Gabriele Steidl
email: steidl@mathematik.uni-kl.de

M Nikolova
email: nikolova@cmla.ens-cachan.fr

Fast Hue and Range Preserving Histogram Specification: Theory and New Algorithms for Color Image Enhancement

Keywords: image enhancement, scaling and shifting methods, hue preservation, gamut problem, exact histogram specification, color perception. I

Color image enhancement is a complex and challenging task in digital imaging with abundant applications. Preserving the hue of the input image is crucial in a wide range of situations. We propose simple image enhancement algorithms, which conserve the hue and preserve the range (gamut) of the R, G, B channels in an optimal way. In our setup, the intensity input image is transformed into a target intensity image whose histogram matches a specified, well-behaved histogram. We derive a new color assignment methodology where the resulting enhanced image fits the target intensity image. We analyze the obtained algorithms in terms of chromaticity improvement and compare them with the unique and quite popular histogram-based hue and range preserving algorithm of Naik and Murthy. Numerical tests confirm our theoretical results and show that our algorithms perform much better than the Naik-Murthy algorithm. In spite of their simplicity, they compete with well-established alternative methods for images where hue-preservation is desired.

the hue and enhancing the brightness, the obtained image will appear more colorful. Examples where the hue is modified are shown in Fig. 1. The range (gamut) preservation is often omitted in works on image enhancement; see, e.g. the recent textbook [2, p. 80]. Each color channel in a digital image can only take a limited number, say L, of integer values, e.g., L = 256 for 8-bit coding. If the enhancement method produces larger or smaller values these are clipped back to the boundary of [0, L -1] which also changes the hue. In Fig. 1 right 36.1 % of the pixels are clipped back to 255 which yields too many yellow pixels. Finally, a low computational complexity of algorithms is particularly important when dealing with "megapixel" images taken by commercial cameras, resources in hardware implementations and extensions to video.

Remark 1. Fully automatic color image enhancement faces (at least) two major limits: i) "The chemical compounds that form color receptors vary among the population. The physical shapes of the receptors vary among the population and within the retina. Thus, the color vision among observers with normal color vision varies significantly" [1, p. 18]. ii) Image enhancement is always driven by an application: typically the user needs specific visual information determined by his/her purpose. Further subjective criteria are of paramount importance [START_REF] Solomon | Fundamentals of Digital Image Processing. A Practical Approach with Examples in Matlab[END_REF]. Consequently, we do not look for fully automatic image enhancement algorithms. Here we focus on histogram based methods. The selection of a suitable target histogram enables the user's needs to be satisfied. Moreover we wish to conceive fast algorithms. In order to achieve our goals, we propose simple algorithms composed of two stages:

(a) the intensity channel of the input RGB image is matched to a specified histogram which gives us the target intensity image; (b) the RGB color values are computed based on the target intensity image so that they satisfy the hue and gamut constraints in an optimal way. These stages are briefly commented below.

Stage (a). Exact histogram specification (HS), also known as histogram matching, of single-valued (gray-valued) images aims to transform an input image to an output image which exactly fits a prescribed target histogram. Histogram equalization (HE) is a particular case of HS where the target histogram is uniform. Usually HE leads to unnatural images and should not be the target of choice. We do not focus on the construction of target histograms. Instead, we adopt a simple approach inspired by [START_REF] Bovik | Handbook of Image and Video Processing (Communications, Networking and Multimedia)[END_REF]. For digital image HS is an ill posed problem [START_REF] Gonzalez | Digital Image Processing[END_REF]. The clue to ensuring exact HS is to obtain a meaningful total strict ordering of all pixels in the input digital image. We perform exact HS using the algorithm in [START_REF] Nikolova | Fast sorting algorithm for exact histogram specification[END_REF] which currently provides the best pixel ordering in terms of quality and speed.

Stage (b). The extension of histogram methods to color images is a quite complex task. The histogram of a grayvalue image is 1D while the histogram of a color image is 3D which gives rise to an under-determined problem. For instance, applying HE to each color channel independently changes the color content (the hue) of the image, see Fig. 1 middle. Further it is not easy to produce color images that respect the range constraints; see Fig. 1 right. As a central result of this paper, we propose a general and optimal hue and range preserving color assignment methodology.

Related Work. Since the inaugural paper [START_REF] Sapiro | Histogram Modification via Differential Equations[END_REF] providing PDE-based and variational formulations for image histogram modifications, these methods were further expanded to deal with color image enhancement; see [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF], [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF], [START_REF] Provenzi | Perceptual color correction: A variational perspective[END_REF], [START_REF] Papadakis | A variational model for histogram transfer of color images[END_REF]. These approaches provide flexible tools to incorporate various knowledge on human visual perceptual phenomena, typically in relation with Retinex theory [START_REF] Land | The Retinex Theory of Color Vision[END_REF]. An automatic color enhancement (ACE) algorithm for digital images, mimicking some characteristics of the human visual system, has been proposed in [START_REF] Gatta | ACE:", in An automatic color equalization algorithm[END_REF] and refined in [START_REF] Rizzi | A new algorithm for unsupervised global and local color correction[END_REF]. A fast implementation of ACE was developed in [START_REF] Getreuer | Automatic color enhancement (ACE) and its fast implementation[END_REF]. A perceptually inspired variational approach allowing a more flexible control of contrast adjustment and attachment to data was proposed in [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF]. A numerical implementation of the gradient descent technique applied to the corresponding energy functionals coincides with the equation of the ACE. Some basic requirements for "perceptually inspired" objectives were formulated in [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] and gave rise to successful algorithms [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF], [START_REF] Bertalmío | Issues About Retinex Theory and Contrast Enhancement[END_REF].

Next we summarize the main approaches via histogram modification of color images following a chronological order. Since the suitably normalized histogram of an image is also the empirical probability distribution of its pixel values, a statistical vocabulary is used in many papers. In [START_REF] Trahanias | Color image enhancement through 3-D histogram equalization[END_REF] a 3D color histogram in the RGB color space was proposed for HE; the resultant images present an excessive brightness for bright pixels, see [START_REF] Han | A novel 3-D color histogram equalization method with uniform 1-D gray scale histogram[END_REF]. A method that preserves both the hue and the range (gamut) constraints was inaugurated by Naik and Murthy in [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF]. Even though this article did not show color image applications, it is a state-of-the-art method applied in many papers; see, e.g., [START_REF] Bassiou | Color image histogram equalization by absolute discounting back-off[END_REF], [START_REF] Han | A novel 3-D color histogram equalization method with uniform 1-D gray scale histogram[END_REF]. As to the choice of the color space, some methods work directly in the RGB space while others operate in transformed color spaces, e.g., LHS, HSI, YIQ, HSV, etc., see [START_REF] Gonzalez | Digital Image Processing[END_REF]. When processing is done in a transform color space, coming back to the original RGB space typically generates a gamut problem, as cautioned in [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF]. Beyond the additional numerical cost, a post-processing in RGB is then needed (often realized using [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF]). Grayvalue grouping was tentatively extended to color HE in [START_REF] Chen | Gray-Level Grouping (GLG): An automatic method for optimized image contrast enhancementpart i: The basic method[END_REF]. In [START_REF] Menotti | A fast huepreserving histogram equalization method for color image enhancement using a bayesian framework[END_REF], a new definition of the histogram of a color image was introduced whose cumulative distribution function (cdf) is the product of the marginal cdf's of each color channel. Then the color values are increased / decreased by the same amount iteratively. This work was refined in a later paper [START_REF] Menotti | Histogram equalization methods for color image contrast enhancement[END_REF]. Another approach, developed in [START_REF] Bassiou | Color image histogram equalization by absolute discounting back-off[END_REF], is to work in the HSI space where the hue and the saturation are equalized and then processed using probability smoothing. All pixels in the RGB space that present gamut problem are corrected using [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF]. A generic brightness preserving dynamic histogram equalization scheme, composed of five steps, was proposed in [START_REF] Kong | Color image enhancement using brightness preserving dynamic histogram equalization[END_REF]. This scheme was applied to color images in several ways, including transforms into other color spaces. The work in [START_REF] Han | A novel 3-D color histogram equalization method with uniform 1-D gray scale histogram[END_REF] demonstrates that the methods in [START_REF] Trahanias | Color image enhancement through 3-D histogram equalization[END_REF], [START_REF] Bassiou | Color image histogram equalization by absolute discounting back-off[END_REF], and [START_REF] Menotti | A fast huepreserving histogram equalization method for color image enhancement using a bayesian framework[END_REF] based on higher dimensional histogram definition, increase the brightness of the image and cannot fit the prescribed uniform histogram. The main conclusion is that only the 1D histogram of the intensity channel can be considered for equalization.

The new color values are then computed using the algorithm in [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF]. The method in [START_REF] Han | A novel 3-D color histogram equalization method with uniform 1-D gray scale histogram[END_REF] was recently improved in [START_REF] Nikolova | A fast algorithm for exact histogram specification. Simple extension to colour images[END_REF]. In order to avoid the excessive contrast enhancement due to HE, a histogram mixing strategy was applied in [START_REF] Arici | A histogram modification framework and its application for image contrast enhancement[END_REF]. There are also many histogram based techniques where the enhancement function is an S-type, or power, or logarithmic transform; see [START_REF] Wang | Fast image/video contrast enhancement based on weighted thresholded histogram equalization[END_REF]- [START_REF] Chien | Color image enhancement with exact HSI color model[END_REF]. In particular, the approach in [START_REF] Sen | Automatic exact histogram specification for contrast enhancement and visual system based quantitative evaluation[END_REF] is based on models for color perception and is automatic. Unfortunately, there are no algorithms nor tests on color images.

Contributions. We propose a general affine model for fast hue and range preserving image enhancement in the RGB space which gives rise to Algorithm 3. Two simple but important instances of this algorithm are the Multiplicative algorithm 4 and the Additive algorithm 5. We show how the outcome of Algorithm 3 can be faithfully approximated as a convex combination of the images obtained by Algorithm 4 and Algorithm 5, which is quite practical. The enhancement performances of our algorithms and the Naik-Murthy algorithm [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF] are analyzed in terms of their chromaticity improvement. In all cases, our algorithms clearly outperform the algorithm in [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF] recently applied to color images in [START_REF] Han | A novel 3-D color histogram equalization method with uniform 1-D gray scale histogram[END_REF]. All numerical tests confirm our theoretical results. Our algorithms are simple and fast. They are really efficient when one wishes to give a better clarity of images (not too altered by artifacts) while preserving the original color ambience.

Outline. In Section II we sketch our HS method and present the Maik-Murthy algorithm [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF]. Section III presents our approach for color image enhancement. In Section IV we evaluate our algorithms and the algorithm in [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF] analytically in terms of saturation as well as qualitatively. Section V presents numerical results. Conclusions and points for future work are drawn in Section VI.

The proofs of all statements are given in the Appendix. II. PRELIMINARIES Let w = (w r , w g , w b ) be an RGB image of size M × N , where w c ∈ {0, . . . , L -1}, c ∈ {r, g, b} are its red, green and blue channels, respectively. For 8-bit images we have L = 256. We reorder each color channel columnwise into a vector of size n := M N and address the pixels by the index set

I n := {1, • • • , n}.

A. Histogram Specification

The intensity of an RGB image w is defined by [START_REF] Gonzalez | Digital Image Processing[END_REF] f (w) := 1 3 (w r + w g + w b ).

(

) 1
Then

f has 3(L-1)+1 different values in 1 3 {0, • • • , 3(L-1)}. Remark 2.
Instead of the intensity f we can also work with other convex combinations of RGB values. E.g., we can use the weights 0.299, 0.587 and 0.114 which are in proportion to the human perception of the RGB channels, see [START_REF] Berns | Billmeyer and Saltzman's Principles of Color Technology[END_REF], [START_REF] Solomon | Fundamentals of Digital Image Processing. A Practical Approach with Examples in Matlab[END_REF].

We want to find an intensity image f with gray values in {0, . . . , L -1} which has a specified (target ,L, where ♯ stands for cardinality. Such exact HS can almost never be achieved for images with a small number of different values compared to the number of pixels using the classical statistical method based on the cumulative density function, see [START_REF] Gonzalez | Digital Image Processing[END_REF]. Instead we will apply a procedure based on a meaningful strict ascending ordering of the pixels in f . Various ordering algorithms for digital images were proposed in the literature, see e.g. [START_REF] Coltuc | Exact histogram specification[END_REF], [START_REF] Wan | Joint exact histogram specification and image enhancement through the wavelet transform[END_REF], [START_REF] Nikolova | Exact histogram specification for digital images using a variational approach[END_REF]. The method in [START_REF] Nikolova | Fast sorting algorithm for exact histogram specification[END_REF], based on [START_REF] Nikolova | Exact histogram specification for digital images using a variational approach[END_REF], provides currently the best way in terms of speed and quality to order the pixels in digital images. The basic idea is to minimize a smoothed ℓ 1 -TV functional by simple fixed point iterations with the original image as initialization. After a few iterations the approximate minimizer has entries which differ (up to very few outliers) pairwise from each other while the ordering of the original gray values is retained. Let ∇ denote the discrete gradient operator (horizontal and vertical forward differences), see [START_REF] Nikolova | Fast sorting algorithm for exact histogram specification[END_REF], ∇ T its transposed and let

) histogram h = ( h 1 , . . . , h L ), i.e., h[k] := ♯ {i ∈ I n : f [i] = k -1}, k = 1, . . .
η(t) := t α + |t| and η -1 (y) = αy 1 -|y| , ( 2
)
where α := 0.05 is the default value. Note that η = θ ′ where θ(t) := |t| -α log(1 + |t| α ), see [START_REF] Nikolova | Fast sorting algorithm for exact histogram specification[END_REF]. Once a strict ordering is obtained, exact HS is direct. Our HS algorithm reads as: Algorithm 1 HS using strict ordering [START_REF] Nikolova | Fast sorting algorithm for exact histogram specification[END_REF] Initialization: u (0) := f , β := 0.1, target histogram h, iteration number K (default K := 5), c 0 := 0. ------------------------1. For k = 1, . . . , K compute

u (k) := f -η -1 ( β ∇ T η(∇u (k-1) )
) .

2. Obtain the ordering {i j } n j=1 of I n from the ascending sort of the entries of u (K) . 3. For k = 0, . . . , L -

1 set c k+1 := c k + h k and f [c k + 1] = . . . = f [c k+1 ] = k.
The importance of a meaningful strict ordering for HS is illustrated in Fig. 2 in the context of HE. The Matlab built-in function histeq does not involve a strict ordering of f and the resulting histogram of f is not uniform. This entails some artifacts shown in (a). Such artifacts are not observed in (b) obtained using our Algorithm 1. The colors in Fig. 2

B. Hue and Range Preservation

Range preservation is a mandatory constraint for all digital imaging devices [START_REF] Bovik | Handbook of Image and Video Processing (Communications, Networking and Multimedia)[END_REF]. A transformed version w of w can be correctly depicted only if

w c [i] ∈ [0, L -1] ∀ i ∈ I n ∀ c ∈ {r, g, b} , (3)
since no more than L digits can be displayed. Otherwise, the obtained image is modified according to the visualization device -which is quite an ad-hoc option; see e.g., Fig. 1 right.

The hue of an image w is given by H(w) = 0 if w r = w g = w b and otherwise by

H(w) := { θ if w b ≤ w g , 360 o -θ if w b > w g , (4)
where

θ := arccos 1 2 ((w r -w g ) + (w r -w b )) ((w r -w g ) 2 + (w r -w b )(w g -w b )) 1 2
, see [START_REF] Gonzalez | Digital Image Processing[END_REF]. Note that the denominator of θ can be rewritten as

( 1 2 ((w r -w g ) 2 + (w r -w b ) 2 + (w g -w b ) 2 )) 1 2 .
Remark 3. The simplest hue and range preserving method is to apply the same affine mapping ξ(w) := aw + b to all pixels, computing a and b so that the least and the largest pixels in ξ(w) are 0 and L -1, respectively. Let

w max := max{w c [i]; c ∈ {r, g, b}, i ∈ I n } and let w min := min{w c [i]; c ∈ {r, g, b}, i ∈ I n }. Then ξ(w) given by ξ(w) := (L -1) w -w min w max -w min (5)
is the desired stretching of w. For example, see Fig. 16, top.

It is easy to see that the hue of the modified image w is also preserved if the color values of each pixel are modified by the same affine transform

w c [i] = a[i]w c [i] + b[i], c ∈ {r, g, b}, (6)
where the constants a[i] and b[i] have to be chosen for any i ∈ I n . Finding other appropriate hue-preserving transforms is an interesting problem. For a[i] = 1, model ( 6) amounts to an additive transform, usually called shifting. For b[i] = 0, it is a linear/multiplicative transform known as scaling. Both scaling and shifting have been introduced in [START_REF] Yang | Efficient luminance and saturation processing techniques for bypassing color coordinate transformations[END_REF], [START_REF] Yang | Saturation clipping in the lhs and yiq color spaces[END_REF]. In general, the result of ( 6) fails the range constraint (3).

The gamut problem was examined by Naik and Murthy in [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF] in the scaling case for a

[i] = f [i]/f [i], where f is a target intensity. If f [i]/f [i] > 1,
the range constraint (3) might not be guaranteed. In such a case the authors propose to avoid the potential problem by switching from the RGB color space to the CMY (Cyan = L -1-R, Magenta = L -1-G, Yellow = L -1-B) space and then to transform back into RGB. This correction step reads for all c ∈ {r, g, b} as

w c [i] = L -1 - L -1 -f [i] L -1 -f [i] (L -1 -w c [i]) if f [i] f [i] > 1.
This formula is equivalent to

w c [i] = L-1-f [i] L-1-f [i] (w c [i] -f [i]) + f [i]
, so that the algorithm in [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF] can be formulated as follows:

Algorithm 2 Naik and Murthy [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF] 1. Compute the intensity f of w and the target intensity f . 2. For i ∈ I n compute

(i) w c [i] := f [i] f [i] w c [i] if f [i] f [i] ≤ 1 (ii) w c [i] := L-1-f [i] L-1-f [i] (w c [i] -f [i]) + f [i] if f [i] f [i] > 1
Algorithm 2 is often used to avoid the gamut problem.

III. NEW AFFINE HISTOGRAM SPECIFICATION MODELS

In this section we develop our affine color enhancement methodology. Given an RGB image w and a target histogram, we compute its intensity f by (1) and then the target intensity image f by Algorithm 1. Our next goal is to transform w into an image w having the following properties: (a) Intensity fit: f = 1 3 ( w r + w g + w b ). (b) Hue preservation: the hue of w and w coincide. (c) Range preservation:

0 ≤ w c ≤ L -1, c ∈ {r, g, b}.
We adopt the hue preserving affine transform [START_REF] Sapiro | Histogram Modification via Differential Equations[END_REF]. Summing up over c in [START_REF] Sapiro | Histogram Modification via Differential Equations[END_REF] shows that property (a) holds if and only if

f [i] = a[i]f [i] + b[i] ⇔ b[i] = f [i] -a[i]f [i]. (7)
Therefore the affine model ( 6) obeys (a) if and only if

w c [i] = a[i](w c [i] -f [i]) + f [i], c ∈ {r, g, b}. (8)
Two particular instances of (6) are the following: -Scaling: For b[i] = 0, model ( 8) reads as

w c [i] = f [i] f [i] w c [i], c ∈ {r, g, b}. ( 9
)
-Shifting: For a

[i] = 1, model (8) becomes w c [i] = w c [i] -f [i] + f [i], c ∈ {r, g, b}.
We have to adapt these models so that they preserve the range. We will use for all i ∈ I n the magnitudes

M [i] := max{w c [i] : c ∈ {r, g, b}}, m[i] := min{w c [i] : c ∈ {r, g, b}} (10)
and similarly M [i] for the maximum and m[i] for the minimum of the RGB components of w [i] given in [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF].

Remark 4. By the definitions of f , m and M we have

0 ≤ m[i] ≤ f [i] ≤ M [i] ≤ L -1.
Further

M [i] = f [i], resp., m[i] = f [i] if and only if w r [i] = w g [i] = w b [i], i.e., w[i] is a gray pixel. A pixel w[i] has an upper gamut problem if M [i] > L -1 and a lower gamut problem if m[i] < 0.
We will treat these gamut problems in an optimal way in the following sense:

• Assume that we have an upper gamut problem, i.e.,

M [i] > L -1 for some i ∈ I n . Then M [i] = w k [i]
for some k ∈ {r, g, b} and the best correction of this pixels is clearly to choose a[i] in (8) so that w k [i] has the closest value in the range, i.e. w k [i] = L -1, see, e.g., [START_REF] Bovik | Handbook of Image and Video Processing (Communications, Networking and Multimedia)[END_REF]. Equivalently,

L -1 = a[i](M [i] -f [i]) + f [i]. (11)
From Remark 4 we know that for non gray-valued pixels

M [i] -f [i] > 0, so that a[i] = L -1 -f [i] M [i] -f [i] ≥ 0.
Thus, for the upper gamut problem, the corrected color values of pixel i are given by

w c [i] = L -1 -f [i] M [i] -f [i] (w c [i]-f [i])+ f [i], c ∈ {r, g, b}. ( 12
)
• Assume we have a lower gamut problem m[i] < 0 for some i ∈ I n . Let k ∈ {r, g, b} be such that

w k [i] = m[i]. Then the optimal correction in (8) obeying (c) is to set w k [i] = 0, i.e., 0 = a[i] (m[i] -f [i]) + f [i]. (13)
By Remark 4 f [i] -m[i] > 0 for non gray pixels, so that

a[i] = f [i] f [i] -m[i] ≥ 0.
Hence for the lower gamut problem, the corrected color value at i is given by

w c [i] = f [i] f [i] -m[i] (w c [i] -f [i]) + f [i], c ∈ {r, g, b}. (14)

A. Affine Algorithm With Optimal Range Preservation

Our affine model is a convex combination of the shifting and scaling models for some λ ∈ [0, 1]:

w c [i] = λ f [i] f [i] w c [i] + (1 -λ) (w c [i] -f [i] + f [i]) (15) = a[i](w c [i] -f [i]) + f [i],
where

a[i] := λ f [i] f [i] + 1 -λ (16)
with upper and lower gamut corrections ( 12) and ( 14) if necessary. Clearly, for λ = 1 we have the scaling model and for λ = 0 the shifting one. Algorithm 2 corresponds to λ = 1 in (15) but the gamut problem is tackled just by thresholding a[i] at one; this appears to be an important drawback.

The next propositions show that correcting the gamut problems using ( 12) or ( 14) does not yield new gamut problems.

0 ≤ w c [i] ≤ L -1, c ∈ {r, g, b}.
Let us mention that a lower gamut problem can obviously not appear for the multiplicative model ( 9) i.e. for λ = 1.

Proposition 2. Let λ ∈ [0, 1). Assume that pixel i ∈ I n in (15) has an lower gamut problem. Then its correction

w c [i] in (14) satisfies 0 ≤ w c [i] ≤ L -1, c ∈ {r, g, b}.
Using Propositions 1 and 2, the optimal range-preserving approximation of our affine model [START_REF] Bertalmío | Issues About Retinex Theory and Contrast Enhancement[END_REF] can be computed in one iteration where all pixels in the input image are modified only once. The algorithm is described below.

a[i] := λ f [i] f [i] + (1 -λ), G λ m [i] := a[i](m[i] -f [i]) + f [i], G λ M [i] := a[i](M [i] -f [i]) + f [i]
and for all c ∈ {r, g, b}:

(i) w c [i] := a[i] (w c [i] -f [i]) + f [i] if G λ m [i] ≥ 0 and G λ M [i] ≤ L -1 , (ii) w c [i] := L-1-f [i] M [i]-f [i] (w c [i] -f [i]) + f [i] if G λ M [i] > L -1, (iii) w c [i] := f [i] f [i]-m[i] (w c [i] -f [i]) + f [i] if G λ m [i] < 0.
Algorithm 3 and the role of λ is illustrated in Fig. 3. The images were computed for Gaussian target histogram with parameters (l, r) = (0.9, 0.1), see [START_REF] Sen | Automatic exact histogram specification for contrast enhancement and visual system based quantitative evaluation[END_REF], Sec. V-A.

B. Multiplicative, Additive Algorithms and Their Combinations

For λ ∈ {0, 1} Algorithm 3 yields two simple range preserving scaling and shifting algorithms called Multiplicative Here the size of the sets U (λ) in [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF] in percent of all image pixels are 0.70, 1.24, 2.12, 3.13, 4.17 and the sets L(λ) are empty. All nuances between the very colorful image w × and the grayish image w + can be also obtained by their convex combinations in [START_REF] Han | A novel 3-D color histogram equalization method with uniform 1-D gray scale histogram[END_REF].

and Additive algorithm, respectively. Observing that

G 0 m [i] = m[i] -f [i] + f [i], G 0 M [i] = M [i] -f [i] + f [i], G 1 M [i] = f [i] f [i] M [i]
these algorithms read as follows:

Algorithm 4 Multiplicative Color Enhancement 1. Compute the intensity f of w and the target intensity f using Algorithm 1. 2. For i ∈ I n compute M [i] by [START_REF] Papadakis | A variational model for histogram transfer of color images[END_REF]

. If f [i] = 0, then w[i] := 0. Otherwise compute G 1 M [i] = f [i] f [i] M [i]
and for all c ∈ {r, g, b}:

(i) w c [i] := f [i] f [i] w c [i] if G 1 M [i] ≤ L -1, (ii) w c [i] := L-1-f [i] M [i]-f [i] (w c [i] -f [i]) + f [i] if G 1 M [i] > L -1.
Algorithm 5 Additive Color Enhancement 1. Compute the intensity f of w and the target intensity f using Algorithm 1.

For i

∈ I n compute M [i] and m[i] by (10). If f [i] = 0, then w[i] := 0. Otherwise compute G 0 m [i] = m[i] -f [i] + f [i] and G 0 M [i] = M [i] -f [i] + f [i]
and for all c ∈ {r, g, b}:

(i) w c [i] := w c [i] -f [i] + f [i] if G 0 m [i] ≥ 0 and G 0 M [i] ≤ L -1, (ii) w c [i] := L-1-f [i] M [i]-f [i] (w c [i] -f [i]) + f [i] if G 0 M [i] > L -1, (ii) w c [i] := f [i] f [i]-m[i] (w c [i] -f [i]) + f [i] if G 0 m [i] < 0.
Let w × be obtained by the Multiplicative algorithm 4 and w + by the Additive algorithm 4. For some λ ∈ [0, 1], consider

w c := λ w × c + (1 -λ) w + c ∀c ∈ {r, g, b}. ( 17
)
Since w c is a convex combination of w × and w + , it obeys all conditions (a)-(c). We want to know if w c can replace the affine Algorithm 3. In order to answer this question, we set

U(λ) := {i ∈ I n : G λ M [i] > L -1}, L(λ) := {i ∈ I n : G λ m [i] < 0}. (18)
Here U(λ) corresponds to the upper gamut step (ii) and L(λ) -to the lower gamut step (iii) in Algorithm 3.

Proposition 3. The sets U(λ) and L(λ) defined in [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF] fulfill L(1) = ∅ and

U(λ 1 ) ⊆ U(λ 2 ), L(λ 1 ) ⊇ L(λ 2 ), 0 ≤ λ 1 < λ 2 ≤ 1.
In particular, [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF] yields

U(1) = { i ∈ I n : f [i] f [i] M [i] > L -1 } , U(0) = { i ∈ I n : f [i] -f [i] + M [i] > L -1 } , L(0) = { i ∈ I n : f [i] -f [i] + m[i] < 0 } . ( 19
)
From Proposition 3 one has U(0) ⊆ U (1). The notation in [START_REF] Bassiou | Color image histogram equalization by absolute discounting back-off[END_REF] enables Algorithms 4 and 5 to be restated as follows:

-Multiplicative algorithm (λ = 1)

(i) w c [i] = f [i] f [i] (w c [i] -f [i]) + f [i] if i ∈ I n \ U (1), (ii) w c [i] = L-1-f [i] M [i]-f [i] (w c [i] -f [i]) + f [i] if i ∈ U(1). ( 20
)
-Additive algorithm (λ = 0)

(i) w c [i] = (w c [i] -f [i]) + f [i] if i ∈ I n \ {U(0) ∪ L(0)}, (ii) w c [i] = L-1-f [i] M [i]-f [i] (w c [i] -f [i]) + f [i] if i ∈ U(0), (iii) w c [i] = f [i] f [i]-m[i] (w c [i] -f [i]) + f [i] if i ∈ L(0). ( 21
)
The relation between w c in [START_REF] Han | A novel 3-D color histogram equalization method with uniform 1-D gray scale histogram[END_REF] and the outcome w of Algorithm 3 for the same λ is described in the next proposition. Proposition 4. Let w be obtained by Algorithm 3 and w by [START_REF] Han | A novel 3-D color histogram equalization method with uniform 1-D gray scale histogram[END_REF]

for the same λ ∈ [0, 1]. Then it holds for i ∈ I n \ { U(1) \ U(0) ∪ L(0) } that w c [i] = w c [i].
The sets U(1), U(0) and L(0) are usually small for reasonable target intensity images f (see Table I) and U(1) \ U (0) contains generally much less pixels than U(1). If we wish to see the enhancement results w of Algorithm 3 for various λ ∈ [0, 1], Proposition 4 justifies to compute instead w by [START_REF] Han | A novel 3-D color histogram equalization method with uniform 1-D gray scale histogram[END_REF] which is much more practical. Thus, by sliding λ in [START_REF] Han | A novel 3-D color histogram equalization method with uniform 1-D gray scale histogram[END_REF], we can easily move between the two models.

IV. COMPARISON OF THE ALGORITHMS A. Saturation Properties

Here we analyze the saturation of images enhanced by our methods and by the Naik-Murthy algorithm. The saturation of an RGB image w in the HSI model [START_REF] Gonzalez | Digital Image Processing[END_REF] is defined by

S(w) :=    1 - min{w r , w g , w b } f (w) if f (w) > 0, 0 if f (w) = 0 . ( 22
)
Proposition 5. Let S(w[i]) and S( w[i]) denote the saturation of pixel i in the input image w and the image w obtained by our Algorithm 3, respectively. If

f [i] ∈ {m[i], M [i]} we have S( w[i]) = 0.
Otherwise the obtained saturation is given by

(i) S( w[i]) = S(w[i]) ( λ + (1 -λ) f [i] f [i]
)

if i ∈ I n \ {U (λ) ∪ L(λ)}, (ii) S( w[i]) = S(w[i]) f [i] f [i] L-1-f [i] M [i]-f [i] if i ∈ U(λ), (iii) S( w[i]) = 1 if i ∈ L(λ).
To clarify the comparison, all magnitudes relevant to Algorithm 2 (Naik and Murthy) hold the superscript •, those relevant to Algorithms 4 (Multiplicative) and 5 (Additive) have the superscripts × and +, respectively. In particular, we obtain:

-Algorithm 4 (Multiplicative) (i) S( w × [i]) = S(w[i]) if i ∈ I n \ U (1), (ii) S( w × [i]) = S(w[i]) f [i] f [i] L-1-f [i] M [i]-f [i] if i ∈ U(1). ( 23
) -Algorithm 5 (Additive) (i) S( w + [i]) = S(w[i]) f [i] f [i] if i ∈ I n \ (U(0) ∪ L(0)), (ii) S( w + [i]) = S( w × [i]) if i ∈ U(0), (iii) S( w + [i]) = 1 if i ∈ L(0). ( 24
) Let us denote V := { i ∈ I n : f [i] f [i] > 1 } . ( 25
)
By [START_REF] Bassiou | Color image histogram equalization by absolute discounting back-off[END_REF] and Remark 4 we find that if i ∈ U(1)

then f [i] f [i] > L-1 M [i] ≥ 1 and that if i ∈ L(0), then f [i] f [i] < 1-m[i] f [i] < 1. Hence V ⊇ U(1) and L(0) ⊂ I n \ V. ( 26
)
Using the notation in [START_REF] Arici | A histogram modification framework and its application for image contrast enhancement[END_REF], case (i) in Algorithm 2 (Naik-Murthy) holds for any i ∈ I n \ V and step (ii) holds for any i ∈ V. The saturation of images enhanced by applying the Naik -Murthy Algorithm 2 is given by the following proposition.

Proposition 6. Let S(w[i]) and S( w • [i]

) denote the saturation of pixel i in the input image w and the image w • obtained by the Naik-Murthy Algorithm 2, respectively. Then

(i) S( w • [i]) = S(w[i]) if i ∈ I n \ V, (ii) S( w • [i]) = S(w[i]) f [i] f [i] L-1-f [i] L-1-f [i] if i ∈ V.
Remark 5. Proposition 3 and [START_REF] Wang | Fast image/video contrast enhancement based on weighted thresholded histogram equalization[END_REF] show that V ⊇ U(1) ⊇ U(0). Note that all these inclusions are almost always strict; see Table I. E.g., in [START_REF] Wang | Fast image/video contrast enhancement based on weighted thresholded histogram equalization[END_REF] we find U(1) = V if and only if

M [i] = L -1 for all i ∈ V.
Using Propositions 5 and 6, the saturation that Algorithms 4, 5 and 2 provide can be rigorously compared.

• Let i ∈ I n \ V. Then i ̸ ∈ U(1) and i ̸ ∈ U(0). Hence S(w[i]) = S( w × [i]) = S( w • [i]) ≤ S( w + [i]),
where the last inequality becomes an equality only for

f [i] = f [i]. Beyond

this case, only the Additive algorithms 5 increases the saturation if f

[i] < f [i].
But since the output intensity is decreased, the perceived colorfulness of the pixel is decreased.

• Let i ∈ V \ U(1) = {i ∈ I n : 1 < f [i] f [i] ≤ L-1 M [i] }. Then i ̸ ∈ (U(0) ∪ L(0)) and consequently S(w[i]) = S( w × [i]) > S( w + [i]) > S( w • [i])
and S( w • [i]) decreases faster than S( w

+ [i]) when f [i] increases because S( w • [i]) S( w + [i]) = M [i] -f [i] L -1 -f [i] < 1.
• Let i ∈ U [START_REF] Berns | Billmeyer and Saltzman's Principles of Color Technology[END_REF]. Then

S(w[i]) > S( w

× [i]) ≥ S( w • [i]),
where the equality is reached if and only if

M [i] = L -1.
But for most of the pixels one has M [i] < L -1. Further,

S(w[i]) > S( w

+ [i]) ≥ S( w • [i]),
where the equality holds if and only if M [i] = L -1 and i ∈ U (0). So the inequality is strict for most of the pixels. In all cases, the images enhanced by the Maik-Murthy algorithm have the weakest saturation. On I n \ V, where the target intensity is less than the input intensity, the Additive algorithm 5 gives a better saturation than the Multiplicative algorithm 4. On V \ U(1) the Multiplicative algorithm 4 gives rise to a better saturation than the Additive algorithm 5.

B. Qualitative Comparison

We begin with a simple but instructive example where we apply Algorithms 4, 5 and 2 to two different "images" each composed of one dark and one bright pixel, resp., [START_REF] Arici | A histogram modification framework and its application for image contrast enhancement[END_REF]48,[START_REF] Nikolova | Fast sorting algorithm for exact histogram specification[END_REF], w bright = (80, 172, 108) having the same hue but different intensities f dark = 35 and f bright = 120. In Figs. 4 and5, the input pixels w are shown on the top row, while the next rows detail the results of the algorithms w.r.t. the target intensity f ∈ {0, . . . , 255} given on the x-axis. By [START_REF] Bassiou | Color image histogram equalization by absolute discounting back-off[END_REF] we see that the pixel belongs to U(1)

w dark =
for f > (L-1)f M =: f U (1) , to U(0) for f > (L -1) -M + f =: f U (0) , to L(0) for f < f -m =: f L(0) and to V for f > f := f V .
The corresponding values for our dark and bright image are given in the following table:

f U (1) f U (0) f L(0) f V w dark
185.9 242 10 35 w bright 177.9 203 40 120

Fig. 4 deals with the dark pixel w dark . The Multiplicative algorithm 4 (i) is applied for f ∈ [0, 185.9]. All color values are multiplied by f /f , where f /f > 1 for f > 35 which yields a clear increase of the distance between all color channels. The third row shows a pleasant enhancement of the dark input pixel. By [START_REF] Kong | Color image enhancement using brightness preserving dynamic histogram equalization[END_REF] the input saturation is preserved. The Additive Algorithm 5 (i) is performed for f ∈ [START_REF] Papadakis | A variational model for histogram transfer of color images[END_REF]242], where all color values are increased by the same amount f -f . Since the input pixel is quite dark, the values w c , c ∈ {r, g, b} and f are relatively close to each other. For this reason, all color channels w + c remain close to each other. On [START_REF] Papadakis | A variational model for histogram transfer of color images[END_REF][START_REF] Yang | Saturation clipping in the lhs and yiq color spaces[END_REF] we have f > f so by [START_REF] Nikolova | A fast algorithm for exact histogram specification. Simple extension to colour images[END_REF], S( w + ) > S(w) and S( w + ) continuously decreases from 1 to S(w) = 0.29. On [START_REF] Yang | Saturation clipping in the lhs and yiq color spaces[END_REF]242], S( w + ) decreases from S(w) = 0.29 to 0.04 according to S(w)f / f . This explains why the colors on the third row remain quite dull, compared to Algorithm 4. For Algorithm 2, case (i) holds only for f ∈ [0, [START_REF] Yang | Saturation clipping in the lhs and yiq color spaces[END_REF], where the input saturation is preserved. If f > 35, step (ii) is performed and S( w • ) decreases much faster than in Algorithm 5. As a consequence, on [START_REF] Yang | Saturation clipping in the lhs and yiq color spaces[END_REF]255] the enhanced colors tend to be nearly equal and the obtained color values are nearly gray, see the third row in the figure .  Fig. 5 shows the performance for the brighter pixel w bright . The Multiplicative algorithm 4 (i) is applied for f ∈ [0, 177.9]. The input saturation is preserved. The Additive algorithm 5 (i) holds for f ∈ [40,203]. On [40, 120] the recovered saturation decreases from 1 to S(w) = 0.33 and on (120, 203] it slowly decreases to 0.6S(w). In Algorithm 2, step (i) holds for f ≤ f = 120 where the input saturation is unchanged. Step (ii) is applied for f ∈ (120, 255] -the interval is not so large as in Fig. 4 and the saturation decreases much less fast to zero. On the 3rd row one sees that the colors obtained with all the three algorithms are quite similar.

Remark 6. From Fig. 4, if a dark pixel has a wrong hue (e.g. due to compression or printing artifacts, noise, color cast, etc.), the Multiplicative algorithm 4 can magnify the intensity of this wrong color. If the input image contains a lot of such pixels, the Additive algorithm 5 can be a better choice.

Our conclusions drawn in Subsection IV-A and our findings for one pixel images are confirmed by our tests on the two images, bungalow (underexposed) and flower (slightly lustreless) depicted in Fig. 6 and mimics the phenomena explained for Fig. 4 and flower those relevant to Fig. 5.

( w × r , w × g , w × b ) ( w + r , w + g , w + b ) ( w • r , w • g , w • b ) 0

V. NUMERICAL RESULTS

Here we demonstrate the performance of our algorithms to render images where we want to preserve the hue.

A. Target Histograms

Our algorithms depend, up to a certain degree, on the choice of a target histogram for the intensity channel. Various target histograms avoiding the drawbacks of HE have been proposed in the literature. Some of them leave gaps in the target histogram which can yeild artifacts as in Fig. 2, see [START_REF] Chen | Gray-Level Grouping (GLG): An automatic method for optimized image contrast enhancementpart i: The basic method[END_REF], [START_REF] Wang | Fast image/video contrast enhancement based on weighted thresholded histogram equalization[END_REF], others preserve the input brightness which limits the enhancement of underexposed images, see,. [START_REF] Kong | Color image enhancement using brightness preserving dynamic histogram equalization[END_REF], [START_REF] Thomas | A modified version of van-cittert's iterative deconvolution procedure[END_REF]. The models proposed in [START_REF] Arici | A histogram modification framework and its application for image contrast enhancement[END_REF] combine the input image histogram and a uniform histogram using various penalties and parameters. Instead, we adopt a simple and intuitive approach.

A common way for histogram based enhancement is to use the histogram of a well exposed example image; see [START_REF] Bovik | Handbook of Image and Video Processing (Communications, Networking and Multimedia)[END_REF]. Commercials in photography and image processing software (e.g., Photoshop) mention that well exposed pictures typically have bell-shaped histograms. Based on these advises we focus on target histograms whose shapes are Gaussian functions h G , with domain [0, L -1], fixed so that

l := h G (0) ≤ 1, max x∈[0,L-1]
h G (x) = 1 and r := h G (L-1) < 1 .

A user has to choose two parameters:

• l ∈ (0, 1] which is the desired portion of dark pixels;

• r ∈ (0, 1] drawing the desired portion of light pixels. Note that one cannot choose l = r = 1. Given l ∈ (0, 1] and r ∈ (0, 1], the shape of the target histogram reads as

h G (x) = exp ( - (x -µ) 2 σ ) , x ∈ [0, L -1], for µ = -(L-1)(ln l- √ ln l ln r) ln r-ln l , σ = (L-1) 2 ( √ -ln l- √ -ln r) 2 (ln r-ln l) 2 . ( 27
)
Finally the target histogram h is normalized according to the number n of pixels in the image:

h(x) = n h G (x) ∑ L-1 x=0 h G (x) ∀ x ∈ {0, • • • , L -1}.
Whenever ( 27) is used, we shall write h G for h.

Remark 7.

If the input RGB image w has no pixel values on an interval [0, L 0 ] for some L 0 ≥ 1 one has to perform the hue-preserving stretching in [START_REF] Gonzalez | Digital Image Processing[END_REF]. The target histogram is chosen based on the stretched histogram and the enhanced image is computed from the stretched image; see Fig. 16.

With this cautionary remark, we can explain how to choose good target histograms using [START_REF] Sen | Automatic exact histogram specification for contrast enhancement and visual system based quantitative evaluation[END_REF]. The input intensity histogram, after stretching if necessary, is denoted by h f .

h mix := 1 2 (h f + h G ) ( 28
)
where the parameters (l, r) for h G are selected following the rules in (i). For example, see Figs. 8, 12 and 15. This remark is illustrated in Fig. 8.

B. Enhancement Tests

We present some results from a large series of test images with the goal to improve the visual quality. The enhanced image should seem natural and an observer should not suspect that it is a post-processing result.

We compare our Multiplicative algorithm 4 (×) with the Naik-Murthey algorithm 2 (NM) and the Additive algorithm 5 (+). The HS in these algorithms is done by Algorithm 1. The histogram of the original intensity image and the target histograms are depicted beneath the images. The percentage of pixels having a gamut problem in Alg. 4 (×), Alg. 5 (+) and Alg. 2 (NM) is contained in Table I. Further we provide comparison results with

• the fast implementation of ACE by [START_REF] Getreuer | Automatic color enhancement (ACE) and its fast implementation[END_REF] available online at http://demo.ipol.im/demo/g_ace/, and • the perceptual color enhancement through variational methods in [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF] and [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF]. ACE has one main parameter, the enhancement strength α whose default value α = 0.5 is often a good choice. For the two perceptual enhancement methods [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF] and [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF], the authors gave us their codes and helped us to tune the parameters.

We present the results for images with different defects. The parameter values for all methods are given in the captions, as well as the image credits. The original images in Figs. 6, 7, 8, 10, 11, 13, 12 and 16 are photos taken by the authors who wanted to improve them. For all these images, we do not have "ground truth". For Fig. 17 we shot an underexposed and a better "example" image which enabled us to compare with the perceptual histogram-based method in [START_REF] Papadakis | A variational model for histogram transfer of color images[END_REF]. Fig. 8 illustrates Remark 8. The image club in (a) is underexposed and its histogram h f does not obey Remark 8(i). The result in (b) is obtained with a Gaussian target histogram h G for parameters following (i). In (c) we use a mixed target histogram h mix as proposed in [START_REF] Thomas | A modified version of van-cittert's iterative deconvolution procedure[END_REF]. This image better shows the ambience of the club. The perceptual method [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] in (d) gives a colder color palette.

Two tests with the image islanda are shown in Fig. 9. The original in (a) is a rather light image. The Multiplicative, the Additive and the NM algorithms with Gaussian target histograms produce quite similar results, which confirms our discussion in Subsection IV-B. Only the issue of our Additive algorithm is shown in (b). The perceptual method [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] and the ACE perform similarly and give a nice, different color content of the image. We depict the result by [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] in (c).

The photo boy-on-stones in Fig. 10(a) was taken in a very sunny day. Due to camera corrections, the picture appears underexposed. Our Multiplicative algorithm gives a realistic result shown in (b). In particular, observe the reflectance of the sunlight on the stones. The NM algorithm yields a grayish result (c). The Additive algorithm (not shown) gives a slightly better enhancement than the NM algorithm which confirms our findings in Subsection IV-B. The perceptional variational algorithms [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF] in (d) and [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] in (e), as well as the ACE in (f), shift the colors towards blue; observe the stones. The cathedral photo in Fig. 11(a) is much too dark. The result of our Multiplicative algorithm in (b) is quite colorful. The same algorithm for the mixed target histogram [START_REF] Thomas | A modified version of van-cittert's iterative deconvolution procedure[END_REF] gives in (c) a darker and still colorful image. The results of [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF] in (d) and [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] in (c) have a darker color palette. The NM algorithm produces a nearly gray value image shown in (f). All results give a different atmosphere.

The photo taken in Jericoacoara, Fig. 12(a), has very dark and also some quite clear areas; see h f in (c). By Remark 8(ii), we use a mixed target histogram h mix . The original has lots of JPEG artifacts so we prefer our Additive algorithm (see Remark 6). The result in (b) is convincing and the details in the dark are clarified. For the ACE in (d) we use a small enhancement, α = 3, in order to limit the false color shift.

The orchid image in Fig. 13(a) has a bad flashlight effect. This artifact is removed by all tested methods and the background of the scene is clear. Our Multiplicative algorithm gives a realistic colorful result, see (b). As in Fig. 6, our Additive algorithm produces a rather pale image (c) while the issue of the NM algorithm in (d) is too gray. The images obtained by [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF] in (e) and by the ACE in (f) exhibit color shifts (see the green leaves on the right and the grass on the bottom left).

The frog image in Fig. 14(a) has an intensity histogram between (i) and (ii) in Remark 8. Indeed, both recipes gave similarly good results. The result with h G is shown in (b). For the ACE in (c) we select a small α = 3 limit the color shift.

The ferrari image in Fig. 15(a) has very dark and very bright areas. Using Remark 8 (ii), we take a mixed target histogram. Our Multiplicative algorithm gives a realistic image shown in (b) with vivid colors that fit the typical red of the brand. The variational methods [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF] in (c) and [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] image. We have applied our Multiplicative algorithm to image (a) using the intensity histogram of (b). The result, shown in (d), is less dull than the example (b). The third row in Fig. 17 depicts a zoom into the area with the raven (right middle). One observes that the bird is fused with part of the background in (c), whereas it is distinguishable in (b) and (d).

♯ U (1) ♯ U (0) ♯ L(0) ♯ V

VI. CONCLUSIONS AND FUTURE WORK

This work provides the first comprehensive and rigorous presentation of the wide family of histogram specification based affine color assignment models. We have proposed a fast hue and range preserving algorithm. We analyzed the performances of this algorithm and two of its important instances as well as the gamut preserving method in [START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF].

Many open questions have been raised that we want to answer in our future research. Since our algorithms are fast, extensions to video should be envisaged. We are aware of the broad literature on color enhancement taking both global and local neighborhood of pixels into account see, e.g., [START_REF] Sapiro | Histogram Modification via Differential Equations[END_REF], [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF], [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF], [START_REF] Rizzi | A new algorithm for unsupervised global and local color correction[END_REF], [START_REF] Kim | An advanced contrast enhancement using partially overlapped sub-block histogram equalization[END_REF]. It will be interesting to incorporate such information into our framework. Moreover, we want to take into account other important properties of the human visual system.

VII. APPENDIX

Proof of Proposition 1: A pixel i ∈ I n in [START_REF] Bertalmío | Issues About Retinex Theory and Contrast Enhancement[END_REF] has an upper gamut problem if

λ f [i] + (1 -λ)f [i] f [i] M [i] + (1 -λ)( f [i] -f [i]) > L -1. ( 29
)
The upper bound follows from the choice of w c [i], c ∈ {r, g, b} in [START_REF] Gatta | ACE:", in An automatic color equalization algorithm[END_REF]. We focus the lower bound.

(L -1 -f [i])(wc[i] -f [i]) + f [i](M [i] -f [i]) ≥ 0, (L -1 -f [i])wc[i] -(L -1)f [i] + f [i]M [i] ≥ 0 which is clearly fulfilled if f [i]M [i] -(L -1)f [i] ≥ 0. ( 30
)
By [START_REF] Chien | Color image enhancement with exact HSI color model[END_REF] we have

f [i]M [i] -(L -1)f [i] > f [i]M [i] - ( λ f [i] + (1 -λ)f [i] ) (M [i] -f [i]) -f [i]f [i] = ( f [i] - ( λ f [i] + (1 -λ)f [i] ) ) (M [i] -f [i]).
By f [i] ≥ f [i] the first factor on the right-hand side is ≥ 0 the second one is > 0. Thus, (30) is satisfied. Proof of Proposition 2: A pixel i ∈ I n in (15) has a lower gamut problem if

λ f [i] + (1 -λ)f [i] f [i] m[i] + (1 -λ)( f [i] -f [i]) < 0. ( 31
)
The lower bound is clear from the construction of w c [i] in [START_REF] Getreuer | Automatic color enhancement (ACE) and its fast implementation[END_REF]. We show the upper bound. Developing [START_REF] Getreuer | Automatic color enhancement (ACE) and its fast implementation[END_REF] yields

wc[i] = f [i]wc[i] -f [i] f [i] + f [i] f [i] -f [i] m[i] f [i] -m[i] = f [i] f [i] -m[i]
(wc[i] -m[i]), c ∈ {r, g, b}. [START_REF] Nikolova | Fast sorting algorithm for exact histogram specification[END_REF] Using ( 31), one has

f [i] λm[i] + (1 -λ)f [i] f [i] + (1 -λ)(m[i] -f [i]) < 0
and hence

f [i] < (1 -λ)f [i] λm[i] + (1 -λ)f [i] (f [i] -m[i]) < f [i] -m[i].
Thus, since 0 < w c [i] -m[i] < L -1, we obtain finally

f [i] f [i] -m[i] (wc[i] -m[i]) < L -1.
Proof of Proposition 3: Let 0 ≤ λ 1 < λ 2 ≤ 1. In the upper gamut case we always have

f [i] > f [i]. Then λ 1 f [i] + (1 - λ 1 )f [i] < λ 2 f [i] + (1 -λ 2 )f [i], hence G λ1 M [i] ≤ G λ2 M [i] and U(λ 1 ) ⊆ U(λ 2 ). Similarly, since f [i] < f [i] in the lower gamut case, λ 1 f [i] + (1 -λ 1 )f [i] > λ 2 f [i] + (1 -λ 2 )f [i] so that G λ1 m [i] ≥ G λ2 M [i]
and hence L(λ 1 ) ⊇ L(λ 2 ). Proof of Proposition 4: Let i ∈ I n \ {U(1) ∪ L(0)}. Then w × c [i] is given by (20)(i) and since U(0) ⊆ U(1), the value w + c [i] is given by (21)(i). Consequently,

wc[i] = λ f [i] f [i] (wc[i] -f [i]) + (1 -λ)(wc[i] -f [i]) + f [i] = wc[i],
where the last equality follows by U(λ) ⊆ U(1) and Algorithm 3(i).

Let i ∈ U(0). Then w + c [i] is given by ( 21

w c [i] = d[i](w c [i] -f [i]) + f [i], (33)
for

d[i] =        λ f [i] f [i] + (1 -λ) if i ∈ In \ {U (λ) ∪ L(λ)}, L-1-f [i] M [i]-f [i] if i ∈ U (λ), f [i] f [i] S(w[i]) if i ∈ L(λ).
Since d[i] > 0 in all cases, we have

S( w[i]) = 1 - 1 f [i] ( d[i] ( min{wr[i]wg[i], w b [i]} -f [i] ) + f [i] ) = 1 - f [i] f [i] d[i] min{wr[i], wg[i], w b [i]} f [i] + f [i] f [i] d[i] -1 = S(w[i]) f [i] f [i] d[i] . ( 34
)
Inserting the above values d[i] into (34) finishes the proof.

Proof of Proposition 6: The case f [i] f [i] ≤ 1 follows just as a special case of Proposition 5(i) for λ = 1. Let

f [i] > f [i].
Then case (ii) in Algorithm 2 can be rewritten as [START_REF] Nikolova | Exact histogram specification for digital images using a variational approach[END_REF] for

d[i] := L-1-f [i] L-1-f [i] .
Combining this with [START_REF] Yang | Efficient luminance and saturation processing techniques for bypassing color coordinate transformations[END_REF] we are done.

Fig. 1 .

 1 Fig. 1. Histogram equalization (HE). Left: Original image onion (Matlab IPT image credits notice). Middle: HE to each color channel independently. Right: Enhancement in three steps following [2]: RGB to HSI transform, HE of the intensity channel, then HSI to RGB transform. Here 36.1 % of the pixels have values in (255, 443.5].

  (a)-(b) were assigned using Algorithm 2 given in the next subsection.

Fig. 2 .

 2 Fig. 2. Illustration of the importance of a meaningful ordering. Top: Original image and application of Matlab histeq. Bottom: Application of Algorithm 1.

Proposition 1 .

 1 Assume that pixel i ∈ I n in (15) has an upper gamut problem. Then its correction w c [i] in (12) satisfies

Algorithm 3

 3 Optimal Range-Preserving Enhancement 1. Compute the intensity f of w by (1) and the target intensity f using Algorithm 1 for given h. 2. For i ∈ I n compute M [i] and m[i] by[START_REF] Papadakis | A variational model for histogram transfer of color images[END_REF]. If f [i] = 0, then w[i] := 0. Otherwise compute

Fig. 3 .

 3 Fig. 3. Input image couple (top left) and its enhancement by our Algorithm 3 for λ = 0, 1 4 , 1 2 , 3 4 , 1.Here the size of the sets U (λ) in[START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF] in percent of all image pixels are 0.70, 1.24, 2.12, 3.13, 4.17 and the sets L(λ) are empty. All nuances between the very colorful image w × and the grayish image w + can be also obtained by their convex combinations in[START_REF] Han | A novel 3-D color histogram equalization method with uniform 1-D gray scale histogram[END_REF].

w 29 AlgFig. 4 .

 294 Fig. 4. Enhancement of a quite dark pixel shown in the first row. Second and third rows: the output intensity f is on the x-axis and the plots depict the results of Algorithms 4, 5 and 2. The second row specify the value of each color channel. The third row shows the resulting color w w.r.t. f and the last row plots the saturation of the output pixel as a function of f .

7 .

 7 The distribution of f [i]/f [i] for these two images is very different -the first one ranges on [0, 18] and the second one on [0, 1.22]. Roughly speaking, bungalow w = (80, 172, 108), f = 120, H(w) = 137.3, S(w) = 0.33 Alg. 4 (×) Alg. 5 (+) Alg. 2 (NM)

Fig. 5 .Fig. 6 .

 56 Fig. 5. Enhancement of a quite bright pixel shown in the first row. Arrangement of images as in Fig. 4.Input intensity histogramTarget histogram

Fig. 7 .

 7 Fig. 7. Original image flower (300 × 400) and enhanced versions. All Algorithms 2, 4 and 5 produce very similar results.

Remark 8 .

 8 The choice of the parameters (l, r) to build h G in[START_REF] Sen | Automatic exact histogram specification for contrast enhancement and visual system based quantitative evaluation[END_REF] depends on the input intensity histogram h f and on the enhancement task. E.g., (l, r) = (1, 0.99) leads to HE.(i) Function h G can be easily adapted to all images whose histogram h f is roughly unimodal -see the original images in Figs.6, 7, 9, 10, 11, 13, 14 and 16. If the pixel values are mainly in the middle of the interval [0, L -1] and decay at the ends (see Figs. 7, 9 and 16), a good enhancement can be done with l r ∈ [0.1, 0.2]. When h f rapidly decays towards L-1, one should choose r ∈ (0, 0.1] (see Figs. 6, 10, 11, 13 and 16). The stronger this decay, the smaller the value of r should be selected (e.g. in Fig. 11, r = 10 -4 .) A too large r should entail artifacts typical for HE. If most of the pixel have small values (underexposed images, see Figs. 6, 10, 11 and 13), it is reasonable to select l ∈ [0.8, 1]. The higher the concentration near 0, the larger the value of l 1 should be taken. (ii) For images with important very dark and very bright areas, function (27) should not work well. Then a good option is to take a mixed target histogram

Fig. 8 .Fig. 9 .

 89 Fig. 8. (a) Image club (1800×3200). Enhancement using our Multiplicative algorithm with: (b) hG for (l, r) = (1, 0.2); (c) h mix by (28) for hG as in (b). (d) Enhancement by the variational method in [8].

Fig. 10 .

 10 Fig. 10. (a) Image boy-on-stones (800 × 800) and plot of h f . Enhancement results: (b) Multiplicative algorithm with hG for (l, r) = (0.8, 0.2); (c) Naik-Murthy algorithm with the same hG; (d) Perceptual variational method[START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF] with data-fitting parameter γ = 0.2; (e) Perceptual variational method[START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] with a symmetric contrast function and slope parameter 10; (f) ACE, α = 5.

Fig. 11 .

 11 Fig. 11. (a) Image cathedral (768×1024) and plot of h f . Enhancement results: (b) Multiplicative algorithm with hG for (l, r) = (1, 10 -4 ); (c) The same algorithm for h mix in Remark 8(ii) with hG in (b); (d) Perceptual variational method [7], default parameters; (e) Perceptual variational method [8] with Michelson's contrast function; (f) NM algorithm with hG in (b).

Fig. 12 .

 12 Fig. 12. (a) Image Jericoacoara (886 × 1181). Enhancement results: (b) Additive algorithm with mixed target histogram h mix for (l, r) = (1, 0.1); (c) Histograms of the input and the target intensities; (d) ACE for α = 3.

  in (d) outperform the ACE (result not shown). The image fields in Fig. 16(a) was taken trough an aircraft porthole. It has no pixels with values in [0, 109]. By Remark

  Perceptual by[START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF] (f) ACE[START_REF] Getreuer | Automatic color enhancement (ACE) and its fast implementation[END_REF]

Fig. 13 .Fig. 14 .

 1314 Fig. 13. (a) Image orchid (768×1024) with a bad flashlight effect. Enhancement results: (b) Multiplicative algorithm with hG for (l, r) = (1, 0.1); (c) Additive algorithm for the same target histogram; (d) NM algorithm[START_REF] Naik | Hue-preserving color image enhancement without gamut problem[END_REF] for the same target histogram; (e) Perceptual variational method[START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF] for γ = 0.2; (f) ACE[START_REF] Getreuer | Automatic color enhancement (ACE) and its fast implementation[END_REF], default parameter.

3 ,

 3 we use in (b) the global hue-preserving stretching. Its further enhancements using our Multiplicative and Additive algorithms give visually the same results, so only the first one is shown in (c). For the ACE in (d) we take α = 8 in order to obtain an enhancement strength similar to (c). The JPEG blue artifacts are stronger in (d) compared to (c).Fig.17

  (a) and (b) show an underexposed and a better exposed "example" image of the same scene flag. In[START_REF] Papadakis | A variational model for histogram transfer of color images[END_REF] the authors propose an algorithm for the color transfer between images (usually of different scenes). Using this algorithm we transferred the colors from the example image (b) to the underexposed one (a). The result in (c) is close to the example

Fig. 15 .

 15 Fig. 15. (a) Image ferrari (courtesy of P. Greenspun) of size 235 × 240 and histogram of its intensity channel. Enhancement results: (b) Multiplicative algorithm with mixed target histogram for (l, r) = (1, 0.1); (c) Perceptual variational method [7] with default parameters (courtesy of the authors); (d) Perceptual variational method [8] with Michelson's contrast function and default parameters (courtesy of the authors).

Fig. 16 .

 16 Fig. 16. (a) Image fields (512 × 512) and histogram of its intensity. Enhancement results: (b) Global affine stretching (Remark 3) and intensity histogram; (c) Multiplicative algorithm applied to (b) with Gaussian target histogram for (l, r) = (0.1, 0.1); (d) ACE for α = 8 applied to (a).

  image

  For our Algorithms 4 (×) and 5 (+) the numbers ♯ U (1), resp., ♯ U (0), ♯ L(0) are very small in all tests; this is not the case for ♯ V in the NM algorithm.

Fig. 17 .

 17 Fig. 17. (a) Input underexposed image flag (1500 × 1125) and (b) example better-exposed flag together with the histograms of their intensities. Enhancement results: (c) the color transfer method [10] and (d) our Multiplicative algorithm with the example target intensity histogram in (b).

  First we see that[START_REF] Chien | Color image enhancement with exact HSI color model[END_REF] impliesf [i] ≥ f [i], since in case f [i] < f [i]we would get by replacing f [i] by f [i] in the denominator of the quotient and the second summand in (29) the contradiction M [i] > L -1. Since M [i] -f [i] > 0 the lower bound holds true if and only if

  )(ii) and since U(0) ⊆ U(1), w × c [i] is given by (20)(ii). We havew + c [i] = w × c [i] which shows by U(0) ⊆ U(λ) that w c [i] = w[i]. Proof of Proposition 5: By (22) we have S(w[i]) = 1-m[i] f [i] , or equivalently, f [i]-m[i] = f [i] S(w[i]). Algorithm 3 computes

TABLE I

 I Percentage of pixels requiring an upper or lower gamut correction.

Acknowledgement:

The authors thank M. Bertalmio, E. Provenzi and N. Papadakis for providing their codes corresponding to [7], [8], and [10] and for the discussion. We want to thank the anonymous reviewers for their constructive remarks.

Her work was supported by the "FMJH Program Gaspard Monge in optimization and operation research", and by the support to this program from EDF.


Cédric Pardanaud

Younès Addab

Céline Martin

Pascale Roubin

Bernard Pegourié

Martin Oberkofler

Martin Köppen

Timo Dittmar

Christian Linsmeier

Raman microscopy as a defect microprobe for hydrogen bonding characterization in materials used in fusion applications

Keywords: Raman microscopy, Thermal stability, Hydrogen release, Plasma/wall interactions, Amorphous carbons

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Context of the study

Understanding and controlling hydrogen isotope retention in plasma-facing components of tokamaks (i.e. machines studying the possibility to make energy from nuclear fusion reactions by confining plasma magnetically), is a crucial safety issue because the international ITER reactor will use T+D, tritium being radioactive [1]. Being able to quantify how the plasma-facing components are processed subsequently to plasma-wall interactions (ionic implantation, erosion, deposition, diffusion, chemical bond formation, material migration... [1][2][3][4]) is then of importance.

To this aim, deuterium retention inside the Tore Supra tokamak (TS), able to perform longtime discharges, has been extensively studied during long time scales of operation (5 years) [5][6][7][8][9][10][11], the main plasma-facing component, called the toroidal pumped limiter, being made of carbon/carbon composite [12]. Post mortem analyses were used to quantify the amount of D trapped inside the materials. These analyses revealed that most of this D was trapped as deuterated amorphous carbon deposits (a-C:D) [13,14], D being involved in chemical bonds such as C(sp 3 )-H and C(sp 2 )-H. However, the amount of trapped D was not the amount estimated by controlling how much D was introduced and how much was pumped out just after plasma discharges (in situ particle balance experiments). Ion beam analyses of many deposits revealed D depletion in the deepest zone of the carbon deposits (i.e. oldest zone of the carbon deposits). To understand this behavior, the importance of a long-term D release has been suggested and then characterized: D was found to be released slowly but significantly with time, this phenomenon strongly depending on the sample temperature [7]. This long term release was then able to reconcile post mortem and particle balance measurements [7,9]. However carbon is no longer planned to be used for the plasma-facing components of ITER. Its plasma-facing components will be made of beryllium and tungsten. Some mixed materials that will modify the initial properties of the pristine materials will be formed due to the presence of:

-oxygen impurities, -neon or nitrogen seeded in the machine, just before confinement is partially lost, to spray the heat loads on higher surfaces and prevent from damaging the plasma facing components, -sputtered species, -hydrogen isotopes… As an example, tungsten oxide may be formed in some zones more or less efficiently, depending on the surface temperature. However, as carbon has been very well documented during the past years, it remains a reference system. In this work we give some examples on how to interpret Raman spectra in order to obtain information on how the hydrogen is bonded in materials. In a first part we compare the thermal stability of reference a-C:H layers to that of TS deposits by means of Raman spectroscopy recorded in situ. In a second part we compare the influence of hydrogen bombardment on pristine graphite, one kind of tungsten oxide and beryllium.

Raman microscopy of a-C:H 2.1 State of the art

The 1000 -1800 cm-1 spectral window of carbonaceous materials containing sp 2 atoms is dominated by two bands: the G and D bands [15]. They contain information on disorder such as the size of aromatic domains, the density of defects… For amorphous carbons (a-C), they can give an indication on the sp 2 /sp 3 ratio, and an estimation of the H content for hydrogenated carbons (a-C:H) [15][16][17]. Spectroscopic parameters generally used to probe the bonding structure are: the width and the position of the G band, denoted Γ G and σ G respectively, the height ratio of the D and G bands, H D /H G , and the m/H G parameter with m being the slope of a superimposed photoluminescence background measured between 800 and 2000 cm -1 . The way they are determined on raw spectra is shown in the figure 1.

Post-mortem analyses of TS deposits

Figure 2 displays the Γ G parameter as a function of the σ G parameter for samples collected on the toroidal pumped limiter and on the neutralizer element of Tore Supra [14]. They display a continuous curve, ranging from nano-crystalline graphite (σ G ≈ 1590 cm -1 , Γ G ≈ 25-50 cm -1 ) to amorphous carbon (σ G ≈ 1510 cm -1 , Γ G ≈ 200 cm -1 ) with various changes in local order in between. Samples from the neutralizer range from nano-crystalline to amorphous whereas samples from the limiter are amorphous. However the former are the most ordered. This can be understood because the limiter is actively cooled during discharges, limiting the surface temperature variations subsequently to heat loads, whereas the neutralizer's deposits are not cooled due to loosely attached and thick deposits, allowing them to reach higher temperatures and then to get a higher degree of organization and a larger H release. As these measurements are rapid compared to ion beam analyses or other relevant techniques, it is possible to perform a statistical treatment on many samples coming from many places of the plasma-facing components. The conclusion is that deuterium retained in deposits found on the neutralizer can be neglected compared to that found on the deposition zones of the limiter. Note that the TS samples' spectroscopic parameters have been compared to those of a-C:H, ta-C:H and ta-C found in literature ( [18] and references therein). [18] and references therein) whereas the samples referred as limiter and neutralizer refer to Tore Supra deposits.

Thermal stability of reference and TS samples

Figure 3 displays the Raman spectra of one a-C:H sample that has been heated from room temperature to 1030°C (H G is normalized to 1 after a linear background subtraction). Temperature increase leads to local organization as evidenced by the G band narrowing and blue shift, and the increase of the D band. Temperature increase also leads to hydrogen release: the inset gives the linear relation we found previously between H D /H G and the H content measured by ion beam analysis [19]. This linear relation was found with all kinds of hydrogen bonding (i.e. H bonded to C(sp 2 ) or to C(sp 3 ) atoms) and can thus be used as a tool to determine the H content. [19]. Figure 4 displays the evolution of m/H G and H D /H G for reference samples compared to TS samples under heating (both linear temperature ramp with α=3K.min -1 and isothermal heating at 500°C [20]). Samples were heated under argon atmosphere, spectra being recorded in situ in real time through a window. We see that data corresponding to the linear ramp and isotherms are superimposed in a quite good agreement. H D /H G probes the hydrogen content whereas m/H G probes successively with increasing temperature: the sp 2 clustering/organization and then the H content of H bonded to C(sp 3 ). Consequently, the data points can be separated in two zones in the m/H G versus H D /H G plot: the upper left corner corresponding mainly to sp 2 clustering and the lower right corner corresponding to the release of H bonded to C(sp 3 ). Note that the curves of the two reference samples have the same shapes in a plot where the x-axis is temperature, but are shifted in temperature which allows them to be clearly distinguished (not shown on this figure). The reference sample data presented in figure 4 were recorded in the thermal range where H comes mainly from the scission of H-C(sp 3 ) bonds. Then, one can see that in the lower right corner of this figure the relation between log10(m/H G ) and H D /H G is linear ; both parameters give the same information, however they do not superimpose as the amount of hydrogen bonded to sp 2 carbons may not be the same. The curve corresponding to the TS sample behaves like that of the as deposited (AD) H/H+C=32% sample concerning the sp 2 clustering, whereas it behaves like the AD H/H+C=37% sample concerning the thermal stability.

Figure 3. Raman spectra of one a-C:H layers heated from room temperature to 1030°C (H/H+C=30% for the as deposited sample). The inset represents the linear relation found between H D /H G and the H content obtained when heating the as deposited sample

Raman microscopy of Be:D and WO 3-x :D

Due to tritium retention, carbon has not been chosen for the design of the ITER wall components. They will be composed of Be and W. First results on the JET ITER like wall have revealed that tritium will be stored mainly in beryllium deposits [21].

Figure 5 displays a comparison of Raman spectra cor-responding to various pristine and D implanted samples, to check the sensitivity of the technique to ion implantation. The samples are Highly Oriented Pyrolitic Graphite (HOPG), tungsten oxide and polycrystalline beryllium. The energies and fluencies of the bombarding D ion were respectively in the range of ≈ 0.5-2 keV and ≈ 10 17 -10 19 ions/cm 2 ( [22] for HOPG and W), depending on the material, allowing ions to be implanted up to a few nm to ≈ 10 nm in depth and creating a sufficiently high amount of defects so that the corresponding Raman signature is intense enough to be detected. One can see that the HOPG becomes amorphous due to implantation, as the spectrum, which has been modified subsequently to ion implantation, looks like that of figure 1, being composed of broad D and G bands. A narrower D band also appears at 1360 cm -1 , related to another kind of defect. The underlying unmodified HOPG signature is also seen [23]. The tungsten oxide sample has been prepared by heating a polycrystalline W sample at 600°C during 60 minutes under 1.5 bar of O 2 . D implantation broadened the stretching (≈ 200-500 cm -1 ) and bending (≈ 600-800 cm -1 ) modes, changed their relative intensity ratio and two new bands appeared at ≈ 400 and 960 cm -1 . The band at 960 cm -1 is known to be a surface W=O bond, suggesting that porosities may have been formed during bombardment.

The Be stretching mode, giving rise to a Raman active mode at 458 cm -1 , is also modified by bombardment. This band is shifted and broadened which is the sign of stress caused by implantation.

Conclusions

Understanding and controlling hydrogen isotope retention in plasma-facing components of tokamaks is important due to safety issue. In this work, we have studied the Raman microscopy ability to detect and characterize the way hydrogen is bonded with elements that have been used for nowadays tokamaks' plasma-facing components (carbon) and that will be used for ITER's plasmafacing components (beryllium and tungsten).

Using continuous changes with temperature of various carbonaceous samples, we have shown how Raman spectroscopic parameters can be used to determine structural and H-content evolution. We have then compared the thermal stability of reference a-C:H deposits to deposits collected in the Tore Supra tokamak to investigate deeper how hydrogen is bonded.

We have evidenced that Raman spectroscopy is a sensitive non contact technique which can give information on how pristine tokamak material have been modified due to plasma/wall interactions. We have then evidenced that implantation modifies the vibrational properties of various materials that may be found in ITER (tungsten oxide and beryllium). We point out that the depth where structural or chemical modification occurs in the material, from a few nm to ≈ 10 nm, can be easily probed by means or Raman spectroscopy. Further studies are now needed to allow Raman spectroscopy to become a H-content quantitative method for tungsten oxide and beryllium.

Figure 1 .

 1 Figure 1. Raman spectrum of a typical plasma-deposited amor-phous carbon. Γ G and σ G are the width at half maximum and the position of the G band, respectively. H D /H G is the height ratio of the D and G bands. m is the slope of the photoluminescence background.

Figure 2 .

 2 Figure 2. G band wavenumber as a function of the G band width for different carbon samples. a-C:H,ta-C:H and ta-C are plasma deposited samples (the symbol t means tetrahedral, see reference[18] and references therein) whereas the samples referred as limiter and neutralizer refer to Tore Supra deposits.

Figure 4 .

 4 Figure 4. m/H G evolution under 500°C isotherm and linear ramp heating plotted against H D /H G for two reference a-C:H samples and compared to TS deposits. AD means As Deposited.

Figure 5 .

 5 Figure 5. Normalized Raman spectra of pristine and D implanted graphite (a), WO 3-x (b) and Be (c).

Acknowledgements. Part of this work has been carried out within the framework of the

EUROfusion Consortium and has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement number 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. We also ac-knowledge the EURATOM program, and the Federation de Recherche FR-FCM. C. P. wants to thank W. Jacob, T. Schwarz-Selinger, C. Hopf and G. Cartry.