Keywords: Configuration knowledge modeling, constraints satisfaction problem, ontology, UML, OWL, rules

A configuration software (configurator) associates a knowledge base (KB) with a knowledge processing unit (PU). The KB describes all possible combinations of components while the PU overlays this knowledge with the customer requirements. Our work deals with the KB and the approaches, models, or tools for modeling configuration knowledge. Our goal is to present a small quantitative literature survey highlighting two work streams: the first one gathers modeling works dealing with constraint-based approaches while the second deals with ontologies, description logic, or object-oriented modeling approach. We will also consider hybrid approaches. We will present a quantitative analysis of published materials in Web of science over the last twenty years. The keywords occurrence versus time will also be studied in detail to identify tendencies in configuration knowledge modeling..

I. INTRODUCTION

Nowadays, in one hand, mass customization plays an important role in competitive environments. In fact, mass customization helps to efficiently answer customers' demands when they want it, and at a competitive price. On the other hand, the context of industry 4.0 implies that companies have to answer quickly to many customers' demands and to offer solutions which match perfectly numerous and often complex requirements. In this context, configuration tools play an important role that will permit to satisfy the customer requirements.

This article deals with product configuration or customization. Configuration is a kind of design activity where products are defined given predefined sets of components linked by compatibility constraints [START_REF] Felfernig | Knowledge-based configuration From Research to Business Cases[END_REF]. Configuration activity is supported by software tools called Product Configuration Software (PCS) [START_REF] Myrodia | Impact of product configuration systems on product profitability and costing accuracy[END_REF]. A PCS gathers (i) a knowledge base (KB) which contains what is called a generic model of the product and (ii) a processing unit (PU) that interacts with the user to assist him during the configuration activity. The generic model contains all component families and product attributes with all compatibility constraints that modulate their possible combinations. One generic model represents one family of products with all possible options and alternatives. The processing unit (PU) is basically responsible for constraint propagation. This means that for each requirement inputted by the user, the constraints are propagated and reduce the definition domain of the other variables linked by the constraints. Consequently, after each requirement input, the solution space or product family size is iteratively reduced and leads to a product solution that satisfies the customer. For more details, do not hesitate to consult [START_REF] Felfernig | Knowledge-based configuration From Research to Business Cases[END_REF].

Many works have been published about the processing unit of PCS (constraint propagation, answer set programming, inference engine, rule-based processing…), which is not the same with the modeling of the generic model necessary to set up product configurators [START_REF] Hotz | Configuration Knowledge Representation and Reasoning[END_REF]. Our purpose is to quantify the amount of work on this subject and what could be interesting tendencies. This is achieved with a quantitative literature survey that deals with two workstreams relevant to this subject: the first one gathers modeling works dealing with constraint approaches while the second deals with ontologies, description logic, or object-oriented modeling approaches. Hybrid approaches that associate will be also considered.

As an example, we will explain several articles in the realm of workstreams: [START_REF] Felfernig | Configuration knowledge representations for Semantic Web applications[END_REF] used description logics-based languages such as Ontology Inference Layer (OIL) and DARPAAgent Markup Language to represent configuration knowledge in the context of the semantic web. Unified Modeling Language (UML) which is a useful modeling language, has also been applied to graphically represent configuration knowledge. [START_REF] Asikainen | A domain ontology for modelling variability in software product families[END_REF] introduced a domain ontology called Kumbang to model variability from both feature and architectural perspectives in software product families. Kumbang can also be used to develop modeling languages, manage variability in software product families based on customer requirements. Natural language and a UML were used to describe Kumbang. [START_REF] Felfernig | Standardized configuration knowledge representations as technological foundation for mass customization[END_REF] used the Unified Modeling Language and the Object Constraint Language (UML/OCL) as standard configuration knowledge representation languages to represent configurable products then integrate them into the Model Driven Architecture (MDA). In the end, UML/OCL has been applied in several industrial projects. The following two articles deal with Hybrid approaches: [START_REF] Yang | Development of a product configuration system with an ontology-based approach[END_REF] represented an ontology-based approach for modeling product configuration knowledge. It used Web Ontology Language (OWL) to define different classes and their relationship to formalize product configuration then used

Twenty Years Of Configuration Knowledge Modeling Reasearch. Main Works, What To Do Next?

Semantic Web Rule Language (SWRL) to define the constraint and after that used JESS. The proposed approach was applied to configuring the ranger drilling machine. Protégé was used in this article which is an OWL-based tool that could be helpful to model and edit ontologies. [START_REF] Yang | Product configuration knowledge modeling using ontology web language[END_REF] represented an Ontology-based method for product configuration knowledge using semantic web technologies. It also used OWL, SWRL, and a rule engine called JESS to improve the product configuration system. In this paper, an actual configuration engine based on derived configuration knowledge was developed as well.

The approach was applied to a case for the personal computer. [START_REF] Zhang | Product configuration: a review of the state-of-theart and future research[END_REF] and [START_REF] Oddsson | From a literature review of product configuration definitions to a reference framework[END_REF] are two good review articles on product configuration that presented various definitions in this area as well as future roadmaps.

The organization of the paper is as follows. We will first present and discuss the main elements of our survey: keywords, way to use keywords, and literature sources. Then we will present and discuss various literature investigations. We will finally conclude on possible ideas for future works.

II. SURVEY MAIN ELEMENTS

This section discusses the selected keywords, various ways to use them, and finally the article's sources.

A. Identifying keywords

We are dealing with knowledge modeling in the context of product configuration. As previously said, we want to consider and differentiate the works relevant to the model structure and variable identification from those that are more centered on relations or dependencies between configuration variables. Furthermore, we want to have keywords with some increasing strength or filtering level. Consequently, we have identified six groups of keywords: 1. Two less restrictive strength groups of keywords:

• Group of keywords 1, kw1= configuration and knowledge • Group of keywords 2, kw2= configuration and knowledge and modeling 2. One group of keywords relevant to model structuring and variable identification:

• Group of keywords 3, kw3= configuration and knowledge and (ontology or UML or description logic) 3. Two groups of keywords more relevant to relations or dependencies between configuration variables:

• Group of keywords 4, kw4= configuration and knowledge and constraint • Group of keywords 5, kw5= configuration and knowledge and (OWL or Rule) 4. One highly restrictive group of keywords that considers simultaneously constraint with another previous keyword:

• Group of keywords 6, kw6= configuration and knowledge and constraint and (ontology or OWL or Rule).

B. Using keywords in queries

Once keywords are defined, we have to decide about how to use them. They can be used to select publications concerning three possible querying fields: article title, article keywords, and article abstract. Another field "article topic" aggregates the three querying previous fields. Of course, abstract is much less restrictive than title or keywords. We will consider mainly the query on field "topic" but in section 4 we will also discuss the results of query on field "title".

C. About literature sources

At the beginning of the work, we have been considering three kinds of sources: (i) the web of sciences, (ii) Mendeley and/or Scopus , (iii) CEUR workshops [11]. The main reason to consider the web of sciences is because of its seriousness in the quality of the selection of the indexed elements. Furthermore, its website is quite well organized to filter and categorize queries.

The reason to consider Mendeley and/or Scopus was to avoid relying on the single WoS knowledge base. However, very quickly we face some difficulties in using and understanding Mendeley's results. Furthermore, if Mendeley is easy to access, this is not the case for Scopus.

The last source we wanted to consider was CEUR proceedings, because the knowledge-based configuration workshops community, that runs for more than twenty years, publishes rather frequently their proceedings with CEUR. But it seems that some years are missing because the workshop has been hosted by some conferences like ECAI or CP that prevent open publications of the workshop proceedings.

Consequently, and this is a drawback of this study, the results will only rely on the WoS database, more exactly on the Science Citation Index [START_REF] Garfield | Citation Indexes for Science: A New Dimension in Documentation through Association of Ideas[END_REF]. In section 4, Science Citation Index and on Conference Proceedings Citation Index queries will be compared.

III. FIRST INVESTIGATIONS

This section presents and discusses the results relevant to the six groups of keywords, with queries on the Science Citation Index considering the field "topic" that gathers: title, keywords and abstract.

A. First raw results

The figure 1 shows the number of articles per year (between 2000 and 2020) on the upper part less restrictive keywords (kw1 and kw2) and on the lower part all the others. In fact, the lower part is just a zoom of the upper part for kw 3,4, 5 and 6.

The upper part dealing with keywords kw1 "configuration and knowledge" and kw2 "configuration and knowledge modeling" show a quite smooth and regular increase. The lower part is not so smooth, but the tendencies for kw4 "configuration and knowledge and constraint" and kw5 "configuration and knowledge and (OWL or Rule)" shows a regular increase. On the opposite, kw3 "configuration and knowledge and (ontology or UML or description logic)" show a very light increase while kw6 "configuration and knowledge and constraint and (ontology or OWL or Rule)" globally remain constant.

B. Discussion about restrictive keywords

Logically, a very large number of articles can be associated with less restrictive keywords. But a quick look shows that many articles come from scientific domains (biology, medicine…) not related to our topic (engineering, design…). However, the rate of increase sounds quite strong for keywords kw1, 2, 3, and 4. Three possibilities to explain that: (i) more people work on this subject, (ii) people publish more, and (iii) Web of Science indexes more journals. It is probably the association of the three phenomena that explains previous growth. However, we compute some kind of yearly increase for our domain and compare it with simple keywords as "Engineering" and "medicine" as shown in table 1.

We can see that the global increase for less restrictive keywords (kw1 and kw2) is with the same order of magnitude with generic keywords as "engineering" or "medicine". All rates are between 9.9% and 10.3%. For other keywords (kw3, 4, and 5), it is clear that the increases are much lower All rates are between 5.2% and 7.4 %. While keyword 6 is not computable. To compare with a quite hot topic of the moment, the computation has been done for the keyword "machine learning" and shows a yearly or rate of 25%.

It seems to be possible to conclude that:

1.

Queries with less restrictive keywords just follow global publication quantitative evolutions.

2. Keywords dealing with relations or dependencies between configuration variables (kw4 and kw5) attract more articles than those dealing with model structuring and variable identification (kw5). 3. Works mixing constraints and model structuring are clearly less frequent.

TABLE 1.

The yearly rate of increase of article quantities IV DISCUSSION ON SOURCES We will first compare; journal and conference proceedings survey then deal with investigations on article title or topic.

A. Comparing journal/conference proceedings

In figure 2 we show the result of a query similar to figure 1 except that it considers WoS proceedings. Figure 1 and 2 should be compared. We can see that the quantity of articles is much lower. This can be explained by the fact WoS-SCI indexes much less conference than journals. The quite regular increases of the number of publications which were seen in the result of journals are not the same in the result of proceedings. If an increase can be seen for less restrictive keywords (kw1 and kw2), this is not the case for all other keywords. Figure 2 shows also two phenomena, the economic crisis of 2008 and the Covid crisis of 2019, that seems to induce clear decreases in the years following them. This is especially true for 2020 with a terrible drop of more than 50%. Consequently, this might have pushed authors to consider more journal publishing than conference publishing.

B. Comparing queries on topic and title.

To study that, we have compared queries on article 'topic" with queries on article "title" on the WoS-SCI journals. Ot shown results show that the quantities of identified articles are roughly divided by an order of magnitude of one hundred. This shows clearly, that making queries on title is too restrictive to derive interesting conclusions.

C. Conclusion on queries parameters

Considering previous results, we only consider queries on journal "articles" or or WoS-SCI made on the "topic" field. However as said in the end of section 3 many papers are not belonging to our subject interest. Next section will therefore add some filters on WoS-SCI queries on "journal" on "topic".

V FILTERING RESULTS

The idea is to remove out a maximum of out-of-scope papers, for example, papers dealing with medicine, biology, law, business… medicine, arts… For this purpose we have used the categories filters proposed by the WoS website, then in a second step, we refine the results with abstract reading. All the results of this section are shown in figure 3.

A. Filtering with WoS categories

Thirteen WoS categories have been selected to filter results of journal articles queries on field "topic" . 3, how the raw results of the left part (equivalent to figure 1 -query on journal and topic) are filtered. The quantities of articles are divided by an order of magnitude of 3. Logically, as the quantities are lower, the curve are less smooth and for more restrictive keywords instabilities are present, see kw3 for example. However, a yearly rate of increase can be computed, for less restrictive keywords (kw1 and kw2) around 8% and between 4% and 6% for kw3, kw4, and kw5. About the most restrictive keywords that interest us, kw6, no yearly increase can be computed, but there is no strong difference with or without WoS category filtering.

An interesting point is that the curves dealing with relations or dependencies between configuration variables (kw4 and kw5) are getting much closer to the ones dealing with model structuring and variable identification (kw3). This could lead to the conclusion that in our domain, works and publications are of the same order of magnitude regarding the three groups of keywords while the last one keeps remaining below.

B. Filtering with with abstract reading

Abstract reading has been done only for the paper relevant to the four strongest restrictive keywords kw3, kw4, kw5, and kw6 so in the lower part of figure 4. This means that around 600 abstracts have been quickly looked. The results are shown in the lower right part of figure 3.

The conclusions of the previous section are confirmed. Now with this filter, there is not any significant difference between the three keyword groups (kw3, kw4, and kw5) while the most restrictive keyword (kw6) remains below.

C. Results discussion

Given the previous element, we have identified the following numbers of articles: • Kw3: configuration and knowledge and (ontology or UML or description logic): 159 articles, • Kw4: configuration and knowledge and constraint: 239 articles, • Kw5: configuration and knowledge and (OWL or Rule): 207 articles, • Kw6: configuration and knowledge and constraint and (ontology or OWL or Rule): 46 articles. The initial tendency about the fact that researchers of the configuration domain rather prefer to work and publish on the subject dealing with "relations or dependencies between configuration variables" rather than those dealing with "model structuring and variable identification" is still lightly present but much less than without the filtering. The "constraint" keyword is the most frequent followed by "OWL or "rule".

Among the 46 articles selected with respect to "constraint" and "ontology or OWL or Rule" (kw6):

• 18 are also selected with respect to "ontology or UML or description logic " (kw3), • 23 are also selected with respect to "OWL or Rule" (kw5), • 4 are also selected by both (kw3 and kw5).

This shows that researchers naturally tend to work with a single model or formalism. It is indeed simpler, more reassuring, and more productive to work on a single model, tool, or approach. As a consequence, many works address separately the approaches based on constraints, ontologies, or rules, but very few combine two of them. As said in [START_REF] Felfernig | Knowledge-based configuration From Research to Business Cases[END_REF] about mixing approaches in what the author call hybrid configuration:" Typically, there is an ontology-like (DL-based) representation for representing components, their compositional relations, and attributes. Additionally, constraints are used for representing n-ary relations between components and for computing and inferring attribute values.".

Associating ontology and constraint approaches to develop configuration software still sounds of interest. Ontology brings the capture of the product concepts hierarchy description and descriptive variable identifications. Constraint diversity allows representing any kind of relation or dependencies that are the key elements of the configuration problem. Since the quite well-cited works of some authors as Dong [START_REF] Yang | Development of a product configuration system with an ontology-based approach[END_REF] and [START_REF] Yang | Product configuration knowledge modeling using ontology web language[END_REF], Asikainen [START_REF] Asikainen | A domain ontology for modelling variability in software product families[END_REF] or Felfernig [START_REF] Felfernig | Configuration knowledge representations for Semantic Web applications[END_REF] and [START_REF] Felfernig | Standardized configuration knowledge representations as technological foundation for mass customization[END_REF] were published more than twelve years ago, this domain definitely needs to be investigated again.

VI CONCLUSION

Our goal was to present a quantitative literature survey on the modeling aspect of configuration problems. Keywords around configuration, knowledge modeling, constraints, ontology have been discussed and selected. Web of Science has also been selected and results with different queries on (i) journal or proceeding, (ii) article topic or article title, have been presented.

A first conclusion is that authors seem to prefer to work on (i) constraints or rules than on (ii) ontology or product model structuring. A second one is that it is much harder to be highly specialized in more than one model or tool and consequently to publish on hybrid solutions associating the two previous work streams.

Knowledge modeling and maintenance is a key issue for configurator system deployment. For future studies, considering more efficient ways to model both product structure, product concepts, and configuration dependencies are still a necessity. Working on the association of ontology, object-oriented and constraint-based approaches seems to be very interesting which will generate with no doubt great improvements for configuration system design and deployment.

Figure 1 .

 1 Figure 1. Journal articles / year -WoS-SCI -on topic

Figure 2 .

 2 Figure 2. Comparing journal and conference proceedings.

INP-ENIT, University of Toulouse, Tarbes, France

IMT Mines Albi, University of Toulouse, Albi, France 2 ISAE-SUPAERO, University of Toulouse, France (maryam.mohammadamini@enit.fr, thierry.coudert@enit.fr, elise.vareilles@isae-supaero.fr, michel.aldanondo@mines-albi.fr)


Philippe Carr

Eric Andres

Discrete analytical Ridgelet Transform

Keywords: Ridgelet Transform, Discrete Analytical Lines, Denoising, Partial Reconstruction

In this paper, we present a new implementation of the Ridgelet transform based on discrete analytical 2-D lines: the Discrete Analytical Ridgelet Transform (DART). This transform uses the Fourier strategy (the projection-slice formula) for the computation of the associated discrete Radon transform. The innovative step of the DART is the construction of discrete analytical lines in the Fourier domain. These discrete analytical lines have a parameter called arithmetical thickness, allowing us to de ne a DART adapted to a speci c application. Indeed, the DART representation is not orthogonal it is associated with a exible redundancy factor (depending on the arithmetical thickness). The DART has a very simple forward/inverse algorithm that provides an exact reconstruction. We discuss the choice of the wavelet transform applied to the Radon projections and illustrate the extension of the DART to a Local-DART (with smooth windowing) and Curvelet-DART (with undecimated quincunx scheme). We apply the DART and its extensions to the denoising and the partial reconstruction of some images. These experimental results show that the simple thresholding of the DART coe cients is competitive or more e ective than the classical denoising techniques.

Introduction

Image analysis is traditionally aimed at understanding digital signals obtained by sensors (in our case cameras). Digital information is considered as sampled continuous information and the theoretical background for it is signal theory. This is sometimes referred to as digital geometry in opposition to discrete geometry for computer graphics. These last ten years, since J-P. Reveill s has introduced it 16], discrete analytical geometry has made an important progress in de ning and studying classes of discrete objects and transformations. This greatly enhanced our understanding of the links between the discrete world Z n and the continuous world R n . In the same time, a new discrete signal decomposition has been developed in image analysis: the wavelet representation. This new representation has many applications such as denoising, compression, analysis, etc. One of the aims of this paper is to apply this new insight in discrete geometry to image analysis and more speci cally to a particular 2-D wavelet transform: the Ridgelet transform (Cand s and Donoho, 1998 5]).

Wavelets are very good at representing point singularities ; however they are signi cantly less e cient when it comes to linear singularities. Because edges are an extremely common phenomena in natural images, an e cient multiresolution representation of images with edges would be quite advantageous in a number of applications. A team of Stanford has recently developed an alternative system of multiresolution analysis, called Ridgelet, speci cally designed to e ciently represent edges in images 5]. The Ridgelet transform can be computed by performing a wavelet analysis in the Radon domain. However, most of the work done with Ridgelets has been theorical in nature and discussed in the context of continuous functions. The important bridge to discrete implementation is tenuous at best. To our knowledge, we can nd in the literature only two solutions for the discrete Ridgelet decomposition [START_REF] Coifman | Translation-invariant de-noising[END_REF][START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF]. These two strategies are recalled in section 2. This paper presents a new approach that aims at representing linear singularities with a discrete Ridgelet transform based on discrete analytical 2-D lines: the Discrete Analytical Ridgelet Transform (DART). The idea behind the associated discrete Radon transform is to de ne each Radon projection by a discrete analytical line in the Fourier domain: L ! p;q] = (x 1 ; x 2 ) 2 Z 2 jqx 1 px 2 j ! 2 with p; q] 2 Z 2 the direction of the Radon projection and !, a function of (p; q), the arithmetical thickness.

There are several advantages at using discrete analytical lines: they o er a theoretical framework for the de nition of the discrete Radon projections. The parameter ! allows us to de ne a DART adapted to speci c application (control of the redundancy factor of the transform). We have used, in this paper, three types of discrete analytical lines: the closed naive lines (DART redundancy factor 2), the pythagorean lines ( DART redundancy factor 2:3) and supercover lines (DART redundancy factor 3). This compares to the transform redundancies of a factor 4 for the Stanford team 18] and a factor 1 (orthogonal transform) for the Lausanne team 9].

The forward DART is based on the extraction of the 2-D Fourier coe cients belonging to the discrete analytical line L ! p;q] . The inverse DART is simply performed by putting the Fourier coe cients back on the exact same place in the Fourier lattice. The DART has therefore a very simple and rapid forward/inverse algorithm. This simple straightforward approach ensures an exact reconstruction without interpolation nor iterative process (that might be sensible to noise).

In order to compare the performances of the DART, we have applied the DART and its extensions (local-DART and Curvelet-DART) to the denoising and the partial reconstruction of some images. We have compared our approach to existing ones. The recent developments have shown that the Ridgelet decomposition has very attractive results for the denoising problem 18]. For this, the Stanford team proposed to apply a thresholding on Ridgelet coe cients. Our experimental results show that the simple thresholding of the DART coecients is competitive or more e ective than the wavelet transform. Moreover the DART denoising results seems to be similar (visually and PSNR measure) than Stanford's strategy but without any iterative reconstruction algorithm and with a more exible de nition. The analysis reported here show how results of discrete analytical geometry can be successfully used in image analysis.

In section 2, we recall the continuous theory of the Ridgelet transform and describe the existing discrete Ridgelet transforms 9,18]. In section 3, we present the Discrete Analytical Radon Transform with some details on the theoretical framework of discrete analytical lines. The Discrete Analytical Ridgelet Transform (DART) is introduced at the end of the section 3. Several illustrations of the DART are proposed in the section 4. In this section, we discuss the choice of the discrete line type and 1-D wavelet transform associated with the DART. We compare also the DART with other Ridgelet transform implementations for the denoising and the partial reconstruction problematics. We present several extensions of the DART to a local-DART and to the Curvelet formalism. We end this section with some thoughts on the extension of the DART to 3-D. We conclude in section 5. The Radon transform of s is de ned as:

The Ridgelet transform

Rs( ; t) = Z R 2 s(x) (x 1 cos + x 2 sin t)dx 1 dx 2 where is the Dirac distribution. The Ridgelet coe cients r s of s are given by the 1-D wavelet transform to the projections of the Radon transform where the direction is constant and t is varying: This is the projection-slice formula that is used in image reconstruction from projection methods. All these relations are resumed in gure 1.

Strategies for Discrete Ridgelet Transform

As we have seen, a basic strategy for calculating the continuous Ridgelet transform is rst to compute the Radon transform Rs and secondly, to apply a 1-D wavelet transform to the slices Rs( ; :). The implementation of the discrete Ridgelet transform can use the same principle.

The discrete wavelet decomposition, associated to a lter bank 15], is easy to implement, is stable and invertible, and can be associated to a discrete orthogonal representation. The discretization of the Radon transform is more di cult to achieve. The majority of methods proposed in the literature have been devised to approximate the continuous formula. None of them, however, were speci cally designed to be invertible transforms for discrete images and can therefore not be used for the discrete Ridgelet transform.

The discrete Radon transform can be computed with one the two following strategies:

Spatial strategy for digital Radon transform. The Radon transform is dened as summations of image pixels over a certain set of lines, that are de ned in a nite geometry. Rs( p; q] ; b) = X (x 1 ;x 2 )2L p;q];b s(x 1 ; x 2 ) with L p;q];b = f(x 1 ; x 2 ) 2 0; N 1] 0; N 1] : qx 1 px 2 b = 0g and (p; q) the normal vector. Fourier strategy for digital Radon transform. The projection-slice formula suggests that approximate Radon transforms for digital data can be based on discrete Fast Fourier transforms (FFT). The Fourier-domain computation of an approximate digital Radon transform is de ned as:

(1) Compute the 2-D FFT of s

(2) Extract Fourier coe cients along the lines L going through the origin.

(3) Compute the 1-D inverse FFT on each line L (de ned for each value of the angular parameter ). This approach can be problematic since step 2 is not naturally de ned on discrete data.

Recently, some articles studied the implementation of the discrete Ridgelet transform. Do and Vetterli proposed in 9] an implementation method of Ridgelet transform based on the use of the nite Radon transform for Z 2 p , the cartesian product of two sets of integers mod p, where p is a prime. This method achieves both invertibility (the inverse transform is stable) and nonredundancy (the associated Ridgelet transform is orthogonal). The obtained denoising results presented in 9] illustrate that this strategy is more e ective than the Wavelet transform in recovering straight edges. This transform is not geometrically faithful (the nite Radon transform has an important wraparound e ect) and is only de ned for image s such that s 2 l 2 0:::p 1] 0:::p 1] where p is a prime number. Moreover this Radon transform integrates over lines that can be rather arbitrarily spread out over the spatial domain and discrete lines in the Fourier domain associated with this decomposition are not closed. This approach is not based on a geometrically faithful notion of Ridgelets and su ers from artifacts (for example in denoising application: gure 10).

In 18], Starck et al. proposed to use an interpolation scheme that substitutes the sampled values of the Fourier transform obtained on the square lattice with sampled values of b s on a polar lattice. They use a pseudopolar Fourier domain that o ers a notion of polar Fourier domain better adapted to digital data (the digital Fourier domain is viewed as a sequence of squares, not circles). The discrete pseudopolar Fourier transform of a digital image s 2 l 2 0:::n 1] 0:::n 1] is de ned by sampling the 2-D Fourier transform at the collection of pseudopolar grid points illustrated by gure 2 (from 3]). The paper 18] follows the strategy based on the pseudopolar grid. However, it uses a simple nearestneighbor interpolation scheme to evaluate pseudopolar grid points in terms of nearby cartesian grid points. More recently Donoho et al. proposed in 3] a fast pseudopolar Fourier transform based on a chirp-Z transform (for the exact evaluation of the 2-D Fourier transform at these non cartesian points). The associated Radon transform (called Slant) is algebrically exact and geometrically faithful. However in order to be invertible this transform requires a xed factor four of redundancy. Moreover, the inverse transform is ill-conditioned in the presence of noise and requires an iterative approximation algorithm.

In this paper, we propose to use the Fourier strategy for the digital Radon transform. Our lines L are de ned with help of the discrete analytical geometry theory in the Fourier domain 2,16]. This solution allows us to have di erent Ridgelet decompositions according to the arithmetical thickness of the discrete lines. This approach presents a limited wrap-around e ect. As for the Donoho's transform 3], our representation is redundant however the degree of redundancy can be adapted by our thickness parameter. Our Radon backprojection is very simply and permits an exact reconstruction (analytical Radon transform followed by backprojection analytical Radon transform is a one-to-one transform). The objects de ned in discrete analytical geometry (3D lines, planes, hyperplanes, ...) allows extensions to 3D and even higher dimensions (see section 4.4).

Multiscale Ridgelet: Curvelet

The Curvelet transform proposed by Donoho 13] allows us to obtain a multiscale Ridgelet analysis with interscale orthogonality. The Curvelet transform is obtained by ltering and then applying a windowed Ridgelet transform on each bandpass image. The motivation behind the Curvelet transform is that by smooth windowing, segments of smooth curves would look straight in subimages, hence they can be well captured by a local Ridgelet transform. Subband decomposition is used to keep the number of Ridgelets at multiple scales under control by the fact that Ridgelets of a given scale exist in a certain subband. The window's size and subband frequency are linked such that Curvelets have support verifying the key anisotropy scaling relation for curves 13]:

width length 2 Starck et al. propose a Curvelet transform that uses an undecimated lter bank 18] for the subband decomposition. They applied their digital Curvelet transform to the denoising of some standard images embedded in white noise. These empirical results reported in 18] and based on simple thresholding of the Curvelet coe cients show that this strategy is very competitive with state of the art techniques based on wavelets.

The study of the DART generalization to the Curvelet transform is not the aim of our article but some examples will illustrate how our Ridgelet tranform can be extended to the multiscale approach.

Discrete Analytical Ridgelet Transform : DART

The idea behind our associated discrete Radon transform is to represent each direction by a discrete analytical straight line. For this we need a discrete straight line that has a central symmetry and that forms a good approximation of the corresponding Euclidean straight line (i.e. direction). Without central symmetry (i.e. if (x; y) and ( x; y) both belongs to the line), the inverse Fourier transform leeds to imaginery values. This excludes the classical Bresenham discrete 2-D line 4]. Instead we chose discrete analytical 2D lines. It de nes not a unique line but a family of lines with a thickness parameter, called arithmetical thickness. The arithmetical thickness provides a control over the transform redundancy factor and properties such as the connectivity of the straight line. This thickness allows us to choose the discrete straight line de nition that ts best a given application. The discrete line is analytically de ned meaning that the discrete line is de ned by equations that allow a precise study of the properties and immediate extensions to higher dimensions. An important body of theory, called discrete analytical geometry, is now available since J-P. Reveill s rst proposed such an approach 16].

De nition and properties of the closed discrete analytical lines

The discrete analytical line we use for our transform are de ned as follows 2]: L ! p;q] = (x 1 ; x 2 ) 2 Z 2 jqx 1 px 2 j ! 2 with p; q] 2 Z 2 the direction of the Radon projection (we have = arctan q p ) and !, a function of (p; q), the arithmetical thickness. J-P. Reveill s introduced the discrete analytical lines in 1991 16], de ned as 0 qx py + r < !. In this paper, since we need central symmetry, we chose a variant of the closed discrete analytical lines, de ned as 0 qx py !; studied in 2].

It is easy to see that the closed discrete analytical lines L ! p;q] have a central symmetry regardless of the value of !. Moreover, the discrete analytical line can easily be extended to higher dimensions as discrete analytical hyperplanes de ned by j P n i=1 q i x i j ! 2 1].

The arithmetical thickness ! is an important parameter that controls, among other things, the connectivity of the discrete lines: let's consider the closed discrete analytical line L ! p;q] and its Euclidean counterpart L p;q] : qx 1 px 2 = 0, then: For ! < max(jpj ; jqj), L ! p;q] is not connected; For ! = max (jpj ; jqj), L ! p;q] is 8-connected. This is called the closed naive line. It is directly related to the distance d 1 since L max(jpj;jqj)

p;q] = M 2 Z 2 d 1 M; L p;q] 1 2 with d 1 (A; B) = x A 1 x B 1 + x A 2 x B 2
For ! max (jpj ; jqj), L ! p;q] is 8-connected. For ! = p p 2 + q 2 , L ! p;q] is 8-connected. This is called the Pythagorean line. This type of line is directly related to the distance d 2 since L p p 2 +q 2 p

;q] = M 2 Z 2 d 2 M; L p;q] 1 2 with d 2 (A; B) = q (x A 1 x B 1 ) 2 + (x A 2 x B 2 ) 2
For ! = jpj + jqj, L ! p;q] is 4-connected. This is called the supercover line and has a theoretical importance 2]. This type of line is directly related to the distance d 1 since L jpj+jqj p;q] = M 2 Z 2 d 1 M; L p;q] 1 2 with d 1 (A; B) = max x A 1 x B 1 ; x A 2 x B 2 . For ! jpj + jqj, L ! p;q] is 4-connected.

These results are direct consequence of a well known result in discrete analytical geometry and more recent studies on distances 16,2]. The fact that these lines can be de ned with help of distances makes a direct link with mathematical morphology 17].

In this paper we studied the DART with the closed naive lines, the pythagorean lines and the supercover lines.

De nition of the discrete analytical Radon transform

We use the Fourier domain for the computation of discrete Radon transform: Fourier coe cients of b s are extracted along the discrete analytical line L ! p;q]

P ! p;q] s = k2Z + b s(f k 1 ; f k 2 ) such that qf k 1 pf k 2 ! 2
and we take the 1-D inverse FFT of P ! p;q] s on each value of the direction p; q]. Formally, our discrete analytical Radon transform is de ned by: R ! s( p; q] ; b

) = K 1 X k=0 P ! p;q] s(k):e 2 j k K b with K length of L ! p;q]
We must de ne the set of discrete directions p; q] in order to provide a complete representation. The set of line segments must cover all the square lattice in the Fourier domain. For this, we de ne the directions p; q] according to pairs of symmetric points from the boundary of the 2-D discrete Fourier spectra. Figure 3a illustrates this choice of angles with the covering of the Fourier domain by the associated Euclidean lines. Notice that this set of angles is not equispaced.

Proposition 1 Let a square lattice be de ned as 2 N = N; N] N; N].

Let us consider the set of directions (p m ; q m ) with 0 m 2N, (p m ; q m ) = (N; m N) and for 2N + 1 m 4N 1, (p m ; q m ) = (3N m; N). The set of all discrete analytical lines de ned by jq m f 1 p m f 2 j !m 2 with ! m sup (jp m j ; jq m j) provides a complete cover of the lattice 2 N .

The proof of this proposition is obvious since we are dealing here with discrete lines that are at least 8-connected as stated in the previous section. As soon as we take ! = sup (jp m j ; jq m j) 1 the lattice is not covered anymore (because of L sup(jpmj;jqmj) 1=0 0;1] and L 0 1;0] ). More complex formulas for ! can probably be propose that provide a complete cover and a lower redundancy. This is still an open and somewhat di cult question. In our applications we chose to work only with connected discrete lines.

Figure 3 illustrates the cover of the Fourier lattice (on the rst octant) for two di erent types of discrete lines. The grey value of the pixel represents the redundancy in the projection (number of times a pixel belongs to a discrete line). One isolated line is drawn to illustrate shape of the discrete lines depending on its arithmetical thickness.

In gure 3b, we show the redundancy for the closed naive lines. They provide a relatively small DART redundancy factor of 2:05. These lines are well In gure 4, we show the result of the discrete Radon decomposition of an array containing a single nonzero entry. We observe that :

In this case, the Radon transform follows a broken line as for the Mathwork transform.

The nite Radon transform of Do and Vetterli has an important wraparound e ect and is di cult to interpret.

Our transform obtains a broken line without wrap-around e ect.

Figure 4d illustrates the in uence of the increasing of the redundancy of the decomposition (in this case, supercover lines are used) : the Radon is an interpolated broken line. We want now to nd for a given coe cient ( p; q]; b) of the Radon transform, which pixels have contributed 3]. For this, we apply the Radon backprojection.

Our analytical reconstruction procedure works as follows:

(1) Compute the 1-D FFT transform for each set R ! s( p; q]; :)

(2) Substitue the sampled value of b s on the lattice where the points fall on lines L ! p;q] with the sampled value of b s on the square lattice. (3) Apply the 2-D IFFT transform.

The previous procedure allows us to obtain an exact reconstruction if the set of directions of lines provide a complete cover of the square lattice. To make a comparison, we use the IRadon procedure of Mathworks that uses the ltered backprojection algorithm (based on an interpolation in the Fourier domain) to perform the inverse Radon transform. Notice that the results obtained with this backprojection are visually similar to Stanford's results. We use also the code of Do and Vetterli. All these backprojections are presented in gure 5.

We can see that our strategy does not exhibit geometrical distortions and a very limited wrap-around artifacts (that could be removed by zero-padding in the Fourier domain). The results are similar for the di erent types discrete analytical lines. . They indicate that experience has shown that compactly-supported wavelets can lead to many visual artifacts when used with nonlinear processing. This is due to the lack of localization of such compactly-supported wavelets in the frequency domain. The rst Stanford implementations have made the choice of bandlimited wavelets, whose support is compact in the Fourier domain. For this, they use the periodic discrete Meyer wavelet that consists of a system of division in the frequential domain. The discrete Meyer wavelet transform is studied in Kolaczyk's thesis 14]. After that, Starck et al. chose a speci c overcomplete system: they de ne the scaling function in the frequency domain as a renormalized B 3 spline and the wavelet function as the di erence between two consecutive resolutions. With this choice, each subband is sampled above the Nyquist rate, hence avoiding aliasing (this phenomenon is present when a nonlinear processing is applied on orthogonal wavelet transforms). Do and Vetterli use classical decimated Symlet Wavelet for the denoising problematic.

In this article, we do not propose a de nitive solution associated with the di erent discrete analytical lines. The choice of the type of 1-D wavelet transform depends of the goal of the transform. This wavelet transform can be decimated or undecimated and the wavelet base can be adapted according to the application, as for the classical wavelet decomposition. In the following sec-tion, we compare the use of Meyer wavelets (whose support is compact in the Fourier domain) and the use of decimated/undecimated compactly-supported wavelets for two applications : denoising and partial reconstruction. [START_REF] Averbuch | Fast slant stack: a notion of radon transform for data in a cartesian grid which is rapidly computible, algebraically exact, geometrically faithful and invertible[END_REF] Illustration of the DART 4.1 naive lines vs pythagorean lines vs supercover lines

A. Denoising

The denoising procedure by Ridgelet transform consists simply in thresholding the Ridgelet coe cients and computing the inverse Ridgelet transform. The thresholding is performed with help of an undecimated method developed for the wavelet decomposition 8,7]. The redundancy of the wavelet decomposition, associated with this method, reduces artifacts that appear after thresholding. Let r ! s be the noisy undecimated Ridgelet coe cients, we use the following hard-thresholding:

r ! s ( p; q] ; a; b) = 8 > < > :
r ! s ( p; q]; a; b) if r ! s ( p; q] ; a; b) 0 otherwise can be de ned as = q 2 log(N) 10]. We can use two strategies for the estimation of the variance of the noisy Ridgelet coe cients :

We consider that the DART is not norm-preserving and the variance depends thus on the projection index. In this case, the individual variance is estimated using the absolute median of the wavelet decomposition's rst scale of each radial projection (if the wavelet decomposition is norm-preserving). By evaluating the DART (de ned with Daubechies D20 wavelets) of a few standard white noise images, we have observed that the variance of noisy coe cients remains constant. The variance can then be estimated before the Ridgelet transform and used for all the Ridgelet transform.

Our experiments have shown that the rst strategy is better for denoising with the DART (the SNR measure of reconstructed image is higher). We use thus this rst method.

In order to illustrate precisely the result of the denoising algorithm with different types of discrete analytical lines we have generated an arti cial image (Figure 6a) and added important white noise (Figure 6b). The SNR1 of the noisy image is equal to 15 dB. Figures 6c, d ande are the results obtained with the three de nitions of discrete analytical lines (naive, pythagorean and supercover) and with the Daubechies D20 wavelets. For a more redundant decomposition (based on supercover discrete lines, gure 6e) the denoising result is better than for a less redundant decomposition ( gure 6c): the edge is reconstructed more precisely and the uniform areas are smoother. As for the wavelet decomposition, overcompletness provides advantages for denoising.

For comparison the result of a denoising carried out by classical decimated wavelet coe cient threshold ( gure 6f) is also given. The result is obtained with a decimated decomposition by Daubechies D20 wavelets and a hard thresholding. The threshold is based on the noise variance, as Donoho et al. introduced in 10]. Noise variance is estimated using the absolute median of the rst decomposition scale. The wavelet method obtains the best SNR measure, but it exhibits numerous blemishes, that are a result of the nonlinear processing with critical sampling. This experiment illustrates the limits of the SNR measure since it does not quantify well this type of artifacts. Contrary to the denoising problematic, for a partial reconstruction, redundancy is of course not interesting. Figure 7 compares partial reconstruction of an arti cial image by using the 30% largest naive-DART coe cients with reconstruction by using the 30% largest supercover-DART coe cients. This illustrates how the arithmetical thickness of the discrete lines employed in our Ridgelet transform in uences the quality of the compressed image. The lower redundancy representation (naive discrete lines) preserves all the features of the original image after a simple thresholding (Figure 7 b). On the other hand, with the higher redundancy representation (supercover lines) we loose features and the image is globally of lower quality. We discuss now the choice of 1-D wavelet transform with the DART. For this, several 1-D wavelet transforms were tested along the radial variable in discrete analytical Radon space (for these experiments, we use only the naive discrete line de nition, the results are equivalent with the others types of lines):

(1) The discrete Meyer wavelet (whose support is compact in the Fourier domain) transform.

(2) The Daubechies D20 wavelet (whose support is compact in the time domain) transform.

A. Denoising

Figure 8 considers an object used by Do and Vetterli in their paper 9], and compares the denoising of this image by thresholding of the DART with different wavelet bases. We use the SNR and the visual analysis to measure the performance. We observe that :

The undecimated methods o er superior performances over orthogonal wavelet decompositions.

With decimated wavelet decomposition, the compactly-supported wavelets in the frequency domain obtain better SNR measures than compactlysupported wavelets in the time domain. The Starck-DART reconstruction and undecimated Daubechies D20 DART reconstruction are very similar.

For denoising with the DART, we can not conclude that the undecimated compactly-supported wavelets in the frequency domain enjoys superior performance over the undecimated compactly-supported wavelets in the time domain. However, in this study, we use time lters (D20 lters) that have very large supports. In this case the lter pro le in Fourier domain converges rapidly to 0 after the cut-o frequency. If the same experiments are done with too small lters, the reconstructed image will contain some disturbing artifacts.

More generally, we have generated a set of noisy images from both Do's Object image and Lenna image. We have then compared the four di erent ltering procedures. These experimental results have con rmed that the Starck-DART reconstruction and undecimated Daubechies D20 DART reconstruction are similar and seem to be the better choice. We now consider the partial reconstruction of Do's object by the 30% largest DART coe cients ( gure 9). Only the decimated wavelet transforms are tested since the redundancy is irrelevant for partial reconstruction. We observe that, in the frequency domain, the more compact the wavelet support is, the better the SNR measure is. These results con rm Stanford's experience. However, comparing the di erent reconstructed images, one can hardly see the di erence. We have generated a set of partial reconstructions from both Do's object and Lenna. These experiments con rm previous visual analyzes: partial reconstructions from D20-DART and Meyer-DART are equivalent.

From these experiments (denoising and partial reconstruction), we propose to use the D20 wavelets for the DART because the D20 basis can be simply associated to an orthogonal transform or an undecimated transform (with the trous algorithm) and moreover this base is norm-preserving. The Ridgelet based on geometrical Radon transform (Mathworks code) and the Starck overcomplete wavelet system. In this case, the noise variance is estimated by evaluating the geometrical Ridgelet transform of standard white noise images (as proposed in 18]). The Do and Vetterli orthogonal Ridgelet.

The DART based on supercover lines and undecimated D20 wavelets.

We illustrate also the results obtain with decimated and undecimated wavelet transform (implemented by Stanford in the toolbox Wavelab).

We observe that :

The geometrical Ridgelet and the DART reconstruction does not contain the many blemishes that one sees in the wavelet reconstructions. The SNR measures of these two transforms are better.

The decimated wavelet transform displays disturbing artifacts. The use of a redundant wavelet transform considerably reduces the number of artifacts. The Do's Ridgelet transform corresponds to a critical sampling. The reconstruction associated with a nonlinear processing contains thus many visual artifacts.

The DART enjoys superior performance over Geometrical Ridgelet transform2 . Notice that the subbands used in the discrete Curvelet transform of continuum functions have the non-standard form h 2 2l ; 2 2l+2 i : Starck et al. propose to use a speci c overcomplete subband ltering: the wavelet function is de ned as the spatial di erence between two consecutive resolutions. We extend our digital Ridglelet transform with the same principle, but we substitute the Starck's overcomplete subband ltering with undecimated quincunux wavelet decomposition in order to preserve the interscale orthogonality. Moreover this wavelet transform preserves the variance of the noise. The comparison between Ridgelet denoising results and Curvelet denoising results is however not the subject of this paper. These experiments are done precisely by Starck et al. (http://www-stat.stanford.edu/~jstarck). We illustrate only how the DART can be extended to the Curvelet formalism.

To illustrate the local DART and the new discrete Curvelet transform based on discrete analytical lines, we have developed an example of denoising algorithm. As for the Ridgelet transform, the denoising by Curvelet transform consists simply in thresholding the Curvelet coe cients and computing the inverse Curvelet transform. For the estimation of the variance of the noisy Curvelet-DART coe cients, we consider that the Curvelet-DART is norm preserving . The noise variance is then estimated before the Curvelet transform.

In our example, a Gaussian noise with a standard deviation equal to 60 was added to the classical Lenna image (Figure 11a). Several methods were used to lter the noisy image: thresholding of the undecimated wavelet transform (Figure 11b), thresholding of supercover DART (Figure 11c), thresholding of local supercover DART (Figure 11d), thresholding of Curvelet DART based on supercover discrete lines and undecimated quincunux scheme (Figure 11e). We observe that the Curvelet DART and Local DART enjoy superior performance over the other transforms. For these decompositions the analysis is better localized in the spatial domain. This property eliminates the parasitic lines present in the DART reconstruction and due to the selection of an important contour in an area of the image. As for the Ridgelet transform, we can use other de nitions of discrete analytical lines for Local DART and Curvelet DART.

To study the dependency of these methods on the noise level, we generated a set of noisy images from both Do's Object and Lenna. We then compared the di erent ltering procedures. This series of experiments is summarized in Figure 12 for Lenna. These experimental results show that the Local DART and Curvelet-DART correspond, in general, to the better results.

In order to make a comparison with Stanford's results, we use one of the example of denoising proposed in 18]: a Gaussian noise with a standard deviation equal to 20 is added to the Lenna image (size: 512 by 512). The reconstruction by local DART and Curvelet-DART are presented gure 13. The results of Stanford's algorithm are visible at the website http://www- Once again, we propose to extract the Fourier coe cients along a 3-D discrete analytical line going through the origin. Once a discrete 3-D line is de ned, the principle of the method is the same as in the 2-D case with the same properties (exact reconstruction, rapidity, exible de nition). This idea works well for the supercover model as the 3-D supercover discrete analytical line is connected (see example in gure 15). Thus the set of 3-D supercover lines covers the 3-D Fourier cubic lattice if we de ne the line directions according to pairs of symetric border points of the lattice. The de nition of a 3-D supercover line of direction p; q; r] is given by 2]: jqx 1 px 2 j jpj + jqj 2 jrx 1 px 3 j jpj + jrj 2 jrx 2 qx 3 j jqj + jrj 2

In the same way, all the supercover of all mats in dimension n are well de ned 2]. This allows us to generalize this approach to 3-D planes (see example in gure 15) and dimension n. For the other models (naive and pythagorean for example) however the de nition and the extension to 3-D lines or higher dimensions is more problematic. For instance, the 3-D naive discrete analytical line is not connected in general and does not ensure a complete cover of the 3-D cubic Fourier lattice. We hope to report on this developpements in a forthcoming paper. The DART algorithm is easy to implement. It provides an exact reconstruction property: the DART followed by a reverse DART is a one-to-one transform. Our experiments have shown that our approach presents a limited wraparound e ect, that does not in uence the denoising or partial reconstruction results. Moreover, by using the analytical formalisation, we de ne a exible Ridgelet transform: we can de ne di erent DART decompositions according to the arithmetical thickness of the analytical discrete lines. As for Stanfords transform, our representation is redundant but the transform redundancy factor depends on the type of the discrete lines used and can be adapted for each application.

Our DART uses a 1-D wavelet transform based on Daubechies D20 wavelet. Our experiments have shown that this basis associated with the analytical Radon transform obtains satisfactory results for the two studied problematics. Moreover, this basis can be simply associated to an orthogonal transform for partial reconstruction or to an undecimated transform for denoising, and it is norm-preserving. We have seen that if, instead, we use a basis that is compactly supported in the Fourier domain (orthogonal or redundant), it does not improve the results of the DART, but the decomposition is less exible.

We have illustrated the performances of the DART for denoising problematics, even with very noisy images (with gaussian noise). This study indicates that the Local-DART and the Curvelet-DART thresholding rivals Stanford's Ridgelet denoising results, that are considered as the reference for Ridgelet's denoising applications, and more generally, classical denoising techniques that have been developped over the last decade. Howerver, contrary to the Stanford strategy, the reverse DART does not require an iterative approximation algorithm and the redundancy factor of the representation is not xed. This work can be extended in several directions. One of the theoretical questions in discrete geometry is the problem of de ning an arithmetical thickness function that provides a smaller redundancy and a cover of the Fourier lattice. This is still an open and di cult arithmetical problem. One of the most important interests of the discrete analytical approach is the possibility to extend easily our work to 3-D. We are currently investigating the application of a 3-D DART to the denoising process of animated video images. In this paper we present some rst thoughts abouth these questions.

We are also considering extending our partial reconstruction algorithm with more sophisticated representations. The ortho-Ridgelet proposed by Stanford seems to obtain good results for partial reconstruction 12] and constitutes an

2. 1

 1 Continuous theory of Ridgelet transform A substantial foundation for Ridgelet analysis is documented in the Ph.D. thesis of Cand s 5]. The continuous Ridgelet transform of s 2 L 2 (R 2 ) is de ned by : r(a; b; ) = Z R 2 a;b; (x)s(x)dx with x = (x 1 ; x 2 ) 2 R 2 and a;b; (x) the Ridgelet 2-D function de ned from a wavelet 1-D function as: a;b; (x) = a 1=2 x 1 cos + x 2 sin b a ! b 2 R is the translation parameter, a 2 R is the dilatation parameter and 2 0; 2 is the direction parameter. The function a;b; is oriented at the angle and is constant along lines x 1 cos + x 2 sin = cst Transverse to these ridges it is a wavelet function. In comparison, the continuous 2-D wavelet function are tensor products of 1-D wavelets a;b de ned by: a;b (x) = a 1 ;b 1 (x 1 ) a 2 ;b 2 (x 2 ) with a 1 ;b 1 (x) transform, the translation parameters (b 1 ; b 2 ) of the 2-D wavelet transform are replaced by the line parameters (b; ). A basic tool for calculating Ridgelet coe cients is to view Ridgelet analysis as a wavelet analysis in the Radon domain: in 2-D, points and lines are related via the Radon transform, thus the wavelet and Ridgelet transforms are linked via the Radon transform.

  r s ( ; a; b) = Z R a;b (t)Rs( ; t)dt The Radon transform can be obtained by applying the 1-D inverse Fourier transform to the 2-D Fourier transform restricted to radial lines going through the origin (this is exactly what we are going to do in the discrete Fourier domain with help of discrete analytical lines): b s(! cos ; ! sin ) = Z R e j!t Rs( ; t)dt with b s the 2-D Fourier transform of s.

Fig. 1 .

 1 Fig. 1. Relation between transforms

Fig. 2 .

 2 Fig. 2. (a) The pseudopolar grid (b) The conversion from cartesian grid to pseudopolar grid (from 3])

Fig. 3 .

 3 Fig. 3. (a) Cover of the Fourier domain with the Euclidean line L p;q] . (b) Redundancy on the cover of the Fourier lattice by closed naive lines, (c) by supercover lines suited for applications such as partial reconstruction where a relatively small redundancy is an asset. in Figure 3c, we show the redundancy for the supercover lines. They provide a more important DART redundancy factor of 3:05. These lines are well suited for applications such as denoising. The pythagorean lines provide an intermediate DART redundancy factor of 2:35. We now give two examples of the discrete analytical Radon transform in action. We propose a comparison with the di erents solutions proposed in the literature. The papers of Stanford 18] reference a Matlab Toolbox developed by Stanford researchers for performing Ridgelet and Curvelet analysis. Although this code is actually not open to the public. In place, we use the Radon transform developped by Mathworks that is based on a geometrically faithful notion of Radon and that does not presents wrap-around e ect. The result of the Mathworks transform is visually similar than the Stanford transform. The code of Do and Vetterli 9] is open to the public (http://lcavwww.ep .ch/~minhdo).In gure 4, we show the result of the discrete Radon decomposition of an array containing a single nonzero entry. We observe that :

Fig. 4 .

 4 Fig. 4. The Radon transform of a point : (a) with a geometrical strategy (b) with the Do and Vetterli approach. (c) with our strategy using the closed na ve lines, (d) the supercover lines

Fig. 5 .

 5 Fig. 5. The Radon backtransform of a point : (a) with a geometrical strategy (b) with the Do and Vetterli approach. (c) with our strategy using the closed naive lines, (d) and the supercover lines

Fig. 6 .

 6 Fig. 6. (a) original image (b) noisy image (to show more precisely the e ect of the noise we plot a vertical line of each image). (c) denoising with naive lines (d) denoising with pythagoricean lines (e) denoising with supercover lines (f) denoising with decimated wavelet transform B. Partial reconstruction

Fig. 7 .

 7 Fig. 7. Partial reconstruction of an arti cial image (a) naive-DART (b) supercover-DART

( 3 )

 3 The undecimated Daubechies D20 wavelet transform. (4) The overcomplete system based on the work of Starck et al. 18] (whose support is compact in the Fourier domain).

Fig. 8 .

 8 Fig. 8. (a) Noisy image (b) denoising with the Meyer-DART transform (c) denoising with D20-DART transform (d) denoising with undecimated D20-DART transform (e) denoising with Starck-DART transform B. Partial reconstruction

Fig. 9 .

 9 Fig. 9. Partial reconstruction of Do's image (a) Meyer-DART (b) D20-DART (c) D8-DART 4.3 DART vs litterature Ridgelet transform

Fig. 10 .

 10 Fig. 10. (a) noisy image ; denoising with (b) the decimated wavelet transform (c) the undecimated wavelet transform (d) the undecimated geometrical Ridgelet (e) the Do and Vetterli approach (f) The supercover-DART B. Extension to Local DART and Curvelet DART The DART can be easily extended to a local transform by a smooth partitioning, or more generally to a Curvelet transform by the following steps: Decompose the image into a set of wavelet bands. Each subband is smoothly windowed into squares. Analyse each square by the DART.

Fig. 11 .

 11 Fig. 11. (a) noisy image (b) denoising with the undecimated wavelet transform (c) denoising with DART based on supercover lines (d) denoising with local DART based on supercover lines (e) denoising with DART-Curvelet based on undecimated quinconx wavelet transform and supercover lines 64 (1) 32 (1)16 (1) Curvelet DART 31.42 31.38 30.61 31.23 Stanford 30.79 30.97 30.87 31.95 Table 1 PSNR measures of DART and Stanford reconstructions. (1) Size of the window

Fig. 12 .

 12 Fig. 12. SNR of reconstructed image versus SNR of noisy image for di erent representation. The panel corresponds to Lenna.

Fig. 13 .

 13 Fig. 13. (a) noisy image with the same level of noise than 18] (b) denoising with local DART based on supercover lines (c) denoising with DART-Curvelet based on undecimated quincunux wavelet transform and supercover lines. These measures con rm the visualy analysis: the Stanford reconstruction and DART reconstruction are very similar. However, as we have said, the backprojection DART algorithm is simpler and the DART representation is more exible (with the variable line thickness). Notice that Color denoising results obtained with the DART are presented in 6].C. Partial reconstructionWe now consider the partial reconstruction of Do's object by the 512 most

Fig. 14 .

 14 Fig. 14. Partial reconstruction of Do's image (512 coe cients) (a) Wavelet (b) Do's Ridgelet (c) Naive DART. Bad result because of the redundancy (d) Na ve ortho-DART

Fig. 15 .

 15 Fig. 15. Discrete analytical 3-D line and plane

  16 (1) Curvelet DART 31.42 31.38 30.61 31.23 Stanford 30.79 30.97 30.87 31.95 Table 1 PSNR measures of DART and Stanford reconstructions. (1) Size of the window stat.stanford.edu/~jstarck. From the gure 13 and the Stanford Web site, one can not see important di erences between the Stanford strategy and the DART strategy. In order to make an quantitative measure we used the PSNR. We indicate in the table 1, The PSNR values 3 after ltering with the local DART and Curvelet-DART. The PSNR values after ltering with the Stanford algorithm indicated in 18].

Preprint submitted to Elsevier Preprint

All the SNR measures are done with the Stanford Matlab function.

We can obtain better results with the Mathworks Radon transform by using an other level of threshold. However, because the Radon and the Wavelet transform associated with this strategy is not norm-preserving, it is di cult to de ne a general level of threshold.

The PSNR measures of DART are done with the Stanford Matlab function.

interesting new area of research.


Emmanuel Letavernier
email: emmanuel.letavernier@aphp.fr

Tilman B Drüeke
email: tilman.drueke@inserm.fr

Kidney toxicity of phosphate: is that crystal clear yet?

Keywords: calcium phosphate, microcrystals, kidney tubule, chronic kidney disease, FGF23, Toll-like receptor-4

Hyperphosphatemia is a well-known feature of advanced chronic kidney disease (CKD) and a major risk factor for cardiovascular events in patients with CKD, inducing calcium phosphate supersaturation and vascular calcifications. [START_REF] Adeney | Association of Serum Phosphate with Vascular and Valvular Calcification in Moderate CKD[END_REF] Although the serum phosphate level is not recognized as an independent risk factor for CKD progression, increased urinary phosphate excretion has been associated with CKD progression in small cohort studies. [START_REF] Kawasaki | Urinary phosphorus excretion per creatinine clearance as a prognostic marker for progression of chronic kidney disease: a retrospective cohort study[END_REF][START_REF] Santamaría | Increased Phosphaturia Accelerates The Decline in Renal Function: A Search for Mechanisms[END_REF] In experimental studies, exposure to a high phosphate diet has been shown to accelerate kidney damage in rats with reduced nephron number induced by partial nephrectomy. [START_REF] Haut | Renal toxicity of phosphate in rats[END_REF] Complementary experiments confirmed that cell toxicity was mediated by the particles and not by phosphate concentration itself. Next, they injected a fluorescent probe that binds to calcium phosphate crystallites in mice fed a high phosphate diet. They observed that the calcium phosphate particles adhered to the apical membrane of proximal tubular cells in vivo, and that this was associated with kidney inflammation and fibrosis. Acidification of urine decreased microparticle formation and kidney lesions, whereas urine alkalinization exacerbated kidney damage by increasing calcium phosphate precipitation. Shiizaki et al.

further observed that Toll-like receptor-4 (TLR-4) expressed in tubular cells is involved in calcium phosphate microparticle tethering. In vitro exposure of HK-2 cells to calcium phosphate particles induced endocytosis of these crystallites, disturbed endosomal trafficking, and activated inflammatory pathways.

The authors then exposed mice with or without unilateral nephrectomy to various phosphateenriched diets to identify the threshold of tubular fluid phosphate concentration above which tubular damage would occur. Because the direct measurement of phosphate concentration in proximal tubular fluid is technically challenging, they used an estimation that was shown to correlate with direct measurement by micropuncture in a small number of animals. They defined a threshold of estimated tubular fluid phosphate concentration above which both circulating FGF23 and markers of tubular damage increased in parallel. In further in vivo experiments, the authors observed a decreasing number of nephrons (indirect estimation) with increasing phosphate excretion per nephron, suggesting that high phosphate load may cause nephron loss. Finally, the authors observed an independent relationship between FGF23 levels and CKD progression in a patient cohort, a finding compatible with the hypothesis that high FGF23-induced phosphate excretion promotes kidney damage.

Why is this study important?

This comprehensive set of experiments shows that an increase in the phosphate concentration of proximal tubular fluid accelerates nephron loss in various CKD models. Moreover, Shiizaki et al. provide original results regarding the role of calcium phosphate microcrystals that accumulate in proximal tubular cells, disturb endosomal trafficking, and eventually induce cell damage, inflammation, tubulo-interstitial fibrosis, and progression of CKD. Until recently, little attention has been paid to a possible effect of phosphate itself on tubular cells.

Renal tubular calcium phosphate crystallization has been described in acute settings when phosphate delivery to the renal tubule is dramatically increased, for instance after phosphatecontaining bowel prep for colonoscopy. [START_REF] Ori | Acute phosphate nephropathy-an emerging threat[END_REF] Calcium phosphate crystals have been previously

shown to be involved in tubular lesions, and calcium phosphate tubular plugs may be associated with an unfavorable renal prognosis in kidney transplant patients. [START_REF] Aihara | Calcium phosphate-induced renal epithelial injury and stone formation: involvement of reactive oxygen species[END_REF][START_REF] Lionet | First investigation on microcrystalline pathologies of kidney allografts through cellular scale physicochemical techniques[END_REF] In patients with CKD, prior studies have shown that urine phosphate excretion is associated with disease progression, but the potential contribution of FGF23 was not assessed. [START_REF] Kawasaki | Urinary phosphorus excretion per creatinine clearance as a prognostic marker for progression of chronic kidney disease: a retrospective cohort study[END_REF][START_REF] Santamaría | Increased Phosphaturia Accelerates The Decline in Renal Function: A Search for Mechanisms[END_REF] The experimental and clinical findings by Shiizaki et al. should prompt further clinical studies. Whether calcium and phosphate concentrations as high as those used in the in vitro experiments are reached in proximal tubule fluid in vivo, and whether intratubular calcium phosphate particles disrupt proximal tubular cell endosomal functions in patients with CKD remains to be seen. The demonstration that serum FGF23 levels are associated with progression of CKD is not definitive proof that the progression is causally linked to increased urine phosphate concentration. Serum FGF23 is tightly associated with glomerular filtration rate (GFR), which is the most potent risk factor for further decline in kidney function. FGF23 may also exert direct harmful effects on various tissues. [START_REF] Musgrove | Regulation and Effects of FGF23 in Chronic Kidney Disease[END_REF] Calcium phosphate supersaturation and crystallization depend not only on urine phosphate concentration, but also on urinary calcium concentration, pH, and various calcification inhibitors. Increases in serum parathyroid hormone levels can modify these parameters, an issue not addressed in the present paper. The dramatic decrease in urine calcium excretion observed in advanced stages of CKD could in theory counteract, at least partly, the risk of

  Prior studies have not considered the role of fibroblast growth factor 23 (FGF23), which inhibits phosphate reabsorption by renal proximal tubules. FGF23 secretion is increased in CKD, which in turn increases phosphate concentration in the proximal tubular fluid. In a recent report in the Journal of Clinical Investigation, Shiizaki et al. hypothesized that increased phosphate concentration in the tubular fluid might be harmful for the kidney, and that patients with CKD might be particularly at risk due to the combination of reduced nephron number and increased FGF23 secretion. 5 What did the study show? Shiizaki et al. began their study with in vitro experiments, exposing HK-2 renal proximal tubular cells to increasing concentrations of extracellular phosphate, from 1.0 mM (control) to 7.0 mM, in the presence of 3.0 mM calcium. Increased phosphate concentrations induced a decrease in cell viability and the formation of calcium phosphate microparticles.

  calcium phosphate microparticle formation in tubular fluid. The findings by Shiizaki et al. advocate for further studies, including detailed assessment of phosphate concentration and calcium phosphate supersaturation in the tubular fluid, together with FGF23 and parathyroid hormone assays, in longitudinal studies dedicated to CKD progression. Modifications in urine pH and electrolyte concentrations in distal parts of the tubule may also play an important role. In the experimental models developed by Shiizaki et al., urine alkalinization worsened kidney function. This observation raises questions regarding the current clinical practice of bicarbonate administration to manage metabolic acidosis, particularly in patients with high phosphate load. Secondary analyses of prior trials of bicarbonate administration could potentially explore the association between urinary phosphate excretion and eGFR decline. Future studies should consider the relevance of these findings to other conditions with high urinary phosphate and bicarbonate excretion, such as proximal tubulopathy induced by light chain deposition or medication toxicity. Finally, it might also be interesting to use imaging techniques in order to directly visualize the induction of microcrystal formation in the proximal renal tubule by phosphate loading in vivo.

Figure legend Figure 1 .

 legend1 Figure legend

Disclosure

The authors declared no competing interests.


F Tortereau
email: flavie.tortereau@inrae.fr

C Marie-Etancelin

P Frutos

J Conington

G Arsenos

I De Barbieri

J H Jakobsen

C Moreno-Romieux

J J Arranz

SMARTER -Which novel traits to improve feed efficiency?

Keywords: small ruminants, sheep, goat, resource use efficiency, novel phenotypes

The SMARTER (SMAll RuminanT breeding for Efficiency and Resilience) H2020 project aims to develop and implement innovative strategies to improve resilience and efficiency (R&E) related traits in sheep and goats. Regarding feed efficiency, the objective is to identify novel traits that would be relevant, easy to measure and cheap enough to be collected in many animals to identify the most efficient individuals. In practice, feed efficiency can be assessed by different criteria such as residual feed intake and feed conversion ratio. Both criteria require feed intake to be measured for each individual which remains an expensive trait to get and hardly feasible under farm conditions. Thus, our objective is to identify novel traits related to feed efficiency and use them as proxies for feed intake and/or feed efficiency. First, novel phenotypes are identified and studied in experimental farms where feed intakes of concentrate and forage are recorded for each animal. Different novel phenotypes are being considered to study feed efficiency in sheep and goats, including (i) biomarkers (from blood or milk metabolomics), (ii) differentially expressed genes in targeted tissues, (iii) genomic polymorphisms, (iv) ruminal microbiota, (v) faecal NIR spectra, (vi) greenhouse gas emissions and body composition traits. These novel traits will be recorded either under classical feeding or under nutritional restriction to quantify, for example, the impact of a shortage of concentrate inputs. Then, the most promising novel traits will be measured in commercial populations. From these larger datasets, we will estimate heritabilities of the novel traits and genetic correlations to other traits in the breeding goal. Finally, for some case studies, we will quantify GxE interactions, particularly by considering the same breed under different breeding systems or regions.

Introduction

Feed efficiency is the ability of livestock to transform feed into food edible by humans. Breeding feed-efficient livestock is of high interest both at environmental and economic levels. Among livestock, ruminants are the only ones able to transform fibers into proteins. There is growing interest in breeding ruminants mainly fed with increasing amounts of forage and decreasing quantities of concentrates. Breeding programs are willing to include feed efficiency in their breeding objectives. Residual feed intake (RFI) is currently one of the most common criteria used to improve feed efficiency. RFI has been shown to be variable and moderately heritable in meat sheep [START_REF] Cammack | Estimates of genetic parameters for feed intake, feeding behavior, and daily gain in composite ram lambs[END_REF][START_REF] Johnson | Preliminary estimates of genetic parameters for residual feed intake in New Zealand maternal sheep[END_REF]Paganoni et al., 2017;Snowder and Van Vleck, 2003;Tortereau et al., 2020) and dairy goats [START_REF] Desire | Genetic parameter estimations for feed intake in UK dairy goats[END_REF]. Selecting on RFI is thus possible, but it requires feed intake to be recorded, which is the major limit to the deployment of this recording on a broader scale, particularly in small ruminants. Therefore, there is an increasing interest in identifying proxies for feed intake and/or feed efficiency directly. In order to be widely recorded, proxies be easy to collect and low in cost. In the SMARTER project, we do not only focus on such new phenotypes, but also on others that are more difficult to collect and/or more expensive to acquire. The overall objective is to understand the biological pathways underlying feed efficiency in small ruminants, which will help in identifying proxies. The objective of this work is to gather data from different experimental and commercial farms of small ruminants to dissect feed efficiency and to propose proxies that could be widely collected to estimate their genetic parameters and their ability to predict feed intake or feed efficiency.

Material and methods

Animals

Two categories of animals are considered: animals from experimental farms and animal from commercial farms. In dairy sheep, experimental protocols rely on 4 different breeds : Chios (n=8), Lacaune (n=62), Frizarta (n=8) and Assaf (n=47), and in commercial farms, ewes from 5 breeds are being phenotyped : Chios (n=250, in 2 farms), Frizarta (n=500, in 2 farms), Lacaune (n=3,972 in 8 farms), Pyrenean breeds (n=876, including ewes from Manech Tête Rousse and Basco-Béarnaises breeds, in 7 farms). An experiment rely on a nutritional challenge in 40 Assaf ewe replacement lambs: half of the animals are subjected, during their prepuberal stage, to a diet contained 42% less crude protein than controls. In both control and restricted group, animals have been chosen from each of the extremes of the distribution of genetic values (paternal average) for milk production. An experiment in Lacaune ewes (n=55) relies on divergent lines selected on milk persistency.

In dairy goats, two experiments are conducted, the first in the Alpine breed (n=110) and the second in a mixed breed (n=3,421) based on Alpine, Saanen and Toggenburg breeds. A total of 11 commercial farms are involved in the collection of fine-phenotypes of goats: 6 farms with Saanen (n=1,678 goats) and 5 farms with Alpine (n=1,176 goats). The experiment in Alpine breed involves divergent lines on longevity. Goats are daughters of bucks selected according to their extremely high/low EBV on that trait.

In meat sheep, experiments are based on 7 different breeds: Romane (n=277), Merino (n=1002), Corriedale (n= 303), Dohne (n= 360), Texel and Texelx Lleyn(n= 2,340) and Scottish BlackFace (n~2,000). In commercial farms, data are being collected in 5 breeds: Mouton Vendéen (n=1,500 ewes in 5 farms), Rouge de l'Ouest (n= 800 ewes in 5 farms), Blanche du Massif Central (n=2,000 ewes in 5 farms), Norwegian White Sheep (n=1,600 ewes in 15 farms) and Texel (n~6,000 animals in 154 flocks). The experiment in the Romane breed relies on two divergent lines on Residual Feed Intake (RFI): male lambs that are phenotyped are selected for their extremely high/low EBV on this trait.

Phenotypes

Different sets of phenotypes and biological samples are being collected, depending on the production type (meat vs. milk) and the farms (experimental vs. commercial). All the phenotypes directly collected in the animals or obtained after analysis of biological samples are described in table 1.

Other phenotypes are calculated from these elementary traits: Average daily gain (ADG), Feed Conversion Ratio (FCR), Residual Feed Intake (RFI). Differences in BCS between two successive physiological stages will also be computed. Milk fine compositions from MIRS were also estimated from previous equations. Microbiota abundances were get from 16S sequencing and metabolome abundances from NMR. In commercial populations, the higher numbers of phenotyped individuals will allow genetic parameters to be estimated in each population. Pedigrees are recorded in the frame of national genetic evaluation programs. These data are currently under collection, so no results are available yet.

Results and Discussion

In dairy ewes from the Chios, Assaf, Lacaune and Frizarta breeds (total=30 ewes), NEFA were used to analyse energy balance: an increase in NEFA being synonymous with a negative energy balance. Association analyses highlighted that a NEFA status of more than 0.3 mmol / L could be predicted by a decrease in Back Fat Thickness (BFT) and in the sum of BFT and Longissimus Dorsi muscle Thickness (LDT) of more than 0.075 mm and 0.350 mm, respectively. Likewise, a NEFA status of more than 0.7 mmol / L could be predicted by a decrease in BFT, in LDT and in their total sum of more than 0.15 mm, 0.065 mm and 0.350 mm, respectively. In the same 30 ewes, a significant positive association of serum albumins with LDT was found. Specifically, a change of serum albumins by 1 g / dL was associated with a change in LDT by 1.27 mm. Such an association could be explained by the fact that serum albumins and LDT are both deposits of amino acids in the tissues.

In meat sheep, Romane from the 3 rd generation of divergent selection on RFI exhibited significant differences (p-value<0.05) in feed intake, with more efficient lambs eating 195g less of concentrate than less efficient lambs, and RFI, with a difference of 1.9 genetic standard deviations between both groups. Live weight differences were observed, with inefficient individuals being heavier (+1.75 kg at 5 months old). Ruminal microbiota analyses highlighted differences in composition, and plasmatic amino acids were also different between both groups: more efficient lambs have lower plasmatic levels than less efficient lambs.

Conclusion

The collection of phenotypes is still on-going. However, first analyses of experimental datasets highlight that some phenotypes are linked to feed intake, feed efficiency or energy status and can be proposed as promising candidates for proxies of those traits.

List of References

Paganoni , B., Rose, G., Macleay, C., Jones, C., Brown, D.J., Kearney, G., Ferguson, M., and Thompson, A.N. (2017). More feed efficient sheep produce less methane and carbon dioxide when eating highquality pellets. J. Anim. Sci. 95, 3839. Snowder, G.D., and Van Vleck, L.D. (2003). Estimates of genetic parameters and selection strategies to improve the economic efficiency of postweaning growth in lambs. J. Anim. Sci. 81, 2704-2713. Tortereau, F., Marie-Etancelin, C., Weisbecker, J.-L., Marcon, D., Bouvier, F., Moreno-Romieux, C., and François, D. (2020). Genetic parameters for feed efficiency in Romane rams and responses to singlegeneration selection. Animal 14, 681-687.

Table 1 :

 1 Phenotypes recorded on experimental and/or commercial dairy ewes, dairy goats and meat sheep.

	Group of traits	Trait	Production Population	Comment
	Intake	Total Feed intake	Meat	Experimental	during given control
		(forage or	Milk	mainly	periods, under different
		concentrate)			diets
	Intake	Concentrate intake	Milk	Commercial	Concentrate supplied in
				Experimental	milking parlour
	Body	Live weight	Meat	Commercial	collected at different ages
	composition		Milk	Experimental	per animal
	Body	Chest width	Meat	Commercial	at different production
	composition		Milk		stages
	Body	Chest depth	Meat	Commercial	at different production
	composition		Milk		stages
	Body	Shoulder height	Meat	Commercial	at different production
	composition				stages
	Body	Muscle depth	Meat	Commercial	Only in experimental farms
	composition	(ultrasound)	Milk	Experimental	for dairy animals and at
					different production stages
	Body	Back Fat Thickness	Meat	Commercial	Only in experimental farms
	composition	(ultrasound)	Milk	Experimental	for dairy animals and at
					different production stages
	Body	Body condition score Meat	Commercial	at different production
	composition		Milk	Experimental	stages
	Udder	Udder conformation Milk	Commercial
	composition			Experimental
	Production	Milk Production	Milk	Commercial
		(MY, FY, PY, FC,		Experimental
		PC)
	Production	Carcass Traits	Meat	Commercial
	Gas emissions	GHG emissions	Meat	Commercial
		(CH4, CO2)		Experimental
	milk samples	gene expression	Milk	Experimental
		(RNA-seq)
	milk samples	epigenetic marks	Milk	Experimental

Feed intakes are obtained with automatic feeders that record each visit of any animal. Feeding behavior traits were calculated from these datasets.

Not all the phenotypes are available for all individuals. However, live weights and body composition traits obtained through ultrasounds are recorded for almost all meat sheep, and milk production traits are recorded for all dairy sheep and goats. Blood samples are obtained for almost all individuals with phenotypes, at least for genotyping purposes.

Statistical analyses

First, each trait will be analysed independently within each experiment. Fixed effects and covariates will be selected according to each trial. For example, for divergent lines experiment, the line will be considered as a fixed effect. Each trait will be considered as a proxy for feed intake or feed efficiency. Combinations of elementary traits listed in table 1 will also be considered as proxies. Proxies will be identified in each protocol and will be tested in the other trials when possible. The quality of a proxy will be assessed through different methodologies, adapted to the different protocol dimension. Here, we analysed an experiment in 30 dairy ewes from 4 breeds (8 Chios, 7 Lacaune, 8 Frizarta and 7 Assaf), with sensitivity/sensibility approach. The receiver operating characteristic analysis was used to define thresholds for changes in body composition traits as predictors of negative energy balance. Moreover, mixed linear models were used to test the association of blood biomarkers with fat and muscle reserves and their mobilization.

Multivariate approaches will be applied for large datasets such as microbiota and metabolomics matrices. The MixOmics package implements such methods and will be used [START_REF] Cao | mixOmics: Omics Data Integration Project[END_REF].