Christophe Charrier

Abdelhakim Saadane

Christine Fernandez-Maloigne

Blind Image Quality Assessment designed by learning-based attributes selection

With the rapid growth of image processing technologies, objective Image Quality Assessment (IQA) is a topic where considerable research effort has been made over the last two decades. IQA algorithms based on image structure have been shown to correlate well with Mean Opinion Scores (MOS). No-Reference (NR) image quality metrics are of fundamental interest as they can be embedded in practical applications This paper deals with a new NR-IQA metric based on natural scenes statistics. It proposes to model the best correlated statistics of seven well known no-reference image quality algorithms by a MultiVariate Gaussian Distribution (MVGD). A part of LIVE database is used with the associated DMOS to fit the MVGD model, namely Model Image Quality Index (MIQI). Hence, the quality of a distorted image is given by the DMOS that maximizes the multivariate Gaussian probability density function. Experimental results demonstrate the method effectiveness for a wide variety of distortions.

Introduction

The rapid development of image and video processing technologies and the exponential increase in the demand of new multimedia services raise the critical issue of assessing the visual quality. To meet this need, reliable methods have been developed. Since the human observer is the ultimate judge, subjective assessment when properly implemented [START_REF] Fernandez-Maloigne | Standardization Sector of ITU, Methodology for the Subjective Assessment of the Quality of Television Pictures[END_REF][START_REF]Standardization Sector of ITU, Methodology for the Subjective Assessment of the Quality of Television Pictures[END_REF] is the most accurate method to evaluate the visual quality but is time consuming and unsuitbale for real-time applications. Objective IQA algorithms [START_REF] Chandler | Seven challenges in image quality assessment: past, present, and future research[END_REF] aim at predicting image quality from objective features extracted from images. Two approaches are generally considered in the design of objective IQA algorithms : (i) the visually based approach that aims to mimic the human visual system behavior and (ii) the signal based approach that extracts and analyses features in image signals. Signal based approaches which represent the context of this paper, present a good tradeoff between performance and complexity. Generally, these approaches require two steps. In the first one, relevant features are extracted while, in the second, these features are pooled in order to produce the quality score of the image under test. The first step has been the subject of several investigations [START_REF] Wang | Image quality assessment: From error measurement to structural similarity[END_REF][START_REF] Shnayderman | An svdbased grayscale image quality measure for local and global assessment[END_REF][START_REF] Wee | Image quality assessment by discrete orthogonal moments[END_REF]. In contrast, the second step still uses conventional combinations. To avercome this drawback, statistical modeling of natural images has been considered and has been shown to correlate well with perceived quality as measured by subjective assessment. Statistical modeling is based on the assumption that natural scenes belong to a specific subspace within which all the images evolve. In the case of no-reference metrics, the features extraction procedure is followed by a learning step.

NR-IQA 1 NR-IQA 2 NR-IQA k      f 1 f 2 . . . f i         f i+1
. . . When a large ground truth is available, the learning methods can efficiently map features and MOS [START_REF] Luo | A training-based no-reference image quality assessment algorithm[END_REF][START_REF] Tsoy | Noreference image quality assessment through interactive neuroevolution[END_REF][START_REF] Babu | No-reference JPEG image quality assessment using GAP-RBF[END_REF][START_REF] Moorthy | Visual importance pooling for image quality assessment[END_REF][START_REF] Moorthy | A two-step framework for constructing blind image quality indice[END_REF]. Instead of looking to define new features, this paper will exploit the features of seven well-known metrics. These features are then sorted regarding their specific correlation coefficient. The most correlated ones are then used by the learning step to fit a multivariate gaussian distribution (MVGD). This MVGD handles the features of a given distorted image to assess its quality. This paper is organized as follows. Section 2 describes the extracted features. Section 3 presents the proposed method and section 4 gives the experimental results.

f i+ j       f l . . .

Features description

All features considered in this paper are extracted from nine commonly used learning-based NR-IQA learning-based metrics : 1) BRISQUE [START_REF] Mittal | No-reference image quality assessment in the spatial domain[END_REF], 2) BLIINDS [START_REF] Saad | A DCT statisticsbased blind image quality index[END_REF], 3) BLIINDS-II [START_REF] Saad | Blind image quality assessment: A natural scene statistics approach in the dct domain[END_REF], 4) NIQE [START_REF] Mittal | Making a "completely blind" image quality analyzer[END_REF], 5) DIIVINE [START_REF] Moorthy | Blind image quality assessment: From natural scene statistics to perceptual quality[END_REF], 6) BIQI [START_REF] Moorthy | A two-step framework for constructing blind image quality indice[END_REF], 7) IL-NIQE [START_REF] Zhang | A feature-enriched completely blind image quality evaluator[END_REF], 8) SSEQ [START_REF] Liu | No-reference image quality assessment based on spatial and spectral entropies[END_REF] and 9) OG-IQA [START_REF] Liu | Blind image quality assessment by relative gradient statistics and ada boosting neural network[END_REF]. A second reason of the choice of those trial algorithms is motivated by the fact that the code of all of them is publicly available.

MIQI NR-IQA method

General Scheme

All the trial NR-IQA methods applies the same principle: after computing a set of different features, a combination of this set is performed in different ways. In this paper, the design a NR-IQA algorithm is based on a new selection process of relevant attributes provided by different common NR-IQA methods. Fig 1 displays the general scheme of the proposed method. From all trial NR-IQA schemes, all features are computed. This yields to a set of n features. From this initial set a subset of M attributes is generated (under the constraint that M ≤ n) in order to keep the most relevant features with respect to some criteria. Then from selected criteria, a MultiVariate Gaussian Distribution (MVGD) is deployed to predict the final score of images.

Selection of features

All previously described features are computed for all original images (and their associated degraded version) of the LIVE IQA database [START_REF]LIVE Image Quality Assessment Database[END_REF]. This database contains 29 original images, each impaired by many levels of 5 distortion types: JPEG2000, JPEG, white noise, Gaussian blur, and fast-fading channel distortions. The total number of distorted images is 779. Then for each feature, the Spearman Rank-Order Correlation Coefficient (SROCC) between values of features and subjective DMOS is computed. Finally, only the highest 10 th percentile is considered to design the NR IQA method. Indeed, it has been previously observed that percentile pooling results in high correlations with subjective perception of quality [START_REF] Pinson | A new standardized method for objectively measuring video quality[END_REF][START_REF] Saad | Blind image quality assessment: A natural scene statistics approach in the dct domain[END_REF]. In addition, percentile pooling is motivated by the fact that the worst distortions in an image dominate subjective impressions. We choose 10% as a round number to avoid the possibility of training.

Table 2 shows the best eighteen features with their corresponding SROCC for the test LIVE image database. It also specifies the NR-IQA algorithms from which, each selected feature is extracted. These features are used to compute MIQI values. We observe that all the 18 selected features only come from four recent NR-IQA methods: BLIINDS-II, IL-NIQE, BIQI and NIQE. The six first parameters show SROCC values greater than 0.7 (0.924 for the higher), that correspond to good to high correlation levels with subjective scores. The remaining features provides SROCC values around 0.5. Thus, all the selected features range from 0.485 to 0.924 including medium to high correlated features with DMOS. One can observe that all features from BLIINDS-II and BIQI have been selected since they present high correlation level with DMOS.

From BLIINDS2, the extracted features are based on a Natural Scene Statistics (NSS) model of DCT coefficients in order to compute the generalized Gaussian Model shape parameter (which allows to capture primary distortion affecting the image quality),the coefficient of frequency variation, the energy subband ratio measure to capture the difference in the local spectral signatures between original and distorted images, and the Orientation Model-Based Feature to capture directional information that may be significant in subjective assessment of quality. THose kind of features are agnostic parameters that do not need any assumption on degradation.

From BIQI, the selected features refer to NSS parameters computed in the wavelet domain to coarsely mimics the scalespace-orientation decomposition that is assumed to occur in area V1 of the primary visual cortex of the human visual system. The selected features are degradation oriented to score the quality of four sets of images : JPEG and JEPG2000 compressed images, Gaussian blurred and white noisy images.

Considering IL-NIQE, three of the five available features are used to defined MIQI. IL-NIQE is based on constructing a collection of "quality aware" features. Those features are derived from the distribution of locally mean subtracted and contrast normalized (MSCN) feature and are statistics of normalized luminance, statistics of MSCN products and statistics of Log-Gabor Filter Responses to mimic orientation and frequency selectivity of neurons in the visual cortex. The two first are spatial features and the third one is computed in the frequency domain.

Regarding NIQE, the used features are space domain NSS, and correspond to the parameters of asymmetric generalized Gaussian distribution along three orientations plus the mean of the distribution. The main idea is to mimic the human judgment that seems to be more heavily weighted from sharp image regions. Those features are computed in the space domain.

Finally, one observes that features from frequency, spatiofrequency and space domain are used to designed MIQI. Another remark relies to the fact that even if many features are agnostic, degradation-based attributes are also used. A combination of the two sets of attributes seems to be necessary to design a NR-IQA index.

Prediction model

Let X i = {a 1 , a 2 , . . . , a k } be the vector of selected attributes, where i is the index of the image to be assessed. In addition let DMOS i be the associated subjective DMOS. To model the distribution of (X i , DMOS i ), a multivariate generalized Gaussian model (MVGD), namely Model Image Quality Index (MIQI), is used to compute the final score as

MIQI (x) = 1 (2π) k/2 |Σ| 1/2 exp -1 2 (x -β ) T Σ -1 (x -β ) (1)
where x = (a 1 , .., a k , DMOS) corresponds to the k most relevant features to which is added the DMOS of training distorted images. β and Σ denote the mean and covariance matrix of the MVGD model and are estimated using the maximum likelihood method.

The probabilistic model is trained on a subset of the LIVE IQA database, for which, one has access to DMOS values. To ensure a robustness of results, multiple training sets were constructed. In each, the image database was subdivided into distinct training and test sets (completely content-separate). For each train set, 80% of the LIVE IQA Database content was chosen, inducing that the remaining 20% is considered for the test set. Specifically, each training set contained images derived from 23 original images, while each test set contained the images derived from the remaining 6 original images. 1000 randomly chosen training and test sets were obtained and the prediction of the quality scores was run over the 1000 iterations.

Experimental results

Experimental setup

To evaluate the performance of the proposed IQA algorithm, two other publicly available databases are used: 1) TID2013 database [START_REF] Ponomarenko | Image database tid2013: Peculiarities, results and perspectives[END_REF] and 2) CSIQ image database [START_REF] Larson | Most apparent distortion: full-reference image quality assessment and the role of strategy[END_REF]. Even if the LIVE database has been used to fit the MVGD model MIQI, TID2013, CSIQ and the 1000 train-test LIVE Images databases will serve as test set.

To perform this evaluation, the SROCC is computed between DMOS values and the scores predicted by the proposed metric and the seven trial NR-IQA algorithms. In this paper, the number of selected features to design MIQI is 18 since it corresponds to the highest 10 th percentile of features.

In Table 3, we report SROCC values between the LIVE DMOS and MIQI scores for the highest 10% and 100% pooled features, respectively. We observe that the correlations are consistently higher when the lowest 10th percentile pooling strategy is adopted. This may be interpreted as further evidence that human sensitivity to image distortions is not a linear function of the distortion. Actually, human being tends to judge small degraded regions in an image more harshly than good ones, and by the way, tends to penalize the whole image quality [START_REF] Pinson | A new standardized method for objectively measuring video quality[END_REF].

In addition, to ascertain which differences between NR-IQA schemes performance are statistically significant, we applied an hypothesis test using the residuals between the DMOS values and the ratings provided by the IQA algorithms. This test is based on the t-test that determines whether two population means are equal or not. This test yields us to take a statistically-based conclusion of superiority (or not) of an NR-IQA algorithm.

Performance evaluation

Table 4 gives SROCC mean values computed between predicted scores and MOS values for the LIVE Images database. For distortions that are similar to the learned ones (JP2K, JPEG, Gaussian blur), using MIQI leads to a better result than using the other NR-IQA algorithms. In all cases, it performs better. Moreover, it achieves a significant performance gain when considering the Gaussian noise distortion.

Table 5 gives SROCC mean values computed between predicted scores and MOS values for the CSIQ Images database. For distortions that are similar to the learned ones (JP2K, JPEG, Gaussian blur), using MIQI leads to a better result than using the other NR-IQA algorithms. In all cases, it performs better. Moreover, it achieves a significant performance gain when considering the Gaussian noise distortion. For non learnt distortions such as Global Contrast Decrement, MIQI remains very interesting since it outperforms trial algorithms. This tends to demonstrate the generalization capability of the proposed method to score the quality of images.

Table 6 presents results when SROCC is computed between predicted scores and MOS values for the TID13 Images database subsets. This will yield us to measure the generalization capability of the proposed method to score the quality of an image with a high level of confidence when distortions have not learnt. When considering each distortion, one can see that the performance of MIQI equals or, in several cases exceeds all of the NR-IQA metrics on 15 distortions. We observe that some of them have not been learnt such as "contrast change", "Change of color saturation", "Chromatic aberrations" or "Sparse sampling and reconstruction". Yet, MIQI is less efficient on nine degradations including learnt artifact such as "Gaussian Blur". When distortions (cumulative subsets line), the overall performance of MIQI is significantly better than that of the other algorithms. The increase of SROCC is about 4% when compared to that of the best algorithm (ILNIQE in this case).

In addition, Table 7 gives obtained results when a One-sided t-test is used to provide statistical significance of NR-IQA/DMOS on the test LIVE database. Each entry in this table is coded using six symbols. The position of each symbol corresponds to one subset of the LIVE database as {jp2k, jpeg, Gaussian noise, Gaussian blur, fast fading, All} . Each symbol gives the result of the hypothesis test on the subset. If the symbol equals '1', the NR-IQA on the row is statistically better than the NR-IQA on the column ('0' means worse, '-' is used when NR-IQAs are indistinguishables).

On can observe that difference between quality scores predicted with MIQI and any trail algorithms is most of the time significant when the entire database is considered or when any subset is used. Even when non learnt distortions are considered, the SROCC difference between MIQI and any trail algorithm is mostly significant.

Table 8 shows similar results when CSIQ database is considered. Most of the the quality scores predicted by MIQI is statistically better than the other trial algorithms. For only one degradation (Gaussian noise), OG-IQA is performs better.

Table 9 presents obtained results when a One-sided t-test is used to provide statistical significance of NR-IQA/DMOS on the TID13 database. Only results for the entire database is given. Except for ILNIQE, difference between the predicted scores provided by MIQI and any other tested NR-IQA method if significant. Considering ILNIQE, the difference is not significant.

Finally, to compare the computational complexity of the proposed algorithm, we measured the average computation time required to assess an image of size 512 × 578 (using a computer with Intel Core-I7 processor at 2.2GHz). Table 10 reports the measurement results, which are rough estimates only, as no code optimization has been done on our Matlab implementations. It can be observed that the proposed method is highly competitive with faster methods.

Conclusion

In this paper, a new NR-IQA metric is proposed, namely MIQI. The main idea applied to design this new metric is the following: recent NR-IQA metrics predict quality with level of

Figure 1 .

 1 Figure 1. General scheme of the proposed NR-IQA algorithm based on combination of features.

Table 1 : NR-IQA metrics considered to investigate the relevance of features.

 1

	NR-IQA algo-	Comments		Features
	rithm
	BRISQUE	Natural scene statistic-based distortion-generic	36	f 1 , . . . , f 36
	BLIINDS	Machine learning-based approach (Probabilistic model)	4	f 37 , . . . , f 40
	BLIINDS-II	Machine learning-based approach (Probabilistic model)	4	f 41 , . . . , f 44
	NIQE	Space domain natural scene statistic model	35	f 45 , . . . , f 79
	DIIVINE	Classification (SVM) and Regression (SVR)	83	f 80 , . . . , f 162
	BIQI		Machine learning-based approach (SVM).	4	f 163 , . . . , f 166
	IL-NIQE	Global Multivariable Gaussian model	5	f 167 , . . . , f 171
	SSEQ	2-stage framework distortion classification and Regression (SVR)	12	f 172 , . . . , f 183
	OG-IQA	AdaBoosting back-propagation neural network	6	f 184 , . . . , f 189
	MIQI feature	Selected from	Initial feature	SROCC
	a 1	BLIINDS-II	f 42	0.924
	a 2	ILNIQE	f 167	0.842
	a 3	ILNIQE	f 168	0.840
	a 4	ILNIQE	f 170	0.815
	a 5	BIQI	f 163	0.749
	a 6	BIQI	f 166	0.714
	a 7	BLIINDS-II	f 43	0.555
	a 8	BLIINDS-II	f 44	0.550
	a 9	BLIINDS-II	f 41	0.547
	a 10	BIQI	f 164	0.543
	a 11	NIQE	f 48	0.516
	a 12	NIQE	f 51	0.509
	a 13	NIQE	f 56	0.505
	a 14	NIQE	f 52	0.505
	a 15	NIQE	f 55	0.505
	a 16	BIQI	f 165	0.503
	a 17	NIQE	f 63	0.499
	a 18	NIQE	f 59	0.485

Table 2 : SROCC mean values of each selected feature to be used by MIQI (LIVE database).

 2

	percentile	10 th	100 th
	MIQI	0.952 0.942

Table 3 : SROCC values computed between subjective DMOS and MIQI scores with highest 10th percentile and 100th per- centile

 3

Table 4 : SROCC values computed between predicted scores and MOS values LIVE Images database. Considering MIQI, median SROCC values are computed between predicted scores and MOS values for the 1000 train-test LIVE Images database.

 4

	Test LIVE subset	BIQI	DIIVINE	BLIINDS	BLIINDS-II	NIQE	BRISQUE	ILNIQE	SSEQ	OG-IQA	MIQI
	JP2K	0.802	0.913	0.912	0.943	0.906	0.938	0.953	0.942	0.937	0.965
	JPEG	0.879	0.910	0.839	0.954	0.847	0.923	0.843	0.951	0.964	0.959
	White Noise	0.958	0.984	0.974	0.980	0.975	0.986	0.972	0.978	0.986	0.985
	Gaussian Blur	0.821	0.921	0.957	0.941	0.945	0.978	0.898	0.948	0.961	0.981
	Fast Fading	0.730	0.863	0.750	0.933	0.882	0.929	0.787	0.904	0.898	0912
	Cumulative subsets	0.824	0.916	0.799	0.912	0.877	0.941	0.468	0.935	0.950	0.952
	CSIQ subset		BIQI	DIIVINE	BLIINDS	BLIINDS-II	NIQE	BRISQUE	ILNIQE	SSEQ	OG-IQA	MIQI
	JP2K		0.708	0.830	0.575	0.895	0.906	0.866	0.906	0.848	0.857	0.914
	JPEG		0.867	0.799	0.264	0.901	0.883	0.903	0.899	0.865.	0.923	0.912
	Gaussian Noise	0.324	0.176	0.293	0.379	0.299	0.252	0.850	0.872	0.877	0.842
	Add. Gaussian Pink	0.879	0.866	0.555	0.801	0.810	0.925	0.874	0.046	0.024	0.933
	Noise
	Gaussian Blur	0.771	0.871	0.774	0.891	0.892	0.903	0.858	0.873	0.893	0.931
	Global	Contrast	0.585	0.396	0.078	0.012	0.232	0.029	0.501	0.200	0.467	0.652
	Decrement
	Cumulative subsets	0.619	0.596	0.170	0.577	0.286	0.566	0.815	0.528	0.559	0.832

Table 5 : SROCC values computed between predicted scores and MOS values for the CSIQ Images database.

 5

Table 6 : SROCC values computed between predicted scores using NR-IQA schemes from which some attributes are extracted to design LEVIQI and MOS values for TID2013 Images database.

 6

	TID2013 subset		BIQI	DIIVINE BLIINDS BLIINDS-II	NIQE	BRISQUE ILNIQE SSEQ	OG-IQA	MIQI
	Additive Gaussian Noise		0.785	0.855	0.515	0.722	0.819	0.852	0.876	0.807	0.809	0.897
	Additive Noise in Color Components	0.541	0.712	0.331	0.649	0.670	0.709	0.815	0.681	0.681	0.815
	Spatially Correlated Noise		0.465	0.463	0.457	0.767	0.666	0.491	0.923	0.635	0.095	0.901
	Masked Noise		0.494	0.675	0.470	0.512	0.746	0.575	0.512	0.565	0.691	0.801
	High Frequency Noise		0.877	0.878	0.677	0.824	0.845	0.753	0.868	0.860	0.834	0.897
	Impulse Noise		0.748	0.806	0.637	0.650	0.743	0.630	0.755	0.749	0.597	0.802
	Quantization Noise		0.389	0.165	0.653	0.781	0.850	0.798	0.873	0.468	0.710	0.859
	Gaussian Blur		0.764	0.834	0.664	0.855	0.795	0.813	0.814	0.858	0.814	0.854
	Image Denoising		0.409	0.723	0.469	0.711	0.590	0.586	0.750	0.783	0.585	0.831
	JPEG Compression		0.857	0.629	0.771	0.864	0.840	0.852	0.834	0.825	0.867	0.879
	JPEG2000 Compression		0.733	0.853	0.740	0.898	0.889	0.893	0.857	0.885	0.882	0.921
	JPEG Transmission Errors		0.304	0.239	0.094	0.117	0.003	0.315	0.282	0.354	0.013	0.2321
	JPEG2000 Transmission Errors		0.367	0.060	0.289	0.620	0.510	0.360	0.524	0.561	0.138	0.665
	Non Eccentricity Pattern Noise		0.007	0.060	0.128	0.096	0.070	0.145	0.080	0.011	0.066	0.127
	Local Block-wise Distortions		0.081	0.093	0.161	0.209	0.127	0.224	0.135	0.016	0.075	0.215
	Mean Shift		0.035	0.010	0.279	0.128	0.163	0.124	0.184	0.108	0.166	0.201
	Contrast Change		0.413	0.460	0.036	0.150	0.018	0.040	0.014	0.204	0.094	0.532
	Change of Color Saturation		0.142	0.068	0.264	0.017	0.246	0.109	0.162	0.074	0.098	0.498
	Multiplicative Gaussian Noise		0.642	0.787	0.479	0.716	0.694	0.724	0.693	0.679	0.682	0.791
	Comfort Noise		0.642	0.116	0.229	0.017	0.155	0.008	0.359	0.033	0.188	0.358
	Lossy Compression of Noisy Images	0.526	0.633	0.641	0.719	0.801	0.685	0.828	0.610	0.656	0.823
	Color Quantization with Dither		0.698	0.436	0.434	0.736	0.783	0.764	0.748	0.528	0.294	0.806
	Chromatic Aberrations		0.544	0.661	0.639	0.539	0.561	0.616	0.679	0.688	0.616	0.692
	Sparse Sampling and Reconstruction	0.760	0.834	0.738	0.816	0.834	0.784	0.865	0.895	0.832	0.903
	Cumulative subsets		0.294	0.355	0.289	0.393	0.311	0.367	0.494	0.332	0.292	0.512
		BIQI	DIIVINE BLIINDS BLIINDS-II	NIQE		BRISQUE	ILNQE	SSEQ	OG-IQA
	MIQI	111111 11-111	11-111	1--101	11-101	11--01	111111 1--1-1 1--1--

Table 7 : Statistical significance matrix of NR-IQA/DMOS on test LIVE database subsets. Each entry in the table is a codeword con- sisting of 1sixsymbols. The position of the symbol represents the tested subsets as {jp2k, jpeg, white noise, Gaussian blur, fast fading, all}. Each symbol gives the result of the hypothesis test on the subset: '1' means that the algorithm for the row is statistically better that the algorithm for the column, '0' means it is worse, and '-' means it is indistinguishable.

 7

Table 8 : Statistical significance matrix of NR-IQA/DMOS on test CSIQ database subsets. Each entry in the table is a codeword consisting of seven symbols. The position of the symbol represents the tested subsets as mentioned in the first column of Table 5. Each symbol gives the result of the hypothesis test on the subset: '1' means that the algorithm for the row is statistically better that the algorithm for the column, '0' means it is worse, and '-' means it is indistinguishable.Table 9 : Statistical significance matrix of NR-IQA/DMOS on the entire TID2013 image database.. Each entry in the table is a symbol. The symbol gives the result of the hypothesis test on the subset: '1' means that the algorithm for the row is statistically better that the algorithm for the column, '0' means it is worse, and '-' means it is indistinguishable.

 89

Table 10 : Comparison of computational time (in sec- ond/image) correlations

 10 with subjective scores, but present high variability of correlation depending on considered degradation that impacts the global correlation rate. Thus selecting highest correlated features with DMOS, one should design one algorithm that provides high correlation value that usual NR-IQA schemes. Thus, MIQI models the best correlated statistics of nine well known algorithms by a multivariate Gaussian distribution.

	Algorihm	BIQI	DIIVINE	BLIINDS	BLIINDS-II
	time (s)	6.95	38.39	509.82	131.25
	Algorihm	NIQE	BRISQUE	ILNIQE	SSEQ
	time (s)	0.83	0.75	20.26	22.24
	Algorihm	OG-IQA	MIQI
	time (s)	12.65	9.11

Author Biography

Christophe Charrier (M'10) received the M.S. degree from the Nantes University of Science and Technology, Nantes, France, in 1993, and the Ph.D. degree from the University Jean Monnet, Saint-Etienne, France, in 1998. He has been an Associate Professor with the GREYC laboratory at the University of Caen, France, since 2001. In 2008, he was a Visiting Scholar with the Laboratory for Image and Video Engineering, University of Texas, Austin. From 2009 to 2011, he was an Invited Professor with the Computer Department, University of Sherbrooke, Sherbrooke, QC, Canada. He is now the head of the E-Payment & Biometrics research team. His current research interests include digital image and video coding, processing, quality assessment, computational vision, and biometrics data quality.

Abdelhakim Saadane received his Ph.D. degree in Signal Processing and Telecommunications from University of Rennes1. He is currently associate professor at the Polytechnic school of Nantes University and a member of the SIC laboratory in XLIM Institute. His research activities are focused on the Human Visual System modeling and its use in applications such as coding, quality assessment, watermarking and user experience. He is a member of French National Color Imaging Group and serves as a member of division 8 of CIE.

Christine Fernandez-Maloigne is currently Vice-Rector of Poitiers University, in charge of International Relations. She is also director of a CNRS research federation (MIRES), which gathers 560 researchers in the South-West of France, in the area of mathematics, image processing, computer graphic, computer science and communication systems. Her research activities are focused on colour imaging, including fundamental researches about introduction of human visual system models in multiscale colour image processes as well as practical applications. She was cofounder member of the French National Colour Imaging Group (GFINC) in 2000, now part of the CFC (Centre Franc ¸ais de la Couleur). Currenlty, Christine Fernandez-Maloigne is French representative of the CIE Division 8 (Commission Internationale de l'Eclairage, Image Technology Division) since 2006 and secretary of this division since may 2015. She is also deputy Editorin-chief of JOSA A.


Yoann Verger

Keywords: Sraffa, non-renewable natural resource, rent, Hotelling, extractive industry JEL classification: B24, B51, Q31 Sraffa, ressourcenaturelle non-renouvelable, rente, Hotelling, industrie extractive

A rich literature exists about the way to handle non-renewable natural resources in the context of classical theory. This article sums up the different approaches that we could consider when we calculate the revenue of the owner of a non-renewable natural resource in a Sraffian framework. It clarifies the concepts of differential rent, depreciation of wasting assets, Hotelling rent, and rent as a share of the product, and links this last concept with some empirical facts about non-renewable natural resource extraction industries.

Introduction

From [START_REF] Parrinello | Exhaustible natural resources and the classical method of long-period equilibrium[END_REF] [START_REF] Parrinello | Exhaustible natural resources and the classical method of long-period equilibrium[END_REF] to Fratini (2016) [START_REF] Saverio | Rent as a share of product and Sraffa's price equations[END_REF], a seemingly never-ending debate runs in the Sraffian literature about the way to handle non-renewable natural resource in the context of classical theory. Sraffa scarcely talks about natural resources, and about the revenue that you can get by owning these resources, in his book. However, it is interesting to note that the three basic commodities that are introduced as an example of an industrial system, in chapter IV of his book are iron, coal, and wheat, i.e. three commodities that needs a direct exploitation of three natural resources to be produced: one renewable resource, land, for wheat, and two non-renewable resources, iron and coal deposits, for iron and coal. 1 Natural resources are presented by Sraffa as key components for the reproduction of the economic system, hence the question of their exhaustion appears truly important. This article presents how the revenue of the owner of non-renewable natural resources is calculated in a Sraffian framework. The difference between renewable and non-renewable resources is actually not straightforward; and one can assume that there should be no difference in kind between the revenue of the owner of a renewable resource and the revenue of the owner of a non-renewable resource (section 2). However, natural resources that are non-renewable may be viewed as wasting assets, and the question of the depreciation of this asset arises (section 3 and 4). Finally, it appears from empirical facts that the revenue of the owner can be presented as a share of the net product of the company mandated for the extraction of the natural resource (section 5).

Differential rent

In chapter XI, Sraffa says that, if the natural resource is scarce, its owner can get a differential rent. The fact that the natural resource is scarce can only be seen by looking at the conditions of production (the equations of production). If one commodity is produced by two distinct processes that use a natural resource, then a differential rent will arise that may be earned by the owner of the natural resource. The two distinct processes may use different qualities of the natural resource, or they may use the same quality, but with different proportions of the means of production and labour. The first case gives rise to the extensive differential rent, the second case to the intensive differential rent. Sraffa only presents an example with a renewable natural resource, land, but he says in the beginning of the chapter that the owner of a non-renewable natural resource can also earn this kind of revenue:

Natural resources which are used in production, such as land and mineral deposits, and which being in short supply enable their owners to obtain a rent, can be said to occupy among means of production a position equivalent to that of 'non-basics' among products. (Sraffa, 1960 [20], § 85). Thus, starting with the example presented by Sraffa in his chapter IV, and then adding two processes in order to express the scarcity of the non-renewable natural resources only, we could get the following system:

The case of the iron industry will lead to the rise of an extensive differential rent, as the technique of production is the same in both processes. The case of the coal industry will lead to the rise of an intensive differential rent, as the techniques of the rent may be sized by the owner of the non In order to calculate the rents, the price system may be written in this way (knowing that for the intensive rent, the process which is more productive per unit of natural resource should need more capital and labour per unit of output, otherwise the rent will be negative):

As for the rent on iron ore, the first process appears to less profitable than the process shall not earn a differential rent, i.e. asserted without knowing the prices, but in the general case, a comparison between the level of the different wages if the rate of profits is fixed from outside, or the different rates of profits if the wages are fixed from outside, will help to find the most profitable process).

Depreciation of wasting assets

Kurz and Salvadori (2015 [START_REF] Heinz | NeriSalvadori: The 'classical' approach to exhaustible resources[END_REF]) report that Sraffa, in a draft of Commodities from 1958, has considered to write a passage, at the end of Chapter XI on Land, § 91, on the fact that wasting assets such as mineral deposits should make allowance of depreciation in the same waythan "live" capital such as machine. The draft is reported to be wri paragraph being the part included in the book, and the second paragraph the passage finally deleted: "Machines of an obsolete type which are still in use are similar to land in so far as they are employed as means of production may apply Marshall's term in a more restricted sense than he gave it) which is received for those fixed capital items which, having been in active use in the past, have now been superseeded but are worth employing for what they can get, is determined precisely in the same way as the rent on land. And like land such obsolete instruments have the properties of non-basics and are excluded from the composition of the Standard product. {...} On the other hand, as in the case of other wasting assets (such as mineral deposits) the annual depletion must be taken into account, which gives rise to as many separate processes as are the years of the asset's prospective residual life, on the same principle as was Chapter X for 'live' fixed capital." ( § 91 and from unpublished Sraffa Papers, folder D3/12/96)

The principle of depreciating fix and the decreasing productivity of the asset, an annual charge is paid for interest and depreciation

The case of the iron industry will lead to the rise of an extensive differential rent, as the technique of production is the same in both processes. The case of the coal industry will lead to the rise of an intensive differential rent, as the techniques of production are different in both processes. In both cases, the rent may be sized by the owner of the non-renewable resources, coal and iron deposits.

In order to calculate the rents, the price system may be written in this way (knowing that for the sive rent, the process which is more productive per unit of natural resource should need more capital and labour per unit of output, otherwise the rent will be negative):

As for the rent on iron ore, the first process appears to less profitable than the process shall not earn a differential rent, i.e. ρ iron ore 1 = 0 (in this particular example, this can be asserted without knowing the prices, but in the general case, a comparison between the level of the different wages if the rate of profits is fixed from outside, or the different rates of profits if the wages ixed from outside, will help to find the most profitable process).

Depreciation of wasting assets

]) report that Sraffa, in a draft of Production of Commodities by Means of 8, has considered to write a passage, at the end of Chapter XI on Land, § 91, on the fact that wasting assets such as mineral deposits should make allowance of depreciation in the same waythan "live" capital such as machine. The draft is reported to be wri paragraph being the part included in the book, and the second paragraph the passage finally deleted: "Machines of an obsolete type which are still in use are similar to land in so far as they are employed as means of production, although not currently produced. The quasi may apply Marshall's term in a more restricted sense than he gave it) which is received for those fixed capital items which, having been in active use in the past, have now been orth employing for what they can get, is determined precisely in the same way as the rent on land. And like land such obsolete instruments have the properties basics and are excluded from the composition of the Standard product. {...} hand, as in the case of other wasting assets (such as mineral deposits) the annual depletion must be taken into account, which gives rise to as many separate processes as are the years of the asset's prospective residual life, on the same principle as was Chapter X for 'live' fixed capital." (Kurz et al., 2015 [START_REF] Heinz | NeriSalvadori: The 'classical' approach to exhaustible resources[END_REF], quotation from § 91 and from unpublished Sraffa Papers, folder D3/12/96)

The principle of depreciating fixed capital is that, depending on the initial price, the expected life and the decreasing productivity of the asset, an annual charge is paid for interest and depreciation

The case of the iron industry will lead to the rise of an extensive differential rent, as the technique of production is the same in both processes. The case of the coal industry will lead to the rise of an production are different in both processes. In both cases, renewable resources, coal and iron deposits.

In order to calculate the rents, the price system may be written in this way (knowing that for the sive rent, the process which is more productive per unit of natural resource should need more

As for the rent on iron ore, the first process appears to less profitable than the second one: thus the first (in this particular example, this can be asserted without knowing the prices, but in the general case, a comparison between the level of the different wages if the rate of profits is fixed from outside, or the different rates of profits if the wages Production of Commodities by Means of 8, has considered to write a passage, at the end of Chapter XI on Land, § 91, on the fact that wasting assets such as mineral deposits should make allowance of depreciation in the same waythan "live" capital such as machine. The draft is reported to be written in this way, the first paragraph being the part included in the book, and the second paragraph the passage finally deleted: "Machines of an obsolete type which are still in use are similar to land in so far as they are , although not currently produced. The quasi-rent (if we may apply Marshall's term in a more restricted sense than he gave it) which is received for those fixed capital items which, having been in active use in the past, have now been orth employing for what they can get, is determined precisely in the same way as the rent on land. And like land such obsolete instruments have the properties basics and are excluded from the composition of the Standard product. {...} hand, as in the case of other wasting assets (such as mineral deposits) the annual depletion must be taken into account, which gives rise to as many separate processes as are the years of the asset's prospective residual life, on the same principle as was done in ], quotation from Sraffa, 1960 [START_REF] Sraffa | Production of Commodities by Means of Commodities[END_REF],

ed capital is that, depending on the initial price, the expected life-span, and the decreasing productivity of the asset, an annual charge is paid for interest and depreciation during all the expected life-span (instead of accounting the price of the fixed the purchase). The method devised in Chapter X to calculate this annual charge is to consider that, at each period of production during the life with a different price, and reused the next year. At the last period, nothing (or only scrap) is jointly produced. Thus a price appears for each dated fixed capital, and the subtraction of the price of the fixed capital jointly produced from the price of the fixed capital u the annual charge to be paid to the producer of the fixed capital.

In order to know the initial price of a "live" capital, it is sufficient to know the means of production and labor required to produce this asset. One reason that could explain why Sraffa did not finally insert the above passage in his book is that, contra exhaustible resources deposits is not a straightforward concept, as such assets are not produced by any human industry. [START_REF] Gibson | Profit and Rent in a Classical Theory of Exhaustible and Renewable Resources[END_REF] account. For instance, if the owner of the natural resource has invested some means of production and labour to find the new mine, he can then sell the right to exploit the mine to an extracting company and the annual charge to be paid to him can be calculated on the same principle as for 'live' fixed capital. profits for all industries), the rentier sells units of his natural exhaustible resource at a price which increases at each period so that he can earn the uniform rate of profits of the economy on his "conservation" activity. I.e, if he decides during one period of production to sell nothing, the value of his stock of natural resource will increase at a rate equal to the ruling rate of profits from this period of production to the next one.

The Hotelling rule

On this regard, two points must be stressed. First, this kind of revenue is a logical consequence of the assumption that we consider a long for an economic actor to earn less (or more) than the ruling rate of profits. Second, to calculate the price of the natural resource, and consequently the revenue of its owner at the present period of production, it is not enough to know the present conditions of production. A back will replace the existing wasting asset must be anticipated. At the period of exhaustion of the wasting asset, the revenue of its owner will equal the differential rent commodities produced with the help of its natural resource, and the value of the commodities produced by the back-stop technology. The revenue of the owner of the natural resource at the present period can be known by going backward from the period of exhaustion, knowing the Hotelling rule (with yt the price of the natural resource at period t and r the uniform rate of profits):

Then it is clear that to calculate this kind of revenue, a "photograph" of the market is not enough. And it is clear that, if the uniform rate of profits is not assumed to be the equilibrium rate of profits of an economy in a long 2 If he earns less than the ruling rate of profits on its conservation industry, he would have an and to sell everything in the present period of production. If he earns more than the rate of profits, he would have an incen to sell nothing in present period of production. Thus, the Hotelling theory states that he mu profits on its conservation activity so that the economy can be in an equilibrium state.

span (instead of accounting the price of the fixed the purchase). The method devised in Chapter X to calculate this annual charge is to consider that, at each period of production during the life-span of the asset, an older fixed capital is jointly produced, rice, and reused the next year. At the last period, nothing (or only scrap) is jointly produced. Thus a price appears for each dated fixed capital, and the subtraction of the price of the fixed capital jointly produced from the price of the fixed capital used during the period of production gives the annual charge to be paid to the producer of the fixed capital.

In order to know the initial price of a "live" capital, it is sufficient to know the means of production and labor required to produce this asset. One reason that could explain why Sraffa did not finally insert the above passage in his book is that, contrary to fixed capital, the initial price of wasting assets such as exhaustible resources deposits is not a straightforward concept, as such assets are not produced by any [START_REF] Gibson | Profit and Rent in a Classical Theory of Exhaustible and Renewable Resources[END_REF] [START_REF] Gibson | Profit and Rent in a Classical Theory of Exhaustible and Renewable Resources[END_REF] suggests, however, that the discovery cost should be taken into account. For instance, if the owner of the natural resource has invested some means of production and labour to find the new mine, he can then sell the right to exploit the mine to an extracting company and l charge to be paid to him can be calculated on the same principle as for 'live' fixed capital.

The Hotelling rule

kind of revenue that frequently appears in the neo-Ricardian literature is also linked to the depreciation of the wasting asset: the Hotelling rent [START_REF] Parrinello | Exhaustible natural resources and the classical method of long-period equilibrium[END_REF][START_REF] Schefold | Mr Sraffa on joint production and other essays[END_REF]

. The Hotelling rule states that, given term equilibrium conditions (i.e. equal rate of profits for all industries), the rentier sells units of his natural exhaustible resource at a price which increases at each period following the rate of profits, so that he can earn the uniform rate of profits of the economy on his "conservation" activity. I.e, if he decides during one period of production to sell nothing, the value of his stock of natural resource will rate equal to the ruling rate of profits from this period of production to the next one.

On this regard, two points must be stressed. First, this kind of revenue is a logical consequence of the assumption that we consider a long-period equilibrium state of the economy, where it is not possible for an economic actor to earn less (or more) than the ruling rate of profits. Second, to calculate the price of the natural resource, and consequently the revenue of its owner at the present period of is not enough to know the present conditions of production. A back will replace the existing wasting asset must be anticipated. At the period of exhaustion of the wasting asset, the revenue of its owner will equal the differential rent that will exist between the value of the commodities produced with the help of its natural resource, and the value of the commodities produced stop technology. The revenue of the owner of the natural resource at the present period can by going backward from the period of exhaustion, knowing the Hotelling rule (with yt the price of the natural resource at period t and r the uniform rate of profits):

Then it is clear that to calculate this kind of revenue, a "photograph" of the market is not enough. And it is clear that, if the uniform rate of profits is not assumed to be the equilibrium rate of profits of an economy in a long-term position, but rather a "conventional" uniform rate of profits, If he earns less than the ruling rate of profits on its conservation industry, he would have an and to sell everything in the present period of production. If he earns more than the rate of profits, he would have an incen to sell nothing in present period of production. Thus, the Hotelling theory states that he mu profits on its conservation activity so that the economy can be in an equilibrium state.

capital only for the year of the purchase). The method devised in Chapter X to calculate this annual charge is to consider that, at span of the asset, an older fixed capital is jointly produced, rice, and reused the next year. At the last period, nothing (or only scrap) is jointly produced. Thus a price appears for each dated fixed capital, and the subtraction of the price of the fixed sed during the period of production gives

In order to know the initial price of a "live" capital, it is sufficient to know the means of production and labor required to produce this asset. One reason that could explain why Sraffa did not finally insert the ry to fixed capital, the initial price of wasting assets such as exhaustible resources deposits is not a straightforward concept, as such assets are not produced by any the discovery cost should be taken into account. For instance, if the owner of the natural resource has invested some means of production and labour to find the new mine, he can then sell the right to exploit the mine to an extracting company and l charge to be paid to him can be calculated on the same principle as for 'live' fixed capital.

Ricardian literature is also linked to the [START_REF] Parrinello | Exhaustible natural resources and the classical method of long-period equilibrium[END_REF][START_REF] Schefold | Mr Sraffa on joint production and other essays[END_REF] ]. The Hotelling rule states that, given term equilibrium conditions (i.e. equal rate of profits for all industries), the rentier sells units of following the rate of profits, so that he can earn the uniform rate of profits of the economy on his "conservation" activity. I.e, if he decides during one period of production to sell nothing, the value of his stock of natural resource will rate equal to the ruling rate of profits from this period of production to the next one. 2 On this regard, two points must be stressed. First, this kind of revenue is a logical consequence of the f the economy, where it is not possible for an economic actor to earn less (or more) than the ruling rate of profits. Second, to calculate the price of the natural resource, and consequently the revenue of its owner at the present period of is not enough to know the present conditions of production. A back-stop technology that will replace the existing wasting asset must be anticipated. At the period of exhaustion of the wasting that will exist between the value of the commodities produced with the help of its natural resource, and the value of the commodities produced stop technology. The revenue of the owner of the natural resource at the present period can by going backward from the period of exhaustion, knowing the Hotelling rule (with yt the Then it is clear that to calculate this kind of revenue, a "photograph" of the market at one point of time is not enough. And it is clear that, if the uniform rate of profits is not assumed to be the equilibrium rate term position, but rather a "conventional" uniform rate of profits, If he earns less than the ruling rate of profits on its conservation industry, he would have an incentive to change its activity and to sell everything in the present period of production. If he earns more than the rate of profits, he would have an incentive to sell nothing in present period of production. Thus, the Hotelling theory states that he must earn exactly the ruling rate of necessary only for the sake of Sraffa's argumentation (see for instance for such a point of view [START_REF] Hicks | Sraffa and Ricardo: a critical view[END_REF][START_REF] Sinha | Listen to Sraffa's silences: an new interpretation of Sraffa's Production of Commodities[END_REF]Sinha, , 2016 [6, 19, 18] [6, 19, 18]), then the logic behind the Hotelling rule falls into pieces. These two points may explain why Sraffa did not introduce this kind of revenue into his book.

Rent as a share of the product

In a recent paper, Fratini (2016) [START_REF] Saverio | Rent as a share of product and Sraffa's price equations[END_REF] investigates rent as a share of the gross product. He shows how, in the Sraffian framework, this view of rent can help to better understand the institutions governing the distribution of the national product (in the line of Adam Smith, Ricardo and Marx), instead of only considering rent per unit of land. In his numerical examples, he does not relate the existence of the rent with the existence of a differential rent. Then the rent a a share of the gross product appears in Fratini as a tax (or a tithe) that could ensure a revenue to the owner of the natural resource even if no differential rent exists for the company extracting the natural resource.

In another paper, Ravagnani (2008) [START_REF] Ravagnani | Classical Theory and Exhaustible Natural Resources: Notes on the Current Debate[END_REF] shows that in the context of oil market in the United States of America, the revenue earned by the owner of the oil deposits was much more related to conventions and relations of power with the company that extracts the deposit than with consideration of the Hotelling rule:

"We therefore submit for discussion, as a constructive contribution to the current debate, the view that royalties on exhaustible resources could appropriately be encompassed within the logical structure of the classical theory of value and distribution as separately determined 'independent variables'. The conventional arrangements based on percentage royalties could be accommodated with no difficulty in the normal positions defined by Sraffa's equations by taking the share of the resource price attributed to landowners as a 'given' coefficient reflecting the (persistent) share paid on average in the actual economy." [START_REF] Ravagnani | Classical Theory and Exhaustible Natural Resources: Notes on the Current Debate[END_REF]) [START_REF] Ravagnani | Classical Theory and Exhaustible Natural Resources: Notes on the Current Debate[END_REF] The study of empirical data (IMF, 2012 [START_REF]IMF: Fiscal Regimes for Extractive Industries: Design and Implementation[END_REF]) on the fiscal regimes imposed on extractive companies by governments owning exhaustible natural resources shows that parameters that determine the conventional arrangements on percentage royalties are: the expected differential rent, uncertainty (on prices, geology, input costs, change in fiscal or political regime, exhaustion of the resource), asymmetric information between the extractive company and the government, and monopolies (either on the extraction of the resource or on the ownership of the resource). Governments have then several possibilities to collect their share of the product. The different mechanisms listed in IMF (2012) [START_REF]IMF: Fiscal Regimes for Extractive Industries: Design and Implementation[END_REF] are: signature and production bonus, royalties (rent as a share of product value), corporate or variable income tax (tax on profits and differential rents), resource rent taxes (tax on differential rents only, i.e. income less the average rate of profits), production sharing (rent as a share of physical product), state participation (rent received as dividends, as any shareholder), and social investments from the extractive company (the company pays the rent through the construction of a road or a school, for instance).

No matter the fiscal regime chosen by the government, the tax rate can be reduced to a share of the net product of the extractive industry, the latter being defined as the sum of the profits and the differential rent earned by the extractive industry. Simulation methods (FARI model, Luca and Puyo, 2016 [START_REF] Luca | Fiscal Analysis of Resource Industries: (FARI) Methodology[END_REF]) and analysis of accounting data shows that the tax rate on the net product t is between 65 and 85 percent for oil and gas industries, and between 45 and 65 percent for other mining industries (IMF, 2012) [START_REF]IMF: Fiscal Regimes for Extractive Industries: Design and Implementation[END_REF].

The introduction of the share of the net product as a "given" coefficient change the price system. In the case of a private company that extracts a natural resource owned by a State, the net product may be shared between the State and the private company. If the sharing occurs after the product has been sold, or if the company does not try to change prices once the tax is announced, the taxation of the net product has no impact on prices. The price system may be represented in this way, with the net product:

If the net product is taxed before the product is sold, or if the private company tries to avoid the effect of tax by increasing the prices (to see an example of this concerning a tax on profits, see [START_REF] Dome | Ricardo's theory of tax incidence: a Sraffian re-interpretation[END_REF]), prices could be impacted. The price system may be repre

It is interesting to notice that in the last system of equations, the differential rent is distributed entirely to the extractive company. As we non-basic commodity, see Sraff company increases, all prices and rents will change a case of the second iron company is an exception, as the increase of the tax will only lower the 3 Only a part of what is effectively produced by the private company is brought to the market. Let us say that of the physical gross product that is taken by the government. The price of the non following form, if the company tries to earn the uniform rate of profits With K the quantity of the input (assuming that there is only one input), and ρ the (possible) differential rent. The net product being defined as the value of the gross product less the value of the input and the value of labour, the share of the net product that goes to the government is defined as: Then the price system can also be written in this way:

Or in this way:

or if the company does not try to change prices once the tax is announced, the taxation of the net product has no impact on prices. The price system may be represented in this way, with If the net product is taxed before the product is sold, or if the private company tries to avoid the effect of tax by increasing the prices (to see an example of this concerning a tax on profits, see ]), prices could be impacted. The price system may be represented in this way:

It is interesting to notice that in the last system of equations, the differential rent is distributed entirely to the extractive company. As we are considering only basic commodities (for a definition of basic and Sraffa, 1960 [START_REF] Sraffa | Production of Commodities by Means of Commodities[END_REF], § 60), as soon as the tax rate on the net product t of any company increases, all prices and rents will change and the maximum rate of profits will decrease. The case of the second iron company is an exception, as the increase of the tax will only lower the Only a part of what is effectively produced by the private company is brought to the market. Let us say that of the physical gross product that is taken by the government. The price of the non-renewable resource will then be of the following form, if the company tries to earn the uniform rate of profits π and to distribute the uniform wage the quantity of the input (assuming that there is only one input), X the quantity of the output (physical gross product), the (possible) differential rent. The net product being defined as the value of the gross product less the value of the and the value of labour, the share of the net product that goes to the government is defined as: Then the price system can also be written in this way:

or if the company does not try to change prices once the tax is announced, the taxation of the net product has no impact on prices. The price system may be represented in this way, with t the tax rate on If the net product is taxed before the product is sold, or if the private company tries to avoid the effect of tax by increasing the prices (to see an example of this concerning a tax on profits, see [START_REF] Dome | Ricardo's theory of tax incidence: a Sraffian re-interpretation[END_REF] sented in this way: 3 It is interesting to notice that in the last system of equations, the differential rent is distributed entirely are considering only basic commodities (for a definition of basic and ], § 60), as soon as the tax rate on the net product t of any nd the maximum rate of profits will decrease. The case of the second iron company is an exception, as the increase of the tax will only lower the Only a part of what is effectively produced by the private company is brought to the market. Let us say that T is the share renewable resource will then be of the and to distribute the uniform wage w: the quantity of the output (physical gross product), the (possible) differential rent. The net product being defined as the value of the gross product less the value of the and the value of labour, the share of the net product that goes to the government is defined as: extensive differential rent earned by this company. 4

Conclusion

This article clarifies the different concepts that we must consider when we calculate the revenue of the owner of a non-renewable natural resource in a Sraffian framework. For a snapshot point of view, what matter to calculate prices and revenues are the conditions of production of the commodities that needs a non-renewable natural resource to be produced, and the distribution of the net product of the companies that produced these commodities. From a dynamic point of view, what matters is the stocks of the natural resource, the evolution of demand and technologies, the possibility of back-stop technologies, and the change of the institutional framework that ensure the distribution of the net product. From a policy point of view, it is important to understand the real cost of production of the commodities that use non-renewable natural resources, so that taxes can be adequately designed depending on the policy chosen (either recovery of maximum revenue, or incentive for private prospecting and exploitation, or protection of the natural resource, etc.).

The debate about non-renewable resources in the context of classical theory largely focused on the Hotelling rule, and more specifically on how to calculate the Hotelling rent. Only recently another line of thought started to emerge, where the revenue of the owner of the natural resource depends on the institutions governing the relations between the owner of the natural resource and the company that extracts the natural resource [START_REF] Ravagnani | Classical Theory and Exhaustible Natural Resources: Notes on the Current Debate[END_REF]Fratini, 2016 [16, 4]). A revenue above uniform profits exists for the extractive company as soon as there is differential rent, and this revenue may be seized by the owner of the natural resource. But empirical data show that in many cases a part of the differential rent is kept by the extractive company. Hence, from the point of view taken in section 5, there exists no "natural law" about who should earn the revenue coming from the extraction of a non-renewable resource, as well as there exists no "natural" regime property for such resources (they could be considered as a common good of humanity, or as a State property, a community property, etc.). This line of research, which could be linked with current heterodox approaches such as Institutional and Marxian, seems a promising way to continue this never-ending debate. It is important to notice that the real cost of production is not changed by the tax, only the distribution is changed. This point is touched upon by Sraffa in one of his unpublished papers: "Pantaleoni, Saggi, I, 146 n.

(quoted by Jannaccone, 342 n) Is real cost of production increased by an increase in a the imposition of a specific tax on product? Yes, says Pantaleoni: the cost per unit of wheat is increased in exactly the same way if 11 hours instead of 10 are required to produced it, or if, having the government taken 10% of the produce, 10 hours now produce only 9⁄10 of a unit. The reply to Pantaleoni isis then the wheat obtained by the Government produced without cost? Special case of taxes (tasse di Einaudi) which are levied by the State to defray cost incurred in order to contribute to the production of the commodity taxed (e.g. taxes on motor cars, affected to road fund, contribute to produce the services of motor cars)." (unpublished Sraffa Papers, folder D1/13: 6 (17); B.III.12.xxv)

4

 4

  Another kind of revenue that frequently appears in the neo depreciation of the wasting asset: Kurz et al., 2000; Kurz et al., 2001; Bidard et al., 2001; Bidard et al., 2001; Kurz et al., 2009; Kurz et al., 2011; Kurz et al., 2015) [15 long-term equilibrium conditions (i.e. equal rate of

  ; Kurz et al., 1997; Kurz et al., 2000; Kurz et al., 2001; Bidard et al., 2001; Bidard et al., 2001; Kurz et al., 2009; Kurz et 15, 17,

  ; Kurz et al., 1997; Kurz et al., 2000; Kurz et al., 2001; Bidard et al., 2001; Bidard et al., 2001; Kurz et al., 2009; Kurz et


Konstantinos Rantos
email: krantos@teikav.edu.gr

Digital Signatures: How close is Europe to truly interoperable solutions?

Keywords: Digital Signatures, interoperability, e-documents, standardisation, eGovernment services

Digital signatures have been a hot topic in the e-government era as a key enabler for e-services provided to business and citizens, and secure exchange of e-documents. When this exchange crosses the borders of closed systems or EU's Member States, several interoperability issues arise. In EU many schemes and solutions have been proposed to overcome some problems, yet there is still more to be done. This paper provides a survey of the actions taken to promote interoperable use of digital signatures and identifies areas where EU has to invest in order to achieve the desired level of interoperability.

Introduction

The penetration of digital signatures across EU has been noticeable the last few years, with more and more digital services adopting them to secure data authentication, integrity and non-repudiation. This flourishing period came after a long "recession" where public key cryptography and its derivatives, such as digital signatures, have been straggling to survive as a technological solution to securing e-services. European initiatives together with work programmes ID-ABC [START_REF]IDABC Work Programme[END_REF] and ISA [2] and the corresponding action plans have been the turning point that triggered EU's member states (MSs) to consider them as one of the key infrastructures in their eGovernment action plans. Nowadays, most of the countries across EU have already deployed digital signature schemes for their two main application fields:

entity authentication, i.e. electronic identity (eid) schemes and data authentication and integrity, i.e. digitally signed documents.

Digital signatures have proven to be the robust solution for secure exchange of e-documents in the context of (cross-border) e-service provision. Although currently provided mainly by governments, they are anticipated to be adopted and widely deployed by other critical sectors, such as banking, to satisfy similar needs. Digital certificates and signatures are expected to be the user's e-passport to e-services.

Digital signatures are well studied and standardised by ETSI (European Telecommunications Standards Institute), CEN (European Committee for Standardisation) and other organisations and initiatives such as the EESSI (European Electronic Signature Standardisation Initiative). These standards have solved technological issues and viable and secure yet, more or less, closed solutions are already deployed and have long been tested. However, the plethora of technical standards resulted in diversified environments related to secure signature creation devices (SSCDs), certificates, and signatures, which bring more obstacles to the road towards interoperable solutions.

As a result, many activities have been taking place in EU aiming to promote digital signatures interoperability at legal, policy and technical level. Such activities include signatures related studies and projects like CROBIES (Cross-Border Interoperability of eSignatures), cross-border demonstrators and other less signature related but with a significant impact on EU's policy formulation on digital signatures, PEPPOL (Pan-European Public Procurement Online), STORK (Secure idenTity acrOss boRders linked), and SPOCS (Simple Procedures Online for Cross-Border Services) and secure documents exchange specific solutions like the Business Document Exchange Network (BUSDOX).

This paper looks at digital signatures interoperability issues and the work that is carried out in the eGovernment services field, emphasizing on e-document signing, yet ommitting application related issues. In this context, this paper provides an up to date mapping of the standards used by solutions deployed across EU and identifies the points that remain problematic and need our attention in order to strengthen weak links in the interoperability chain. It also aims to assist those that design digital signature enabled applications and have to make important interoperability decisions and those planning to develop signature generation and validation tools as the ISA programme requires.

The paper is organized as follows. Section 2 provides a quick overview of the activities that take place in EU regarding digital signatures interoperability and a thorough mapping of signature and e-document related standards. Section 3 sets the cornerstones of interoperability and defines the three levels of interoperability that seek viable solutions while section 4 identifies those areas where EU has to invest for truly interoperable solutions.

Current activities and standards

The EU's intentions to promote secure exchange of e-documents in the context of eGovernment services and to use electronic identities (eIDs) across all MSs, have boosted the use of digital signatures across the EU. This cross-border vision, however, has brought up many interoperability problems for digital signature enabled applications. With a plan on e-signatures in 2008 [START_REF]Action Plan on e-signatures and e-identification to facilitate the provision of crossborder public services in the Single Market[END_REF], Europe sets the cornerstones for the actions that have to be taken towards MSs acceptable and interoperable e-signatures. These plans are reinforced by the Commission's decision to define "interoperability of public key infrastructures (PKI)" as one of the key activities of the ISA programme [2] defined in actions 1.8 and 1.9 regarding e-documents exchange and verification of e-signatures respectively.

Prior to start implementing EU's policy most of the MSs had justifiably designed their eID and e-services solutions considering merely their own needs and based on the applicable legal framework, usually overlooking other MSs needs and specifications. Figure 1 depicts the current status of standardisation in EU regarding digital signatures and e-documents. It is an update to the mapping published by EU [START_REF] Mandate | Standardisation Mandate to the European Standardisation Organisations CEN, CENELEC and ETSI in the field of Information and Communication Technologies applied to electronic signatures[END_REF] enhanced by the strong dependencies between those standards. The already complicated map coupled with the wide variety of options that each of these standards offers and the diversities among the adopted solutions reveal the reason why these e-signature related standards hinder the provision of interoperable solutions. TSLs also contain mandatory information one should otherwise find on a certificate, according to the corresponding standards, e.g. information whether the certificate resides on an SSCD or not (ETSI 101 862 [9]) but the CSP did not include it in the certificate in a standardized and commonly accepted manner. Moreover, they complement digital certificates mechanism and the trust model they offer in a way that raises some questions as dependence on a list implies that user's trust circle has to be expanded to also cover mechanisms and procedures that the supervising authority employs for managing and publishing this list. -EC has also adopted Decision 2011/130/EU [5], which defines common specifications regarding the format of signed documents to facilitate cross-border exchange within the context of service provision. These specifications promote the standardised CAdES [START_REF]Electronic Signatures and Infrastructures (ESI); CMS Advanced Electronic Signatures (CAdES)[END_REF], XAdES [START_REF]XML Advanced Electronic Signatures (XAdES)[END_REF], and the recently ETSI adopted PAdES [START_REF]Electronic Signatures and Infrastructures (ESI); PDF Advanced Electronic Signatures (PAdES)[END_REF] formats for documents signed by competent authorities. Although these standards offer many options to cover a wide range of security concerns, only *-BES (Basic Electronic Signature) and *-EPES (Explicit Policy Electronic Signature) profiles are currently adopted by EU. These flavours have limited features and do not secure the provision of additional information required for the critical long term archiving and time of signature generation.

Defining interoperability

When implementing e-services that entail secure exchange of e-documents, or when signing documents, there are several questions that someone has to answer, including:

-What should the signed document format be? -What types of electronic signatures (advanced, advanced based on qualified certificate, qualified) is the relying party willing to accept? In practice, what are the means that the signer has in possession and what is the risk the relying party is willing to accept in terms of signer's commitment and authorization? -What are the acceptable key sizes, signature algorithms or hash functions? -What additional information does the verifier need to verify and accept the provided signature, including timestamps and validation data?

The answers to these questions disclose the interoperability problems that digital signature deploying services are likely to face and can be mapped to the following interoperability levels.

Legal interoperability

The foundations of the legal framework at European level were laid by Directive 1999/93/EC [7], which defined the types of digital signatures and the requirements for the provision of certification services. Although it provided a solid basis for the use of digital signatures, it was only the basis on top of which many other structural components had to be built by MSs. Since Directive 1999/93/EC several enhancements took place by corresponding decisions and plans for the wider deployment of e-services supported by the use of electronic services. Many studies have been conducted within the IDABC programme regarding legal interoperability [START_REF]IDABC Work Programme[END_REF], the details of which are out of the scope of this paper.

Interoperability at policy level

Although law is one of the factors that will affect the validity of a signature, policy restrictions is another and is related to the security considerations of the service for which signatures are deployed. For instance, while an advanced signature based on a qualified certificate might suffice for a document used within a specific service, the same document for a different type of service might require a qualified signature, i.e. the one based on a SSCD [7].

Policy restrictions should be considered by the verifier prior to accepting a digital signature and could include issues related to certificate validity, certificate suitability (e.g. requirements for trust establishment, use of SSCDs, acceptable SSCDs, CSP's auditing), signature validity (e.g. algorithms and key sizes, timestamps) and document signature properties (e.g. signature format, signature placement).

Current practices and EU adopted mechanisms do not secure policy requirements satisfaction leaving verifiers with the difficult task of collecting the necessary information or deciding ad-hoc about the acceptance or not of a signed document (assuming that the verifier has policy requirements).

Technical interoperability

Legal and policy restrictions might affect technical decisions on document, signature and certificate formats and characteristics. If signer and verifier cannot support the same algorithms, protocols and mechanisms, signature verification is bound to fail. Technical interoperability problems typically can be bypassed if solutions, algorithms and mechanisms adhere to specific standards and alternatives are narrowed down to a limited set of options.

In this section key areas that are anticipated to play an important role in future e-services seeking maximum interoperability are defined.

Time stamping

Time stamps are the means to provide assurances about the existence of data before a particular time [START_REF]Electronic Signatures and Infrastructures (ESI); Policy requirements for time-stamping authorities[END_REF]. While in some transactions is important to mark the time of data receipt, e.g. in e-procurement services, there are other examples where time-stamps have to be provided by the signer, e.g. a certificate issued by a competent authority some time in the past, in which case the verifier has to establish a relationship with the time-stamping authority (TSA) to accept both the time-stamp and the signature.

According to Directive 1999/93/EC and ETSI 102 023 [START_REF]Electronic Signatures and Infrastructures (ESI); Policy requirements for time-stamping authorities[END_REF] a TSA can be considered as a CSP which issues time stamp tokens. Based on EU's trust model, solid trust relationships between TSA and verifier can only be established if the TSA is a qualified one and therefore, supervised/accredited by the MS's competent authority and if the services are listed in the MS's TSL, given that these are included in a voluntary basis [3].

Trust establishment is essential for the verifier to be assured about the procedures, practices and the security measures taken by the authority to provide this service. Otherwise the validity of the time stamp is jeopardized. Time-stamping is a service that has to be put into the picture to enhance digital signature enabled services.

Signatures with validation data and long term archiving

Although currently adopted formats of *-BES and *-EPES seem satisfactory for a very basic exchange of signed e-documents, they do not suffice for stronger requirements set by the receiving entity in a transaction.

Verification information, all or part of it, as previously mentioned, can be provided by the signer, which bears the risk of deploying non-interoperable solutions. To avoid this unpleasant situation various solutions adopt alternatives for secure signature validation, such as validation points proposed by Decision 2011/130/EU [5] and used by large scale pilots, such as PEPPOL and SPOCS. Such ad hoc approaches however do not provide robust solutions to signed document validation.

Moreover, long-term archiving data are needed for the verification of signer's signature at a time when the signer's certificate might be revoked or will have expired. The aforementioned reliable timestamping is much related to the information that supplements the signature for long-term archiving. Such data will prove essential in the long run when governments go all digital, hence all archives are in digital form. In that case keeping all the necessary verification information with the signed document will be vital.

Policy restrictions

As the number of e-services is anticipated to grow in the years to come, more diversified and open environments will come into picture with their own requirements and restrictions regarding signature generation and validation. Currently accepted standards allow the signer only to disseminate his/her signature policy using the *-EPES format. This approach, although straightforward for the signer, requires the verifier to adapt his/her requirements on what the signer provides and decide about the validity on an ad-hoc basis. It does not consider the verifier's requirements regarding signature and, as a result to that, document acceptance.

A more appropriate solution would also allow the verifier provide his/her own requirements and establish a kind of agreement with the signer on the corresponding needs prior to signer's commitment. EU should elaborate on the formulation of an EU wide policy format and MSs must formulate their policies on e-signatures based on this for all cross-border services considering the following.

-Signer's policy adopted format should be unambiguously interpretable by the verifier, and automatically processed to relieve the end user from this very complicated task. ETSI 102 038 [START_REF]TC Security -Electronic Signatures and Infrastructures (ESI)[END_REF] and ETSI 102 045 [START_REF]Electronic Signatures and Infrastructures (ESI); Signature policy for extended business model[END_REF] standards can form the basis towards this achievement. -Promote policies recording and mapping based on commonly accepted standards to achieve the much desirable interoperability at policy level. -Work on schemes that will facilitate policy agreements between signer and verifier prior to signer's commitment.

Commonly adopted standards for certificate formats

Current standards provide robust solutions for certificates and facilitate unambiguous interpretation of the type of certificate, its properties and security characteristics. However, not all of them are adopted by CSPs who issue qualified certificates, leaving critical information out of the certificate or included in a non standardized manner. Adoption of commonly accepted profiles based on specific standards, such as those included in Figure 1, would allow certificates to carry all the necessary information that is now "delegated" to other schemes. Such an approach would help overcome the aforementioned problem caused by the issuance of certificates by different CSPs under different procedures, specifications and with a different perspective on the format that a certificate should have. CROBIES has already proposed an interoperable certificate profiles that can be adopted by the EU [START_REF]CROBIES: Interoperable Qualified Certificate Profiles[END_REF].

Although adoption of common profiles is not an easy task, it will secure the wide acceptance of certificates and convert them to the e-passport that EU envisioned and tries to promote through corresponding plans and actions.

The wide variety of standards on digital signatures complemented by the large number of options they offer, have resulted in many cross-border interoperability problems in EU.This paper provided an up to date mapping of digital signature and e-documents related standards and identified the key areas that still need to be considered by the EU in order to achieve truly interoperable solutions. In contrast to EU's approach of introducing new ad-hoc mechanisms and solutions to old problems the paper suggests the use of existing standards assuming that the options they offer are narrowed to a commonly accepted subset, an approach that is also adopted by Decision 2011/130/EU [5]. Adopting a more strict set of rules will simplify this complicated environment and will pave the way to a wider deployment of digital signatures, even for digitally analphabetic users.

Fig. 1 .-

 1 Fig. 1. E-signatures standards map

Services Directive 2006/123/EC eInvoicing 2006/112/EC Public Procurement 2004/18/EC, 2004/17/EC CMS Advanced Electronic Signatures (CAdES)

					technically
					specified by
	basis for	technically specified by Use of Electronic Signatures: Legal and extended Electronic Signature formulated General guidelines for electronic signature Application of formulated by Technical Aspects verification CWA 14365 -Part 1 CWA 14171 ... harmonized by	Security Requirements for Cryptographic Internet X.509 PKI Certificate and CRL Profile RFC 5280 Modules FIPS 140-2 ETSI 102 437 Natural Persons ETSI 102 280 X.509 V.3 Certificate Profile for Certificates Issued to depends on based on Qualified Certificate Profile RFC 3739 ETSI 101 862 Guidance on TS 101 456 Module Policy requirements for adopts certification authorities Algorithms and issuing qualified certificates ETSI 101 456 CWA 14167-3,4 Cryptographic Parameters for complemented Guidelines for the implementation of SSCDs CWA 14355 amplified by by adopts guided by Secure guided Electronic by Signatures Guidance on ETSI 102 176 -TS 101 456 ETSI 102 437 Parts 1-2
		Standards in Europe
		ETSI 102 438				CMS	based on
				ETSI 101 733	, RFC 5126	extends	CMS -Cryptographic	PKCS#7 Cryptographic
		amended Trust-Service Status Information based on XML Advanced Electronic Signatures (XAdES) ETSI 101 903 by Commission Decision Trusted Lists utilises 2010/425/EU	used by	XML format for signature policies ETSI 102 038 amplified by extended by	Message Syntax RFC 2315 extended by Message Syntax RFC 5652 Profiles of CMSAdvanced ETSI 102 734 by RFC 5035 based on TS101 733 (updated ) RFC 2634 Electronic Signatures Services for S/MIME Enhanced Security
		1,2,4,5,6 ETSI 102 778 -Parts Signatures (PAdES) Electronic PDF Advanced PDF PDF Advanced ETSI 102 231 PAdES: Usage and ETSI 102 923 guidelines implementation Registered Email: used by Electronic Signatures (PAdES) ETSI 102 778 -Part 3 guided by ETSI 102 605 ETSI 102 640 -Parts 1-5 by complemented	ETSI 102 904 Profiles of XML Advanced eSignatures based on amplified by	XML Syntax and Processing XML Signature W3C -XML Signature (XMLDSig) extended by amplified by Profiles of XML Advanced Electronic Signatures ETSI 102 904 based on TS101 903	profiles ETSI 101 861 RFC 3161 ` Time stamping profile ETSI 102 023 authorities Signature policy for extended business model ETSI 102 045 for time-stamping TSA Policy requirements

European Union Directive 1999/93/EC Legal framework for eSignatures and Certification Service Providers Commission Decision 2003/511/EC Generally Recognised Standards for Signature Products fostered by Commission Decision 2009/767/EC Trusted Lists Commission Decision 2011/130/EU (Adoption of X/C/PAdES BES/EPES) Trustworthy systems managing certificates CWA 14167-1,2 SSCD EAL4+ (ISO 15408) CWA 14169 recognises recognises Application interface for smart cards used as SSCDs CWA 14890-1,2