Mireille El Gheche
email: mireille.el-gheche@u-bordeaux.fr

Member, IEEE Jean-Franc ¸ois Aujol
email: jean-francois.aujol@math.u-bordeaux1.fr.

Member, IEEE Yannick Berthoumieu
email: yannick.berthoumieu@ims-bordeaux.fr.

Charles-Alban Deledalle
email: charles-alban.deledalle@math.u-bordeaux1.fr.

Texture Reconstruction guided by a High-Resolution Patch

Keywords: Super-Resolution, Texture Synthesis, Texture Reconstruction, Wasserstein distance, Histograms, Nonlocal regularization, Proximal algorithms, Nonconvex optimization. I

In this paper, we aim at super-resolving a lowresolution texture under the assumption that a high-resolution patch of the texture is available. To do so, we propose a variational method that combines two approaches, that are texture synthesis and image reconstruction. The resulting objective function holds a nonconvex energy that involves a quadratic distance to the low-resolution image, a histogram-based distance to the high-resolution patch, and a nonlocal regularization that links the missing pixels with the patch pixels. As for the histogram-based measure, we use a sum of Wasserstein distances between the histograms of some linear transformations of the textures. The resulting optimization problem is efficiently solved with a primaldual proximal method. Experiments show that our method leads to a significant improvement, both visually and numerically, with respect to state-of-the-art algorithms for solving similar problems.

extract a cylindrical sample of rock from a reservoir, scan it through computed tomography (CT), and obtain a stack of texture images, which can be then analyzed to find visual cues related to some petro-physical parameters. A typical example concerns the segmentation and classification of textures, which in turn relate to the porosity levels of the underlying material.

Due to the large size of a rock sample, the CT scanner produces a low-resolution image of the reservoir texture, leading to a loss of important details for the estimation of petro-physical parameters. Fortunately, the resolution can be improved by zooming into a specific part of the sample through a micro-CT scanner. Therefore, in order to make a highresolution image of the reservoir texture, one can perform two CT acquisitions of the same rock sample. The first one supplies a low-resolution image of the entire texture, while the second one provides a high-resolution image of a small part of the texture. The same approach can be used for the analysis of micro-structure materials, whose scanned volume resolution should go to one nanometer to measure its porosity [START_REF] Wildenschild | X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems[END_REF], or when using a digital camera for texture defect detection [START_REF] Xie | A review of recent advances in surface defect detection using texture analysis techniques[END_REF], which plays an important role in automatic inspection of most production processes (food, textile, steel industries, . . . ).

The main focus of this paper is on the super-resolution of a texture image, under the assumption that a small patch of the sought texture is available in high-resolution (the considered setting is illustrated in Fig. 1). Note that this problem can be related to:

• texture synthesis from a high-resolution patch, guided by a low-resolution texture; • image reconstruction from a low-resolution texture, guided by a high-resolution patch;

However, neither of them exactly describes the considered problem. Indeed, by texture reconstruction, we refer to the problem of generating a high-resolution image not necessary pixel-wise identical to the unknown underlying image, but only identical in terms of textural statistics. Obviously, the sought image should satisfy the constraint that its low-resolution version is faithful to the observed low-resolution one (at least up to the noise).

A. Related work a) Reconstruction: The simplest way to upscale a lowresolution image is by mean of linear scaling (such as bi-cubic interpolation [START_REF] Keys | Cubic convolution interpolation for digital image processing[END_REF]) or image sharpening methods [START_REF]Image upsampling via imposed edge statistics[END_REF]. However, these methods introduce high frequency concealing, which blur the image and erase small texture details.

A number of single-image reconstruction methods are based on the assumption that a dataset of high-resolution textures are available, from which they learn a dictionary of lowresolution patches, a dictionary of high-resolution patches, and a correspondence map between the two dictionaries [START_REF] Yang | Coupled dictionary training for image super-resolution[END_REF], [START_REF] He | Single image super-resolution using gaussian process regression[END_REF], [START_REF] Zhang | Single image super-resolution with non-local means and steering kernel regression[END_REF]. However, the results depend on the estimated dictionaries, and thus the reconstruction of true details is not guaranteed if the target texture does not appear in the training dataset. A similar reconstruction method was described in [START_REF] Chang | Super-resolution through neighbor embedding[END_REF], where the k-nearest neighbors are searched into an external patch database. Instead of using external data, the authors in [START_REF] Glasner | Super-resolution from a single image[END_REF] extract correspondences at different scales directly from the low-resolution image. Some recent approaches learn mapping functions from external low-and high-resolution exemplar pairs [START_REF] Dong | Learning a Deep Convolutional Network for Image Super-Resolution[END_REF], [START_REF] Schulter | Fast and accurate image upscaling with super-resolution forests[END_REF], [START_REF] Dai | Jointly optimized regressors for image super-resolution[END_REF]. Nevertheless, these techniques are not very robust to noise.

A different approach consists of introducing a suitable regularization into the reconstruction problem formulation, with the aim of conveying some prior knowledge about the signal to be recovered. In this context, Total Variation (TV) [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF], [START_REF] Combettes | Image restoration subject to a total variation constraint[END_REF], [START_REF] Farsiu | Fast and robust multiframe super resolution[END_REF], [START_REF] Mitzel | Video super resolution using duality based tv-l 1 optical flow[END_REF] has emerged as a simple regularization, consisting in penalizing the gradient coefficients. However, TV fails to preserve textures, details, and structures, because they are hardly distinguishable from noise. To improve this behavior, the TV model has been extended by using higher-order spatial differences [START_REF] Bredies | Total generalized variation[END_REF], [START_REF] Ono | Second-order Total Generalized Variation constraint[END_REF], higher-degree directional derivatives [START_REF] Hu | Higher degree total variation (HDTV) regularization for image recovery[END_REF], [START_REF]Generalized higher degree total variation (HDTV) regularization[END_REF], or the nonlocality principle [START_REF] Peyré | Non-local regularization of inverse problems[END_REF], [START_REF] Gilboa | Nonlocal operators with applications to image processing[END_REF], [START_REF] Protter | Generalizing the nonlocal-means to super-resolution reconstruction[END_REF], [START_REF] Freedman | Image and video upscaling from local self-examples[END_REF], [START_REF] Maggioni | Nonlocal transform-domain filter for volumetric data denoising and reconstruction[END_REF], [START_REF] Couprie | Dual constrained TV-based regularization on graphs[END_REF], [START_REF] Chierchia | A nonlocal structure tensor based approach for multicomponent image recovery problems[END_REF], [START_REF] Chierchia | Epigraphical projection and proximal tools for solving constrained convex optimization problems[END_REF]. The latter approach leads to the so-called nonlocal total variation (NLTV) regularization.

b) Texture synthesis: Texture synthesis techniques [START_REF] Dai | The synthesizability of texture examples[END_REF] can be broadly categorized into region-growing local methods and optimization-based global methods. Local approaches grow the texture one pixel (or patch) at a time, while maintaining the spatial coherence with nearby pixels by modeling the neighborhoods with Markov fields and fractal models [START_REF] Cross | Markov random field texture models[END_REF]. A weakness of these methods is that the spatial coherence between pixels is enforced at a local scale. A possible approach to circumvent this limitation consists of resorting to a small patch from which to grow the texture, following a procedure that sequentially processes the pixels [START_REF] Efros | Texture synthesis by non-parametric sampling[END_REF], [START_REF] Wei | Fast texture synthesis using tree-structured vector quantization[END_REF], [START_REF] Lefebvre | Parallel controllable texture synthesis[END_REF], [START_REF] Hertzmann | Image analogies[END_REF], [START_REF] Darabi | Image Melding: Combining inconsistent images using patch-based synthesis[END_REF]. In such methods, however, small errors can accumulate over large distances, leading to inconsistencies in the synthesized texture. Another possible scenario is to use several exemplars describing the texture at various scales and hence generate textures with arbitrary amount of details [START_REF] Han | Multiscale texture synthesis[END_REF], [START_REF] Eisemann | ZIPMAPS: zoom-into-parts texture maps[END_REF].

Global methods process the entire texture as a whole, using some criteria for measuring its similarity with a small texture patch. For example, the latter can be modeled with a statistical descriptor based on histograms [START_REF] Heeger | Pyramid-based texture analysis/synthesis[END_REF], [START_REF] Kaspar | Self tuning texture optimization[END_REF], wavelet coefficients [START_REF] Portilla | Image denoising via adjustment of wavelet coefficient magnitude correlation[END_REF], or Fourier coefficients [START_REF] Galerne | Random phase textures: Theory and synthesis[END_REF]. A similar approach was recently proposed in [START_REF] Tartavel | Constrained sparse texture synthesis[END_REF], [START_REF] Tartavel | Variational texture synthesis with sparsity and spectrum constraints[END_REF], which introduces a preliminary step of dictionary learning for exploiting the given patch, and (not least) the Wasserstein distance for comparing the histograms of the entire texture with an extended version of the small patch. While the Wasserstein distance is well-known in image processing and computer vision under the name "earth mover distance" [START_REF] Rubner | The earth mover's distance as a metric for image retrieval[END_REF], it was only recently expanded to the context of texture synthesis [START_REF] Rabin | Wasserstein regularization of imaging problem[END_REF], [START_REF] Tartavel | Variational texture synthesis with sparsity and spectrum constraints[END_REF].

Regarding the problem considered in this paper (recall Fig. 1), an extension of NLTV regularization was recently proposed in [START_REF] Hidane | Image zoom completion[END_REF]. This prior consists in building a pointwise estimate of the image, where each missing pixel is connected to a local neighborhood and to the most similar pixels in the given high resolution patch. The author in [START_REF] Fablet | Missing data super-resolution using non-local and statistical priors[END_REF] proposed a statistical prior in addition to the standard NLTV regularization. This prior is based on a nonparametric spatial covariance structure from empirical estimation, and a parametric generalized Gaussian model learned from the high-resolution patch.

B. Contributions

In this paper, we follow a variational approach that aims at super-resolving a low-resolution texture (possibly degraded by blur and zero-mean additive Gaussian noise) by explicitly taking into account a high-resolution patch. To do so, we propose an algorithm for optimizing a nonconvex energy that involves two problem-specific terms in addition to the standard quadratic distance to the low-resolution image. The first term is a sum of Wasserstein distances to the histograms of the high-resolution patch, of its gradient, and of its Laplacian. The second term is a NLTV regularization that measures the distance of each pixel with a selected subset of pixels in the high-resolution patch. The variational approach adopted in this paper was inspired from [START_REF] Tartavel | Modèles variationnels pour les textures : applications à la synthèse et à la restauration[END_REF], which uses the Wasserstein distance for comparing the histograms of the entire texture with a parametric model learned from the small patch, and introduces a preliminary step of dictionary learning for exploiting the given patch, in order to perform texture synthesis and image restoration.

There exist few methods in the literature that consider the texture synthesis problem guided by a low-resolution texture [START_REF] Hidane | Image zoom completion[END_REF], [START_REF] Fablet | Missing data super-resolution using non-local and statistical priors[END_REF]. These approaches have some similarities with the proposed one with notable important differences:

• In [START_REF] Hidane | Image zoom completion[END_REF], the problem is solved by taking into account a nonlocal model. The approach consists in connecting each missing pixel to a set of neighboring pixels and some other pixels in the high-resolution patch. However, this approach neglects the statistical information that can be extracted from the high-resolution patch. The main novelty of the proposed approach w.r.t. [START_REF] Hidane | Image zoom completion[END_REF] lies in the introduction of statistical prior that preserves the texture details, as well as the design of a new nonlocal graph that provides better connections between the missing pixels and the high-resolution patch pixels. • To interpolate the missing data, the work in [START_REF] Fablet | Missing data super-resolution using non-local and statistical priors[END_REF] exploits a nonlocal regularization and a Generalized Gaussian distributions model of the texture gradients, whose parameters are learned on the high-resolution patch. Differently from [START_REF] Fablet | Missing data super-resolution using non-local and statistical priors[END_REF], we enforce a histogram prior using the Wasserstein distance, linking our approach with the optimal transport [START_REF] Villani | Topics in optimal transportation[END_REF]. The originality of our technique consists of the ability to consider multi-histogram priors (such as the intensity values, the gradients and the Laplacians histograms [START_REF] Tartavel | Modèles variationnels pour les textures : applications à la synthèse et à la restauration[END_REF]Chapter 5]) without being constrained to a parametric model, regardless of the histogram shapes (unimodal, bimodal or multimodal mixture).

The paper is organized as follows. Section II describes the degradation model, the histogram-based distance, and the new graph for the NLTV regularization. Section III presents an algorithm for solving the proposed optimization problem via proximal tools. Section IV provides an experimental validation in the context of texture images. Finally, the conclusion is given in Section V.

C. notation

Let • be the standard Euclidean norm, and let Id be the identity matrix. The domain of a function f :

R N → ] -∞, +∞] is dom f = {x ∈ R N |f (x) < +∞}. Γ 0 (R N ) is the class of lower semi-continuous convex functions from R N to ] -∞, +∞] such that dom f = ∅. Let f ∈ Γ 0 (R N ). The conjugate of f is the function f * ∈ Γ 0 (R N ) defined by f * : R N →] -∞, +∞] : u → sup x∈R N x u -f (x). When f is Gâteaux-differentiable at y ∈ R N , ∂f (y) = {∇f (y)} where ∇f (y) is the gradient of f at y. A differentiable convex function has β-Lipschitz continuous gradient ∇f if (∀(x, y) ∈ R N × R N ) ∇f (x) -∇f (y) ≤ β x -y , where β ∈ ]0, +∞[. Let C be a nonempty subset of R N , then ι C is the indicator function of C, equal to 0 on C and +∞ otherwise.

II. PROPOSED APPROACH

The high-resolution signal of interest is denoted by x ∈ R N , which generally corresponds to an image of size

N = N 1 ×N 2 .
The degradation model that we consider is the following:

z (1) = DBx + η 1 , z (2) = Mx. (1)
Hereabove, z (1) ∈ R Q is the complete low-resolution image, B ∈ R N ×N is a linear operator modeling some blur, D ∈ R Q×N stands for spatial down-sampling by a dyadic factor in each direction, yielding Q = 2 -r N , and η 1 ∈ R Q is a realization of an additive zero-mean white Gaussian noise with standard deviation τ . Moreover, z (2) ∈ R M denotes a small patch of the high-resolution image x, and M ∈ R M ×N is a selection operator that extracts the high-resolution patch from x.

We propose to recover x from the observations z (1) and z (2) through a variational approach that leads to solving the following optimization problem:

minimize x∈R N DBx -z (1) 2 + H(x, z (2) ) + λ R(x, z (2) ) s. t. Mx = z (2) , (2)
where λ > 0 is a regularization parameter. Beside the data fidelity terms w.r.t. the observations z (1) and z (2) , we use two additional pieces of information: a term H(•, z (2) ) conveying some histogram-based statistics, and a term R(•, z (2) ) enforcing a new type of nonlocal regularization. The histogram-based term is modeled through the Wasserstein distance discussed in Section II-A, while the regularization is grounded on the nonlocal approach presented in Section II-B.

The presented approach takes into account an image regularization and a global statistical information in one model. Note that, minimizing the term R(•, z (2) ) entails spatial regularization, whereas H(•, z (2) ) involves statistical information that is not spatially indexed. However, taking into account the statistical information through histogram priors, the proposed approach infers the correct structures and details of the highresolution patch to the estimated structure, leading to improved results w.r.t. the classical regularizations considered alone.

A. Wasserstein distance to the high-resolution patch

Wasserstein distance is a well-known dissimilarity measure between probability distributions, as it is easier to optimize than other distances, such as the ϕ-divergences (Kullback-Leibler, Hellinger, Jeffreys-Kullback, • • • ) [START_REF] Rabin | Wasserstein regularization of imaging problem[END_REF], [START_REF] Tartavel | Variational texture synthesis with sparsity and spectrum constraints[END_REF].

The Wasserstein distance between the histograms 1 of two images u ∈ R N and v ∈ R N is defined as follows:

W 2 2 (ν u , ν v ) = min σ∈Σ N u -v • σ 2 , (3)
where ν u and ν v are the normalized histograms of u and v, the symbol v • σ denotes a permutation of the vector v, and Σ N is the set of all the permutations of N -length vectors. For grayscale images, the optimal permutation σ * is computed as

σ * = σ v • σ -1 u , (4)
where σ v (resp. σ u ) denotes the permutation operator that arranges the pixels of v (resp. u) in ascending order (see Fig. 2). In our case, however, the two images have a different number of pixels (u ∈ R N and v ∈ R M , with M < N ). Hence, we replicate the patch v so as to obtain a larger image ṽ ∈ R N such that the normalized histogram ν ṽ is equal to ν v . 2 Although the Wasserstein distance is nonconvex (due to the histogram transformation), its gradient is Lipschitz-continuous and takes the following form [START_REF] Tartavel | Constrained sparse texture synthesis[END_REF], [START_REF] Tartavel | Modèles variationnels pour les textures : applications à la synthèse et à la restauration[END_REF]Chapter 5]

∇ u W 2 2 (ν u , ν ṽ ) = 2(u -ṽ • σ ṽ • σ -1 u ). (5)
This property allows us to employ the Wasserstein distance into the optimization algorithm presented in Section III. Our first contribution is to use the Wasserstein distance for comparing the histograms of the intensity values, their gradient, and their Laplacian, in order to ensure that the synthesized image has similar details as the high-resolution patch. In other words, we want to restore the gray levels, the gradients and the Laplacian values of the textures. To do so, for every s ∈ {1, . . . , 4}, let L s be a matrix in R Ns×N defined as follows.

• L 1 : the identity matrix (N 1 = N ).

• L 2 : the concatenation of the horizontal and vertical difference operators (N 2 = 2N ). • L 3 : the concatenation of the diagonal difference operators (N 3 = 2N ). • L 4 : the isotropic Laplacian operator (N 4 = N ). We define the term H in (2) as

H(x, z (2) ) = 4 s=1 α s W 2 2 (ν Lsx , ν z(2) s ) (6)
where, for every ∀s ∈ {1, . . . , 4}, α s > 0, z

s = L s z (2) and z(2) s is the extension of z (2) s (2)
(after the linear transformation).

B. Nonlocal regularization

The quality of the images reconstructed with the nonlocal regularization highly depends on the graph used to model the similarity between pixels. Remarkable results can be obtained when a good estimate of the true similarity graph is available. However, in the context of super-resolution, this task presents some challenges, due to the missing information in the low-resolution image. One way to circumvent this issue consists of estimating the standard nonlocal graph over an interpolated version of the low-resolution image. Unfortunately, this approach performs poorly on texture images, as the graph connections are limited to local neighborhoods. A better solution was proposed in [START_REF] Hidane | Image zoom completion[END_REF], where the missing pixels in the low-resolution image are linked (by patch similarity) to a set of pixels in the high-resolution patch. Since the original pixels are blurry, the pixels to which they are connected are also blurry, because the quadratic distance used to evaluate the similarity is not robust to this kind of degradation. Hence, the image reconstructed with such a nonlocal graph is necessarily blurred. The proposed approach is different, since each pixel is connected to the similar ones in the high-resolution patch, which are not blurry. Hence, the similarity scores are more likely to be reliable, resulting in a denoised image of better quality.

In the following, we present the way to build the new nonlocal graph (see Fig. 3), and we review two state of the art approaches: the conventional one and the one proposed in [START_REF] Hidane | Image zoom completion[END_REF]. 1) The proposed graph: we define the function R in (2) as follows

R(x, z (2) ) = N i=1 j∈Ni w i,j x i -z (2) j 2 , (7)
where N i ⊂ {1, . . . , M } is a subset of positions w.r.t. the support of z (2) , and w i,j > 0. For every i ∈ {1, . . . , N }, the support N i and the weights (w i,j ) j∈Ni are computed as explained in the following.

• Firstly, we select the pixel z

(i) that is the most similar to x i . However, since the high-resolution image x is not available at this stage, we interpolate the low-resolution image z (1) ∈ R Q with a bicubic approach, so as to obtain a new image z (1) ∈ R N that approximates x without loosing its spectral structure. Once we have z (1) , we find the index j (i) according to the criterion

j (i) = argmin m∈{1,...,M } p i ( z (1) ) -p m (z (2) ) 2 , (8)
where p i ( z (1) ) is a block extracted from z (1) and centered around the position i, and p m (z (2) ) is defined similarly. • Secondly, we define N i as the set of positions j 1 , . . . , j K such that z

(2) j k 1≤k≤K are the K most similar pixels to z (2) j (i) . To do so, for every m ∈ {1, . . . , M }, we compute the Euclidean distance between the blocks p j (i) (z (2) ) and p m (z (2) ), and we select the positions (j k ) 1≤k≤K associated to the K lowest distances, which of course includes j (i) itself (i.e., j 1 = j (i) ).

• Finally, we compute the weights w i,j as follows

(∀j ∈ N i ) w i,j = e -δ p j(i) (z (2) )-pj (z (2) ) 2 , ( 9
)
where δ is a positive constant. The nonlocal energy proposed in [START_REF] Zhang | Single image super-resolution with non-local means and steering kernel regression[END_REF] differs from the standard nonlocal regularization, as it takes into account the information carried by the high-resolution patch. To clarify this concept, we now review two nonlocal approaches related to our work.

2) The conventional approach: The classical NLTV consists of computing, for every i ∈ {1, . . . , N }, the most similar pixels lying in a local neighborhood, yielding the energy

NLTV(x) = N i=1 j∈Ni⊂Wi w i,j (x i -x j ) 2 , ( 10
)
where W i is the set of positions located into a Q × Q window W i ⊂ {1, . . . , N }\{i} centered at i. For every i ∈ {1, . . . , N }, the support N i includes the positions of the K most similar pixels to x i that are located within W i . Since the image x is not available, one can interpolate z (1) ∈ R Q and obtain z (1) ∈ R N , which is used for building N i and for computing the associated weights

(∀j ∈ N i ) w i,j = e -δ pi( z (1) )-pj ( z (1) ) 2 . ( 11
)
3) The approach in [START_REF] Hidane | Image zoom completion[END_REF]: This approach consists of connecting each pixel (x i ) i∈{1,...,N } to its K-nearest observed pixels in the high-resolution patch z (2) . The nonlocal regularization is expressed as

NL-HR(x) = N i=1 j∈Ni w i,j (x i -z (2) j ) 2 , (12)
where N i ⊂ {1, . . . , M } includes the positions of the pixels in z (2) that are the K most similar to x i . Just like before, the interpolated image z (1) ∈ R N is actually used for building N i and for computing the associated weights

(∀j ∈ N i ) w i,j = e -δ pi( z (1) )-pj ( z (1) ) 2 . ( 13
)
Note that the approach in [START_REF] Hidane | Image zoom completion[END_REF] builds the similarity graph over an interpolated version of the low-resolution image. Conversely, we propose to build the graph by only using the high-resolution patch (see Fig. 3). This makes a very important difference in the quality of reconstructed images, as shown in Section IV.

III. OPTIMIZATION

The solution of Eq. ( 2) requires an efficient algorithm for dealing with problems involving nonsmooth functions and linear operators that are non-necessarily circulant. Recently, it has been shown experimentally that primal-dual proximal methods [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF], [START_REF] Briceño-Arias | A monotone + skew splitting model for composite monotone inclusions in duality[END_REF], [START_REF] Combettes | Primal-dual splitting algorithm for solving inclusions with mixtures of composite, lipschitzian, and parallelsum type monotone operators[END_REF], [START_REF] Vũ | A splitting algorithm for dual monotone inclusions involving cocoercive operators[END_REF], [START_REF] Condat | A primal-dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms[END_REF], [START_REF] Komodakis | Playing with duality: An overview of recent primal-dual approaches for solving large-scale optimization problems[END_REF], which were originally designed for convex optimization, can be also applied to nonconvex problems in some circumstances [START_REF] Storath | Jump-sparse and sparse recovery using Potts functionals[END_REF], [START_REF] Strekalovskiy | Real-Time Minimization of the Piecewise Smooth Mumford-Shah Functional[END_REF], [START_REF] Möllenhoff | Energy Minimization Methods in Computer Vision and Pattern Recognition: 10th International Conference[END_REF], [START_REF] Möllenhoff | The primaldual hybrid gradient method for semiconvex splittings[END_REF]. In the convex setting, the key tool of these methods is the proximity operator [START_REF] Moreau | Fonctions convexes duales et points proximaux dans un espace hilbertien[END_REF] of a lower semicontinuous convex function ϕ : R N →] -∞; +∞], defined as

(∀y ∈ R N ) prox ϕ (y) = argmin z∈R N ϕ(z)+ 1 2 z -y 2 . ( 14
)
The proximity operator can be interpreted as an implicit subgradient step for the function ϕ, since p = prox ϕ (y) is uniquely defined through the inclusion y -p ∈ ∂ϕ(p). Proximity operators enjoy many properties [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF]. In particular, they generalize the notion of projection onto a closed convex set C, in the sense that prox ι C = P C . Hence, proximal methods provide a unifying framework that allows one to address a wide class of convex optimization problems involving nonsmooth penalizations and hard constraints. Among the wide array of existing proximal algorithms, we employ the Forward-Backward Primal Dual method (FBPD) [START_REF] Condat | A primal-dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms[END_REF] reported in Algorithm 1. The operators required by this algorithm are detailed below.

• The nonlocal regularization can be expressed as the 1,2 -norm composed with a discrete difference operator, yielding R(x, z (2)

) = Tx 1,2 , (15)
where

Tx =     w 1,j (x 1 -z (2) j ) j∈N1 . . . w N,j (x N -z (2) j ) j∈N N     } ∈ R K . . . } ∈ R K . ( 16
)
The proximity operator of the 1,2 -norm, that is prox • 1,2 in Algorithm 1, can be found in [START_REF] Combettes | Signal recovery by proximal forwardbackward splitting[END_REF]. • The projection onto the convex set associated to the constraint Mx = z (2) is expressed as

P {M•=z (2) } (x) = x + M (z (2) -Mx). ( 17
)
• The gradient of the sum of the remaining terms, that is

f (x) = DBx -z (1) 2 + 4 s=1 α s W 2 2 (ν Lsx , ν z(2) s ), reads ∇f = 2 B D (D Bx -z (1) ) + 2 4 s=1 α s L s (L s x -z(2) s • σ z(2) s • σ -1 Lsx ), ( 18
)
where ∇f is β-Lipscitz with β = 2(1 +

4 1 α s L s 2 ).
Although there is no theoretical guarantee about the estimate produced by Algorithm 1, in our experiments we observed that it always converges to a stable solution.

Algorithm 1 FBPD [54] INITIALIZATION       Choose x [0] , y [0] ∈ R n × R Kn set τ > 0 and ω > 0 such that τ β/2 + ωλ T 2 < 1 FOR l = 0, 1, . . .          x [l] = ∇f (x [l] ) + T y [l] x [l+1] = P {M • =z (2) } x [l] -τ x [l] y [l] = T 2x [l+1] -x [l] y [l+1] = prox ωλ • * 1,2 y [l] + ω y [l]
IV. EXPERIMENTAL RESULTS

The numerical analysis is decomposed in three parts. Firstly, we assess the performance achieved by only using the nonlocal regularization proposed in Section II-B. Secondly, we extend our analysis to the histogram-based distance proposed in Section II-A, by evaluating its impact on the performance. Finally, we compare the full approach with several state-ofthe-art methods.

As shown in Fig. 4, all the test images are grayscale textures, such as Nature (Grass, Leaves, Leaf, Flowers), Forest (Bark, Wall), Hair, Biology (Cell), and Petrology (CT scan). [START_REF] Hidane | Image zoom completion[END_REF], and the classical NLTV [START_REF] Chierchia | Epigraphical projection and proximal tools for solving constrained convex optimization problems[END_REF].

A. Error measures

The simplest and the most popular quality measure is the Mean Squared Error (MSE) along with the related quantity of Peak Signal-to-Noise Ratio (PSNR). However, these indexes may not be good evaluation metrics for the considered problem, since we are dealing with texture reconstruction, and not with image reconstruction. In this paper, we consider the visual inspection to asses the image quality in terms of the downsampling factor and the high-resolution patch size. In order to provide an objective measure, we will evaluate the results with the SSIM index (the closer to 1, the better the quality) [START_REF] Malpica | Encyclopedia of Multimedia[END_REF] and we propose to add a new statistical error measure Err x, defined as

Err x = 4 s=1 α s N s W 2 2 (ν Ls x, ν Ls x), (19)
where x denotes an estimate of x. This error aims at showing how much the resulting image is statistically close to the true high-resolution image. Indeed, the closer Err x to 0, the better the results.

To allow a reliable assessment, we will add the PSNR indices to the numerical evaluation. The aim of this step is twofold: to be homogeneous to the state-of-the-art methods evaluation and to prove that the PSNR might not reflect the visual quality of reconstructed images.

B. Nonlocal regularization

In this section, we test in isolation the proposed regularization. To do so, we consider the texture denoising problem arising from (2) by setting D = I, B = I, and (α s ) {1≤s≤4} = 0, resulting in the following optimization problem (2) .

minimize x∈R N x -z (1) 2 + λ R(x, z (2) ) s. t. Mx = z
The 8-bit grayscale images are of size 256 × 256, and z (2) is the half of the original image x. The observation z (1) is generated by degrading x with an additive Gaussian noise whose standard deviation is equal to 50. The optimization is performed using Algorithm 1. The parameter λ is hand-tuned so as to obtain the best visual results. For the nonlocal graph detailed in Section II-B, K has been fixed to 14, Q to 25 and the blocks to 5 × 5 pixels.

Fig. 5 gives an example of natural textures, whose elements hold a rough, irregular and nonsmooth patterns. These delicate textures are difficult to recover when corrupted by noise, due to the concealing effect on the small details. Having a noiseless patch, we aim at recovering these details using a nonlocal regularization, whose graph is constructed as proposed in Section II-B. The visual inspection of the results shows the interest of considering the proposed nonlocal regularization R(•, z (2) ) for texture denoising. Indeed, R(•, z (2) ) proves to be more effective than NL-HR [START_REF] Hidane | Image zoom completion[END_REF], which in turn outperforms the classical NLTV [START_REF] Chierchia | Epigraphical projection and proximal tools for solving constrained convex optimization problems[END_REF] that tends to average the pixel values. The better performance of R(•, z (2) ), w.r.t. NL-HR, seems to be related to its ability to better choose connections to the noiseless patch, which in turn preserve the edges and the thin structures.

C. Discussion of the proposed approach

In this section, we compare the approach proposed in (2) w.r.t. the following methods

• when λ = 0 (without the nonlocal regularization)

minimize x∈R N DBx -z (1) 2 + H(x, z (2) ) s. t. Mx = z (2) , (20)
• when, for every s ∈ {1, • • • , 4}, α s = 0 (without the statistical prior)

minimize x∈R N DBx -z (1) 2 + λ R(x, z (2) ) s. t. Mx = z (2) . (21)
The parameters λ and (α s ) 1≤s≤4 are set with the following strategy:

In these experiments, x corresponds to a 256 × 256 8-bit grayscale image. The observed image z (1) is generated by degrading x with a convolution operator B, which is equal to a truncated Gaussian function with standard deviation 1.2 and kernel size 3 × 3, the blurry result is then downsampled by a factor r = 2 in each direction, and finally we add to this latter a white Gaussian noise of standard deviation τ = 10. The observed high-resolution patch z (2) amounts to 25% of the total image x. The quality of the results is evaluated in term of PSNR, SSIM and Err x.

Fig. 6 gives two examples of texture images: a leaf and a biological cell. One can observe that the visual results achieved with Problem (2) are better than the ones obtained with the standard formulations [START_REF]Generalized higher degree total variation (HDTV) regularization[END_REF] and [START_REF] Peyré | Non-local regularization of inverse problems[END_REF]. Indeed, without the nonlocal regularization, we do not have a spatial prior. As the problem is nonconvex, we may get stuck in a local minimum that is near to the patch z (2) , but far from the true image x. However, without the statistical prior, even if the nonlocal regularization performs quite well to recover the smooth areas and to reconstruct the edges, it fails to infer the correct image structures. Using the proposed formulation (2), we gain the best visual results and the best SSIM -Err x, which indicates that the nonlocal regularization pushes the local minima of (20) closer to x. While the best PSNR is given by the results of Eq. ( 21), this is not reflected on the visual inspection. Such a behaviour can be explained by the fact that the SSIM and Err x measures take in consideration some features, while the PSNR fails to illustrate them (see Fig. 6).

In Table I, we evaluate the experiments on several images: Wall, Flowers, Leaves, Leaf, Hair, Cell 3 and Bark 2 (see Fig. 4). We report the PSNR, SSIM and the error measure to the true image. We present the numerical results obtained with [START_REF]Generalized higher degree total variation (HDTV) regularization[END_REF] (without the nonlocal prior) and (2) (with the nonlocal prior). As one can see the latter approach achieves the best performance in term of PSNR and SSIM, as well as the smallest error in term of W 2 2 (ν Ls x, ν Ls x) for almost every s ∈ {1, • • • , 4}, and the better mean weighted value Err x regardless of the degradation conditions. The best values for each case is put in boldface for the sake of clarity.

D. Comparison to the state-of-the-art methods

We now compare our results with the following approaches:

• Dong et al. [START_REF] Dong | Image deblurring and super-resolution by adaptive sparse domain selection and adaptive regularization[END_REF]: the method is based on the assumption that a dataset of high-resolution textures is available, from which it is learned a dictionary of low-resolution patches, a dictionary of high-resolution patches, and a correspondence map between the two dictionaries. • Hertzmann et al. [START_REF] Hertzmann | Image analogies[END_REF]: given an image A and its transformation A', the method provides a new image B' from an image B, such that the relationship between B and B' is the same as the one between A and A'. • Portilla et al. [START_REF] Portilla | A parametric texture model based on joint statistics of complex wavelet coefficients[END_REF]: this is a texture synthesis method based on a variational approach and the wavelet transform. • The convolution operator is equal to a truncated Gaussian function with standard deviation 1.2 and kernel size 3 × 3. • The down-sampling factor r = 4 in Fig. 7 (r = 2 in Fig. 8). • Noise standard deviation τ = 10.

• z (2) matches 25% of x in Fig. 7 (6.25% in Fig. 8) Fig. 7 gives two texture examples where the down-sampling factor is equal to 4, and the high-resolution patch amounts to 25% of the total image. As illustrated by these experiments, the proposed approach leads to a better texture reconstruction in the synthesized images, while the state-of-the-art super-resolution method [START_REF] Dong | Image deblurring and super-resolution by adaptive sparse domain selection and adaptive regularization[END_REF] tends to smooth the textures, and the state-of-theart image analogies technique [START_REF] Hertzmann | Image analogies[END_REF] does not preserve textures and produces a final image that is highly noisy. With regard to texture synthesis approach [START_REF] Portilla | A parametric texture model based on joint statistics of complex wavelet coefficients[END_REF], it produces a high-resolution image (by replicating the patch) without exploiting information provided by the low-resolution image. Hence, the results may be far from the ground truth. The improvement of the proposed method is confirmed both visually and numerically by the SSIM and Err x. However, the PSNR fails to assess the image quality.

Fig. 8 gives a similar example where the down-sampling factor is equal to 2 and the high-resolution patch amounts to 6.25% of the total image. The proposed approach gives the best visual results compared to the state-of-the-art methods. This is also confirmed by looking at the reconstructed textures and the SSIM -Err x indexes. Again, the PSNR fails to assess the image quality.

V. CONCLUSIONS AND FUTURE WORK

In this paper, we have proposed a variational approach to solve the problem of texture synthesis guided by a lowresolution image and a high-resolution patch. To do so, we have exploited two different sources of information: a nonlocal smoothness prior and a distance between histograms. The resulting model is well adapted to textures with sharp edges and small patterns, as shown in Fig 8 . We have investigated a primaldual proximal algorithm, which allows us to consider various convex and nonconvex functions. Interesting perspectives include a better modeling of the textures, possibly through constraints on the power spectrum of images as in [START_REF] Tartavel | Variational texture synthesis with sparsity and spectrum constraints[END_REF], [START_REF] Fablet | Missing data super-resolution using non-local and statistical priors[END_REF], as well as the use of a multi-scale strategy for a down-sampling factor greater than 4. Another perspective is to explore the convex relaxation of the histogram prior in order to have a convex minimization problem [START_REF] Swoboda | Convex variational image restoration with histogram priors[END_REF]. Finally, it could be interesting to explore the scenario in which the high-resolution patch is replaced by an intermediate image obtained with some existing state-of-art texture synthesis algorithm.

Fig. 1 .

 1 Fig. 1. The left figure shows a complete low-resolution acquisition, which was up-sampled by a factor 2 for visualization purposes. The middle picture displays a high-resolution patch of the target texture. Our goal is to combine both observations in order to get a high-resolution texture (right figure).

1 uFig. 2 .

 12 Fig. 2. A simple example showing histogram transfer from v to u.

Fig. 3 .

 3 Fig. 3. Brodatz Bark image: bicubic interpolation of the low resolution image with (a) the high-resolution patch, (b) the location of the high-resolution patch (black square). Dashed Lines are the transitional links to build the final graph (straight lines).

3

 3

Fig. 4 . 20 Fig. 5 .

 4205 Fig. 4. Test images. From the left to the right, top line: Grass, Bark 1, Bark 2, Wall, Hair and Leaves. From the left to the right, bottom line: Leaf, Flowers, Cell 1, Cell 2, Cell 3, and CT image scan of a petroleum reservoir.

Fig. 7

 7 Fig.7and 8 present a comparison of the proposed solution with the state-of-the-art results for the following data:• x corresponds to a 256 × 256 8-bit grayscale image.

z ( 1 ) z ( 2 ) 35 Fig. 6 .

 12356 Fig.6. PSNR -SSIM -Errx. From the left to the right: low-resolution image (down-sampling factor r = 2, noise standard deviation τ = 10, blur 3 × 3), high-resolution patch (25% of the total image), true image, the proposed approach (with the statistical prior and the nonlocal regularization), the proposed approach without the nonlocal regularization, and the proposed approach without the statistical prior.

z ( 1 ) 53 Fig. 7 . 50 Fig. 8 .

 1537508 Fig. 7. PSNR -SSIM -Errx. From the left to the right: low-resolution image (down-sampling factor r = 4, noise standard deviation τ = 10, blur 3 × 3), high-resolution patch (25% of the total image), true image, proposed approach, Dong et al. [64], Hertzmann et al. [34] and Portilla et al. [65].

TABLE I ERROR

 I MEASURES ON THE PROPOSED APPROACH WITH AND WITHOUT THE NONLOCAL PRIOR R(•, z (2) ) (DOWN-SAMPLING FACTOR r = 2, NOISE STANDARD DEVIATION τ = 10, BLUR 3 × 3). THE IMAGES ARE EVALUATED USING THE PSNR, SSIM, AND THE WASSERSTEIN DISTANCE BETWEEN INTENSITY VALUE HISTOGRAMS (s = 1), HORIZONTAL-VERTICAL GRADIENT HISTOGRAMS (s = 2), DIAGONAL GRADIENT HISTOGRAMS (s = 3), OR LAPLACIAN HISTOGRAMS (s = 4), AND THE ERROR TO THE TRUE IMAGE Errx (19).

	Images	Regularization	PSNR SSIM		W 2 2 (ν Ls x, ν Ls x)
					s = 1	s = 2	s = 3	s = 4	Errx
	Wall	without R(•, z (2) ) with R(•, z (2) )	28.02 28.84	0.863 0.876	10.26 5.68	2.47 0.99	8.30 3.17	5.27 5.19	8.20 4.56
	Flowers	without R(•, z (2) ) with R(•, z (2) )	26.83 27.53	0.832 0.835	14.49 12.43	0.45 0.48	2.71 2.24	0.20 0.25	3.45 2.97
	Leaves	without R(•, z (2) ) with R(•, z (2) )	23.57 24.06	0.892 0.895	45.49 21.03	13.66 12.56	26.74 22.77	63.21 42.19	20.96 12.15
	Leaf	without R(•, z (2) ) with R(•, z (2) )	24.02 25.29	0.839 0.845	37.92 8.64	0.79 0.48	3.73 2.35	3.03 5.36	4.91 1.70
	Hair	without R(•, z (2) ) with R(•, z (2) )	19.43 19.61	0.679 0.682	45.12 40.03	10.87 10.74	26.51 26.10	39.84 39.45	5.49 5.47
	Cell 3	without R(•, z (2) ) with R(•, z (2) )	21.99 22.36	0.763 0.769	25.08 3.81	4.25 2.92	6.70 8.04	3.68 2.17	2.43 0.82
	Bark 2	without R(•, z (2) ) with R(•, z (2) )	24.77 25.79	0.858 0.859	47.99 21.94	1.32 0.56	5.09 2.01	0.84 2.34	20.14 9.38

The histogram of a signal refers to a histogram of the pixel intensity values.

Another solution consists in oversampling the sorted sequence v • σv using a nearest-neighbor approximation.

The authors would like to thank "Sismage" team from TOTAL Group for providing us the CT data.

• The data fidelity term is normalized by the noise standard deviation τ ; • λ is set to 100;• For every s ∈ {1, • • • , 4}, α s = τs τ4, where τ s is the variance of L s z[START_REF] Xie | A review of recent advances in surface defect detection using texture analysis techniques[END_REF] .

This work was supported by the French National Research Agency through the Investments for the Future Programme IdEx BordeauxCPU under Grant ANR-10-IDEX-03-02. The work of J.-F. Aujol was supported by the Institut Universitaire de France.


Marcel M Lambrechts
email: marcel.lambrechts@cefe.cnrs.fr

Aude E Caizergues

Charles Perrier

Anne Charmantier

Samuel P Caro

Surface temperatures of non-incubated eggs in great tits (Parus major) are strongly associated with ambient temperature

Keywords: Parus major, Ectotherm eggs, Nest temperature

Temperature is one of the best investigated environmental factors in ecological life-history studies and is increasingly considered in the contexts of climate change and urbanization. In avian ecology, few studies have examined the associations between thermal dynamics in the nest environment and its neighbouring air. Here, we placed avian nests and non-incubated eggs inside nest boxes at various air temperatures that ranged from 0.3 to 33.1 °C, both in the field and in laboratory conditions. We measured how the design of the boxes, their compass orientation and their location in more or less urbanized environments affected the surface temperature of nests and eggs. We also assessed whether covering the eggs with lining material influenced their surface temperature. Overall, across all performed tests, we found that the surface temperature of nests and eggs strongly reflected the air temperature measured outside of the nest boxes. While the design of the nest boxes had little influence on the temperature of nests and eggs, orienting the nest boxes to the north or to the west significantly decreased their surface temperature. The presence of lining material also kept eggs slightly warmer when air temperatures were low. Altogether these results suggest that nonincubated eggs are not well protected against extreme air temperatures prior to the onset of incubation. From an evolutionary point of view, producers of ectotherm eggs need therefore to time egg-laying appropriately in order to avoid unfavourable thermal nest environments.

Introduction

Temperature is amongst the most extensively investigated environmental factors in ecological life-history studies, especially since the recent overwhelming increase of the biological consequences of climate change (IPCC Report 2014; [START_REF] Gaughan | Sixty years of animal biometeorology[END_REF]. Temperature is also a rising concern in the field of urban ecology and urban evolutionary biology, with particular focus on the heat island effect of urbandwelling organisms [START_REF] Bornstein | Observations of the urban heat island effect in New York City[END_REF][START_REF] Diamond | Evolutionary consequences of the urban heat island[END_REF][START_REF] Imhoff | Remote sensing of the urban heat island effect across biomes in the continental USA[END_REF][START_REF] Seress | Habitat urbanization and its effects on birds[END_REF]. Long-term wildlife studies report associations between air temperature and reproductive performance [START_REF] Visser | Global climate change leads to mistimed avian reproduction[END_REF][START_REF] Mainwaring | Climate change and nesting behaviour in vertebrates: a review of the ecological threats and potential for adaptive responses[END_REF][START_REF] Andreasson | Experimentally increased nest temperature affects body temperature, growth and apparent survival in blue tit nestlings[END_REF]. Over evolutionary timescales, many species have evolved proximate responses to environmental factors in order to regulate reproductive functions, so that offspring are reared in optimal breeding conditions [START_REF] Baker | The evolution of breeding seasons[END_REF]. Breeding performance is therefore expected to be adjusted to changes in air temperature. In addition, breeding is often proximately inhibited by factors reflecting energetic constraints associated with extremely low air temperature [START_REF] Glądalski | Hatching delays in great tits and blue tits in response to an extreme cold spell: a longterm study[END_REF].

The location and design of bird nests are expected to create a suitable micro-habitat for developing eggs and offspring, with suboptimal nest conditions adversely affecting breeding performance. Nest structure can influence the micro-climate surrounding the eggs [START_REF] Britt | First-egg date and air temperature affect nest construction in blue tits Cyanistes caeruleus, but not in great tits Parus major[END_REF][START_REF] Crossman | Variation in the structure of bird nests between Northern Manitoba and Southeastern Ontario[END_REF][START_REF] Deeming | Local temperature and not latitude determines the design of blue tit and great tit nests[END_REF][START_REF] Mainwaring | Latitudinal variation in blue tit and great tit nest characteristics indicates environmental adjustment[END_REF][START_REF] Mainwaring | The design and function of birds' nests[END_REF][START_REF] Heenan | The influence of climate on avian nest construction across large geographic gradients[END_REF]. Earlier studies also claimed that temperatures are higher and more stable inside nest chambers than outside of nests [START_REF] Mainwaring | Climate change and nesting behaviour in vertebrates: a review of the ecological threats and potential for adaptive responses[END_REF]. However, to our best knowledge, studies that quantitatively examined associations between the temperature of the nest environment and the temperature dynamics close to the nest remain rare [START_REF] Gibb | The breeding biology of the great and blue titmice[END_REF][START_REF] Kluijver | Daily routines of the great tit, Parus m. major[END_REF][START_REF] Hafthorn | Incubation during the egg-laying period in relation to clutch-size and other aspects of reproduction in the great tit Parus major[END_REF][START_REF] Maziarz | Microclimate in tree cavities and nest-boxes: implications for hole-nesting birds[END_REF].

Here, we examine associations between surface temperatures of nests and non-incubated eggs from great tits (Parus major) placed inside nest boxes and external surface or air temperatures measured close to boxes. Surface temperatures of nests and eggs were first monitored in field conditions in boxes permanently attached to tree trunks and in which birds regularly breed (Fig. 1), both in forests and in urban areas including streets and parks. Urban areas consist in highly modified environments characterized by numerous new conditions such as high proportion of impervious surfaces, high concentrations in air pollutants as well as modified densities of trees [START_REF] Gil | Avian urban ecology[END_REF], which might affect egg temperature. We then used different set-ups in the laboratory to further explore the relationships between inside and outside temperatures over a large range of experimental conditions. In particular, the same set of boxes, nests and egg clutches were monitored both indoors and outdoors; in settings replicating more and less urbanized environments; in various compass directions; and with and without lining material protecting the eggs (Fig. 1). In this way, the same sets of nests and eggs were confronted with temperature regimes that might occur at different locations in the great tit distribution range, in natural and urbanized areas, or during different study periods in long-term studies throughout Europe. We predicted significant positive associations between surface temperatures measured inside the nest and surface or air temperatures measured outside the nest box. We also expected that the design of the boxes and their orientation would modulate these relationships.

Methods

We measured simultaneously surface temperatures inside nest boxes and surface or air temperatures around nest boxes in environmental conditions that varied in space and time. In the field, we measured the surface temperatures of box floors inside empty boxes, rims of nests, eggs and surface temperature of the tree trunk on which each box was placed (measured in the shade). In the laboratory, we measured the surface temperature of nest rim and eggs, and air temperatures within 1 m from the nests (Fig. 1). The measurements of air temperature were taken with an Ebro thermometer (TFH620, Ebro Electronic, Ingolstadt, Germany), whereas surface temperatures were measured with a Powertec Energy infrared thermometer (Kaleas, Denzlingen, Germany). To validate the values given by the two instruments, we correlated the air temperature measured in the shade close to the boxes with the Ebro thermometer, with the surface temperature of the front door of the same boxes with the Powertec thermometer. There was indeed a strong positive correlation between the

Field study

In the context of a long-term field study on great tits (Parus major), we monitored nest boxes in five geographically separated study plots within the urban area of Montpellier [START_REF] Demeyrier | Experimental demonstration of an ecological trap for a wild bird in a man-transformed environment[END_REF][START_REF] Lambrechts | Nest design in a changing world: great tit Parus major nests from a Mediterranean city environment as a case study[END_REF]. A gradient of urbanization was measured in [START_REF] Demeyrier | Experimental demonstration of an ecological trap for a wild bird in a man-transformed environment[END_REF], integrating vegetation cover, artificial night lighting and air and noise pollutions within a 50-m-radius disk around each nest box. We provided three box types made from European larch (Larix decidua) that differed in the bottom surface (small, 6 cm × 6 cm = 36 cm 2 vs. medium, 11 cm × 11 cm = 121 cm 2 vs. large, 14.5 cm × 14.5 cm = 210 cm 2 ) and, by extension, differed in the volume of the nest chamber under the entrance hole (648, 2178 and 3780cm 3 , respectively) [START_REF] Lambrechts | Nest design in a changing world: great tit Parus major nests from a Mediterranean city environment as a case study[END_REF]. In addition, we provided cylindrical wood-concrete Schwegler B1 boxes (internal diameter, 10.5 cm leading to a bottom area of ca. 113 cm 2 ; volume of the nest chamber under the entrance hole, 1525 cm 3 ), a box type used in long-term studies [START_REF] Lambrechts | The design of artificial nestboxes for the study of secondary hole-nesting birds: a review of methodological inconsistencies and potential biases[END_REF].

The contents of the boxes were monitored with standardized protocols from the second half of March until mid-July. Briefly, monitoring consists in quantifying nest building activity (a few pieces of moss, crown of moss, nest without lining, nest with lining), measuring the height of the nest rim, counting the number of eggs, the number and estimated age of the chicks. These monitoring protocols have been applied for several decades in Mediterranean southern France [START_REF] Blondel | A thirty-year study of phenotypic and genetic variation of blue tits in Mediterranean habitat mosaics[END_REF][START_REF] Lambrechts | The design of artificial nestboxes for the study of secondary hole-nesting birds: a review of methodological inconsistencies and potential biases[END_REF][START_REF] Demeyrier | Experimental demonstration of an ecological trap for a wild bird in a man-transformed environment[END_REF]. Box contents, mainly egg sizes, and presence of nestlings and adults were also used to identify the breeding species [START_REF] Blondel | A thirty-year study of phenotypic and genetic variation of blue tits in Mediterranean habitat mosaics[END_REF][START_REF] Demeyrier | Experimental demonstration of an ecological trap for a wild bird in a man-transformed environment[END_REF].

During the 2017 breeding season (25 April-12 June 2017), we measured the surface temperature inside 106 boxes that were either empty, contained a nest but no eggs (nests measured before laying or after fledging) or contained a nest and non-incubated eggs (eggs prior to incubation or deserted eggs). During each box visit, we first measured the surface temperature of the trunk on which each box was placed (measures done in the shade). We then rapidly removed the front panel of the box to measure the surface temperature of (a) the internal box floor (in empty boxes), (b) the nest rim or (c) the eggs. Some boxes were measured on several occasions, in different days and/or at different breeding stages (with and without nests and eggs). We chose surface temperature of trunks as a measure of environmental temperature, instead of air temperature like in the laboratory experiments below, mainly to minimize disturbance time at the boxes that were occupied by breeding birds. Before opting for tree trunk surface temperature, we checked that it was well correlated with other kinds of surface (vegetation and box entrance, all R 2 > 0.8, p < 0.0001, n = 424 measures) and with ambient air (R 2 = 0.6, p < 0.0001, n = 72 measures) temperatures.

Laboratory experiments

General procedures

In tits, the nest insulation capacity during incubation and after fledgling is positively correlated across nests that differ in size [START_REF] Cruz | Nest insulation capacity during incubation and after fledgling are related[END_REF], and does not differ significantly (Lambrechts et al., unpublished data). We collected six nests from first breeding attempts that were deserted during the nestling stage or after chicks had successfully fledged and we transferred them to the laboratory. The collected nests varied between 14 and 64 g (table S1). This variation in nest mass was larger than that reported in many field studies [START_REF] Deeming | Functional properties of nests[END_REF].

The collected nests were inserted into two of the box types that were used for the field study (see above): three of the large wooden boxes made from Larix decidua (floor area 210 cm 2 , see above) and three wood-concrete boxes (floor area ca. 113 cm 2 , see above). Nests from the larger wooden boxes were substantially larger and heavier than nests from the smaller wood-concrete boxes [START_REF] Lambrechts | Nest design in a changing world: great tit Parus major nests from a Mediterranean city environment as a case study[END_REF], table S1). In those experiments, box type therefore reflected not only differences in nest chamber design but also differences in nest size per se. All laboratory tests were conducted with the same six nests and six boxes. Positions of the boxes were alternated between successive temperature measurements, so that all individual boxes and box types had on average the same spatial position. Temperature was monitored with an interval of at least 1 h in between two successive measurements.

We also collected 30 eggs from deserted great tit nests in 2017 that were brought back to the laboratory and kept at room temperature before the start of the study. The eggs were randomly selected to constitute six 4-egg clutches that combined eggs from different nests of origin. A clutch size of four eggs per nest facilitated the measures of egg surface temperatures, and also allowed to control for a possible effect of clutch size on egg temperatures [START_REF] Cooper | Seasonal and latitudinal trends in clutch size: thermal constraints during laying and incubation[END_REF][START_REF] Nord | Context-dependent costs of incubation in the pied flycatcher[END_REF]Deeming and Reynolds 2015). Deserted eggs had different stages of development and included eggs with yolk and eggs with embryos. Because the thermic properties of eggs are associated with egg development [START_REF] Cooper | Avian incubation patterns reflect temporal changes in developing clutches[END_REF], eggs were switched between boxes so that different boxes received the same eggs and/or the same clutches.

The associations between temperatures measured inside and outside the nest boxes were investigated in different conditions that are illustrated at Fig. 1. These conditions included comparisons of boxes placed indoors and outdoors (test 1a), with eggs that were or were not covered with wool (test 1b), and boxes that were placed in different compass directions (test 2), in more or less urbanized environments (test 2).

Effect of indoor and outdoor environments on nest and egg temperatures (test 1a)

In a set-up that controlled for box orientation, six boxes (three large wooden boxes and three wood-concrete boxes) were placed side by side with entrance holes all oriented in the same direction (N-E). Boxes and their nests were first at the Redon CNRS campus in Montpellier (indoors, 30 October-4 December 2017) and then transferred to a suburban garden at 15-km flight distance from the Redon campus (outdoors, 5-11 December 2017) (test 1a on Fig. 1). Temperatures were monitored with an interval of at least 1 h between 6:30 and 22:30. Outdoor conditions reflected dynamic weather conditions with exposure to wind and sun and lower air temperatures, whereas indoor conditions reflected more stable conditions without wind and sun (one closed window not facing the sun) and higher air temperatures.

Effect of lining material on egg temperatures (test 1b)

In addition to comparing temperatures between indoor and outdoor conditions, we also examined the impact of lining material on surface temperatures of eggs. Great tits indeed cover unheated eggs with material, called lining material, which can be animal-based (hair, feathers, etc.) or artificial (for example, wool), depending on what is available around their territory [START_REF] Kluijver | Daily routines of the great tit, Parus m. major[END_REF][START_REF] Perrins | British tits[END_REF][START_REF] Surgey | Opportunistic use of a woollike artificial material as lining of tit (Paridae) nests[END_REF]. We monitored uncovered eggs (the same data as in test 1a) and the same eggs fully covered with a fabric of pure wool (thickness ca. 3.5 mm, surface ca. 40 cm 2 ) (7 sampling days; test 1b on Fig. 1). We did not monitor nest rim temperatures as only eggs were covered by the wool fabric in this test.

Effect of nest chamber orientation on nest and egg temperatures in more or less urbanized environments (test 2)

Previous studies suggested that the orientation of the entrance hole exposes nest chambers to more or less sunshine and/or wind, which might influence the micro-climate inside the nest [START_REF] Wilkin | The use of GIS in estimating spatial variation in habitat quality: a case study of lay-date in the great tit Parus major[END_REF][START_REF] Goodenough | Nestbox orientation: a species-specific influence on occupation and breeding success in woodland passerines[END_REF]. To test the impact of box orientation on the micro-climate inside the box, we oriented the entrance holes of six boxes (three large wooden boxes and three wood-concrete boxes) in four compass directions (north, south, west, east) (Fig. 1). Compass directions were determined with a Pointe'S@t Optex device (Optex Electronique, Savines-le-Lac, France). All boxes from all compass directions were placed against a wooden beam (height 22 cm, thickness 11.5 cm) to simulate boxes attached to tree trunks. The presence of a wooden beam might prevent sunshine touching the boxes, depending on the time of the day. This experiment was repeated in two different set-ups, situated 30 m apart that simulated two different levels of urbanization. At one site (suburban area with ground covered with tiles and surrounded by pine trees and evergreen oaks, at 15 km N-E from the Redon CNRS campus), six boxes were placed on a concrete table covered with tiles simulating a more urbanized environment (23-28 January 2018, 4 sampling days). At the other site (the same suburban garden, but on a spot covered with lawn and surrounded by pine trees and evergreen oaks), the same six boxes were placed on a wooden table on a natural soil simulating green space in a less urbanized environment (29 January-4 February 2018, 7 sampling days).

Statistical analyses

For field data, we applied mixed model procedures (type 3 tests of fixed effects; SAS 9.4, data not transformed) and ran three models with surface temperature of (a) internal box floor (n = 267 measures), (b) nest rim (n = 40 measures) or (c) non-incubated eggs (prior to incubation or deserted, n = 27 measures) as dependent variables, and considered box type (four different box types: three wooden boxes of different sizes and one wood-concrete box) and surface temperature of the tree trunks as independent variables. Box identity (i.e. box number) and field site (five geographic areas) were added as random factors. Results of these tests are summarized in Table 1.

For laboratory data, we ran different models using the surface temperature of (a) the nest rim or (b) eggs, as dependent variables. For test 1a, location (indoors vs. outdoors), box type (wood vs. wood-concrete) and their interaction were used as fixed effects. We also ran two other models (one for each dependent variable) in which we replaced location (categorical variable) by the air temperatures measured outside of the box (< 1 m from nest) (continuous variable). For test 1b, we used the same main effects as those in test 1a, added lining material (presence vs. absence of wool) and only used the twoway interactions that contained lining material (box type × lining and location × lining, or air temperature × lining). For test 2, we used location (urbanized vs. less urbanized), box type (wood vs. wood-concrete), box orientation (compass directions N, E, S, W) and the two-way interactions involving box orientation (location × orientation and box type × orientation) as fixed effects. As for tests 1a and 1b, we also ran two tests in which location was replaced by the air temperatures measured outside of the box (< 1 m from nest). For all statistical tests, nest identity (6 different nests) and measure day (in Julian dates) were added as random factors. Non-significant interactions were removed from the models. Results of these laboratory tests are summarized in Table 2.

Results

Field study

In the field, outside temperature varied between 8.2 and 33.1 °C. For each of the three mixed models that analysed separately the field data, surface temperatures measured on the tree trunks on which the boxes were attached strongly predicted the surface temperature of the box floors, nest rims, and eggs (all p < 0.001, Table 1). Thus, surface temperatures of box floors, nests without eggs and non-incubated eggs were higher when outside surface temperatures were higher. In

Laboratory experiments

Effect of indoor and outdoor environments on nest and egg temperatures (test 1a)

During this test, nests and eggs were subjected to air temperatures that varied between 1.2 and 23.3 °C, with air temperatures that were approximately 10 °C higher in the indoor conditions than in the outdoor conditions. Surface temperatures of the nest rims and the eggs were strongly influenced by the location of the boxes (p < 0.001, Table 2), with higher surface temperatures of nest rims and eggs measured indoors compared with outdoors. There was no effect of the type of nest box on nest rim and egg temperature in this model (p > 0.2, Table 2). Replacing location by measured air temperature outside of the boxes led to similar results, except that in this analysis, eggs were found to be significantly cooler in large wooden boxes than in small wood-concrete boxes (p = 0.006, Table 2).

Effect of lining material on egg temperatures (test 1b)

Similarly to results from test 1a, the surface temperature of eggs was strongly influenced by the location of the boxes, with higher surface temperatures measured indoors compared with outdoors (p < 0.001, Table 2). However, when location was replaced by the exact air temperature measured outside of the boxes, we also found that the effect of air temperature on egg temperature was modulated by the presence of lining material (interaction air temperature × lining, p = 0.005). In particular, the presence of lining material kept eggs warmer at lower air temperatures (Fig. 2).

Effect of nest chamber orientation on nest and egg temperatures in more or less urbanized environments (test 2)

During test 2, nests and eggs were subjected to air temperatures that varied between 0.3 and 17.2 °C, with air temperature that was on average 3.5 °C higher in the urban than in the less urban environment. Surface temperatures of nest rims and eggs were not reliably influenced by the location of the boxes (urban or less urban, Table 2), presumably because of the moderate 3.5 °C temperature difference between the two sites.

Nest box orientation however influenced the surface temperature of nest rims (p = 0.049), with cooler rims in boxes oriented to the north and west (mean ± SD: north = 9.5 ± 3.7 °C; west = 9.9 ± 3.9 °C; south = 10.5 ± 4.5 °C; east = 10.7 ± 4.1 °C). Replacing location by the exact air temperatures measured again confirmed that air temperature strongly influences surface temperature of nest rims and eggs (both p < 0.001, Table 2). In this analysis, the surface temperature of eggs was also influenced by the orientation of the boxes, with cooler eggs in boxes oriented to the north and west (mean ± SD: north = 9.4 ± 3.8 °C; west = 9.7 ± 3.8 °C; south = 10.1 ± 4.3 °C; east = 10.2 ± 3.9 °C), like it was already the case for nest rims in the previous analysis.

Discussion

Avian populations are monitored in different locations at large spatial scales and are therefore exposed to different air temperature regimes that might influence local life-history decisions during the nesting period. However, the relative importance of nest placement and nest design influencing the thermal nest environment cannot be investigated when a nonmanipulated nest has a fixed spatial placement and is exposed to a localization-specific micro-climate. In our field and laboratory study, nests and eggs from a model species were exposed to different temperature regimes that varied between 0 and 33 °C, covering temperatures that might occur in the species-specific Western Palaearctic distribution range [START_REF] Harrison | An atlas of the birds of the Western Palaearctic[END_REF]. In addition, the same sets of nests and eggs were exposed to more stable indoor environments without wind and more dynamic outdoor environments with wind. This switch in environmental conditions simulated natural conditions where nests and eggs can suddenly be exposed to a drop in air temperature [START_REF] Dhondt | Laying interruptions in tits Parus spp[END_REF]. Finally, in the laboratory conditions, nest placement was frequently changed so that nests with different designs had on average the same spatial position. In all settings, we found a close match between the surface temperatures of nests or eggs on the one hand and the temperatures measured outside the nest chambers on the other hand. Differences in temperatures of nests, eggs and air during a visit to a box were small compared with the huge fluctuations in temperature throughout the day and/or across days. We expected that the placement and orientation of nests would influence the micro-climate inside nests. For instance, for nest chambers exposed to sunshine, we observed that eggs could occasionally be ca. 5 °C above air temperature. This can be seen in Fig. 2, where the largest temperature differences between air and eggs occurred outdoors, especially in the few cases when nest boxes were exposed to the sun, while the lowest temperature variation occurred indoors where sun and wind were absent. The significant effects of location and orientation on the surface temperatures of nest rims and eggs were perhaps caused by different proximate underlying mechanisms. The simplest explanation is that heat produced by sunshine makes nest chambers warmer, and that exposure to sunshine, clouds or wind differed between sampling days, study locations and orientations of boxes. In addition, our outdoor study locations, which were less than 30 m apart, differed in the placement and presence of evergreen tree canopy (Aleppo pine, Pinus halepensis, stone oak) so that the extent of exposure to sunshine differed between neighbouring locations. Future studies of micro-geographic variation in nest and egg temperatures therefore should focus not only on seasonal and yearly changes in local weather conditions but also on the micro-structure of the vegetation exposing nest chambers more or less to sunshine or shade. Different territories might therefore be exposed to different micro-climatic conditions and females might take these local micro-climatic conditions into account to adjust life-history decisions during the nesting season.

Surprisingly, we found that nest box type (and by extension nest size and mass) rarely influenced nest or egg temperatures. This would imply that prior to the onset of incubation, larger and heavier nests do not offer more efficient protection for eggs against extreme fluctuations in air temperature than smaller and lighter nests. What are the possible implications for how ectotherm eggs cope with aspects associated with cold spells or climate change? First, in great tits, [START_REF] Dhondt | Laying interruptions in tits Parus spp[END_REF] reported that females do not start laying eggs until a cold spell is over, and [START_REF] Schaper | Increasing temperature, not mean temperature, is a cue for avian timing of reproduction[END_REF] showed that a temperature increase is a trigger for the onset of egg-laying. [START_REF] Meijer | Temperature and timing of egg-laying of European starlings[END_REF] also showed that female European starlings (Sturnus vulgaris) need a temperature rise to initiate laying. In our study, we observed that eggs that are not protected by parents could freeze, whatever the size of the nest. When timing their reproduction, parents therefore not only have to anticipate favourable periods to rear offspring (cf. [START_REF] Baker | The evolution of breeding seasons[END_REF][START_REF] Visser | Global climate change leads to mistimed avian reproduction[END_REF]) but perhaps also need to avoid unfavourable periods for breeding, e.g. to prevent egg freezing or to reduce physiological costs of incubation in cold nest environments. Second, small passerines, like tits, cover their eggs with animal-based material during the egg-laying period [START_REF] Kluijver | Daily routines of the great tit, Parus m. major[END_REF][START_REF] Perrins | British tits[END_REF]. Hiding eggs could be a protection strategy against nest predators but also might protect unheated eggs against low air temperatures. Although we found that eggs that were covered with a layer of wool were slightly warmer at low air temperatures, there was still a close match between the temperatures of the eggs and the air temperature outside the nest chamber (Fig. 2). This implies that hiding non-incubated eggs in the absence of the parents does not completely protect the eggs against important fluctuations in air temperature. Kluijver (1950, p 115-118) noticed that at night great tit females can sit on the lining material used to cover eggs before the onset of incubation. Night roosting on eggs during the egg-laying period has also been reported by [START_REF] Hafthorn | Incubation during the egg-laying period in relation to clutch-size and other aspects of reproduction in the great tit Parus major[END_REF] and more recently by [START_REF] Pendlebury | Night-time behaviour of egg-laying tits[END_REF]. This behaviour might in some conditions be a strategy to prevent egg freezing during the egg-laying period, especially at night when air temperatures and associated nest rim temperatures are low. In that case, the presence of a roosting female together with a layer of lining material and a thick nest could more strongly buffer the temperature decrease at night. Females may also sample surface temperatures prior to egglaying to reduce risks of egg freezing or risks associated with incubation in cold nest environments. This might explain why the nest construction period can last 2-3 weeks, despite the fact that females have the physical ability to construct a nest in a single or a few days [START_REF] Lambrechts | Nest size and breeding success in first and replacement clutches: an experimental study in blue tits Cyanistes caeruleus[END_REF]). Fig. 2 Interaction between ambient air temperature and presence/absence of lining material on surface temperature of non-incubated great tit eggs. At low air temperatures, eggs are slightly warmer when covered with lining material. On the highest range of outdoor temperature, eggs sometimes become much warmer than air temperature (see "Discussion" for details). On the other hand, eggs can also sometimes be cooler than air temperature, presumably due to a short time-lag between egg temperature and warming air temperature in the morning. The black dashed and grey dotted lines are for illustration purposes only. The dashed line depicts the line of equality (x = y), and the dotted line shows temperature measurements that were taken indoors and outdoors

To conclude, our results indicate that nests and their design, assumed to be adjusted to anticipate fluctuations in weather or changes in climate [START_REF] Deeming | Local temperature and not latitude determines the design of blue tit and great tit nests[END_REF][START_REF] Mainwaring | Latitudinal variation in blue tit and great tit nest characteristics indicates environmental adjustment[END_REF], do not protect unheated eggs against extreme fluctuations in air temperature. However, we do not exclude that nests are efficient tools to protect eggs against cold after the onset of incubation, e.g. to slow down egg cooling when females briefly leave incubated eggs. Responses of nests to cooling could be experimentally examined by exposing nests to a cold source (cf. [START_REF] Nager | Energetic limitation in the egglaying period of great tits[END_REF] or a heat source (e.g. an infrared lamp), and quantify the dynamics of egg temperatures after exposure to cold or heat.

Fig. 1

 1 Fig. 1 Experimental design of the study of the influence of ambient temperature on unattended great tit (Parus major) nests and eggs. In the field, we assessed the relationships between the surface temperature of tree trunks and the surface temperature of (a) bottoms of empty nest boxes (n = 267), (b) nests that contained no eggs (n = 40) or (c) eggs (n = 27), in four types of boxes. In the laboratory, we related ambient air temperature

Table 2

 2 Effects of location (indoors vs. outdoors in tests 1a [n = 289] and 1b [n = 168]; urban vs. less urban in test 2 [n = 522]), box type (wood vs. wood-concrete), orientation (N, S, E, W), lining (presence vs. absence) and/or air temperature measured outside of the boxes, on surface temperatures of nest rims and eggs. Nest identity and day of measure were included as random factors. Non-significant interactions were removed from the statistical models

	Temp. nest rims		Temp. eggs
	F values	p values	F values	p values

Table 1

 1 Effects of box type (wood small, wood medium, wood large, wood-concrete) and tree trunk surface temperature (measured in the shade) on surface temperatures of box floors (n = 267), nest rims (n = 40) and eggs (n = 27). Box identity and field site were included as random factors

		Temp. box floors		Temp. nest rims		Temp. eggs
		F values	p values	F values	p values	F values	p values
	Trunk temp.	2779.29	< 0.001	466.32	< 0.001	52.02	< 0.001
	Box type	2.02	0.117	0.92	0.411	2.17	0.163
	In bold are p values < 0.05

Acknowledgements This work was supported by the city of Montpellier, the University of Montpellier, OSU-OREME, and the Mediterranean Centre for Environment and Biodiversity (LabEx CeMEB). Nicolas Lambrechts helped with monitoring. We thank Arnaud Grégoire, Stéphanie Grosset, Thierry Noell, Pascal Pagano, Vincent Perret and Gilles Tessier for their logistic support. We thank three anonymous reviewers for their very constructive comments on earlier versions of this manuscript.