Leonardo Trujillo

Gustavo Olague
email: olague@cicese.mx

Pierrick Legrand

Evelyne Lutton

A new regularity based descriptor computed from local image oscillations

This work presents a novel local image descriptor based on the concept of pointwise signal regularity. Local image regions are extracted using either an interest point or an interest region detector, and discriminative feature vectors are constructed by uniformly sampling the pointwise Hölderian regularity around each region center. Regularity estimation is performed using local image oscillations, the most straightforward method directly derived from the definition of the Hölder exponent. Furthermore, estimating the Hölder exponent in this manner has proven to be superior when compared to wavelet based estimation. Our detector shows invariance to illumination change, JPEG compression, image rotation and scale change. Results show that the proposed descriptor is stable with respect to variations in imaging conditions, and reliable performance metrics prove it to be comparable and in some instances better than SIFT, the state-of-the-art in local descriptors.

Introduction

The feature extraction problem, in the domain of image analysis systems, poses to main research questions. How can distinctive areas within an image be identified? How can distinctive areas be represented in such a way as to facilitate their identification? The main concepts to be taken from those questions are: identification and representation. Concerning the former, the identification problem, a mainstay in vision systems are interest point or interest region extraction algorithms. These techniques search for image pixels, or image regions, that exhibit high signal variations with respect to a particular local measure. Solutions have been designed based on studying intrinsic properties of 2D-signals [START_REF] Moravec | Towards automatic visual obstacle avoidance[END_REF], and more recently by solving a properly framed optimization problem [START_REF] Trujillo | Synthesis of interest point detectors through genetic programming[END_REF][START_REF] Trujillo | Using Evolution to Learn How to Perform Interest Point Detection[END_REF]. In response to the second question, dealing with the concept of representation, different techniques hanve been proposed that encode the information within these so called interesting regions. Discriminative feature are constructed that uniquiley characterize each interest regions. This in turn allows for efficient feature matching in a wide range of imaging problems. Currently, the SIFT [START_REF] Lowe | Distinctive Image Features from Scale-Invariant Keypoints[END_REF] descriptor has proven to be the most discriminative local descriptor in machine vision literature, and shows the highest performance with respect to the current set of benchmark tests [START_REF] Schmid | A Performance Evaluation of Local Descriptors[END_REF].

This paper presents a novel region descriptor based on the concept of Hölderian regularity. By approximating the pointwise Hölder exponent, also known as the Lipschitz exponent, using local signal oscillations around each image point, we are able to construct discriminative feature vectors. Our proposed descriptor is invariant to several types of changes in viewing conditions, exhibiting high and stable performace. As such, the main contribution of this work is that it introduces novel concepts to the field of feature extraction algorithms, using formal mathematical tools and corroborated by high performance on standard tests.

The rest of this paper is organized as follows. Section 2 gives a brief overview of related work. Section 3 presents the concept of Hölderian regularity and how to estimate it. Section 4 introduces our local descriptor based on pointwise Hölder exponents. Later in Section 5, experimental results are provided. Finally, in Section 6 we give conclusions and outline possible future work.

Related Work

It is not our intention to give a comprehensive summary on the subject of local descriptors, such a discussion can be found in [START_REF] Schmid | A Performance Evaluation of Local Descriptors[END_REF]. Hence, we will only focus on presenting the basic strategies followed by the most common type of region detectors, distribution based descriptors, and discuss the SIFT strategy.

Currently, most state-of-the-art local descriptors use a distribution based approach. These techniques characterize image information using local histograms of a particular measure related to shape or appearance. The most simple would be using histograms of pixel values, while more complex representations could be values representing texture characteristics. The most successful descriptor currently available in computer vision literature is SIFT, developed by David Lowe [START_REF] Lowe | Distinctive Image Features from Scale-Invariant Keypoints[END_REF], which builds an histogram of gradient distributions within an interest region. The descriptor builds a 3D histogram of gradient locations and orientations, weighted by the gradient magnitudes. Although SIFT combines both a scale invariant detector with the gradient distribution descriptor, only the latter has proven to outperform other types of techniques, and it is possible to replace the former with a more reliable region detector.

Hölder Regularity

One of the most popular ways to measure a signals regularity, be it pointwise or local, is to consider Hölder spaces. Hence, we will present the concept of regularity expressed through the Hölder exponent.

DEFINITION 1. Let f : ℜ → ℜ, s ∈ ℜ + * \ N and x 0 ∈ ℜ. Then, f ∈ C s (x 0 ) ⇔ ∃η ∈ ℜ + * , a polynum P of degree < s and a constant c such that ∀x ∈ B(x 0 , η), | f (x) -P(x -x 0 )| ≤ c|x -x 0 | s . ( 1
)
The pointwise Hölder exponent of f at x 0 is α p = sup s { f ∈ C s (x 0 )}, see Figure 1. The concept of signal regularity, characterized by the Hölder exponent, has been widely used in fractal analysis [START_REF] Falconer | Fractal geometry, Mathematical Foundations and Applications[END_REF][START_REF] Evertsz | Multifractal Measures[END_REF]. With regards to image analysis, the Hölder exponent provides a great deal of information related to the local structure around each point. Hence, it has been applied to such tasks as edge detection [START_REF] Véhel | Fractal Approaches in Signal Processing[END_REF], image denoising [START_REF] Legrand | Evolutionary denoising based on an estimation of Hölder exponents with oscillations[END_REF] and image interpolation [START_REF] Legrand | Local regularity-based interpolation[END_REF]. Furthermore, because most local image descriptors are fundamentally attempting to describe local image variations and overall structure, it is a natural conclusion to expect that Hölderian regularity will prove to be a useful tool in this task. Now, we are left with the task of accurately estimating the pointwise Hölder exponent.

Estimating the Hölder Exponent with oscillations

The most natural way to estimate the Hölder exponent, because it follows from its definition, consists in studying the oscillations around each point. This method gives accurate results, better then those obtained using wavelet analysis [START_REF] Legrand | Debruitage et interpolation par analyse de la regularite Hölderienne[END_REF], hence it will be the technique of choice to compute our proposed descriptor. A brief description of this technique will now be given, for a more detailed analysis please see [START_REF] Tricot | Curves and Fractal Dimension[END_REF]. It is pointed out that the Hölder exponent of function

f (t) at t is α p ∈ [0, 1], if a constant c exists such that ∀ t in a vicinity of t, | f (t) -f (t )| ≤ c|t -t | α p . (2)
In terms of signal oscillations, this condition can be written as:

a function f (t) is Hölderian with exponent α p ∈ [0, 1] at t if ∃c ∀τ such that osc τ (t) ≤ cτ α p , with osc τ (t) = sup |t-t |≤τ f (t ) -inf |t-t |≤τ f (t ) = sup t ,t ∈[t-τ,t+τ] | f (t ) -f (t )|. (3)
An estimation of the regularity will be built at each point by computing the slope of the regression between the logarithm of the oscillation and the logarithm of the dimension of the neighborhood at which one calculates the oscillation. From an algorithmic point of view, it is preferable not to use all sizes of neighborhoods between two values τ min and τ max . Hence, we calculate the oscillation at point t only on intervals of the form [t -τ r : t + τ r ], where τ r = base r . Here, we use least squares regression, with base = 2 and r = 1, 2, . . . , 7. For a 2D signal, t defines a point in 2D space and τ r a radius around t, such that the Euclidian distances d(t ,t) and d(t ,t) are ≤ τ r . We can visualize this process in Figure 2. The method of estimation with oscillations will give good results under three conditions: that α p < 1, the regression converges, and the regression converges towards a valid slope.

Hölder Descriptor

Now that we have described a method to accurately characterize the pointwise signal regularity, we can now move on to describe how we use this information to build our local descriptors. The process, described in Figure 3, is as follows.

First, a set Λ of regions of interest are extracted from an image. Second, the dominant gradient orientation φ λ is computed, this preserves rotation invariance. Finally, our feature vector δ λ contains the Holder exponent α p of the region center and of 128 concentric points, orderd according to φ λ .

Region Extraction: The first step in the process requires stable detection of prominent image regions. The type of regions to be extracted will depend on the requirement of the higher level application with respect to invariance. For instance, an interest point detector will suffice when the scale of the imaged scene is not modified. In our work, we use a detector optimized for geometric stability and global point separability, the IPGP2 detector which is the determinant of the Hessian matrix smoothed by a 2D Gaussian [START_REF] Trujillo | Synthesis of interest point detectors through genetic programming[END_REF][START_REF] Trujillo | Using Evolution to Learn How to Perform Interest Point Detection[END_REF]. All regions extracted with an interest point detector are assigned the same scale, w λ = 2.5 pixels. For images where scale is a factor, we use the Hessian-Laplace detector presented in [START_REF] Mikolajczyk | Scale and affine invariant interest point detectors[END_REF], which searches for extrema in a linear scale space generated with a Gaussian kernel. After this step we are left with a set Λ of circular image regions, where the scale is set to s λ = 5 • w λ , and w λ is the scale given by the detector.

Dominant Orientation: In order to preserve rotation invariance, the dominant gradient orientation is computed and used as a reference for the subsequent sampling process. For the scale invariant detector, all image regions are normalized to 41x41 bit size using bicubic interpolation. An orientation histogram is constructed using gradient orientations within the interest region, similar to what is described in [START_REF] Lowe | Distinctive Image Features from Scale-Invariant Keypoints[END_REF]. The histogram peak is obtained and thus ∀λ ∈ Λ a corresponding φ λ is assigned. In this way, each region is described by a set λ = {x λ , y λ , s λ , φ λ }, the image center, scale and orientation of the region.

Hölder Descriptor: Now that regions are appropriately detected and described with λ , we can now continue to construct our region descriptor δ λ ,∀λ ∈ Λ. Our sampling process is simple, see Figure 3, the first element of δ λ is the Hölder exponent α p computed at the region center (x λ , y λ ). Next, the Hölder exponent of points on the perimeter of four concentric rings are sampled, with radii of 1 4 • s λ , 1 2 • s λ , 3 4 • s λ and s λ respectively. A total of 32 points on each ring are sampled, starting from the position given by φ λ , uniformly spaced and ordered counterclockwise. Hence, our feature vector δ λ has 129 dimensions, compared to the 128 of SIFT.

Experimental Results

In order to effectively evaluate and compare our results, we use standard image sequences provided by the Visual Geometry Group [14]. From each image sequence there is one reference image and a set of test images, since we know beforehand the transformation between the reference and test images we are able to quantify a matching score for our descriptor. For image sequences where there is no scale change, we use threshold based matching, and for images with scale change we use neareast neighbor distance ratio matching. The former, is a strategy where two image regions λ 1 and λ 2 are matched if the following relation holds d(δ λ 1 , δ λ 2 ) < t.

While the latter strategy assigns a match between regions if

d(δ λ 1 ,δ λ 2 ) d(δ λ 1 ,δ λ 3
) < t, where λ 2 is the near- east neighbor of λ 1 , and λ 3 is the second nearest. In both cases, the value of t is varied to obtain the performance curves. Two types of curves are presented: one plots recall versus 1-precision, characterizing the matching between one test image and the reference image [START_REF] Schmid | A Performance Evaluation of Local Descriptors[END_REF]; the other is a double y-axis plot, one axis for recall and the other for 1-precision, that characterizes the performance of the descriptor on an entire image sequence. The second type of plot, includes errorbars in order to visualize the stability of the descriptor. Recall and 1-precision are defined as in [START_REF] Schmid | A Performance Evaluation of Local Descriptors[END_REF]: recall = #correctmatches #correspondences , and 1-precision = # f alsematches #correctmatches-# f alsematches . For comparison, the performance of our descriptor is plotted with that of SIFT. To compute SIFT descriptor, the Harris and Harris-Lapplace detectors were used to extract image regions, as suggested in [START_REF] Schmid | A Performance Evaluation of Local Descriptors[END_REF]; executables for SIFT and the Harris detectors were obtained from [14].

Conclusions and Future Work

Results show very promising performance, in general we can appreciate how the regularity based descriptor is more stable and achieves equal or better performace than SIFT for image sequences without scale change. Even do this is not the case for scale change transformations, we can still appreciate competitive performace up to a reasonable change in scale. The performance drop-off in this circumstances is expected to be directly related to the method of Hölder exponent estimation. For this reason an appropriate modification of the oscillations method is necessary in order to obtain a more efficient scale invariance for our Hölder descriptor.

Fig. 1 .

 1 Fig. 1. Hölderian envelope of signal f at point x 0 .

Fig. 2 .

 2 Fig. 2. Estimating the Hölder exponent with oscillations. Left: the region of interest λ , and three of the seven neighborhoods around point t, when r = 1, 2, • • • , 7. Center: the neighborhood of radius τ 5 = 32 pixels, with base = 2' Right: computing the supremum of the differences within radius τ 5 , where d denotes the Euclidian distance.

Fig. 3 .

 3 Fig. 3. Descriptor building process.

Fig. 4 .

 4 Fig. 4. Columns, from left to right: 1)Rotation (36 images in sequence), 2)Illumination change (10 images), 3)JPEG compression (6 images), and 4)Scale change (first 6 images of sequence). Rows, from top to bottom: 1)Reference image, 2)Test Image, 3)Performace between test and reference with Hölder-Green and SIFT-Red, 4)SIFT average performance on entire set, and 5)Hölder average performance.


Anna Staerz

Arne Kobald

Tamara Russ

Udo Weimar

Anne Hémeryck

Nicolae Barsan

THERMAL WATER SPLITTING ON THE WO 3 SURFACE-EXPERIMENTAL PROOF

Keywords: WO3, Gas Phase Water Splitting, DRIFT Spectroscopy, DFT, Hydrogen Evolution, Nano Material

Here, the splitting of water from humidity at the surface of WO 3 is reported. The production of hydrogen occurs under mild conditions, at 300 °C in the presence of humidity. Using operando diffuse reflectance infrared Fourier transform spectroscopy and DC resistance measurements complemented with DFT calculations, it was possible to elucidate the temperature dependent surface reactions of WO 3 with humidity. By examining multiple different nanomaterials at 300°C, it was found that the oxidation of the surface by humidity is generally valid for WO 3 . The evolution of hydrogen was detected using catalytic conversion measurements. In addition to the evolution of hydrogen, understanding the interaction between the surface and humidity is a prerequisite for the optimized use of WO 3 for a wide array of applications, such as optical sensors or memristors.

Introduction WO 3 is an n-type semiconductor with a number of interesting characteristics, making it a versatile material that is widely used. Already in the 1960s, Deb et al. found that WO 3 is electrochromic. 1 The electrochromic process of WO 3 is the change in color associated with the loss/gain of a proton/electron due to a small applied voltage. 1 In addition, WO 3 is known to be gasochromic; through modification with noble metals the direct insertion of hydrogen via a spillover effect is reported in literature. The resulting formation of tungsten bronzes, A x WO 3 where A + is a small ion, e.g. H + , leads to the reduction of W 6+ to W 5+ . 2 As a result of the changed oxidation state of tungsten, the bronzes adsorb photons in the visible region. The degree of coloration is found to be dependent on the atmospheric concentration of hydrogen. 3 This has been used for the development of optically-based hydrogen sensors. 3 The effect of atmospheric humidity on the coloration of WO 3 is, however, poorly understood and reports are even conflicting. 4,5 While it is thought that adsorbed water is key for its photochromic properties (light activated chromism), humidity reportedly decreases the sensitivity of optical hydrogen detectors. 4,6,7 Also, the memristive qualities of WO 3 are found to increase in the presence of background humidity. [START_REF] Zhou | Modulating Memristive Performance of Hexagonal WO3 Nanowire by Water-Oxidized Hydrogen Ion Implantation[END_REF] Although it is usually assumed that the memristive properties of WO 3 is induced by the migration of oxygen vacancies, Zhou et al. hypothesize purely based on I-V curves that the enhancement is a result of H + ions stemming from oxidized water drifting on the surface of a WO 3 nanowire via the Grotthuss mechanism. [START_REF] Zhou | Modulating Memristive Performance of Hexagonal WO3 Nanowire by Water-Oxidized Hydrogen Ion Implantation[END_REF] Despite the vast amount of research done on WO 3 across the many application fields, the reactions between the surface and the atmosphere are still poorly understood. Specifically, despite its omnipresence in applications, the interaction between humidity and the WO 3 surface remains unclear. It is known for porous films made of semiconducting metal oxide (SMOX) nanopowder, in general, that the reaction between the atmos-pheric gas and the surface of the grains results in a detectable change of the resistance. [START_REF] Brattain | Surface Properties of Germanium[END_REF][START_REF] Seiyama | A New Detector for Gaseous Components Using Semiconductive Thin Films[END_REF] In the case of an n-type material like WO 3 operated at 300 °C, an increase of the resistance is expected for oxidizing gases and a decrease in the presence of humidity. [START_REF] Taguchi | Gas-Detecting Device[END_REF][START_REF] Barsan | Conduction Model of Metal Oxide Gas Sensors[END_REF][START_REF] Heiland | Physical and Chemical Aspects of Oxidic Semiconductor Gas Sensors[END_REF] For porous films of WO 3 , however, the increase of the baseline resistance, at an operation temperature of 300 °C in the presence of humidity, has been reported. [START_REF] Staerz | WO3 Based Gas Sensors : Identifying Inherent Qualities and Understanding the Sensing Mechanism[END_REF][START_REF] Staerz | The Oxidizing Effect of Humidity on WO3 Based Sensors[END_REF] Humidity reduces the response of WO 3 chemiresistive gas sensors to oxidizing gases, e.g. NO 2 and ozone. [START_REF] Staerz | The Oxidizing Effect of Humidity on WO3 Based Sensors[END_REF][START_REF] Pokhrel | Mechanism, and Gas-Sensing Application of Surfactant Tailored Tungsten Oxide Nanostructures[END_REF][START_REF] Utembe | An Ozone Monitoring Instrument Based on the Tungsten Trioxide Semiconductor[END_REF] Based on preliminary measurements, the competing reactions of humidity and oxidizing gases has been given as a tentative explanation. [START_REF] Staerz | The Oxidizing Effect of Humidity on WO3 Based Sensors[END_REF][START_REF] Neri | First Fifty Years of Chemoresistive Gas Sensors[END_REF][START_REF] Pokhrel | Mechanism, and Gas-Sensing Application of Surfactant Tailored Tungsten Oxide Nanostructures[END_REF] A similar mechanism was recently reported by Daeneke et al. for the interaction of molybdenum sulfide and titanium oxide ink with atmospheric humidity under light irradiation. [START_REF] Daeneke | Surface Water Dependent Properties of Sulfur-Rich Molybdenum Sulfides: Electrolyteless Gas Phase Water Splitting[END_REF] As the ink can be coated onto any substrate, the hydrogen production from humidity is arguably easy and inexpensive. [START_REF] Daeneke | Surface Water Dependent Properties of Sulfur-Rich Molybdenum Sulfides: Electrolyteless Gas Phase Water Splitting[END_REF] WO 3 is also a promising material for use in photoelectrochemical or photocatalytic water splitting. In the context of this application, Albanese et al. and Teusch et al. have theoretically examined the interaction of water with WO 3 . Both very recent studies, find that molecular H 2 O easily adsorbs onto surface tungsten sites. Albanese et al. concluded that the reaction of the WO 3 surface with hydrogen could easily result in surface oxygen vacancies. [START_REF] Albanese | H2O Adsorption on WO3 and WO3-x (001) Surfaces[END_REF] Similarly, Teusch et al. found that the water splitting process does not take place without external influences due to the strong adsorption of hydrogen. [START_REF] Teusch | Understanding the Water Splitting Mechanism on WO3(001) -A Theoretical Approach[END_REF] Kishore et al. modelled the oxygen evolution reaction (OER), which is found to account for most of the overpotential in photoelectrochemical water splitting with WO 3 . [START_REF] Kishore | Electrochemical Water Oxidation on WO3 Surfaces: A Density Functional Theory Study[END_REF] Although these studies offer an interesting basis for understanding the interaction of water with the WO 3 surface, they are limited to theory.

The studies were also tailored to photoelectrochemical or photocatalytic water splitting, where WO 3 is immersed in an electrolyte solution. As a result, the interaction of the surface with molecular oxygen and gaseous water from the atmosphere is not considered. Additionally, only the low temperature γ-phase of WO 3 was considered. Based on their results Teusch et al. even concluded that the conditions examined in the study presented here are more fitting to those most likely required for water splitting on WO 3 , at an elevated temperature (ca. 300 °C, β-phase) and in humidity (gaseous water). [START_REF] Teusch | Understanding the Water Splitting Mechanism on WO3(001) -A Theoretical Approach[END_REF] The presented work serves as a crucial missing link between existing theoretical and experimental literature. Water splitting on the surface of WO 3 was experimentally identified using operando diffuse reflectance infrared Fourier transform (DRIFT) spectroscopy and resistance measurements coupled with exhaust measurements. Through a systematic temperature study, complemented with information from DFT calculations, it is possible to clarify several aspects of the interaction between the WO 3 surface and humidity.

The justification of this research is twofold. The evolution of hydrogen from WO 3 at 300 °C (much lower than the temperature typically used for thermochemical water splitting) as a result of humidity exposure would be remarkable. [START_REF] Mao | Hydrogen Production via a Two-Step Water Splitting Thermochemical Cycle Based on Metal Oxide -A Review[END_REF][START_REF] Abanades | Metal Oxides Applied to Thermochemical Water-Splitting for Hydrogen Production Using Concentrated Solar Energy[END_REF] In addition, however, it is vital for a wide array of applications, to fully understand the interaction between the WO 3 surface and humidity.

Experimental Film Preparation: Films were made based on commercially available nanopowder (< 100 nm, Sigma Aldrich). The powders were mixed with 1,2-propanediol (Sigma-Aldrich, 99.5+% ACS reagent) and ground with mortar and pestle into a viscous paste which was printed (EKRA Microtronic II) onto alumina substrates containing platinum electrodes and a backside platinum heater (300 µm electrode gap, 300 µm width and 5 µm thickness; Ceramtec AG). The films were dried at room temperature for 1 h, overnight at 80 °C (Heraeus UT12) and then annealed for 10 min each at 400-500-400 °C in a tubular furnace (Heraeus ROK 6/30). Based on the geometry of the screen, a final film thickness of 50 µm is achieved. The sensitive layers were characterized using a JOEL JSM-6500F Scanning Electron Microscopy (SEM) with an acceleration of 15 kV, a probe current of 7 and a stage height of 10 mm.

The X-ray diffraction measurements of the heated films under humidity exposure were done using a BRUKER D8 discover GADDS microdiffractometer equipped with a Co-X-ray tube, HOPG-primary monochromator, X-ray optics and a large two dimensional VANTEC-500 detector covering 40° 2Θ, 40° 2θ and 40° ψ allowing fast and locally resolved measurements of the film. [START_REF] Staerz | The Oxidizing Effect of Humidity on WO3 Based Sensors[END_REF][START_REF] Berthold | Fast XRD2 Microdiffraction with Focusing X-Ray Microlenses[END_REF] DC Resistance Measurements and Operando DRIFT Spectroscopy: The films are mounted into a homemade Teflon chamber and heated using an Agilent E3614A DC power supply. The gas flow was regulated using a computer monitored mass flow controller unit. The resistance was monitored using a Keithley 6517B electrometer. In order to ensure that the desired conditions were created, the exhaust gas of the resistance measurements was monitored a HYT939 humidity and temperature monitor (IST AG, Switzerland). An example of the received output is plotted in S1Error! Reference source not found..

For the operando DRIFT spectra, a Vertex80v with a midband mercury cadmium telluride (MCT) detector (Bruker, Billerica, MA, USA) and a spectral resolution of 4 cm -1 was used. The films were mounted in a homemade chamber with a KBr window. The films were heated and their resistance was monitored simultaneously. A single-channel spectrum was recorded every 15 min. The absorbance spectra was calculated using the relation Absorbance= -log ( single channel test gas single channel reference

)

eq. 1

In this case the ratio between the single channel in a test gas to one taken during exposure to a reference background is considered to be the relative transmission. [START_REF] Olinger | Quantitative Effects of an Absorbing Matrix on Near-Infrared Diffuse Reflectance Spectra[END_REF] Operando diffuse reflectance infrared Fourier transform spectroscopy, is widely used in the catalysis and gas sensor field for the insitu studies of surface chemistry. By referencing a spectrum taken during exposure to a certain atmosphere (e.g. oxygen containing synthetic air) to a spectrum taken of the sample while in an atmosphere lacking the target gas (e.g. nitrogen), it is possible to identify the vibrational wavenumbers of surface species which change as a result of the gas exposure. In their study, Sirita et al. examined the various functions derived from the signal collected by DRIFTS used to evaluate the surface coverage of adsorbates. The authors found that the method of Olinger and Griffiths, yields the most accurate results when dealing for systems showing baseline drifts. [START_REF] Olinger | Quantitative Effects of an Absorbing Matrix on Near-Infrared Diffuse Reflectance Spectra[END_REF][START_REF] Sirita | Quantitative Analysis of Adsorbate Concentrations by Diffuse Reflectance FT-IR[END_REF] The ratio between the reflected intensity of the sample divided by the intensity of the reference state (I∞(sample)/ I∞(reference) is then taken as an approximate value for absolute reflectance R∞. A linear representation of an adsorbate concentration is attained with log(1/R), noted "absorbance" (analogy with the absorbance log (1/T)). [START_REF] Olinger | Quantitative Effects of an Absorbing Matrix on Near-Infrared Diffuse Reflectance Spectra[END_REF][START_REF] Sirita | Quantitative Analysis of Adsorbate Concentrations by Diffuse Reflectance FT-IR[END_REF] Theoretical Calculations: The periodic spin polarized calculations were performed using DFT based on the generalized gradient approximation (GGA) employing the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional as implemented in the plane-waves program Vienna Ab initio Simulation Package (VASP). [START_REF] Kresse | Ab Initio Molecular-Dynamics Simulation of the Liquid-Metal-Amorphous-Semiconductor Transition in Germanium[END_REF][START_REF] Kresse | Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set[END_REF][START_REF] Perdew | Rationale for Mixing Exact Exchange with Density Functional Approximations[END_REF][START_REF] Wicker | Ambient Humidity Influence on CO Detection with SnO2 Gas Sensing Materials. A Combined DRIFTS/DFT Investigation[END_REF] The projector-augmented wave (PAW) potentials were applied for the core electron representation. [START_REF] Joubert | From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method[END_REF][START_REF] Blochl | Projector Augmented-Wave Method[END_REF] A converged value of E cut = 500 eV was used as the cutoff energy of the plane wave. The integration in reciprocal space was performed with a Monkhorst-Pack grid. [START_REF] Monkhorst | Special Points for Brillouin-Zone Integrations[END_REF] A k-point mesh of (4 × 4 x 1) was used. The stoichiometric (001) β-WO 3 surface was modeled by the periodic slab shown in Figure 4. The p(√2x√2)-R45° cell is composed of a total number of 128 atoms with 32 W-and 96 O-atoms, corresponding to a ~10.5 Å x 10.5 Å surface and four layers. A vacuum zone of 15 Å in the z/c direction was used to create a surface effect. As detailed in Albanese and coworkers' paper [START_REF] Albanese | H2O Adsorption on WO3 and WO3-x (001) Surfaces[END_REF] all the atoms are fully relaxed. We checked that bond length variations fit with trends given in the work of Albanese et al. [START_REF] Albanese | H2O Adsorption on WO3 and WO3-x (001) Surfaces[END_REF] The energy of oxygen vacancy formation ΔE O was calculated using the following relation

∆E O = E vac + 1 2 E O 2 -E surface eq. 2
and the adsorption energies are calculated using the following formula:

∆E ads = E adsorbed molecule -E surface -E gaseous molecule eq. 3
where E vac is the energy of the surface with one terminal oxygen vacancy, E O 2 is the energy of the isolated spin polarized O 2 , E surface is the total energy of the relaxed surface, E gaseous molecule is the total energy of the isolated gaseous molecule and E adsorbed molecule is the total energy of the system with a gaseous molecule adsorbed.

To calculate atomic charges and charge transfers, Bader's analysis was adapted. [START_REF] Tang | A Grid-Based Bader Analysis Algorithm without Lattice Bias[END_REF] The charge transfer is the sum of changes in charge for all atoms associated with the reference surface. Here, the reference is the slab with a terminal oxygen vacancy, V 0 C . The common DFT+U approach was found to have an insignificant effect on the charge transfers (see S2).

The visualization of the cell was done using VMD. [START_REF] Humphrey | Visual Molecular Dynamics[END_REF] Pictures were then rendered using POV-Ray. [START_REF]Persistence of Vision Raytracer[END_REF] Catalytic Conversion Measurements: To gain information about the catalytic activity of WO 3 with humidity, 1 g of the commercially available nanopowder from Sigma Aldrich (<100 nm) was filled into a quartz tube (fixed with quartz wool). The tube was then heated to 300 °C using an oven (Thermicon controlled Hereaus reactor). Using a computer monitored mass flow controller unit, dry synthetic air was passed through the heated WO 3 for 3 h, followed by air at 10 % RH (attained using a water bubbler) for 2 h at a constant total flowrate of 150 ml/min. A schematic of the catalytic conversion setup is shown in Figure 1. The measurement was repeated two times. The exhaust of the measurement was then dried using silica gel and molecular sieve before being analyzed by a metal oxide based gas sensor that is highly sensitive to H 2 (TGS2600, FIGARO Engineering Inc.). The sensor's response to hydrogen was verified by exposing the sensor to different H 2 concentrations. It was verified that the drying process was efficient and that the sensor's response was not affected by residual humidity as no response was seen if WO 3 was replaced by heated quartz wool.

Results

It is known that the reaction between the atmosphere and the surface of the grains results in a detectable change of the resistance. 9,10 Simplistically, under normal operation conditions, it is considered that atmospheric oxygen reacts with the surface of the metal oxides. The oxygen then traps electrons at the surface of the grains, resulting in an increase of the resistance for n-type semiconducting metal oxides. For WO 3 , this mechanism can be verified using operando DRIFT spectroscopy. Simultaneously to resistance measurements, infrared spectra are taken of a heated sample during exposure to different atmospheres. In order to gain insight into the surface reaction, the single channel spectrum taken in synthetic air (taken during the fifteen minutes highlighted in blue in Figure 2) is then referenced to a spectrum taken of the sample in nitrogen (taken during the fifteen minutes highlighted in red in Figure 2). According to the work of Olinger and Griffiths, this ratio represents a relative transmission and can be converted into the absorbance by taking the negative logarithm, see Figure 2. In the absorbance spectra, bands that are decreasing are associated with surface groups that are leaving the surface as a result of the gas while increasing bands belong to surface groups which are more readily present. A schematic showing how the absorbance spectrum for synthetic air was attained is shown in Figure 2. The absorbance spectrum verifies the assumed surface reaction of the WO 3 film with atmospheric oxygen. In literature the bands at 2063 cm -1 and 1854 cm -1 are attributed to tungsten -oxygen bonds. [START_REF] Kanan | Identification of Surface Sites on Monoclinic WO3 Powders by Infrared Spectroscopy[END_REF] In the absorbance spectra, these bands are increasing indicating, as expected, that the surface is more oxidized in synthetic air than in nitrogen. Additionally the band at 1421 cm -1 , is found to be increasing and is therefore attributed to tungsten-oxygen, in line with a large DRIFT study on films produced from differently prepared WO 3 samples. [START_REF] Staerz | WO3 Based Gas Sensors : Identifying Inherent Qualities and Understanding the Sensing Mechanism[END_REF] Overall, the DRIFT spectrum in synthetic air supports the general assumption that the WO 3 surface is oxidized by atmospheric oxygen. The results are helpful for the interpretation of the absorbance spectra under humidity exposure.

For most n-type semiconducting metal oxides, e.g. SnO 2 and In 2 O 3 , humidity is known to result in a decrease of the films resistance at 300 °C. In the case of WO 3 , however, an increase of the baseline resistance in the presence of humidity has been reported. [START_REF] Staerz | WO3 Based Gas Sensors : Identifying Inherent Qualities and Understanding the Sensing Mechanism[END_REF][START_REF] Staerz | The Oxidizing Effect of Humidity on WO3 Based Sensors[END_REF][START_REF] Pokhrel | Mechanism, and Gas-Sensing Application of Surfactant Tailored Tungsten Oxide Nanostructures[END_REF] When referenced to nitrogen, the bands attributed to tungsten-oxygen bonds are increasing during humidity exposure, similarly to dry synthetic air, for the WO 3 film at 300 °C see Figure 3. [START_REF] Humphrey | Visual Molecular Dynamics[END_REF] Furthermore, the formation of hydroxyl groups is not visible.

This finding indicates a direct oxidation of the WO 3 surface by humidity. In order to verify these findings and gain insight into how water oxidizes the WO 3 surface, a temperature study was done. The resistance measurements were done on four identically prepared WO 3 films. The films were heated between 100 and 300 °C. Unexpectedly, under 175 °C, the resistance of the films significantly decreases in the presence of humidity (Figure 3a, blue region). At mid-range temperatures around 200 °C (Figure 3a, purple region), the effect of humidity on the resistance is negligible. Above 250 °C, the resistance of the films is found to increase in the presence of humidity (Figure 3a, red region). Operando DRIFT spectra taken in 10 % RH referenced to dry synthetic air at the different temperatures provide insight into why the effect of humidity on the resistance changes over the temperature range. The most significant change is seen between 2500 and 1200 cm -1 , for a complete overview of the spectra please see supplementary information S2.

Overall the change of the bands attributed to the tungsten oxygen bonds correlates well to the change in resistance. At 300 °C, these bands are increasing associated with an increase in the resistance. At mid-range temperatures, around 200 °C, the resistance is largely humidity independent and no significant changes in the bands belonging to tungsten-oxygen are visible. In line with the significant decrease in the resistance of the material, at 125 °C the bands (2063 and 1854 cm -1 ) attributed to tungsten oxygen bonds are decreasing. The formation of hydroxyl groups is not visible in the DRIFT spectrum, S3. An increasing band at around 1617 cm -1 is, however, visible. In literature this band is attributed to the bending mode of surface adsorbed water. [START_REF] Kanan | Identification of Surface Sites on Monoclinic WO3 Powders by Infrared Spectroscopy[END_REF] This designation was verified using a H 2 O/D 2 O experiment. A DRIFT spectrum taken during a pulse of D 2 O after H 2 O exposure was referenced to dry air. As expected, the exchange of H 2 O with D 2 O has no effect on the bands attributed to tungsten-oxygen. The band at 1617 cm - 1 , however, decreases and a new increasing band at 1185 cm -1 becomes visible. This shift is in line with surface adsorbed H 2 O being replaced by D 2 O.

DRIFT spectroscopy is known to be a very surface sensitive method and at the examined film temperatures, it is probable that the observed reactions are surface based. [START_REF] Roedel | On the Local Sensitivity of Different IR Techniques: Ba Species Relevant in NO(x) Storage-Reduction[END_REF] It is known, however, that water can be integrated into the lattice of WO 3 resulting in the formation of hydrates . [START_REF] Daniel | Infrared and Raman Study of W03 Tungsten Trioxide and W03x H2O Tungsten Trioxide Hydrates Trioxides[END_REF] , Here, no hydrate formation is visible in the in operando XRD, see S4Error! Reference source not found., even after hours at high humid-ity. This verifies that the process doesn't occur in the bulk of the material but is surface based.

In order to understand the charge transfer associated with the identified surface reactions, DFT calculations were done. It is known that WO 3 has different temperature dependent crystal structures. [START_REF] Arai | Raman Studies of Phase Transitions in Gas-Evaporated WO3 Microcrystals[END_REF][START_REF] Righettoni | Annealing Dynamics of WO3 by in Situ XRD[END_REF][START_REF] Salje | The Orthorhombic Phase of WO3[END_REF][START_REF] Loopstra | Neutron Diffraction Inverstigation of WO3[END_REF] For bulk WO 3 the phase transition between the monoclinic γ-phase and the orthorhombic β-phase occurs at temperature above 400 °C. The transition, however, has been found to occur at lower temperatures (237 °C) for nanomaterials. [START_REF] Salje | The Orthorhombic Phase of WO3[END_REF][START_REF] Lu | Raman Study of Thermochromic Phase Transition in Tungsten Trioxide Nanowires[END_REF] The XRD spectra of the γ-and β-phase are very similar, but a splitting of the reflex at ca. 39° could indicate that the presence of both phases, see S5. To address both phases the results attained here for the stoichiometric (001) β-WO 3 surface were compared to the theoretical work of Albanese et al., Teusch et al. and Jin et al. for the γ-phase. [START_REF] Albanese | H2O Adsorption on WO3 and WO3-x (001) Surfaces[END_REF][START_REF] Teusch | Understanding the Water Splitting Mechanism on WO3(001) -A Theoretical Approach[END_REF][START_REF] Jin | Insight into the Mechanism of CO Oxidation on WO3(001) Surfaces for Gas Sensing: A DFT Study[END_REF] Here, an oxygen terminated surface is considered, in which every other tungsten is 5-fold-coordinated (W 5c ). In their work, Teusch and Klüner only considered the adsorption of water onto W 5c . Albanese et al. attributed the n-type conductivity of WO 3 to surface oxygen vacancies. [START_REF] Albanese | H2O Adsorption on WO3 and WO3-x (001) Surfaces[END_REF] In the relaxed slab, the bond between the terminal oxygen and tungsten has a length of 1.71 Å, see Figure 4. This oxygen is stable singly charged with one excess electron. [START_REF] Albanese | H2O Adsorption on WO3 and WO3-x (001) Surfaces[END_REF][START_REF] Jin | Insight into the Mechanism of CO Oxidation on WO3(001) Surfaces for Gas Sensing: A DFT Study[END_REF] In literature, the dominant surface vacancy (V 0 C ) is found to result from the loss of O 1C

C . [START_REF] Albanese | H2O Adsorption on WO3 and WO3-x (001) Surfaces[END_REF][START_REF] Jin | Insight into the Mechanism of CO Oxidation on WO3(001) Surfaces for Gas Sensing: A DFT Study[END_REF] The WO 3 films show n-type behavior, and the surface of the films is in constant equilibrium with the atmosphere. In other words, the V 0 C vacancies are in constant equilibrium with gaseous species in the atmosphere, e.g. water and oxygen. For completeness, the adsorption of oxygen and water molecularly onto both the W 5c and into the V 0 C is therefore considered. The adsorption energy and charge transfer of water and oxygen were calculated using the surface with a V 0 C (formation The adsorption of oxygen into the vacancy is energetically favorable in both positions. In line with the findings of Jin et al., the parallel adsorption is more favorable than the perpendicular position. [START_REF] Jin | Insight into the Mechanism of CO Oxidation on WO3(001) Surfaces for Gas Sensing: A DFT Study[END_REF] The adsorption of oxygen on W 5c , however, is energetically unfavorable. The adsorption of molecular oxygen on both W 5c and the neighboring V 0 C is energetically close to just the respective adsorption of a single molecule in the vacancy. Similar to the results of Jin et al., climbing image nudged elastic band calculations suggest that the activation barrier for the rotation of oxygen from the perpendicular mode into the parallel mode is insignificant. Therefore, a two-step desorption process of the parallel adsorbed molecule via the perpendicular mode can be considered. The rate determining step is then likely governed by the activation energy which equals the difference in energy of these two modes, ∆E=0.8 eV.

The adsorption of molecular water on both the V 0 C and W 5c is found to be energetically favorable and additionally more favorable than the perpendicular adsorption of oxygen. It is favorable if water adsorbs onto the W 5c and oxygen adsorbs molecularly into the V 0 C , both in a parallel (-1.87 eV) and perpendicular (-1.04 eV) mode. The adsorption of molecular oxygen onto the W 5c and the adsorption of water into the V 0 C , is significantly less favorable (-0.69 eV). If water adsorbs onto both sites, the adsorption energy is additive, i.e. the total adsorption energy is the sum of the individual ones. These findings indicate that when considering the interaction between humidity and the WO 3 surface under real conditions, both W 5c and the V 0 C are possible adsorption sites.

The DRIFT spectra considered in the context of these calculations indicate that at 125 °C, the molecularly adsorbed water most likely disturbs the equilibrium of the surface with atmospheric oxygen. The detected decrease of the resistance as a result of humidity exposure at low temperatures can be explained in several ways. Considered using V 0 C as a reference, it is calculated that molecularly bound oxygen in the parallel position has a similar charge transfer as if the vacancy was healed (0.7), see Figure 5. Also in the perpendicular position, molecularly adsorbed oxygen in the vacancy removes charge from the lattice. Contrarily, the molecular adsorption of water into the vacancy is even found to push charge in, which should result in a decrease of the resistance on the surface. Additionally as a result of the adsorption of water into the surface vacancy, the overall re-oxidation process by atmospheric oxygen could be hindered. Molecularly adsorbed water onto the W 5c also pushes charge towards the surface. Even if water adsorbs into the W 5c position next to molecularly adsorbed oxygen in the vacancy, then the charge transfer to the surface is overall lower than only in the presence of the oxygen.

In the spectra at 300 °C, the bands attributed to the overtones and combinations of the WO 3 lattice, 2063 and 1854 cm - 1 , are increasing. At this temperature, the bands attributed to molecularly adsorbed water are barely visible. This, in addition to the detected increase of the resistance, indicates that the tungsten oxide surface is more oxidized in the presence of humidity than in dry synthetic air. An explanation for the spectrum would be the healing of the oxygen vacancies by water followed by the subsequent release of hydrogen. In their work, Teusch and Klüner theoretically examined the splitting of water on WO 3 . [START_REF] Teusch | Understanding the Water Splitting Mechanism on WO3(001) -A Theoretical Approach[END_REF] In their suggested path, the formation of hydroxyl groups is an intermediary step. Their work was however only theoretical. Uniquely, this paper combines experimental results with calculations in order to understand the temperature dependent evolution of hydrogen from humidity. In the DRIFT spectra, unlike other metal oxides, e.g. SnO 2 [START_REF] Wicker | Ambient Humidity Influence on CO Detection with SnO2 Gas Sensing Materials. A Combined DRIFTS/DFT Investigation[END_REF] , there is no hydroxyl group formation visible at any temperature. This experimental result indicates that after adsorption of water molecularly into a vacancy (dominantly visible at low temperatures) the direct oxidation of the surface takes place without the formation of intermediaries (e.g. hydroxyl groups). The experimental measurements were, however, done under very high humidity conditions (10 %RH).In their study, Teusch and Klüner, only consider the W 5c as an adsorption site for water. Based on the findings here, and the work of Albanese et al., it is clear that adsorption onto V 0 C must also be considered. It is therefore possible that in reality, the level of water adsorption on the surface is more than the monolayer (all W 5c covered) examined by Teusch and Klüner. [START_REF] Teusch | Understanding the Water Splitting Mechanism on WO3(001) -A Theoretical Approach[END_REF] In this case, the evolution of hydrogen, for example could be a result of an interaction between neighboring water molecules. A large theoretical study coupled with additional measurements is currently being run to further examine the role of surface coverage on the mechanism.

In summary, if the temperature study using DRIFT spectroscopy and DC resistance measurements is considered in the context of the literature and the DFT calculations, the following surface reactions can be identified. At low temperatures water can adsorb molecularly into V 0 C or onto W 5c . This results in a significant decrease of the resistance, because the reoxidation of the surface through atmospheric oxygen is hindered and molecularly adsorbed water pushes charge into the surface. At 200 °C, the degree of molecular water adsorption and the subsequent surface oxidation/ release of hydrogen are similar. As a result the change of the film's resistance is also low. In this case, no change of the bands attributed to tungsten-oxygen bonds is visible in the DRIFT spectra. Additionally, the bands associated with adsorbed water are lower than at 125 °C. At 300 °C, hydrogen is released from the molecularly adsorbed water. The formation of intermediary hydroxyl groups is not visible in the DRIFT spectra. An overall reaction can be formulated as

V O C⋅⋅ + 2e -+ H 2 O -0.65 eV → 2H -O O C⋅⋅ +2e -1.22 eV → O O C + H 2
eq. 4 where 1.22 eV is the energy required to split an adsorbed water mocule to form hydrogen. The adsorption of hydrogen on the reoxidized surface is depicted in Figure 6. It is important to note that this process doesn't cease as the surface is in constant equilibrium with the atmosphere. Namely, oxygen vacancies are created and healed by de-and adsorption of oxygen as described by the following reaction: 49 In order to verify the evolution of hydrogen at 300 °C, catalytic conversion measurements were done. One gram of WO 3powder was heated to 300 °C in a reactor. The powder was first exposed to dry synthetic air and then to 10% RH. The exhaust was monitored using a highly sensitive H 2 sensor (TGS2600, Figaro Engineering Inc., Japan). 50 The sensor's response during the exhaust measurement is shown in Figure 7. The red and black lines are measurements done with one gram of WO 3 in the reactor heated to 300 °C. The measurements were done twice with two different samples (fresh powder) to show that the results are reproducible and that the process is reversible. The blue line is the same measurement done without WO 3 (only a heated tube) to show that the sensor is not responding to residual humidity. The green line is a measurement done during gas pulses from a hydrogen bottle, in order to show the response of the sensor to hydrogen. This shows that hydrogen is not eliminated by the molecular sieve or the silica gel. It also indicates that based on the response of the TGS2600 to the exhaust, that the reaction between WO 3 and humidity results in a production of over 1 ppm of H 2 .The measurement indicates that the hydrogen evolution is constant and that once humidity is removed the process stops. Repetition of the measurement using the same sample showed the same response indicating that the process is fully reversible. Overall, the presented results reveal that at 300 °C water is split on the surface of WO 3 and there is a constant evolution of hydrogen. This process appears to be of catalytic nature and the proposed mechanism is shown in Figure 8.

Conclusion

The value of the systematic study presented here is twofold. It is remarkable that the evolution of hydrogen from humidity was identified under extremely mild conditions, 300 °C in humidity. In addition, with the gained insights into the reaction mechanism, the interference of humidity in the various applications of WO 3, can now be understood. For example, it is probable that the sensor signal of the chemiresistive WO 3 gas sensors operated at 300 °C decreases to oxidizing gases because humidity itself also oxidizes the surface. The findings support the explanation of Gavrilyuk that adsorbed water decreases the photochromism of pure WO 3 by blocking the surface adsorption centers. 6 The findings, however, also give an explanation for the remarkable promotion of the oxidation reaction rate by humidity described by Matsuyama et al. for fiber optic gas sensors based on noble loaded WO 3 . 4 The remarkable ability of water to oxidize the surface, may also offer an alternative explanation for the enhanced memristive behavior of WO 3 in the presence of humidity. In other words, instead of the mechanism proposed by Zhou et al., one driven by the incorporation of oxygen from the water molecules is probable. A similar mechanism was proposed by Heisig et al. for SrTiO 3. [START_REF] Heisig | Oxygen Exchange Processes between Oxide Memristive Devices and Water Molecules[END_REF] Additionally, although recently there has been a high level of interest in modeling the surface reaction with humidity, this paper identifies several aspects that must be additional considered, e.g. higher level of water coverage and competing reaction with atmospheric oxygen.

ASSOCIATED CONTENT

Supporting Information. DRIFT spectra during 10 %RH. Temperature dependent XRD of the film based on SA. Operando XRD of a SA sensor operated at 125 °C during an initial 3 h exposure to dry synthetic air and then to 90 %RH.

S-2

Figure 1 .

 1 Figure 1. Setup of the catalytic conversion measurements. 1 g of WO3 powder was heated to 300 °C and then exposed to 10% RH. The exhaust was dried using silica gel and molecular sieve and then monitored using a TGS2600 (Figaro Engineering Inc., Japan) metal oxide sensor. Reprinted with permission from Anna Staerz, from ref 39.

Figure 2 .

 2 Figure 2. Resistance measurement and simultaneously attained single channels spectra of a WO3 film during exposure to dry synthetic air (blue) and nitrogen (red) at 300 °C. By referencing the single channel spectrum in dry synthetic air to the one in nitrogen, the absorbance spectrum was attained.

Figure 3 .

 3 Figure 3. (a) The relative change in the resistance as a result of exposure to 10 % RH. The average of four identically prepared films was taken and the standard deviation was calculated. (b) The DRIFT spectra taken of the film at three different temperatures (125, 200, and 300 °C) during exposure first to 10 % RH (H2O, light red) followed by 10 % RH (D2O, dark red). The reference state was the spectrum taken in dry synthetic air.

Figure 4 .

 4 Figure 4. (a) Top-and (b) side view of the optimized (001) β-WO3 surface with a 𝐕 𝟎 𝐂 represented in black and an adsorption position over the W5c marked in gray. A cutout of the examined W5c and 𝐕 𝟎 𝐂 of the cell (c) perfect surface and with (d) vacancy. Tungsten atoms are ochre and oxygen atoms red. Bond lengths and distances are given in Å.

  89 eV) as the reference point. Jin et al. found that the surface adsorption of molecular oxygen into V 0 C in a parallel orientation, via both oxygen atoms, is energetically more favorable than a perpendicular orientation via only one atom. Both adsorption orientations were considered here, see Figure5(a) and (b).

Figure 5 .

 5 Figure 5. View of a section taken out of the optimized (001) β-WO3 surface, (a) perpendicular adsorption of O2 into 𝐕 𝟎 𝐂 (b) parallel adsorption of O2 into 𝐕 𝟎 𝐂 , (c) adsorption of O2 onto W5c (d) perpendicular adsorption of O2 into 𝐕 𝟎 𝐂 and adsorption of O2 onto W5c. (e) parallel adsorption of O2 into 𝐕 𝟎 𝐂 and adsorption of O2 onto W5c, (f) adsorption of H2O into 𝐕 𝟎 𝐂 (g) adsorption of H2O onto W5c (h) adsorption of H2O into 𝐕 𝟎 𝐂 and adsorption of H2O onto W5c, (i) parallel adsorption of O2 into 𝐕 𝟎 𝐂 and adsorption of H2O onto W5c, (j) perpendicular adsorption of O2 into 𝐕 𝟎 𝐂 and adsorption of H2O onto W5c, and (k) adsorption of O2 onto W5c and adsorption of H2O into 𝐕 𝟎 𝐂 . CT denotes the charge transfer in electrons and the adsorption energy is the change referenced to the surface with a 𝐕 𝟎 𝐂 . W atoms are colored ochre and O atoms red. If electronic charge is transferred to the surface, when compared to the surface with a terminal vacancy, then a blue arrow pointing towards the surface is used. If electronic charge is transferred from the surface, in comparison to a terminal vacancy, then the arrow is red and points upwards. Bond lengths and distances are given in Å.

Figure 6 .

 6 Figure 6. View of a section taken out of the optimized (001) β-WO3 surface on which a surface vacancy was healed and hydrogen was released.

Figure 7 .

 7 Figure 7. The resistance of the TGS2600 sensor. The red and black lines are measurements done of a heated gram of WO3. The blue line is the same measurement done without WO3 (heated empty tube). The green line is the resistance change during exposure to different concentrations of hydrogen taken from a gas bottle.

Figure 8 .

 8 Figure 8. Proposed catalytic cycle of the continuous water splitting on the WO3 surface.

S2.

  To account for strongly self interacting electrons the DFT+U approach was also considered here. Similar to Kishore et al. the Hubbard U parameter for W was chosen as 2.0 eV. 1,2 However, the charge transfers were found to not differ significantly. S3. Total DRIFT spectra of the film during exposure to H 2 O and subsequent exposure to D 2 O.

ACKNOWLEDGMENT

This work is conducted in the frame of the France Campus -PHC PROCOPE 2107 -Project N°37744TG. The computer simulations were performed using HPC resources from CALMIP (Grant 16038).

AUTHOR INFORMATION Corresponding Author

* nb@ipc.uni-tuebingen.de; Tel.: +49 7071 29 78761

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

SUPPORTING INFORMATION THERMAL WATER SPLITTING ON THE WO 3 SURFACE-EXPERIMENTAL PROOF

Anna Staerz 1 , Arne Kobald 1 , Tamara Russ 1 , Udo Weimar 1 , Anne Hémeryck 2 and Nicolae Barsan


D C Johnson

I M Ballard

A Bessière

K W J Barnham

J P Connolly

M Mazzer

C Calder

G Hill

J S Roberts

Observation of

Strain-balanced quantum well solar cells (SB-QWSC) have exhibited efficiencies similar to GaAs p-n devices [START_REF] Ekins-Daukes | Strain-balanced GaAsP/InGaAs quantum well solar cells[END_REF] and are proving to be a viable path to achieving efficiencies beyond the long-standing GaAs single junction world record [START_REF] Green | Solar cell efficiency tables (version 26)[END_REF]. Growing strain balanced quantum wells in the intrinsic region of a GaAs p-i-n solar cell increases the short-circuit current (J SC ), due to the lower absorption threshold energy in the well. The increase in J SC exceeds the reduction in open-circuit voltage (V OC ) provided the compressively strained InGaAs wells are balanced by the tensile strained GaAsP barriers [START_REF] Ekins-Daukes | Strain-balanced GaAsP/InGaAs quantum well solar cells[END_REF]. Further efficiency enhancement has been achieved by growing the devices on a distributed Bragg reflector (DBR) such that incident photons, which are not absorbed by the device initially, are reflected back into the device. The second pass absorption further increases J SC [START_REF] Johnson | Advances in Bragg stack quantum well solar cells[END_REF]. Another benefit of growing SB-QWSCs on a DBR substrate, as we report here, is that the device dark current is suppressed when compared to devices grown on GaAs substrates. Reduction in dark current is only observed when the device is operating in a regime where radiative recombination is the dominant recombination mechanism [START_REF] Barnham | Observation of ideal dark-current behaviour in strain-balanced quantum well solar cells[END_REF]; this behaviour is indicative of photon recycling (PR) or photon self-absorption effects. In this letter we will present experimental and theoretical evidence of PR in SB-QWSCs.

PR occurs when photons that are emitted from a semiconductor device are subsequently re-absorbed. Considering that this process may occur several times for a single photon, the effect is often observed experimentally as an increase in the minority carrier lifetime [START_REF] Ahrenkiel | Ultralong minority-carrier lifetime epitaxial GaAs by photon recycling[END_REF][START_REF] Lush | A study of minority carrier lifetime versus doping concentration in n-type GaAs grown by metalorganic chemical vapour deposition[END_REF]; therefore, one can also expect to observe PR as a reduction in the dark current. Dumke [START_REF] Dumke | Spontaneous radiative recombination in semiconductors[END_REF] was one of the first to describe PR quantitatively while more recently Asbeck's photon recycling factor [START_REF] Asbeck | Self-absorption effects on the radiative lifetime in GaAs-GaAlAs double heterostructures[END_REF] has been used widely both theoretically and experimentally. Evidence of PR has been readily demonstrated in semiconductor devices such as double-heterostructures [START_REF] Kuriyama | Effect of photon recycling on diffusion length and internal quantum efficiency in GaAs-Alx Ga1-xAs heterostructures[END_REF][START_REF] Schnitzer | Ultrahigh spontaneous emission quantum efficiency, 99.7% internally and 72% externally, from AlGaAs/GaAs/AlGaAs double heterostructures[END_REF] or light emitting diodes (LED) [START_REF] Schnitzer | 30% external quantum efficiency from surface textured, thin-film light-emitting diodes[END_REF][START_REF] Schubert | Resonant cavity light-emitting diodes[END_REF][START_REF] Dupont | Efficient GaAs light-emitting diodes by photon recycling[END_REF], whereas, results from solar cells have been rather more elusive [START_REF] Zahraman | High efficiency Al0.2Ga0.8As/Si stacked tandem solar cells using epitaxial lift-off[END_REF]. Patkar et al [START_REF] Patkar | Characterization of photon recycling in thin crystalline GaAs light emitting diodes[END_REF] were one of the only groups to demonstrate PR as a reduction of the dark current, however, this was in LEDs rather than solar cells. Here we present the first direct evidence for dark-current reduction due to PR in solar cells and confirm this through the observation of changes in the electroluminescence (EL) spectra.

Results presented here have been observed in devices with three different SB-QWSC designs (referred to as Design 1, 2 and 3) which vary only in the number and depth of the quantum wells. Each SB-QWSC device design was gown on both a DBR and a GaAs substrate for comparison. To ensure that the wafers with and without DBRs were grown in as similar conditions as possible a pre-grown DBR wafer and a n-type GaAs substrate were placed down-and up-stream, respectively, in the metal organic vapour phase epitaxy (MOVPE) reactor and both overgrown in the same gas flow. The quantum wells are grown with alloys of In X Ga 1-X As from x = 0.11 to x = 0.22 and the barriers are grown with alloys of GaAs y P 1-y where y = 0.91. The DBRs were grown using alternating layers of material with low (Al 0.8 Ga 0.2 As) and high (Al 0.13 Ga 0.87 As) refractive indices. The DBR layer thicknesses were designed to maximize absorption of normal incident light by the quantum wells. To reduce the series resistance presented by the DBR, intermediate layers of Al 0.5 Ga 0.5 As are grown between each DBR layer. Further details of the DBR design can be found in Ref 3.

Devices were processed into fully metalised, 800um diameter, mesa structures for dark current characterization as well as concentrator devices with 3.7mm diameter mesas and 11% metalisation [START_REF] Maroto | suns GaAs concentrator solar cell with efficiency over 23%[END_REF] for EL characterisation. Fully metalised devices are used such that series resistance effects on the ideality n = 1 behaviour of the devices can be kept to a minimum while operation under illuminated conditions can still be predicted [START_REF] Tibbits | Demonstration of additivity in strainbalanced quantum well solar cells and efficiency at high concentration[END_REF]. All devices were grown and processed at the EPSRC National Centre for III-V Technologies.

A computer controlled Keithley 238 SMU was used to measure the dark current response of the fully metalised devices. Measurements were made using the four point method and pulsed current sourcing to reduce heating. All measurements were taken at 25° C using a Peltier cooled stage. The measurements made in the high current region of 20 control and DBR devices of Design 2 are presented in Fig. 1 The measurements were subsequently fit to a two exponential model, based on the ideal Shockley diode equation [START_REF] Shockley | The theory of p-n junctions in semiconductors and p-n junction transistors[END_REF], shown below:

) /( ) ( 02 ) /( ) ( 01 2 1 T k n R J V q T k n R J V q s s e J e J J ⋅ ⋅ ⋅ - ⋅ ⋅ ⋅ ⋅ - ⋅ ⋅ + ⋅ =
(1) where J 01 and J 02 are reverse saturation currents, n 1 and n 2 are ideality factors, V is the applied bias, R S is the series resistance, k is the Boltzmann constant and T is the temperature. For the purposes of fitting, the ideality factor n 1 is fixed at 1. The first term in eqn. 1 dominates at high bias and corresponds to quantum well radiative recombination, which we have shown to be more important than the ideal Shockley recombination in the field-free p and n regions [START_REF] Barnham | Observation of ideal dark-current behaviour in strain-balanced quantum well solar cells[END_REF]. The ideality n 2 in the second term is fit to a value of ~ 2 which corresponds to non-radiative Shockley-Read-Hall (SRH) recombination.

By fitting the measured data to eqn 1 we obtain values for J 01 , J 02 , n 2 and R S . The average J 01 and J 02 values obtained by fitting results on ~ 15 devices for each design are listed in Table I. The ideality n = 1 reverse saturation current (J 01 ) obtained from the fit is reduced by >25% for all three device designs and by 33% for Designs 1 and 3. The latter reduction would result in absolute efficiency enhancement of ~0.3% at concentration levels of ~500x entirely due to PR as a result of the DBR substrate. One can see that while J 01 for the DBR cell is reduced compared to the control in all cases, the non-radiative contribution J 02 is largely unchanged for Designs 1 and 2; the third design exhibited irregular behaviour in the ideality n ~2 region, which is why the J 02 values could not be fit.

To validate the observed reduction in the J 01 reverse saturation current a photon recycling model has been developed. The EL spectra and, therefore, dark current are calculated using the method outlined in Refs [START_REF] Nelson | Observation of suppressed radiative recombination in single quantum well p-i-n diodes[END_REF][START_REF] Araujo | Absolute limiting efficiencies for photovoltaic energy conversion[END_REF]. The net photon flux exiting the surface of a semiconductor device can be described using the generalised Planck equation for a non-black body. Integrating the photon flux density over energy, one can obtain the radiative dark-current for a given bias or quasi-Fermi level separation. This method requires calculation of device absorptivity. The absorptivity of an arbitrary device design is calculated using a complex refractive index which is calculated from an experimental quantum efficiency and device thickness. The resulting refractive index is then used along with the matrix method [START_REF] Macleod | Thin Film Optical Filters[END_REF] to calculate the absorptivity of a stack of thin films including anti-reflection coatings, the window layer, active layer and DBR. This method takes into account multiple passes of photons and interference effects in the structure; these effects are important to consider in SB-QWSCs since the quantum wells have relatively low absorptivity, which can result in multiple reflections before emitted photons are re-absorbed. All calculations are done using the low-AOD spectrum (913W/m 2 ) while assuming ideal behaviour with zero series resistance and 5% optical shading.

EL spectra are measured by driving devices into forward bias using a Keithley 238 source measure unit (SMU); while maintaining the device at 25° C using a Peltier cooled stage. The resulting emission spectrum is measured using a computer controlled charge-coupled device. Fig. 2 shows a comparison between the resulting measured EL spectrum and the modeled spectrum of a SB-QWSC grown on a DBR (Design 1). In both the measured and modeled spectrum one can see the distinct Fabry-Perot oscillations in the emission and that the results from our model fit the experiment quite well. By integrating the EL spectrum over energy we are also able to calculate the expected radiative contribution to the device dark current. Fig. 1 shows both the measured dark currents for the control and DBR devices of Design 2 as well as the predicted radiative contribution to the dark current as calculated from our model. The reduction of the dark current is evident in both the measurements and the calculated values from our model. Table I compares the calculated reverse saturation currents with the average measured parameters for all three designs. One can see that the model quite accurately describes the ideality n = 1 component of the measured dark currents in both relative and absolute terms.

Given close agreement between the model and experiment we have calculated DBR designs in an effort to achieve the highest overall device efficiency. As we have seen previously, a DBR can enhance the J SC of a SB-QWSC considerably [START_REF] Johnson | Advances in Bragg stack quantum well solar cells[END_REF]. Similarly, we have shown here that a SB-QWSCs grown on a DBR benefits from an increase in V OC due to PR. However, to increase both J SC and V OC it is desirable to maximise the DBR optical "stop band" magnitude and width while reducing the angular dependence of the reflectivity. Given the DBR material (Al 1-X Gas X As) and series resistance limit imposed on the number of DBR periods, the DBR reflectivity can be optimised in one of two ways -growing multiple DBRs with varying stop band wavelengths and growing chirped DBRs (CDBR) [START_REF] Saka | Bragg Reflector of GaAlAs/AlAs Layers with Wide Bandwidth Applicable to Light-Emitting-Diodes[END_REF].

Using the model, we calculated the efficiency of a 50 shallow quantum well CDBR (20½ period 1% chirp) SB-QWSC at 350x concentration while varying the initial thickness of the DBR layers. One can see in Fig. 3 that there are two distinct efficiency maxima. The bottom left (indicated by the letter A) corresponds to a DBR optimised to reflect normal incident photons of energies which are readily absorbed by the quantum wells and so the efficiency enhancement is primarily due to an increase in J SC . On the other hand, the peak at the bottom right of the plot (indicated by the letter B) corresponds to a DBR which primarily reflects photons of energies which are absorbed by the quantum wells at off-normal incidence resulting in PR. The critical angle for photons to escape from the GaAs-Air surface is ~16 degrees and so any photons emitted towards the front surface at an angle greater than this will be reflected back into the device. If the reflected photons are not re-absorbed by the quantum wells, they may also be reflected at the back of the device by the off-normal incidence reflectivity of the DBR. Therefore, photons emitted outside of the GaAs-Air escape cone may then be re-absorbed by the quantum wells and thus result in a reduction of the device dark current. By designing a device with two 10½ period 1% chirped DBRs each of which is designed for one of the peak efficiencies in Figure 3, the calculated device J SC and V OC values were both enhanced. Table II summarises the results of calculated efficiencies for devices 50 well SB-QWSC devices grown with no DBR, and grown on CDBRs optimized for V OC , J SC , and both. An further increase in efficiency can be realised by growing the SB-QWSC on two CDBRs with different reflectivity.

In conclusion PR in SB-QWSCs has been demonstrated in both the dark current and the EL for the first time in a solar cell. Both effects are also well described by a model based on radiative recombination. We have also demonstrated that PR can have a considerable impact on the overall conversion efficiency of a SB-QWSC. We expect optimized device and DBR designs to enhance the PR effect which should, in conjunction with increased normal incidence absorption, result in >2% (absolute) increase in efficiency. This work not only demonstrates experimental evidence of PR in SB-QWSCs but also the effect of DBR design on PR and ultimately overall device efficiency.

Figure 1 .

 1 Figure 1. Measured dark current of shallow quantum well SB-QWSC grown on DBR (grey) and GaAs (black) substrates. Also shown are the calculated ideality n = 1 dark current components for the DBR (dotted) and Control (dashed).

Table I .

 I Reverse saturation currents as calculated and fitted from measurement.

		In fraction Calculated	Measured
	Design #	x	J 01 (x10 -21 A) J 02 (x10 -13 A) J 01 (x10 -21 A)
	1 Control 0.11	1.7	3.2±0.4	1.5±0.2
	1 DBR	0.11	1.2	2.9±0.7	1.0±0.2
	2 Control 0.17	6.8	4.5±0.5	8.9±0.3
	2 DBR	0.17	5.1	5.1±0.2	6.7±0.1
	3 Control 0.22	1.8	NA	1.5±0.2
	3 DBR	0.22	1.3	NA	1.0±0.1

Table II .

 II Calculated V OC , J SC and efficiency (η), at 350x concentration (low-AOD spectrum), of a 50 shallow well SB-QWSC grown on a GaAs substrate, a single 20½ period DBR optimised to increase J SC , a single 20½ period DBR optimised to increase V OC , and two 10½ period DBRs one of which is optimised for J SC and the other for V OC enhancement. All DBRs are chirped at 1% per period.

	10000	Control Ideality n=1
	2 )	DBR Ideality n=1
	DBR Design None Single CDBR J SC optimised Single CDBR V OC optimised 100 1000 Current density (mA/cm	V OC (V) J SC (A/m2) FF (%) 1.15 257 90.1 1.16 267 89.8 1.19 259 88.9 Control DBR Ideality n=1 reduction	η (%) 29.2 30.5 30.1
	Double CDBR J SC and V OC 0.95	1.17 1.00	266 1.05	90.2 1.10	30.7 1.15
			Bias (V)

  Figure 2. Measured (empty squares) and modeled (solid line) electroluminescence spectra of shallow quantum well DBR SB-QWSC.Figure3. Modeled device efficiency for a 50 well CDBR SB-QWSC at 350x (low-AOD spectrum) concentration in the radiative limit, as a function of the initial layer thicknesses of a Al 0.8 Ga 0.2 As / Al 0.13 Ga 0.87 As DBR chirped at 1% per period. The high efficiency in the lower left (point A) is due to an increase in J SC while the lower right efficiency peak (point B) is due to an increase in V OC .

	0.1 1 80 100 120 140 160 180 200 220 Emission (a.u.) 0.87 As Layer Thickness 20 40 60 Al 0.13 Ga	A		Measured EL Calculated EL B	28.6% 28.7% 28.9% 29.0% 29.2% 29.3% 29.5% 28.3% 28.4%
	50	100	150	200	250
	1.32 1.34 1.36 1.38 1.40 1.42 1.44 Al 0.80 Ga 0.20 As Layer Thickness
			Energy (eV)


Pierrick Legrand

Evelyne Lutton

Gustavo Olague

Evolutionary denoising based on an estimation of Hölder exponents with oscillations

, which necessitates the design of ad-hoc genetic operators.

Introduction

In the past years many different signal and image denoising techniques have been proposed, some of them being even based on artificial evolution [START_REF] Olague | Evolutionary signal enhancement based on hölder regularity analysis[END_REF][START_REF] Grenier | An interactive ea for mulifractal bayesian denoising[END_REF]. The basic notations are the following. One observes a signal or an image Y which is some combination F (X, B) of the signal of interest X and a noise B. Making various assumptions on the noise, the structure of X and the function F , one then tries to obtain an estimate X of the original signal, optimal in some sense. We consider denoising as equivalent to increasing the Hölder function αY (see section 2 for definitions) of the observations. Indeed, it is generally true that the local regularity of the noisy observations is smaller than the one of the original image, so that in any case, αX should be greater than αY . In this paper, section 2 recalls some basic facts about Hölder regularity analysis. We describe in section 3 how oscillations are used to provide an estimator of the Hölderian regularity. The new denoising method is explained in section 4 and the evolutionary algorithm, with its ad-hoc genetic operators, are detailed in section 5. Numerical experiments are presented in section 6.

To simplify notations, we deal with 1D signals, and we assume that signals are nowhere differentiable. Generalisation to differentiable signals simply requires to introduce polynomials in the definitions [START_REF] Meyer | Wavelets, Vibrations and Scalings[END_REF]. Below the definitions of the pointwise and local Hölder exponents are given. Let α ∈ (0, 1), and

x0 ∈ K ⊂ R. A function f : K → R is in C α x 0 if for all x in a neighbourhood of x0, |f (x) -f (x0)| ≤ c|x -x0| α (2)
where c is a constant. The pointwise Hölder exponent of f at x0, denoted αp(f, x0), is the supremum of the α for which (2) holds. Let us now introduce the local Hölder exponent: Let α ∈ (0, 1), Γ ∈ R.

One says that f ∈ C α l (Γ ) if ∃C : ∀x, y ∈ Γ : |f (x)-f (y)| |x-y| α ≤ C (3). Let α l (f, x0, ρ) = sup{α : f ∈ C α l (B(x0, ρ))}. The local Hölder exponent of f at x0 is α l (f, x0) = lim ρ→0 α l (f, x0, ρ).
Since αp and α l are defined at each point, we may associate to f two functions x → αp(f, x) and x → α l (f, x) which are two different ways of measuring the evolution of its regularity. The quality of a denoising technique based on these exponents, strongly relies on the quality of an estimator of these quantities. In [START_REF] Olague | Evolutionary signal enhancement based on hölder regularity analysis[END_REF], the estimation was performed by a wavelet technique. We will see in the sequel that a better estimation of the Hölder exponent can be obtained by measuring the oscillations of the function.

Estimation by oscillations

The estimation based on oscillations measurements is a direct application of the local Hölder exponent definition (see [START_REF] Tricot | Curves and Fractal Dimension[END_REF]). The condition (3) can be written as:

A function f (t) is Hölderian with exponent α ∈ [0, 1] at t if there exists a constant c, for all τ : oscτ (t) ≤ cτ α with oscτ (t) = sup |t-t ′ |≤τ f (t ′ ) -inf |t-t ′ |≤τ f (t ′ ) = sup t ′ ,t ′′ ∈[t-τ,t+τ ] |f (t ′ ) -f (t ′′ )|.
At each point we estimate the pointwise Hölder exponent as the slope of the regression of the logarithm of the oscillation versus the size of the window τ . As we see in figure 1, the estimation by oscillations provides better results than an estimation by wavelets.

Method

According to the notation of section 1, we seek a denoised version X of the observed signal Y that meets the following constraints: 1) X is close to Y in the L 2 sense.

2) The Hölder function of X is prescribed.

If αX is known, we choose α X = αX . In some situations, αX is not known but can be estimated from Y , see [START_REF] Legrand | Débruitage et interpolation par analyse de la régularité hölderienne. application à la modélisation du frottement pneumatique-chaussée[END_REF]. Otherwise, we just set α X = αY +δ, where δ is a user-defined positive function, so that the regularity of X will be everywhere larger than the one of the observations. Two problems have to be solved in order to obtain X. First, a procedure that computes the Hölder function of a signal from discrete observations is needed.

Second, we need to be able to manipulate the data so as to get a specific regularity. To solve the first problem, the estimation method of section 3 is used, and for the second problem, an evolutionary algorithm has been designed.

Evolutionary algorithm

We consider that an individual is a signal (1D or 2D). On the contrary to [START_REF] Olague | Evolutionary signal enhancement based on hölder regularity analysis[END_REF] where an individual was made of a subset of wavelet coefficients, a direct encoding of the signal in the genome has been used. Initialisation: As the search space is extremely large, a direct search starting from a random set of initial signals has no chance to provide a good denoising in a reasonable time. However, many initial guess are available, including the noisy signal itself. We actually use several deterministic denoising methods to provide the initial population. These methods are the Multifractal Bayesian Denoising [START_REF] Vehel | Bayesian multifractal denoising[END_REF] and the Multifractal Pumping [START_REF] Legrand | Débruitage et interpolation par analyse de la régularité hölderienne. application à la modélisation du frottement pneumatique-chaussée[END_REF], and depend from a parameter setting. The parameters are generated randomly. Fitness: An important point of the method is that the fitness calculation is based on two kind of fitness function. A pointwise fitness has been defined for each point of the signal as a combination between the quality of the individual in term of regularity and in term of distance to the noisy signal. As said above, it is based on the estimation by oscillations. The pointwise fitness is then combined to provide a local fitness. The local fitness is the sum of the pointwise fitness on a given segment (or window). The local fitness is used in the crossover and in the mutation operators. We compute the global fitness when we perform this sum on the full signal. This fitness is used for the selection and for the ranking. Crossover: A simple ranking selection mechanism with selective pressure 2 is used to select two individuals. Random crossing points are then selected. For images, a set of random rows and columns is chosen. The local fitness on each resulting segment is then used to select the best parts of the two individuals as the corresponding segment of the child. Mutation: In a similar way, each segment (or image window) is muted using a probability law inversely proportional to the local fitness. For each individual we consider the worst local fitness wlf i.e. the fitness of the worst segment. Let lf (j, i) the local fitness of the i th segment of the j th individual. The probability of mutation for this segment is P m(j, i) = lf (j,i) wlf .

Numerical Results

For the first example (see figure 2), the original signal is a Generalized Weierstrass function with a regularity α(X, t) = t with t ∈ [0, 1]. This signal is corrupted by a white Gaussian noise (standard deviation equal to 0.3). We use a synthetic image to perform an experiment in 2 dimensions. The figure 3 shows the original image, the noisy one, a denoising by Soft Thresholding and by our method. The second row displays the corresponding Hölder functions. As in the previous examples, our method allows to obtain a denoised version of the signal with the prescribed regularity.

Conclusion

We have experimented in this paper a new scheme for a multifractal denoising technique. It is based on a more precise and more complexcomputation of the Hölder exponent of a signal. This work is actually a first attempt to use an estimation of Hölder exponent based on oscillations for signal enhancement. Preliminary experiments yield satisfactory results, with a more precise control of the reconstructed regularity, which has to be considered as a major advantage for this type of techniques. Moreover, the evolutionary engine that has been designed has the following interesting characteristics: it performs a basic hybridisation with two other denoising techniques (Multifractal Bayesian Denoising and Multifractal Pumping for the initialisation step), and uses locally optimised genetic operators. Further work will first consist in a more precise analysis of the locally optimised genetic operators in comparison with classical "blind" ones. Second, the hybridisation scheme has to be investigated as it may be a good solution to reduce computation costs of the method. Additionally, the availability of a pointwise and local definition of the fitness opens the way to "Parisian" evolution implementations for the genetic engine. This may be another solution to reduce computational expenses of the method, see for example [START_REF] Raynal | Polar ifs + parisian genetic programming = efficient ifs inverse problem solving[END_REF][START_REF] Dunn | Automated photogrammetric network design using the parisian approach[END_REF].

Fig. 1 .Fig. 2 .

 12 Fig.1. 10 multifractional Brownian Motions have been built with a regularity H evolving like a sine. The 2 methods of estimation of the Hölderian regularity have been applied: a wavelet-based (W1) and the method by oscillations (OSC). After an optimisation of the parameters of the 2 methods in term of risk, the means of the estimated Hölder functions are displayed.

Fig. 3 .

 3 Fig. 3. Original image, the noisy one, a denoising by Soft Thresholding and by our method (100 ind, 500 gen). The second row displays the corresponding Hölder functions.