptual quality. On the other hand, HDR values are related to the scene intensities. Thus, there is a unique white point for each scene and HDR content is often described as scenereferred. Such scene-referred visual signals tend to be visually more appealing as they can represent the dynamic range of the visual stimuli present in the real world. Not surprisingly, the emergence of HDR is seen as an important step towards improving the visual quality of experience (QoE) of the end users.

While HDR imaging offers obvious advantages over the traditional LDR contents in terms of better visual quality of experience (QoE), it comes with the price of much larger storage space requirements as compared to an LDR file. For instance, an HDR image may occupy 4 times the space needed by an LDR version of the same image [START_REF] Banterle | Advanced High Dynamic Range Imaging: Theory and Practice[END_REF] . So there is need for research into effective HDR compression schemes and this therefore has been an important research area. A crucial and related issue is that the existing coding architectures have become widely adopted standards supported by almost all software and hardware equipment dealing with digital imaging. As a result, it will be of great interest to design HDR compression schemes that are compatible with existing coding architectures. Not surprisingly, substantial research effort has been put into designing HDR compression systems that are backward compatible [START_REF] Ward | JPEG-HDR: A Backwards-Compatible High Dynamic Range Extension to JPEG[END_REF][START_REF] Sugiyama | HDR Compression Using Optimized Tone Mapping Model[END_REF][START_REF] Mantiuk | Backward Compatible High Dynamic Range MPEG Video Compression[END_REF] with the standard image (e.g. JPEG and JPEG 2000) and video coders (e.g. H.264/AVC).

Due to the requirement of backwards-compatibility, HDR compression typically introduces artifacts due to three reasons. First, tone mapping is often exploited to reduce the dynamic range of HDR in a typical backward-compatible HDR compression pipeline. This causes loss of visual details. Second, the compression algorithm (eg. JPEG, MPEG) itself leads to loss of visual quality (eg. JPEG can introduce blockiness). Lastly, the inverse tone mapping is employed to rescale the dynamic range of the compressed bit-stream data. Again, inverse tone mapping being a lossy process can damage the perceptual quality. Thus, the decompressed HDR signal undergoes several processes all of which potentially decrease visual quality. This gives rise to the need of proper validation of perceptual quality in order to provide the endusers with minimum acceptable quality HDR content.

BACKROUND

Even though subjective assessment of visual quality remains the 'gold' standard, its deployment is difficult in some situations (eg. real-time HDR compression). Thus, there is obviously a strong need to develop objective computational models that can predict the perceptual quality of HDR signals in an objective manner. Such models will be extremely useful in an HDR processing pipeline for predicting the visual quality of processed HDR images/videos. Unfortunately, the conventional objective visual quality prediction methods do not take into account the luminance range and typically assume that the input pixel values are perceptually uniform. As a result, these cannot be used in case of higher luminance conditions as is usually the case with HDR visual signals. Recently, the HDR-VDP-2 algorithm 5 has been proposed. It is an extension of the Visible Differences Predictor (VDP) algorithm. The HDR-VDP-2 uses an approximate model of the human visual system (HVS) derived from new contrast sensitivity measurements. Specifically, a customized contrast sensitivity function (CSF) was employed to cover large luminance range as compared to the conventional CSFs. HDR-VDP-2 is essentially a visibility prediction metric. That is, it provides a 2D map with probabilities of detection at each pixel point and this is obviously related to the perceived quality because a higher detection probability implies a higher distortion level at the specific point. Nevertheless, in many cases, it is crucial to know an overall quality score (rather than just the local distortion visibility probability). Pooling is a crucial aspect in converting local error distribution into a single score that denotes the perceptual quality and the human visual system (HVS) can very easily do that accurately. But it is much more difficult to realize that in an objective quality prediction model given the underlying complexities and lack of knowledge of the HVS's pooling mechanisms. It is believed that multiple features jointly affect the HVS's perception of visual quality, and their relationship with the overall quality is possibly nonlinear and difficult to be determined apriori. Therefore, the approach that HDR-VDP-takes is that finding the pooling parameters via optimization of correlation with subjective scores.

In its original implementation, the authors of HDR-VDP-2 tried over 20 different combinations of aggregating (or pooling) functions. These included maximum value, percentiles (50, 75, 95) and a range of power means (normalized Minkowski summation) with the exponent ranging from 0.5 to 16. The aim was to maximize the value of Spearman's correlation coefficient in order to find the best pooling function and its parameters. While HDR-VDP-2 is fairly comprehensive method for HDR quality assessment, there is an issue with regards to pooling in HDR-VDP-2. This is related to parameter optimization. That is, the parameters of the pooling function in HDR-VDP-2 were found by maximizing (optimizing) correlation using existing LDR image databases. Therefore, its effectiveness in predicting the visual quality of HDR images is questionable given the different characteristics LDR and HDR images especially in terms of distortion visibility and overall visual appeal. To address that, we propose to compute the pooling parameters via optimization using HDR content. In the following, we first describe the development of a comprehensive HDR database and use it for parameter optimization.

SUBJECTIVE DATABASE FOR HDR VISUAL QUALITY

In this section, we will give a brief description of how we developed the HDR quality database. This will be used important for parameter optimization in HDR-VDP-2 as explained in the next section. Further, the HDR database will be the test bed for evaluating and comparing the performances of objective quality prediction methods. For developing the HDR database, we considered a total of 10 reference HDR scenes and two types of distortions: JPEG and JPEG 2000 compression artifacts. To our knowledge, our efforts are amongst the first ones to introduce a comprehensive HDR image database with subjective scores. This will be of immense value to the research community given the lack of publicly available databases for HDR content quality evaluation.

Test Material Preparation

First, we generated the HDR stimuli with JPEG distortions. For that we chose 10 reference (i.e. undistorted) HDR scenes, 7 compression bit rates so that the resulting visual quality covers the entire range i.e. from excellent (rating 5) to bad (rating 1). Since HDR compression involves tone mapping operator (TMO), we employed the image color appearance model iCAM06 algorithm [START_REF] Kuang | iCAM06: A refined image appearance model for HDR image rendering[END_REF] . Also, two optimization criteria were used. As a result, we obtained a total of 140 compressed HDR images (10 reference images × 1 TMO × 2 optimization criterion × 7 bit rates). With the inclusion of 10 reference scenes, we have a total of 150 images, i.e. 150 conditions = 10 reference images × 15 conditions per reference image, to be evaluated by subjects. The keen reader is also referred to our previous work [START_REF] Narwaria | Tone mapping Based High Dynamic Range Image Compression: Study of Optimization Criterion and Perceptual Quality[END_REF] for further details.

For JPEG 2000 distorted content, we chose 6 reference HDR scenes. In this case, we selected 5 TMOs: 3 local and 2 global ones. The local TMOs include the ones proposed by Ashikmin 9 , Reinhard 10 and Durand [START_REF] Durand | Fast bilateral filtering for the display of high-dynamic range images[END_REF] . For global TMOs, we chose the logarithmic TMO and the global version of the TMO proposed by Reinhard. Seven bit rates were chosen such that the resulting visual quality covers the entire range i.e. from excellent (rating 5) to bad (rating 1). As a result, we obtained a total of 210 decompressed HDR images (6 reference scenes × 5 TMOs × 7 bit rates). With the inclusion of the 6 reference scenes, we obtained a total of 216 still HDR images, i.e. 216 conditions = 6 reference scenes × 36 conditions per reference image, to be evaluated by subjects.

Subjective Testing

Observers were seated in a standardized room conforming to the International Telecommunication Union Recommendation (ITU-R) BT500-13 recommendations 12 . For displaying the HDR images, SIM2 HDR47E S 4K display 13 was used. The HDR47E S 4K is a 47-inch, 1080p LCD TV with maximum displayable luminance of 4000 cd/m². The viewing distance was set to three times the height of the screen (active part), that is approximately 178 cm and the room illumination was set to 130cd/m².

For rating the decompressed HDR images, we adopted the absolute category rating with hidden reference (ACR-HR) which is one of the rating methods recommended by the International Telecommunication Union (ITU) in Rec. ITU-T P.910 [START_REF]Subjective video quality assessment methods for multimedia applications[END_REF] . For rating overall quality, a five-level scale is used: 5 (Excellent), 4 (Good), 3 (Fair), 2 (Poor) and 1 (Bad). A total of 27 observers (16 males and 11 females) were employed for JPEG while 29 observers (14 males and 15 females) subjectively evaluated the visual quality for the case of JPEG 2000. All observers naive (not expert in image or video processing) for the purpose of the study. We also employed post-experiment screening of the subjects in order to reject any outliers in accordance with the Video Quality Experts Group (VQEG) multimedia test plan [START_REF] Hands | Video Quality Experts Group (VQEG) Multimedia Group Test Plan[END_REF] . Analysis per processed image and per source (i.e. reference) image was performed and in our case, none of the observers was rejected. The mean opinion score (MOS) for each stimuli was obtained by averaging the scores for that stimuli from all the observers. The keen reader is also referred to our previous works 18, [START_REF] Narwaria | Impact of Tone Mapping in High Dynamic Range Image Compression[END_REF] for further details on the test material preparation and the subjective experiments.

IMPROVING QUALITY PREDICTION WITH HDR-VDP-2

In this section, we first give brief and relevant details of HDR-VDP-2. Then, we will outline the method to improve prediction performance based on optimization with HDR content.

Brief review of HDR-VDP-2

The HDR Visual Difference Predictor (HDR-VDP-2) algorithm is primarily designed for predicting the visibility of distortions in HDR images. To that end, HDR-VDP-2 provides a 2D map with probabilities of detection at each point and this is obviously related to the perceived quality because a higher detection probability suggests a higher distortion level at the specific point. Nevertheless, as an extension to provide an overall quality score, HDR-VDP-2 also employs pooling strategy so that the detected features can be pooled (fused) into a single number that denotes the overall quality scores for the image. Towards that end, the authors of HDR-VDP-2 tried over 20 different combinations of aggregating (or pooling) functions [START_REF] Mantiuk | HDR-VDP-2: A Calibrated Visual Metric for Visibility and Quality Predictions in All Luminance Conditions[END_REF] . These included maximum value, percentiles (50, 75, 95) and a range of power means (normalized Minkowski summation) with the exponent ranging from 0.5 to 16. The aim was to maximize the value of the Spearman's correlation coefficient in order to find the best pooling function and its parameters. The resulting expression to predict quality score Q was defined as:

             F f O o I i f i o f D I w O F Q 1 1 1 2 ) ]( , [ 1 log . 1  (1)
where i is the pixel index,  = 10 -5 is a constant to avoid singularities when D is close to 0, and f, o are respectively the spatial frequency band and orientation indices of the steerable pyramid. I is the total number of pixels and the per-band weighting f w was found by maximizing the correlation with an LDR image quality database.

Improved Optimization of Pooling in HDR-VDP-2

As mentioned, the per-band weighing f w was obtained by optimizing with an LDR database. This is problematic because the characteristics of LDR content are different from those of HDR especially with regards to perceptual quality. More specifically, the influence of spatial frequencies on the perceptual quality can be different in HDR and LDR.

Consequently, it is necessary to find the per-band weighting using HDR content. To that end, we employed JPEG compressed HDR images and their corresponding ratings. Because the subjective ratings and the HDR-VDP-2 predictions are not in the same range, a logistic mapping function of the following form was employed before computing the RMSE:

4 3 2 1 )) ( exp( 1 1 2 1                  Q Q l (2)
where Q denotes the objective score and l Q represents the logistically transformed value and

     N k k k l w S Q f 1 2 , min (3)
To solve for f w by minimizing the above function, we employed the Nelder-Mead simplex algorithm 6 which is widely used for minimizing real-valued functions. The Nelder-Mead method attempts to minimize a scalar-valued nonlinear function of n real variables using only function values, without any derivative information (explicit or implicit). It maintains at each step a nondegenerate simplex, a geometric figure in n dimensions of nonzero volume that is the convex hull of n + 1 vertices. Each iteration of a simplex-based direct search method begins with a simplex, specified by its n + 1 vertices and the associated function values [START_REF] Lagarias | Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions[END_REF] . One or more test points are computed, along with their function values, and the iteration terminates with bounded level sets. The optimized weights f w obtained were then used to predict the quality scores for JPEG 2000 compressed images. Therefore, the content employed for optimization is different from the testing set. Note that there are a total of 216 HDR images for this condition. Another reason for using these sets of images for performance evaluation is related to their processing. Recall that the database for JPEG 2000 compressed HDR images the perceptual quality is not only affected by the compression rate but also depends on five tone mapping operators.

PERFORMANCE EVALUATION ON HDR DATABASE

Even though HDR-VDP-2 employs the pooling function in (1) to predict quality, to our knowledge, it has not been evaluated on a comprehensive set of distorted HDR images with MOSs since the original HDR-VDP-2 paper was more focused on visibility predictions rather than overall quality assessment. In fact, the quality prediction performance was tested only on a set of LDR images (from TID2008 database [START_REF] Ponomarenko | TID2008 -A Database for Evaluation of Full-Reference Visual Quality Assessment Metrics[END_REF] ). Hence, it will be interesting to assess the performance of HDR-VDP-2 for quality prediction of HDR images and examine its effectiveness for the task of prediction (which is not entirely the same as detection). As mentioned, currently there is no publicly available HDR database with subjective quality ratings. Thus, the performance of HDR-VDP-2 and even conventional LDR metrics has not been evaluated with HDR content except our previous study [START_REF] Narwaria | Tone mapping Based High Dynamic Range Image Compression: Study of Optimization Criterion and Perceptual Quality[END_REF] in which we evaluated the performance for JPEG compressed HDR images. In this paper, we further validate the performance of objective methods on HDR images affected by JPEG 2000 compression errors as well as distortions due to tone mapping.

Qualitative analysis

The experimental results are reported in terms of four criteria commonly used for performance comparison, namely: the Pearson linear correlation coefficient C P (for prediction accuracy), the Spearman rank order correlation coefficient C S (for monotonicity), the Kendall rank correlation coefficient C K and the Root Mean Squared Error (RMSE) between the MOS and the objective predictions. For a perfect match between the objective and subjective scores, C P = C S = C K = 1 and RMSE=0.We not only evaluate the overall prediction accuracies but also report the results for two cases: (a) percontent prediction accuracy, (b) accuracy based on tone mapping operator (TMO). The former provides more information on how different objective methods perform for different content while the latter gives insights into method performance for predicting quality affected by TMO. In this paper, we considered 3 LDR objective methods namely Mean Squared Error (MSE), Structural Similarity Index Measure (SSIM) [START_REF] Wang | Image quality assessment: From error visibility to structural similarity[END_REF] and the scalable image quality measure (SIQM) [START_REF] Narwaria | Fourier Transform Based Scalable Image Quality Measure[END_REF] . For HDR methods, we evaluated the HDR-VDP-2 with original parameter values and the modified values based on optimization with HDR content via (3). These cases are respectively denoted as HDR-VDP-2 (original) and HDR-VDP-2 (modified). The results for the per-content evaluation are given in Tables 1-5 from which we can make the following observations:

1. The overall prediction performance of the three LDR methods is very poor as compared to the two version of HDR-VDP-2 with SIQM performing the best. Such poor performance of LDR methods is however not entirely unexpected. This is because these methods typically assume perceptually scaled pixel value representation of the image signal. But with HDR, the pixels values are represented in terms of physical luminance values. Another possible reason for such poor performance is related to the high luminance conditions with HDR. Consequently, more distortions might be visible on an HDR display as compared to conventional LDR displays. This in effect can reduce the effectiveness of contrast sensitivity models that LDR methods in general might directly or indirectly employ (of course MSE does not use such models).

2. While the LDR methods perform quite poorly, their best performance occurs for 'Forest_path' content. As explained in our previous work [START_REF] Narwaria | Impact of Tone Mapping in High Dynamic Range Image Compression[END_REF] , the subjective ratings for this scene processed by the five TMOs were quite close. That is, despite the scene being processed by different TMOs, the resultant HDR qualities were judged by subjects as being quite close. We attributed this to the fact that the scene 'Forest_path' has mainly bright regions and so the TMOs yield very similar visual qualities. This can also be used to explain why LDR methods perform the best for this scene. Because of the absence of very dark regions, the overall luminance is spread in a more uniform manner. Therefore, this is more similar to an LDR content but with brighter luminance leading to better quality prediction by LDR methods.

3. The two versions of HDR-VDP-2 perform much better than all the three LDR methods. The proposed optimization indeed improves the overall performance of HDR-VDP-2. However, the improvement is not statistically significant as verified in the next section. We suspect that the performance can be further improved by calibration of other HDR-VDP-2 parameters (other than pooling ones like the peak sensitivity parameter [START_REF] Mantiuk | HDR-VDP-2: A Calibrated Visual Metric for Visibility and Quality Predictions in All Luminance Conditions[END_REF] ).

Further evaluation results for each TMO are reported in Tables 5678910. One can notice improvement in the prediction performance for each TMO. The biggest improvement is for SIQM which in some cases performs closer to HDR-VDP-2. On the other hand, the performance of HDR-VDP-2 (both versions) is similar to the per-content case. However, HDR-VDP-2 (modified) is still overall better although the performance is degraded for Reinhard_local TMO. The marked improvement in case of LDR methods indicates that within the same distortion (we can assume that each TMO is a source of distortion), LDR methods can predict quality more reliably. But with a more complex scenario (images processed by different TMOs), the performance of LDR methods starts to degrade rapidly. Overall, HDR-VDP-2 and its modified version clearly outperform the LDR methods.

Statistical analysis

In this section, we evaluate the statistical significance of the overall prediction performance of different objective methods. To that end, an F-test [START_REF] Montgomery | Applied Statistics and Probability for Engineers[END_REF] was performed on the prediction residuals between the objective predictions (after applying the logistic mapping) and the subjective scores. The test is based on an assumption of Gaussianity of the residual differences. Therefore, we first need to check if the residuals can be assumed to be Gaussian or not. For that, we used the Kolmogorov-Smirnov (KS) test [START_REF] Massey | The Kolmogorov-Smirnov Test for Goodness of Fit[END_REF][START_REF] Marsaglia | Evaluating Kolmogorov's Distribution[END_REF] and Table 11 lists the results and the corresponding test statistics. The critical value which is computed based on the number of residuals (in this case 216) was 0.0916. For determining normality, the KS test compares the test statistic with the critical value and a smaller test statistic value (as compared to the critical value ) implies normality. In Table 11, 0 for the KS test implies that the null hypothesis cannot be rejected at 5% significance level and therefore implies normality. One finds the residuals SIQM and the two HDR-VDP-2 versions are normally distributed. However, the test statistic of the remaining residuals is also not too large as compared to the critical value. This means that those residuals (from MSE and SSIM) can be taken to be approximately Gaussian. This was further confirmed by the skewness and kurtosis values which are also reported in Table 11. Since the Gaussian distribution has K value of 3, commonly, K values between 2 -4 can be deemed Gaussian approximately. Further given that S = 0 for normal distribution, we could assume approximate normality if S values are close to 0. We therefore find that the assumption of Gaussianity of residuals of all the five objective methods holds (or nearly holds). Assuming that

2 MSE  , 2 SSIM  , 2 SIQM  , 2 ) ( 2 orginal VDPP HDR    , and 2 )
( 2 modified VDPP HDR   
denote the variances of the residuals from the respective objective quality assessment algorithms, a measure known as the F-value can be defined as

         2 1 2 2 Method Method F   for the F-test to ascertain statistical significance critical F F  critical F F   1 1 1   F F critical critical F F 1  Method2 has



In Table 13, we present the F-values when comparing two objective methods. In this table, the (refer to Table 12), we can see from Table 13 that HDR-VDP-2 (original) and HDR-VDP-2 (modified) are statistically better than the LDR methods. Moreover, the three LDR methods lead to statistically indistinguishable performances. This once again confirms with statistical evidence that LDR methods cannot be used for HDR visual quality measurement. The statistical results also reveal that the wo HDR-VDP-2 versions are statically indistinguishable but HDR-VDP-2 (modified) performs better overall (F > 1). This has been highlighted in Table VII by bold-face F values for the corresponding cases. On the other hand, all the LDR based methods SIQM, SSIM and MSE are statistically indistinguishable from each other.

CONCLUSIONS

This paper has dealt with HDR visual quality assessment evaluation both from subjective and objective viewpoints. To that end, we first introduced an HDR database with JPEG and JPEG 2000 compression distortion as well as TMO induced distortions. We then used the HDR database for improving the prediction performance of HDR-VDP-2 by finding better pooling parameters. This was done by minimizing the error between the logistically transformed predicted values and the subjective ratings. The performance of three LDR methods namely MSE, SSIM and SIQM and the two versions of HDR-VDP-2 was evaluated on a set of 216 HDR images. The use of HDR images for parameter optimization lead to an overall better performance. We also expect that calibration of several other parameters in HDR-VDP-2 (eg. parameters controlling the peak sensitivity, visual contrast masking) with the HDR database will improve the prediction accuracy of HDR-VDP-2 further.

  k S denote the logistically transformed HDR-VDP-2 score and the subjective score for the k th image and assume there are N images. The function to be minimized can be obtained as

  of the residuals from the two objective methods which need to be compared. The F value is then compared with a critical value denoted as critical F to establish statistical difference between the two methods.

criticalF

  is computed based on the number of residuals and the desired confidence level.

F

  the method indicated in each row is Method1 while the one in the column is Method2. With 216 residuals and 95% confidence level we have critical 8000. Keeping in mind the implications of the F values as compared to critical F

Table 1 .

 1 Per-content prediction performance of MSE

	Measure/	C P	C S	C K	RMSE
	HDR content
	Office_ivc	0.2466	0.0115	0.0734	1.0451
	Carpark_ivc	0.4875	0.2261	0.8022	0.2016
	Bausch_lot	0.1734	0.2325	0.2424	0.9010
	Forest_path	0.5537	0.5927	0.4637	0.8903
	Lake	0.5452	0.5636	0.4427	1.0353
	Moto	0.5145	0.5174	0.3725	0.9374
	Overall results	0.1356	0.1482	0.1098	1.1238
		Table 2. Per-content prediction performance of SSIM
	Measure/	C P	C S	C K	RMSE
	HDR content
	Office_ivc	0.1479	0.0095	0.0797	1.0665
	Carpark_ivc	0.5062	0.2187	0.1984	1.0392
	Bausch_lot	0.1330	0.2325	0.2424	0.9068
	Forest_path	0.6314	0.6597	0.5147	0.8291
	Lake	0.4734	0.5387	0.4172	1.0879
	Moto	0.4871	0.5219	0.3725	0.9548
	Overall results	0.1053	0.1466	0.1070	1.1280
		Table 3. Per-content prediction performance of SIQM
	Measure/	C P	C S	C K	RMSE
	HDR content
	Office_ivc	0.1851	0.1269	0.1467	1.0597
	Carpark_ivc	0.8328	0.8699	0.6944	0.6671
	Bausch_lot	0.4008	0.4384	0.3445	0.8382
	Forest_path	0.5470	0.5947	0.4861	0.8951
	Lake	0.5011	0.4700	0.3535	1.0687
	Moto	0.4934	0.5260	0.3725	0.9508
	Overall results	0.3720	0.3034	0.2145	1.0529
		Table 4. Per-content prediction performance of HDR-VDP-2 (original)
	Measure/	C P	C S	C K	RMSE
	HDR content
	Office_ivc	0.5818	0.6030	0.4556	0.8771
	Carpark_ivc	0.8797	0.8909	0.7296	0.5731
	Bausch_lot	0.4852	0.5487	0.4358	0.8019
	Forest_path	0.6464	0.7234	0.5721	0.8157
	Lake	0.9600	0.9650	0.8567	0.3459
	Moto	0.9183	0.9657	0.8649	0.4328
	Overall results	0.7009	0.7389	0.5616	0.8090

Table 5 .

 5 Per-content prediction performance of HDR-VDP-2 (modified)

	Measure/	C P	C S	C K	RMSE
	HDR content
	Office_ivc	0.7433	0.7240	0.5901	0.7214
	Carpark_ivc	0.7610	0.7556	0.5792	0.7817
	Bausch_lot	0.6735	0.6303	0.4944	0.6801
	Forest_path	0.7460	0.7647	0.6040	0.7120
	Lake	0.9420	0.9434	0.7962	0.4145
	Moto	0.9123	0.9260	0.7818	0.4476
	Overall results	0.7201	0.7499	0.5620	0.7871

Table 9 .

 9 Prediction performance of HDR-VDP-2 (original) for each TMO

	Measure/	C P	C S	C K	RMSE
	TMO
	Ashikmin	0.7677	0.7777	0.6104	0.7460
	Durand	0.6192	0.7025	0.5519	0.7479
	Log	0.7624	0.7919	0.6157	0.6187
	Reinhard_global	0.8112	0.8197	0.6655	0.6716
	Reinhard_local	0.7635	0.7835	0.6240	0.5384

Table 10 .

 10 Prediction performance of HDR-VDP-2 (modified) for each TMO

	Measure/	C P	C S	C K	RMSE
	TMO
	Ashikmin	0.8029	0.7832	0.6174	0.6940
	Durand	0.5805	0.6414	0.4726	0.7755
	Log	0.8390	0.8543	0.6717	0.5202
	Reinhard_global	0.7905	0.8105	0.6515	0.7034
	Reinhard_local	0.6665	0.6339	0.4704	0.6215

  Table 12 summarizes the implications of different ranges of F values.

Table 11 .

 11 Test of normality for the residuals (difference between logistically transformed objective predictions and MOSs) from the 5 methods namely MSE, SSIM, SIQM, HDR-VDP-2 (original) and HDR-VDP-2 (modified). '0' implies that the null hypothesis cannot be rejected at 5% significance level and implies normality while '1' denotes the opposite case.

		MSE	SSIM	SIQM	HDR-VDP-2	HDR-VDP-2
					(original)	(modified)
	KS test (0/1)	1	1	0	0	0
	Test statistic	0.1226	0.1301	0.0881	0.0893	0.0811
	Skewness	-0.4676	-0.4783	-0.5728	-0.4811	-0.3090
	Kurtosis	2.0666	2.0585	2.3360	2.9551	2.8186
		Table 12. Interpretation of F-values

Table 13 .

 13 F-test result for the four objective methods. The F values Method1' while the method in each column denotes 'Method2'. The boldface values imply statistically significant difference between the two objective methods.

	significantly	Since F > 1 Method1 performs	Since F < 1 Method2 performs	Method2 has significantly
	larger residuals than		better than Method2 but both	better than Method1 but both	smaller residuals than
	Method1, so Method1 is		are statistically	are statistically	Method1, so Method1 is
	statistically better than		indistinguishable because	indistinguishable because	statistically worse than
	Method1.		F 	critical F	.	F		critical F 1	.	Method2.
						     2 2 Method 1 2 Method F  	   	are computed such that the method in each row
	is 'MSE	SSIM	SIQM			HDR-VDP-2	HDR-VDP-2
	MSE SSIM SIQM HDR-VDP-2 (original)	 0.9926 1.1392 1.9295	1.0074  1.1477 1.9439	0.8778 0.8713  1.6937			(original) 0.5183 0.5144 0.5904 	(modified) 0.4905 0.4869 0.5588 0.9465
	HDR-VDP-2	2.0386	2.0538	1.7895			1.0565
	(modified)

ACKNOWLEDGMENT

This work has been supported by NEVEx project FUI11 which is an FUI (Fond Unique Interministériel) financed project recognized by the Images & réseaux cluster.