Gwénolé Quellec

Mathieu Lamard

Lynda Bekri

Member, IEEE Guy Cazuguel

Fellow member, IEEE Christian Roux

Béatrice Cochener

Medical case retrieval from a committee of decision trees

Keywords: information retrieval, decision trees, CBIR, CAD, medical databases

A novel content-based information retrieval framework, designed to cover several medical applications, is presented in this paper. The presented framework allows the retrieval of possibly incomplete medical cases consisting of several images together with semantic information. It relies on a committee of decision trees, decision support tools well suited to process this type of information. In our proposed framework, images are characterized by their digital content. It was applied to two heterogeneous medical datasets for computer aided diagnosis: a diabetic retinopathy follow-up dataset (DRD) and a mammography screening dataset (DDSM). Measure of precision among the top five retrieved results of 0.788±0.137 and 0.869±0.161 was obtained on DRD and DDSM, respectively. On DRD for instance, it increases by half the retrieval of single images.

I. INTRODUCTION

M EDICAL experts base their diagnoses on a mixture of textbook knowledge and experience acquired through real-life clinical cases, hence the growing interest in Case-Based Reasoning (CBR) [START_REF] Aamodt | Case-based reasoning: Foundational issues, methodological variations, and system approaches[END_REF] for computer aided diagnosis systems [START_REF] Bichindaritz | Case-based reasoning in the health sciences: What's next?[END_REF]. CBR assumes that analogous problems have similar solutions: interpreting a new situation involves retrieving similar cases in a case database. Relevance is usually modeled via a similarity measure between a query (a new medical case analyzed by a medical expert) and each case in a reference database. The retrieved cases are then used to help interpreting the new case [START_REF] Aamodt | Case-based reasoning: Foundational issues, methodological variations, and system approaches[END_REF]. CBR was originally designed to process structured cases such as regular feature vectors. However, information required by physicians to diagnose some pathologies are more complex. To diagnose Diabetic Retinopathy (DR) for instance, physicians analyze series of images together with -usually structuredcontextual information, such as the patient age, sex and medical history [START_REF] Wilkinson | Proposed international clinical diabetic retinopathy and diabetic macular edema disease severity scales[END_REF], [START_REF] Davis | Risk factors for high-risk proliferative diabetic retinopathy and severe visual loss: Early treatment diabetic retinopathy study report 18[END_REF]. Consequently, medical CBR systems should be able to manage both symbolic information such as clinical annotations, and numerical information such as images. Some existing systems were designed to manage symbolic information [START_REF] Cauvin | Computer-assisted diagnosis system in digestive endoscopy[END_REF]. Some others, relying on Content-Based Image Retrieval (CBIR) [START_REF] Nastar | Indexation d'images par le contenu: un état de l'art[END_REF], [START_REF] Smeulders | Content-based image retrieval at the end of the early years[END_REF], were designed to manage digital images [START_REF] Müller | A review of content-based image retrieval systems in medical applications -clinical benefits and future directions[END_REF], [START_REF] Tourassi | Computer-assisted detection of mammographic masses: A template matching scheme based on mutual information[END_REF], [START_REF] Alto | Content-based retrieval and analysis of mammographic masses[END_REF]. However, there were only few attempts to merge these two Gwénolé approaches. One existing system linearly combines a text based and an image based similarity measure into a common similarity measure [START_REF] Shao | Medical image retrieval based on visual contents and text information[END_REF]; however, this approach does not apply to structured textual information. Another system lets the user restrict a CBIR search to images acquired from the same localization and/or with the same device [START_REF] Bozec | Towards content-based image retrieval in a HIS-integrated PACS[END_REF]. More generally, another system lets the user restrict a CBIR search to images whose contextual information match an SQL query specified by the user [START_REF] Antani | A biomedical information system for combined content-based retrieval of spine X-ray images and associated text information[END_REF]; however, he/she is assumed to know which queries are relevant: it is likely not the case if such a system is needed for diagnosis aid. As a consequence, we believe heterogeneous information retrievali.e. information retrieval based on both clinical descriptors and digital image features -is still an open issue. A novel CBR approach that fuses these two types of information is presented in this paper. In the proposed framework, heterogeneous attributes (digital images, nominal and continuous variables) have to be aggregated and the value of some of these attributes is possibly unknown. To solve this generalized CBR problem, the use of decision trees (DTs) is proposed [START_REF] Quinlan | C4.5: Programs for Machine Learning[END_REF], [START_REF] Breiman | Classification and regression trees[END_REF]. A novel indexing scheme based on DTs is introduced; for improved retrieval efficiency, several DTs can be used. To that purpose, a randomized decision tree learning algorithm is applied so that several DTs can be generated. Finally, a boosting strategy is proposed to handle unbalanced classes [START_REF] Freund | Experiments with a new boosting algorithm[END_REF]. The proposed framework has another advantage: the time required for a user (e.g. a medical expert) to query the reference database can be reduced. A procedure is proposed to update the retrieval list as new attributes are inputted by the user. As a consequence, the user can decide to stop inputting attributes if he/she is satisfied with the results. Moreover, each time he/she inputs an attribute, a second procedure identifies the remaining attributes likely to be the most discriminant; in other words, a fast path towards satisfactory results is suggested. The paper is organized as follows. Section II presents decision trees and their advantages for heterogeneous information retrieval. Section III explains how images can be included in a decision tree. The proposed decision tree based retrieval framework is presented in section IV. Section V describes the medical datasets used for evaluation. Results are given in section VI and we end with a discussion and conclusions in section VII.

II. DECISION TREES

A. Description

A decision trees (DT) [START_REF] Quinlan | C4.5: Programs for Machine Learning[END_REF], [START_REF] Breiman | Classification and regression trees[END_REF] is a decision support tool relying on a set of rules dividing a population of cases into homogenous groups. Each rule associates a conjunction of tests on some attributes with a group (for instance "if sex=male and age<40 then the case belongs to group 3"). In our case, attributes are either images or contextual information. These rules are organized as a tree; the structure of this tree can be interpreted as follows (see Fig. 1):

• each non-terminal node represents a test on a single attribute (e.g. what is the patient sex ?) • each edge represents a test outcome (e.g. male)

• each leaf represents a cluster of cases that provided a similar answer to each test (e.g. males younger than 40) DTs were first designed to segment a population of nominal attribute vectors (each test outcome corresponds to an attribute value or group of values). Quinlan [START_REF] Quinlan | Learning with continuous classes[END_REF] extended them to continuous attributes (training cases are grouped by attribute value ranges). More generally, DTs can process any attribute, so long as we can provide a way to cluster cases with respect to that attribute. Since each test is performed on a single attribute, DTs are well suited to process heterogeneous cases. DTs are generally used as classifiers: an unlabeled case is first associated to a group, and then it is assigned to the most frequent class in that group. In the presented application, we won't use DTs as classifiers; we will use them to define a similarity measure between two cases.

B. Learning

To build a DT in an automatic fashion, we search for the most discriminant attributes and divide the population into homogenous groups according to the value of those attributes (see Fig. 2). This process is supervised and then requires classified cases. In the medical datasets we considered, the disease severity level was used as a class label. Before learning the tree, the dataset has to be divided into three subsets:

• a learning set L, actually used to learn the DT (at the end of the learning process, each case in this set is assigned to the leaves of the tree), • a validation set V , used to assess the performance of the DT with different parameter settings, • a test set T , used to assess the final performance of the DT, using the optimal parameter setting Note that cases assigned to V and T are not used to learn the DT, and T is not used to tune the system at all.

At the beginning of the learning process, the tree consists of a single node containing the whole learning set L. Then each leaf l of the growing tree is recursively divided. In that purpose, the most discriminant attribute f among the population P ⊂ L assigned to leaf l is searched for. P is then divided into new child nodes, one for each possible answer to the test on f . In the Fig. 2. Illustration of the learning process. At each step, a group of medical cases is divided into subgroups, according to the value of the most discriminant attribute within that group.

proposed method, the discriminant power of a test is measured by the Shannon entropy gain G obtained when dividing a node v 0 into its child nodes v n , n=1..N (c4.5 algorithm [START_REF] Quinlan | C4.5: Programs for Machine Learning[END_REF], see equation ( 1)).

G = N n=1 I n -I 0 I n = - C c=1 p cn log p cn , n = 0..N (1)
where p cn is the percentage of cases assigned to class c in node v n , c = 1..C, I 0 is the entropy in node v 0 (before it is split) and I n is the entropy in the n th child node v n , n = 1..N . Entropy measures the homogeneity of each node with respect to class label. If no test can improve the entropy enough or if population P is too small, l is not divided. The learning algorithm can manage missing information: we describe herein the mechanism provided by c4.5 algorithm [START_REF] Quinlan | C4.5: Programs for Machine Learning[END_REF]. Suppose that the value of an attribute f , tested at a node v 0 , is missing for a case. Then this case is assigned to each child v n of v 0 with a weight w(e 0n ), 0 ≤ w(e 0n ) ≤ 1, where e 0n denotes the edge from v 0 to v n . w(e 0n ) is the percentage of cases in v 0 , whose value for f is known, assigned to v n (see Fig. 3). In other words, w(e 0n ) approximates the prior probability for a case in v 0 to belong to v n . Consequently, at the end of the learning process, each learning case c i is assigned to each leaf l j , j = 1..M , with a weight w ij such that M j=1 w ij = 1 (w ij =0 or 1 if each tested attribute is known for c i , 0 < w ij < 1 otherwise).

III. INCLUDING IMAGES IN A DECISION TREE

To include images in a DT, the principle of Content-Based Image Retrieval (CBIR) is applied [START_REF] Smeulders | Content-based image retrieval at the end of the early years[END_REF]. CBIR involves 1) building a feature vector characterizing each image -this feature vector is referred to as signature -, and 2) defining a distance measure between two signatures. In the proposed framework, images are Fig. 3. Managing missing information. If the tested attribute is unavailable for some medical case (?), this case is assigned to all subgroups with a weight equal to the prior probability to be assigned to that subgroup. characterized by their wavelet transform [START_REF] Quellec | Wavelet optimization for content-based image retrieval in medical databases[END_REF]. Then, measuring the distance between two images comes down to measuring the distance between their signatures. Similarly, when building a DT, we use the distance measure between image signatures to divide a population of images into subgroups. An unsupervised classification algorithm is used to cluster similar image signatures, as described in section III-C. By this process, image signatures can be included in a DT like any other attribute.

A. Image signature

In previous studies on CBIR, we decided to extract signatures for images from their wavelet transform [START_REF] Quellec | Wavelet optimization for content-based image retrieval in medical databases[END_REF]. Using the wavelet transform for database management is convenient: images can be compressed in JPEG-2000 format [START_REF] Taubman | Image Compression Fundamentals, Standards and Practice[END_REF], which relies on the wavelet transform, and their signature can be extracted directly in the compressed domain. Moreover, wavelet-based image signatures have shown their superiority over other image signatures [START_REF] Quellec | Wavelet optimization for content-based image retrieval in medical databases[END_REF]. The proposed signatures model the distribution of the wavelet coefficients in each subband of the wavelet decomposition; as a consequence, a multiscale description of images is obtained. To characterize the distribution of wavelet coefficients in a given subband, Wouwer's work was applied [START_REF] Van De Wouwer | Statistical texture characterization from discrete wavelet representations[END_REF]: Wouwer showed that this distribution can be modeled by a generalized Gaussian function (see equation ( 2)).

p(x; α, β) = β 2αΓ( 1 β ) e -( |x| α ) β (2)
Γ(z) = ∞ 0 e -t t z-1 dt, z > 0 (3)
The maximum likelihood estimators (α, β) of the wavelet coefficient distribution in each subband are used as a signature [START_REF] Do | Wavelet-based texture retrieval using generalized gaussian density and kullback-leibler distance[END_REF]. These estimators can be computed directly from JPEG-2000 compressed images, which can be useful when a large number of images have to be processed. Any wavelet basis can be used to decompose images. However, the effectiveness of the extracted signatures largely depends on the choice of this basis. For this reason, we proposed to search for an optimal wavelet basis within the lifting scheme framework [START_REF] Quellec | Wavelet optimization for content-based image retrieval in medical databases[END_REF], which is at the core of the JPEG-2000 compression standard.

B. Distance Measure

Do and Vetterli proposed the use of the Kullback-Leibler (see equation ( 4)) divergence between wavelet coefficient distributions in each subband to define a distance measure between signatures [START_REF] Do | Wavelet-based texture retrieval using generalized gaussian density and kullback-leibler distance[END_REF].

D(p(X; θ 1 )||p(X; θ 2 )) = p(X; θ 1 ) log p(X; θ 1 ) p(X; θ 2 ) dx (4)
Kullback-Leibler divergence is not symmetric, which is a requirement of clustering algorithms. A symmetric version of the divergence, Ds, is used instead (see equation ( 5)).

Ds(p(X; θ 1 )||p(X; θ 2 )) = (D(p(X;θ1)||p(X;θ2))+D(p(X;θ2)||p(X;θ1))) 2

(5)

By injecting equation ( 2) in ( 5), we obtain the expression of the distance measure between two wavelet coefficient distributions (see equation ( 6), the expression in the asymmetrical case is given in [START_REF] Do | Wavelet-based texture retrieval using generalized gaussian density and kullback-leibler distance[END_REF]).

Ds(p(X; α 1 , β 1 )||p(X; α 2 , β 2 )) = α1 α2 β2 Γ( β 2 +1 β 1 ) Γ( 1 β 1 ) + α2 α1 β1 Γ( β 1 +1 β 2 ) Γ( 1 β 2 ) -1 β1 -1 β2 (6)
Finally, the distance between two images is a weighted sum of these symmetric divergences over the subbands [START_REF] Quellec | Wavelet optimization for content-based image retrieval in medical databases[END_REF]. The ability to select a weight vector and a wavelet basis makes this image representation suitable for specialized medical datasets.

C. Signature Clustering

Thanks to the image signatures and the associated distance measure above, a population of cases can be divided into subgroups using an unsupervised classification algorithm, provided that a custom distance measure can be specified. Because it is simple and fast, the Fuzzy C-Means algorithm (FCM) [START_REF] Bezdek | Fuzzy mathemathics in pattern classification[END_REF] was used for this purpose; the Euclidian distance was replaced in FCM by the proposed distance between signatures. Finding the right number of clusters is generally a difficult problem. However, when the data is labeled, mutual information between cluster and class labels can be used to determine the optimal number of clusters K [START_REF] Strehl | Relationship-based clustering and cluster ensembles for highdimensional data mining[END_REF] (see equation [START_REF] Smeulders | Content-based image retrieval at the end of the early years[END_REF]).

K = argmax K C c=1 K k=1 p(c, k) log C+K p(c, k) p(c)p(k) (7)
where c = 1..C are the class labels, p(c, k) is the joint probability distribution function of the class and cluster labels, p(c) and p(k) are the marginal probability distribution functions.

IV. DECISION TREE BASED RETRIEVAL

A. Objective

Let c q be a case placed as a query by a user. The objective of the proposed framework is to retrieve the R most similar cases in a reference database. For diabetic retinopathy follow-up, the number of cases retrieved by the system is set to R = 5, at ophthalmologist's request; they consider this number sufficient for time reasons and in view of the results provided by the system. Consequently, the satisfaction of the user's needs is assessed by the precision at R, denoted π R , defined as the percentage of cases relevant for c q among the topmost R results.

B. Single Tree Based Indexing

To find the R most similar cases, we need to compute a similarity measure between c q and each case c i in the reference database. To that purpose, we propose to compare their assignment weights to each leaf l j : w qj and w ij , j = 1..M . These weights have been computed for each learning case (subset L of Fig. 4. Illustration of the retrieval process. In (b), where attribute 'late angio.' is missing, the leaves for Groups 1 to 3 will be browsed, whereas for (c), where this attribute is available, only the leaves for Groups 1 and 3 will be browsed. the reference database) while building the tree (see section II-B). They can be computed a posteriori for each remaining case in the reference database -in particular those added after the learning phase -and for the query c q . In that purpose, the weight w(e) of each edge e in the tree is stored (see section II-B). The retrieval system is illustrated on an example in Fig. 4.

A similarity measure S ab between two cases c a and c b is defined in equation ( 8); S ab relies on the assignment weight (w aj ) j=1..M (resp. (w bj ) j=1..M ) of c a (resp. c b ) to each to each leaf l j , j = 1..M .

S ab = M j=1 w aj w bj (8)
This similarity measure, the scalar product of (w aj ) j=1..M and (w bj ) j=1..M , maps to [0;1]. It is maximal when a and b are completely assigned to the same leaf. It is minimal when there is no leaf to which both cases are at least partially assigned.

The similarity measure between the query c q and each case c i in the reference database can be computed very quickly. It does not require browsing the entire reference database:

• For each leaf l j in the DT, a list L j containing every case c i such that w ij = 0 is built during the learning process. These lists are updated each time a new case is included in the reference database. • At the beginning of the retrieval process, each similarity measure S qi is set to 0. • For each leaf l j such that w qj = 0, L j is browsed: for each case c i ∈ L j , S qi is increased by w qj w ij .

C. Multiple Tree Based Indexing

Because of the hierarchical architecture of the system above, some attributes might be given too much weight. In the example of Fig. 4 for instance, a male and a female both aged 30 would be regarded as completely dissimilar, because of their different sex, whereas age might play a significant role. To solve this problem, we propose a retrieval system relying not only on one DT but rather on several (say τ ) DTs. Retrieving similar cases from a single DT or from several DTs can be done similarly: instead of computing the scalar product between the assignment weights to the leaves of one DT, we simply compute the scalar product between the assignment weights to the leaves of each of these τ DTs. The expression of the new similarity measure S ′ ab is given in equation 9.

S ′ ab = τ t=1 Mt j=1 w atj w btj (9)
where w atj is the assignment weight of case c a to the j th leaf of the t th tree and M t is the number of leaves in the t th tree.

Several methods have been proposed in the literature to generate sets, or committees, of DTs: Random Forests [START_REF] Breiman | Random forests[END_REF] or randomized c4.5 [START_REF] Dietterich | An experimental comparison of three methods for constructing ensembles of decision trees: bagging, boosting, and randomization[END_REF] for instance. They usually perform better as classifiers than single DTs. To generate DT committees, the learning algorithm is randomized as follows: to decide which test should be selected for dividing a tree node, the k most discriminant attributes, according to the entropy measure (see equation ( 1)), are identified one of them is picked randomly with uniform probability.

D. Retrieval System Boosting

When applied to unbalanced datasets, DTs tend to be biased towards the largest classes [START_REF] Guo | Learning from imbalanced data sets with boosting and data generation: the databoost-im approach[END_REF]. If DTs are used as classifiers, this problem can be alleviated thanks to boosting [START_REF] Freund | Experiments with a new boosting algorithm[END_REF]. Boosting algorithms typically build a DT committee in iterations, by incrementally adding weak classifiers (i.e. with a predictive accuracy at least better than chance) to a final strong classifier. At each iteration k, a weak classifier h k is learnt from the learning set with respect to a distribution (learning cases are assigned more or less weight); the weight distribution is initially uniform. The weak classifier is then added to the final strong classifier and the learning cases are reweighted: misclassified cases gain weight and correctly classified cases lose weight. We followed the example of Adaboost [START_REF] Freund | Experiments with a new boosting algorithm[END_REF], the most popular boosting algorithm, to define a boosting strategy for our retrieval system. In our application, h k denotes a set of DTs used as a "weak retriever" (see section IV-C). At each iteration k, the weight d k (c i ) of case c i is updated as follows:

1) the weighted average retrieval error ǫ k of h k is computed:

ǫ k = 1 -i d k (c i )π k R (c i ) 2) the weight α k of h k is updated: α k = 1 2 ln 1-ǫ k ǫ k 3) a variable γ k (c i ) indicating whether d k (c i ) should be increased (γ k (c i ) > 0) or decreased (γ k (c i ) < 0) is computed: γ k (c i ) = 1 -2π k R (c i ) 4) d k (c i ) is updated according to α k and γ k (c i ): d k+1 (c i ) ∝ d k (c i )e α k γ k (ci)
The final "strong retriever" H is thus a set of DT sets h k weighted by α k . Equivalently, H is a DT set in which each tree t in h k is assigned a weight α t = α k in H. Consequently, the final similarity measure becomes S ′′ ab , given in equation [START_REF] Alto | Content-based retrieval and analysis of mammographic masses[END_REF].

S ′′ ab = τ t=1 Mt j=1 α t w atj w btj (10)

V. APPLICATION TO TWO MEDICAL DATASETS

The proposed framework has been applied to two heterogeneous medical datasets. The first dataset (DRD) is being built at the LaTIM laboratory (Inserm U650), in collaboration with ophthalmologists from Brest University Hospital. The second one (DDSM) is a well-known public access dataset [START_REF] Heath | Current status of the digital database for screening mammography[END_REF].

A. Diabetic Retinopathy Dataset (DRD)

Diabetic retinopathy is damage to the retina caused by complications of diabetes, which can eventually lead to blindness. The diabetic retinopathy dataset contains retinal images of diabetic patients, with associated anonymized information on the pathology. The dataset consists of 86 patient files containing 1399 photographs altogether. Patients have been recruited at Brest University Hospital since June 2003 and images were acquired by experts using a Topcon Retinal Digital Camera (TRC-50IA) connected to a computer. Images have a definition of 1280 pixels/line for 1008 lines/image. They are lossless compressed images. An image series is given in Fig. 5. The contextual information available is the patients' age and sex and structured medical information (see table I). If patients records were comprehensive, they would consist of 10 images per eye (see Fig. 5) and of 13 contextual attributes. However, in our dataset, 11.9% of images and 39.7% of contextual attribute values are missing. The disease severity level, according to ICDRS classification [START_REF] Wilkinson | Proposed international clinical diabetic retinopathy and diabetic macular edema disease severity scales[END_REF], was assessed by one three-year experienced expert for each patient. The distribution of the disease severity among the above-mentioned 86 patients is given in table II.

B. Digital Database for Screening Mammography (DDSM)

The DDSM project [START_REF] Heath | Current status of the digital database for screening mammography[END_REF], involving the Massachusetts General Hospital, the University of South Florida and the Sandia National laboratories, has built a mammographic image database for research on breast cancer screening. It consists of 2277 patient files. Each one includes two images of each breast, along with some associated patient information (age at time of study, subtlety rating for abnormalities, American College of Radiology breast density rating and keyword description of abnormalities) and image information (scanner, spatial resolution, etc.). The following contextual attributes were included in the system:

• age at time of study • breast density rating • digitizer The remaining attributes were not used, either because they are regarded as useless (date of study, date digitized, etc.) or because they require advanced expert interaction (the description of the lesions visible in images). Images have a varying definition, of about 2000 pixels/line for 5000 lines/image. Each patient file has been graded by a physician. Patients are then classified in three groups: normal, benign and cancer. The distribution of grades among the patients is given in table II.

C. Attributes of a medical case

In those datasets, each patient file consists of a mixture of digital images and contextual information. Contextual attributes (13 in DRD, 3 in DDSM) do not require advanced preprocessing: textual attributes (such as "treatments" in DRD) were translated into codes and processed as nominal attributes; numerical contextual attributes (such as "breast density rating" in DDSM) did not require any preprocessing at all. Images, on the other hand, require advanced preprocessing: numerical attributes were extracted to characterize them. A usual solution to extract numerical attributes from images is to segment these images and extract domain specific information. However, this approach requires expert knowledge, so it is not sufficient for generic database management. Another solution was proposed: images were indexed by their digital content (i.e. a generic signature was extracted -see section III-A). An image signature was computed for each type of images (10 for DRD, 4 for DDSM). To push performance further, it is possible to include domainspecific knowledge in the proposed framework, in addition to the generic signatures. In DRD, for instance, we counted the number of microaneurysms (the most frequent lesion of diabetic retinopathy) detected by the algorithm described in [START_REF] Quellec | Optimal wavelet transform for the detection of microaneurysms in retina photographs[END_REF]. We are not mammography experts, so we did not include any expert knowledge in the system for DDSM.

D. Retrieval system calibration

To maximize the retrieval precision of the proposed system, the following parameters had to be tuned:

• the wavelet basis used to decompose images and the weight vector used in the distance measure between signatures (p 0 ), • the number p 1 of generated trees in each weak retriever h k ,

• the random parameter p 2 = k (the number of attributes among which the tested attribute is selected at each node), • the FCM parameter p 3 (the fuzzy coefficient [START_REF] Bezdek | Fuzzy mathemathics in pattern classification[END_REF]) System performance was assessed by 10-fold cross-validation. Let N be the number of cases in a reference database (N =86 for DRD, N =2277 for DDSM). For each fold, each reference database was divided randomly into two sets: T (the test set -0.1N cases) and T (0.9N cases). The wavelet basis and the weights (p 0 ) were trained on T for each "image attribute" [START_REF] Quellec | Wavelet optimization for content-based image retrieval in medical databases[END_REF]. For each element in the product space P 1 × P 2 × P 3 , a DT committee was built using T . Each tree in this committee was learnt using a learning set L (0.8N cases) selected at random in T ; V = T \ L was used for validation. The precision at R of this committee for a case c i in V (i.e. π R (c i )) was computed over the DTs learnt when c i was in V . The score of this committee is the average π R (c i ), c i ∈ T . The DT committee of maximal performance for the current fold, according to T , was then assessed using the test set T .

The search for the best element in the product space P 1 × P 2 × P 3 has been sped up by a genetic algorithm [START_REF] Goldberg | Genetic Algorithms in Search, Optimization and Machine Learning[END_REF].

E. Robustness to information incompleteness

Robustness to information incompleteness has been assessed as follows:

1) For each case c i in the test set T , 100 new cases have been generated as described in 2). 2) Let n i be the number of attributes inputted for c i , each new case has been obtained by removing a number of attribute values randomly selected in {0, 1, ..., n i }. 3) Robustness to information incompleteness is visually assessed by plotting the average precision at five with respect to the number of available attributes, using the cases generated as described in 1)-2).

F. Baseline heterogeneous information retrieval method

To evaluate the contribution of DTs for heterogeneous and incomplete case retrieval, the proposed approach has been compared to a weighted sum of heterogeneous distance functions, managing missing values [START_REF] Wilson | Improved heterogeneous distance functions[END_REF]. This method was used as a reference for being the natural generalization of CBR. We extended it to cases containing images thanks to the distance measure between image signatures defined in section III-B.

VI. RESULTS

A precision at R=5 of 0.788±0.137 (resp. 0.869±0.161) was measured on DRD (resp. DDSM) using the process described in section V-D. It means that, on average, approximately four cases among the five cases retrieved for a query are relevant. The best set of parameter values, obtained for each dataset by the process described in section V-D, is given in table III. Because of the limited number of images per class in DRD (see table II), retrieval performance necessarily drops with increasing Rs: a precision at R=10 of 0.681±0.133 and a precision at R=20 of 0.571±0.129 were obtained on this dataset. Retrieval performance also decreases on DDSM: a precision at R=10 of 0.819±0.157 and a precision at R=20 of 0.756±0.163 were obtained on this dataset. To bring out the discrimination ability of each attribute, we report in table IV the precision at 5 of a retrieval system that simply finds the 5 most similar cases with respect to that attribute. More generally, to estimate the contribution of numerical (image series signatures) and contextual information, DT sets were learnt using numerical or contextual information alone. On DRD, retrieval precision based on all attributes is significantly higher than retrieval precision based on numerical attributes alone, at the 90% confidence level (but not on DDSM). To evaluate the contribution of boosting, the average precision at five over each class, with or without boosting, is given in table V. Robustness to information incompleteness is assessed in Fig. 6. Whereas a precision at five of 0.788±0.137 (resp. 0.869±0.161) was obtained on DRD (resp. DDSM) with the proposed approach, a precision at five of 0.531±0.185 (resp. 0.709±0.251) was obtained on DRD (resp. DDSM) with a usual CBIR approach (i.e. each case simply consists of one image), using the same image signatures; a comparison with other image signatures is provided in [START_REF] Quellec | Wavelet optimization for content-based image retrieval in medical databases[END_REF]. The precision at five obtained with the baseline heterogeneous information retrieval method described in section V-F is 0.553±0.178 on DRD and 0.739±0.182 on DDSM. Finally, the average computation time required to retrieve the 5 most similar cases, using the settings of table III, is given in table VI. Clearly, most of the time is spent while image signatures are computed. Once the distances between images have been computed, the learning process in itself only takes 0.8 seconds (resp. 80 seconds) per DT for DRD (resp. DDSM). Experiments were performed using an AMD Athlon 64-bit based computer running at 2 GHz.

VII. DISCUSSION AND CONCLUSIONS

A novel medical information retrieval framework has been presented in this paper: it supports queries consisting of image series with contextual information. The framework uses decision 6. Robustness to information incompleteness. Good retrieval performance can be obtained even if many attributes are unavailable: on DRD, a precision at five of 60% (resp. 70%) can be reached even if less than 40% (resp. 55%) of the attributes are available.

taken into account [START_REF] Nastar | Indexation d'images par le contenu: un état de l'art[END_REF]. On this dataset, the proposed framework outperforms the retrieval of single images by a factor of 1.48 in precision. This stands to reason since an image alone is generally not sufficient for experts to correctly diagnose the disease severity level of a patient. Nevertheless, as table IV shows, images are discriminant attributes, red free photographs (Fig. 5 (b)) and angiographs (Fig. 5 (d)-(j)) in particular. These two image modalities are indeed the most useful for physicians to follow up diabetic retinopathy. On DDSM, the superiority of images over nominal attributes is even more obvious, as illustrated in table IV. This table also shows that using images series without contextual information, instead of single images, increases by itself the average precision at five by a factor of 1.32 on DRD. Adding contextual information increases precision further. Besides, this non-linear retrieval method is 1.43 times more precise than a linear combination of heterogeneous distances on DRD. The improvement brought by heterogeneous information retrieval is more moderate for DDSM (0.869±0.161 as opposed to 0.709±0.251). Performance increase can be explained by the combination of evidence from four images instead of one and by a fine segmentation of the feature space into homogeneous groups provided by DTs, which helps us better separate the classes.

Boosting does not lead to a significant increase of precision over the entire dataset, however it increases precision for rare classes: in DRD, precision significantly increases for class 1, the rarest class, at the 90% confidence level (see table V).

The proposed retrieval system is fast: most of the computation time is spent during the image processing steps. Moreover, it is not necessary to process every image. The first reason is that the retrieval system only needs to characterize attributes tested at nodes browsed by the query case; as a consequence, certain images do not need to be processed. The second reason is that sufficient precision can be reached before every attribute has been inputted by the user. Provided that the retrieval list is updated each time an attribute is updated, the user can stop formulating his/her query when he/she is satisfied with the results. On DRD for instance, a precision at five of 60% can be reached by inputting less than 40% of the attributes (see Fig. 6): with this precision, the majority of the retrieved cases (3 out of 5) belong to the correct class.

Another interest of the proposed framework is its generality: any multimedia database may be processed so long as a process to cluster cases is provided for each new modality (sound, video, etc). This paper reports promising results about the use of data mining techniques to combine numerical and contextual information in a medical retrieval framework; we are now focusing on alternative data mining algorithms to improve performance.

Fig. 1 .

 1 Fig. 1. Toy example of decision tree. Late angiographs are images obtained from one modality (late angiography -see section V-A): in this example, these images are clustered into 2 groups.

Fig. 5 .

 5 Fig. 5. Photograph sequence of a patient's eye. Photographs (a), (b) and (c) were obtained with different color filters. Photographs (d) to (j) constitute a temporal angiographic series: a contrast product is injected and photographs are taken at different stages (early (d), intermediate (e), (g)-(j) and late (f)). (g)-(j) are images from the periphery of the retina.

  Quellec, Guy Cazuguel and Christian Roux are with INSTITUT TELECOM/TELECOM Bretagne, Dpt ITI, Brest, F-29200 France. Mathieu Lamard and Béatrice Cochener are with University of Bretagne Occidentale, Brest, F-29200 France.

Gwénolé Quellec, Mathieu Lamard, Lynda Bekri, Guy Cazuguel, Christian Roux and Béatrice Cochener are with Inserm, U650, Brest, F-29200 France. Lynda Bekri and Béatrice Cochener are with CHU Brest, Service d'Ophtalmologie, Brest, F-29200 France.

TABLE V

 V

		PRECISION AT FIVE FOR EACH CLASS
	dataset	DRD	DDSM
	class	no boosting	boosting	no boosting	boosting
	1	0.307±0.178	0.597±0.150	0.862±0.173	0.873±0.156
	2	0.705±0.166	0.802±0.146	0.818±0.183	0.872±0.164
	3	0.870±0.096	0.835±0.092	0.839±0.180	0.864±0.162
	4	0.719±0.163	0.793±0.156
	5	0.677±0.170	0.758±0.166
	6	0.847±0.128	0.809±0.133
	entire set	0.742±0.157	0.788±0.137	0.840±0.179	0.869±0.161


Gurvan Uguen

* -Gilles Coppin

Claire Lassudrie -

Philippe Lenca

A social approach of the individual decision making process by the search for a dominance structure: risk management by a business process manager

Keywords: Risk Management, Multicriteria Decision Making, Decision Process, Dominance Structure, Constructive Approach MOTS-CLÉS : Management des risques, Décision multicritère, Processus décisionnel, Structure de dominance, Approche constructive

We focus on the way a business process manager treats the risks affecting a supervised business process. A descriptive model of the individual decision making process is proposed. This model mainly results from the integration, in the model of decision making by the search for a dominance structure, of the social aspect of the business process manager activity. In order to do this, we adopt a constructive and constructivist approach concerning the definition by the business process manager of his representation of the risk treatment strategy that has to be implemented.

RÉSUMÉ. Nous nous intéressons à la manière dont le responsable d'un processus métier traite les risques affectant un processus métier non complètement automatisé. Un modèle descriptif du processus individuel de décision est proposé.

Il résulte principalement de l'intégration, dans le modèle de décision basé sur la recherche d'une structure de dominance, de la composante sociale de l'activité d'un responsable de processus. Nous adoptons pour cela une approche constructive et constructiviste concernant la définition par le responsable de processus de sa représentation de la stratégie de traitement du risque à mettre en oeuvre.

Introduction

In this paper, we address the problem of designing a strategy to mitigate a risk in a supervised business process. Reasons for risk mitigating can come from the discovery of a new risk, due to previous incomplete risk assessments or due to activity changes. It also can come from the acquisition of new information implying the management of a known risk. Our aim is to provide a decision support system to a business process manager (BPM), that takes into account organizational aspects to allow him to more accurately control his risk management task, risk treatments and therefore the business process evolution. In order to reach this goal, we define here the descriptive model of the BPM decision making process. This model will allow us to specify which kind of support can be provided to the BPM. In particular, the present model takes care of the multicriteria and social aspects of such an organisational problem.

In section 2, we give a description of the BPM's activity. The section 3 focuses on the social dimension of the BPM's risk management task. The section 4 presents a model of individual decision making corresponding to our hypothesis concerning the way the BPM treats information in order to decide. The last section presents the adaptations of this decision making model in order to take care of all of our hypotheses.

The business process manager's activity

We first introduce our hypotheses on the BPM's activity. We then introduce three main aspects we have to take into account in order to understand the way the BPM manages risks.

Managing relationship between people and managing the business process model

The role of the BPM is to ensure the quality of the operations of his business process according to fixed business process objectives. In order to reach this goal, the BPM has to take into account various people involved in the part of the organization activity he is responsible of. These people can be for example suppliers, customers, employees of the organization or seniors in rank. All of them express specific needs, wishes, expectations, constraints, etc., that the BPM has to take care of in order to ensure the right operation of the organization activity he is responsible of. We suppose that the resulting structure of the organization activity is described in a business process model that the BPM must maintain up to date. This business process model describes the activities of the business process and the objectives, occurrence conditions, ressources and employes' roles associated to each activity, or at least to the whole business process. So, the complexity of his task lies in the need to influence course of things while giving the possibility for the business process actors to initiate new courses of action during process occurrences. This task implies for the BPM to take into account technical and social aspects.

The BPM's task can be related to the crew management task of cockpit commanders. Ginnett notices that commanders appointed as excellent leaders by crews and inspectors of aviation spend more time structuring the team [START_REF] Ginnett | Crews as Groups: Their Formation and Their Leadership[END_REF]. During this, they devote much time to the unusual tasks, and little time on routine tasks that the crew may be confronted with. They open the group to actors who play a role in his performance, not only the cockpit crew. They clearly explain that each member of the crew must contribute to the safety of all and support effective communication and cooperation. They can play all types of leadership behaviours, from a democratic mode during consultations to an autocratic style during operating phases. Thus, we consider that a BPM should be helped in order to express such leadership abilities.

Dealing with risk

As Slovic notices, in the modern world we can deal with risk in three fundamental ways: risk as feeling, risk as analysis and risk as politics [START_REF] Slovic | Risk As Analysis and Risk As Feelings Some Thoughts About Affect, Reason, Risk, and Rationality[END_REF].

The first way deals with human factors involved in the emotional and automated schematic cognitive treatment level and in biases generated by wrong uses of information processing heuristics [START_REF] Kahneman | Judgement under uncertainty: Heuristics and biases[END_REF].

The second way deals with human factors involved in the rationalizing analytical cognitive treatment level and with the use of risk assessment methodologies, models or theories. We have pointed out the fact that faced with various risks, the BPM may use various existing methods adapted for specific types of risk [START_REF] Uguen | Le management continu de l'incertitude par stratégies[END_REF].

The third way deals with the fact that, faced with risk representation in complex systems, experts can provide conflicting points of view. Communication and negociation are the central aspect of this approach. The need to consider risk as politics is particularly true when dealing with major risks [START_REF] Peretti-Watel | Sociologie du risque[END_REF]. In the next section, we extend this third way to all risks perceived by the BPM. So, our problematic addresses the three ways distinguished by Slovic. We focus on a unique individual who has specific abilities and knowledge, can use various tools to assess a situation and is politically, or generally speaking, socially situated. In particular we are interessed in the problem of knowledge transfer between two activities. Indeed, on the one hand we consider the BPM as an expert of past encountered risks and problems that lead to the actual business process model. On the other hand, the BPM identifies the need to focus on one risk. He therefore points out his difficulty to manage the new or too complex situation involved by the identification of this risk. So, he points out his need to change his mental representation of the situation, to be able to design an efficient strategy to treat this risk.

As a result, we have been interested in designing a descriptive model of individual decision making that would take care of the social dimension of the BPM activity.

The social dimension of the BPM risk management task

In this section, we highlight the need to take into account the social dimension of the BPM activity as a main aspect of the way he deals with risks. Then, we show how our approach can be linked with the work of two authors from the domains of sociology of organization and psychosociology of organization.

Communicating with agents

Generally speaking, during his activity, the BPM has to deal with various points of view. These points of view can be expressed by various agents: single persons or groups of persons providing subjective, expert or not, judgments, or results of the application of a risk assessment technique. These agents can express hierarchical (top-down or bottom-up), societal or legal constraints, or just express an advice. More over, as they may not have a clear or complete representation of the situation, these agents and the BPM can express multiple points of view. The need for the BPM to consider various constraining agents' point of view comes from two reasons. First he has to clarify his representation of the situation in order to obtain a unique coherent personal point of view. Second, these agents are implicated in the risk scenarios or in one of the possible treatment strategies. For example, although a treatment strategy may satisfy the BPM's senior in rank, if it doesn't also satisfy the employees of the business process, the strategy will never be applied as it is designed in the business process model. As a result, contexts of the risk assessment and of business process occurrences play an important part. Therefore, the re-use of past strategies and the design of generic strategies must be made carefully. Consequently, we identified the need to adopt a constructive approach for the representation of a risk and the design of a risk treatment strategy (cf. 5.3.1.).

Sociology of organization and power games

We consider that the individual decision criteria and objectives of the BPM strongly reflect the social context of the organisation he works in. Our approach can be related to the notion of power games introduced by Crozier and Friedberg [START_REF] Crozier | Actors and Systems: The Politics of Collective Action[END_REF].

Crozier and Friedberg start from the individual in order to explain organizational phenomena. They define the notion of power as the ability of an agent to act on another agent or, at the opposite, to refuse to do or provide what the other agent asks. It's therefore an attribute of a relation between two agents rather than a constitutive attribute of an agent or an agent's role. The four main sources of power are linked to:

-Agents competences; -Relations between the organization and its environments; -Communication and information; -Organizational rules.

As the BPM has the power to modify organizational rules through the business process model, the uncertainty zone notion explained by Crozier and Friedberg can be faced with objectives defined as value intervals on decision criteria. A specific risk management can be seen as a new power game, and the initial risk description can be seen as the result of past and present power games in which the BPM took or takes part in. Consequently, the BPM doesn't first try to treat a risk. He treats the power game resulting from the risk consideration and this treatment can pass through risk specific treatments, or not. We have already pointed out this aspect distinguishing two main types of risk treatment strategies: perceived uncertainty control strategies and perceived uncertainty modification strategies [START_REF] Uguen | Le management continu de l'incertitude par stratégies[END_REF]. Uncertainty control strategies deal with business process modifications in order to avoid or limit the risk appearance (ex: potection and prevention) , while perceived uncertainty modification strategies (monitoring, acceptance and assessment) deal with changes of the perception of uncertainty. However we also here highlight the fact that the acceptance of risk can pass through compensation mechanisms of risks taken by some agents.

Psychosociology of organization and sensemaking process

As March summarizes it [START_REF] March | Decisions and Organizations[END_REF], various authors argue that:

-Decisional preferences are often incoherent, unstable, and defined by external influences;

-Decisions and actions are slightly coupled and bound in an interactive way rather than a linear way; -The past is not very reliable as a guide for the present and the future; -Political considerations and symbolic systems occupy a central part in decision making.

Weick identifies three ways to deal with decision making that propose some answers to such a review [START_REF] Weick | The Collapse of Sensemaking in Organizations: The Mann Gulch Disaster[END_REF]):

-Naturalistic decision making models and the sensemaking process; -Models based on the notion of power; -Models that better take account of social relationships.

Weick has worked for years on the notion of sensemaking process (Vidaillet, 2003). From his point of view, the sensemaking process starts with the unexpected, is directed toward action, and is context sensitive. People try to represent things under a rational way for themselves and for others. Problems come from the possibility of various personal or inter-individual interpretations of the situation rather than the lack of information about the situation. Weick highlight the importance of the social dimension in information processing at a collective level but also at the individual level.

The collective sensemaking process in organizations is a process aiming at collectively reducing perceived equivocity of a situation. This is an activity allowing members of the organization to establish priorities and preferences between actions and, thus, to constitute a system of organized actions. However, the collective construction of sense does not lead to the construction of a collective sense. Maintaining a system of collective actions does not require for each member of the team to understand behaviours of other members. Each one only has to foresee these behaviours.

We consider that during the sensemaking process, the BPM is faced with various decisional tasks aiming at partially removing the perceived uncertainty, so as to be able to give sense for him to the situation while preserving freedom margins for him or other agents, and thus, to be able to act.

The individual decision making process

In this section, we present a model of decision making that is in agreement with our hypothesis of a multicriteria descriptive model of individual decision making. This model can be related to naturalistic decision making [START_REF] Lipshitz | Taking Stock of Naturalistic Decision Making[END_REF].

The search for a dominance structure

We consider the decision making task as a decision process in the same way as Montgomery: "the decision process is seen as a search for good arguments […] as a search for a dominance structure, i.e. a cognitive structure in which one alternative can be seen as dominant over the others. […] the final choice will follow in a selfevident way from the given structure" [START_REF] Montgomery | From cognition to action: The search for dominance in decision making[END_REF].

From this consideration, Montgomery proposed a multicriteria decision making process based on the use of various decision rules (conjunctive rule, disjunctive rule, addition of utilities rule…) in order for a promising alternative to satisfy the dominance rule. The dominance rule is expressed as follow: "Choose alternative A 1 over A 2 if A 1 is better than A 2 on at least one attribute and not worse than A 2 on all other attributes" [START_REF] Montgomery | Decision rules and the search for a dominance structure: toward a process model of decision making[END_REF]. This approach can be related to the work of Payne, Bettman and Johnson on the adaptive selection of heuristic choice strategies in decision making [START_REF] Payne | The Adaptive Decision Maker[END_REF]. A recent overview of the cognitive approach to decision making, following Montgomery's work, can be found in [START_REF] Barthélemy | Cognitive approach to decision making and practical tools[END_REF].

The model of Montgomery

The use of various rules during the decision making process has been more precisely formalized by Montgomery. For a choice task, he proposed a dominance search model compounded of four main phases (Figure 1).

The process starts with a pre-editing phase consisting in selecting the considered attributes and alternatives from the whole initial sets of attributes and alternatives. The second phase consists in finding a promising alternative that is tested against other alternatives during the third phase, in order to identify whether or not it fulfills the dominance rule. If the promising alternative is found dominant over all other alternatives, the decision making process ends with the choice of the promising alternative. Otherwise the decision maker gets into a fourth phase labeled dominance structuring. During this phase, the decision maker tries to eliminate or neutralize the identified dominance violations. Four main strategies are described by Montgomery for this phase:

-De-emphasizing the importance of an attribute or of differences across alternatives on a particular attribute; -Bolstering the promising alternative by enhancing its positive aspects or enhancing negative aspects associated with non-promising alternatives;

-Cancelling an attractiveness difference in one attribute by the attractiveness difference in another attribute; -Collapsing two or more attributes into a new and more comprehensive attribute.

If dominance violations are not eliminated, the decision maker goes back to the search of a new promising alternative.

To finish, it is interesting to note that the two first phases of the process consist in information processing relative to all alternatives whereas the two last phases consist in information processing relative to the specific promising alternative. [START_REF] Montgomery | Decision rules and the search for a dominance structure: toward a process model of decision making[END_REF]

Figure 1. A dominance search model of decision making

Adaptation of the dominance search model

We propose some adaptations to this dominance search model of decision making in order to take into account our task specificities previously presented. We especially consider this process as a socially situated one. This approach can be linked with the considering of decision making as a social representation [START_REF] Laroche | From Decision to Action in Organizations: Decision-making as a Social Representation[END_REF].

Tacking stock of agents' points of view

Agent and point of view

We have already indicated that we use the notion of agents rather than individual in order to socially situate the decision making process (cf. 3.1.). Any agent can provide multiple points of view to the BPM. However, at a given moment, the BPM personal points of view are only the risk representation, his objectives and a promising strategy. His work on alternative strategies is seen as a work on strategies of some agents.

The notion of point of view can either refer to a risk treatment strategy, an objective, or an evaluation of the risk. Agents' points of view can express advices or constraints on the structure of criteria to consider or on the promising strategy. However, all of them are described as strategies in the model of decision making. Each strategy is described by a set of values on a specific set of criteria. The initial set of important criteria to consider comes from the identification of unfulfilled objectives of the BPM (cf. 5.2.1.).

In order to be implemented, the promising strategy must satisfy the different structures of the various constraining points of view (those that come from an agent that the BPM must take care of, i.e. an agent who express constraints). Therefore, the promising strategy can drastically evolve during the decision process by aggregating other strategies or contextualizing process modifications that it describes. The BPM may also have to convince of or to impose reviewed objectives on conflicting points of view during the dominance structuring phase.

Point of view assessment

The assessment of a point of view can be made thanks to either a mental simulation by the BPM of the point of view of an agent, a direct or indirect consultation of the agent's point of view, or the application of an assessment method that does not require stressing the assessed agent.

In order to be able to test the dominance of the promising strategy of the BPM, the result of a point of view assessment is reformulate according to the set of the criteria used to describe the promising strategy. If some criteria cannot be reformulated, the following dominance test fails and the criteria can be temporarily integrated during the dominance structuring phase (cf. 5.3.3.). The BPM can also go back to the beginning of the process in order to integrate the new criteria completely.

"Pre-editing" and "Finding a promising alternative" phases inversion

In our model, the pre-editing phase of the model of Montgomery corresponds to the phase consisting in the definition of the points of view of one agent. As agents are identified thanks to the BPM's promising strategy, the promising strategy identification phase must happen before the pre-editing phase.

Suppression of the exhaustiveness test of accounted information

We consider that the exhaustiveness test of accounted information (Figure 1: "All relevant information evaluated?") arise from the processing of all relevant agents, and is a constitutive test of definition of BPM's strategies and dominance structuring phases. Thus, this test no more appears as a step in the model.

Risk representation

Unfulfilled objectives

A risk perceived by the BPM can be represented as a set of aspects defined on a subset of the criteria of BPM's objectives. This subset of criteria correspond to the criteria on which BPM's objectives seems unfulfilled. Unfulfilled BPM's objectives correspond to the aspects of the ongoing strategy before the identification of the risk. So it is also the representation of the BPM's targeted strategy. These unfulfilled objectives allow the BPM to identify a set of important criteria at the beginning of the decision making process. Therefore, as we consider that these objectives results from the integration of various agents' constraints, it also allow the BPM to initially identify the specific agents he has to take care of.

Agents' objectives are also represented as strategies in the model. As any other point of view, objectives can be modified during structuring phases.

The passive acceptation strategy

The BPM doesn't have systematically a fixed set of alternative strategies among which he has to make his choice. He may not be able to provide more than one alternative strategy or even no other strategy that the passive acceptation of the risk: neglecting the risk and continuing to act as previously. Let us note that the description of this strategy correspond to the description of the current strategy from the identification of the risk, that is also the description of the new identified risk [START_REF] Uguen | Le management continu de l'incertitude par stratégies[END_REF]. Thus, we do not distinguish the "decision" output of the decision making process from the "give up or postpone decision" output of the model of Montgomery.

A constructive and constructivist approach

Constructive approach and new information integration

As the model of Montgomery is defined for a choice task, it requires knowing all the decision criteria and all the possible risk treatment strategies from the beginning of the decision process. The model doesn't explicitly prevents from adding new information but it doesn't explain how criteria, thresholds and aspects are obtained and how new information should be integrated. Although collapsing allows defining a new criterion, it consists in a reformulation of available information rather than the integration of new ones.

However, one cannot suppose that the BPM will be able to recover all the desired points of view in a first phase without reacting to it progressively. Strategies can evolve with the acquisition of new information about the risk, due to external events or new risk assessments. Some strategies can become obsolete, appear, or be reconsidered. Criteria, thresholds and aspects of points of view, including the promising strategy, can evolve all along the decision process, as the BPM and other agents acquire new knowledge and abilities specific to the management of the risk.

The decision making process must be seen as a multi-period decision making process. Indeed, the BPM has other tasks than the resolution of a problem or risk and has thus a limited amount of time dedicated to this specific resolution. Moreover, agents may not be available for a meeting with the BPM, and some agents may need some time to emit a point of view (some time in order for them to proceed the same way as the BPM to make a decision, or in order to operate a specific risk assessment method).

Thus, the use of an evolving promising strategy in a dominance search model of decision making can be linked with the notion of sensemaking process (cf. 3.3.).

Constructivist approach

The constructivist approach [START_REF] Landry | A Note on the Concept of "Problem[END_REF] argue that representations are neither completely objective nor completely subjective but result from the interaction between the subject and the situation; i.e. result from the processing of elements of the subject experience and of the environment, assembling them in a form with a specific sense and labelled "problem".

The process of dominance search follows this constructivist approach: the chosen alternative is not intrinsically dominant; it has to be made dominant over other alternative thanks to dominance structuring strategies that include subjective knowledge integration. Another example is Rousseau's work on territory management [START_REF] Rousseau | Comparaison de points de vue pour la formulation de problèmes[END_REF]. Contrary to us, he distinguishes an individual decision making process cycle from a collective decision making process. Indeed, as we focus on an individual, the proposed decision making process is an individual one, composed of two parts, one focusing on his points of view and the other one focusing on the integration of agents' points of view. Thus, we do not consider any collective decision making process. Collective sense emerges from the need for any individual to obtain a unique coherent point of view from other agents. This need is stressed by the necessity for any individual to make sense of the situation in order to act, i.e. the necessity to have a unique personal point of view on the situation.

Temporary modifications

The model of Montgomery shortly described in section 4 exhibits two main information processing stages: one relative to all alternative and one focusing on the promising alternative. These two main stages can also be distinguished in our adaptation of this model as information processing stages relative to the BPM for the first one and to a specific agent for the second one.

Therefore, we highlight the distinction between temporary modifications of criteria and BPM's promising strategy during the dominance structuring phase (second stage), and the definition of criteria and BPM's strategies during the first stage. Indeed, these temporary modifications can be useful only for the processing of the points of view of the current considered agent.

Description of the model

The resulting adaptation of the dominance search model is presented in the figure 2. The model starts from the identification of a new risk. This identification presupposes at least two strategies:

-The initial representation of the risk, represented as the new ongoing strategy from the risk identification, and labelled BRS (BPM's risked strategy) (cf. 5.2.2.); -The initial BPM's objectives, represented as the ongoing strategy before the risk identification and seen as a BPM's targeted strategy (BTS) (cf. 5.2.1.).

Initially these strategies are represented on a unique set of criteria.

Then, the BPM choose one of these two strategies as his promising strategy (BPS). From the this promising strategy, he identifies an agent to take care of and assesses the agent's strategies (AS). This assessment can be a BPM's mental simulation, an analytical method, or a consultation of the agent's point of view. Then, the BPM testes the dominance of his promising strategy over all agent's strategies. If the promising strategy is dominant, the BPM can then try to identify another agent. Otherwise, the BPM tries a dominance structuring phase. He can try to modify the problematic agent's strategy and criteria, or try to temporary modify the promising strategy or the criteria he use. If the dominance structuring phase fails, the BPM go back to the first phase of the problem in order to modify his problem representation and his promising strategy. The BPM's promising strategy can therefore no longer be one of the BPM's targeted strategy or BPM's risked strategy: the promising strategy can partially or completely integrate the problematic agent's strategy.

Finally, the strategy to implement is the BPM's promising strategy obtained at the end of the decision making process. This end is reached when all constraining points of view identified by the BPM have been taken into account. This is reached when the promising strategy and the targeted strategy are equals. If the BPM's risked strategy is also equal to his promising strategy and his targeted strategy, then the implemented strategy is the passive acceptation strategy (cf. 5.2.2.).

Figure 2. A socially situated dominance search model of decision making

Conclusion

The proposed individual decision making model was initially designed in order to describe how a BPM decide to implement a specific risk treatment strategy. However, the model results from considerations from domains such as psychology of organization, sociopsychology of organisation and a naturalistic decision making model. These considerations lead to the definition of a process addressing not only the strategy choice but also the processing of the evolving representations of a risk and objectives.

Complementary works may focus on various points.

First of all our hypotheses and model have to be validated through tests on real risk management cases in a business process.

Then, we have hypothesized that the individual decision making process is socially situated. However, as the notions of agent and point of view used in this article aren't restricted to individuals, these notions address a stronger hypothesis. It may relate our work to the domain of situated cognition.

Another point is the definition of the ways the BPM identifies agents that he should or must take care of. This model may also be used in major risk management. Indeed, major risks management needs to make a distinction between political decision maker, experts as scientists and the population. Our model can therefore be used as a descriptive model of the political decision maker and his need to take care of conflicting points of view. However, major risks often come from the combination of various minor risks. As the presented model addresses the way a decision maker treat one risk, such combinatorial aspect should be deepened.

Finally, this descriptive model is defined in order to identify how we should assist the BPM during his risk management task. We hypothesize that this task is a kind of sensemaking process, and that we must focus on a constructive and constructiviste approach of the BPM's situation representation and preferences. Therefore, it should be more interesting to try first to identify information concerning the agents that the BPM should take care of, rather than information about stabilized preferences and representations. Moreover, as we focus on risks, this is more the ways he processes new or usual information than preference structures that may be stable.

MASHS 2007 -Brest, France, May 2007


F Frappart

G Ramillien

P Maisongrande
email: philippe.maisongrande@cnes.fr

M.-P Bonnet

Denoising Satellite Gravity Signals by Independent Component Analysis

Keywords: Filtering technique, gravimetry from space, hydrology, independent component analysis (ICA)

Independent component analysis (ICA) is a blind separation method based on simple assumptions of the independence of sources and the non-Gaussianity of observations. An approach based on ICA is used here to extract hydrological signals over land and oceans from the polluting striping noise due to orbit repetitiveness and present in the gravity anomalies detected by the Gravity Recovery and Climate Experiment (GRACE) satellites. We took advantage of the availability of monthly level-2 GRACE solutions from three official providers (i.e., CSR, JPL, and GFZ) that can be considered as different observations of the same phenomenon. The efficiency of the methodology is demonstrated on a synthetic case. Applied to one month of GRACE solutions, it allows for clearly separating the total water storage change from the meridional-oriented spurious gravity signals on the continents but not on the oceans. This technique gives results equivalent to the destriping method for continental water storage.

I. INTRODUCTION

The Gravity Recovery and Climate Experiment (GRACE) geodetic mission provides a global mapping of the time variations of the gravity field [START_REF] Tapley | GRACE measurements of mass variability in the Earth system[END_REF]. So far, raw GRACE orbit observations are used to produce ten-day and monthly geoids in terms of Stokes coefficients (i.e., dimensionless spherical harmonics of the geopotential) up to degree 60-150. Solutions are corrected from atmospheric and oceanic mass variations using ad hoc reanalysis and models, but these data, which represent mainly continental hydrology and ocean masses (plus other unmodeled phenomena), contain meridian-oriented spurious gravity signals or stripes. This stripping is mainly due to resonances at degree 15, 30 . . . and insufficient forward modelling (principally ocean tides and atmospheric and oceanic nontidal mass variations). Several strategies for filtering GRACE solutions have been proposed to remove these northsouthoriented features and extract useful hydrological signals on continents. They include classical isotropic Gaussian filter [START_REF] Wahr | The accuracy of GRACE mass estimates[END_REF], various optimal filtering decorrelations of GRACE errors [START_REF] Han | Non-isotropic filtering of GRACE temporal gravity for geophysical signal enhancement[END_REF]- [START_REF] Kusche | Approximate decorrelation and non-isotropic smoothing of time-variable GRACE-type gravity field models[END_REF], and statistical constraints on the time evolution of GRACE coefficients [START_REF] Davis | A statistical filtering approach for Gravity Recovery and Climate Experiment (GRACE) gravity data[END_REF] or from global hydrology models [START_REF] Ramillien | Time variations of land water storage from an inversion of 2 years of GRACE geoids[END_REF]. Nevertheless, these approaches are insufficient as the results still suffer from aliasing, leakage, and noise contamination. We undertake a different strategy. Instead of comparing the GRACE solutions from official providers [Center for Space Research (CSR), GeoForschungsZentrum (GFZ), and Jet Propulsion Laboratory (JPL)], we assume that any CSR, GFZ, and JPL solution contains joint information.We combine GRACE solutions from these groups and use an independent component analysis (ICA) approach to extract signals that are statistically independent as the mixture of space and time decorrelated geophysical signal and noise, without subjective a priori constraints. ICA is a fast linear method which has been successfully applied in different domains like separating independent sources from microphone records [START_REF] Stone | Independent Component Analysis: A Tutorial Introduction[END_REF]. Our objective is to separate the different independent components (i.e., the sources according to ICA terminology), combining the level-2 GRACE products from CSR, GFZ, and JPL (i.e., the observations) and expecting the ICA-estimated components to match interpretable geophysical signals. This approach is first tested on simulated GRACE observations and then applied to real GRACE products. We compared our results with those from the empirical destriping and smoothing method [START_REF] Chambers | Evaluation of new GRACE time-variable gravity data over the ocean[END_REF].

II. ICA

ICA is a powerful method for multivariate data analysis, particularly for blind signal separation, and has various practical applications, including communications, medical signal processing, speech signal processing, and so on [START_REF] Comon | Independent component analysis: A new concept?[END_REF]- [START_REF] Lathauwer | An introduction to independent component analysis[END_REF]. A typical example of ICA is isolating speech signals from recordings of people talking simultaneously in a room ("cocktail party problem" [START_REF] Haykin | The cocktail party problem[END_REF]), under the assumption of no time delays or echoes. An important consideration is that if N sources are present, at least N observations (e.g., from microphones or receivers) are needed to retrieve the original source signals, with ICA remaining applicable for square or overdetermined problems. ICA proceeds by maximizing the statistical independence of the estimated components. As a first step of the method, non-Gaussianity of the input signals must be verified. The central limit theorem is used for measuring the statistical independence of the components. Typical algorithms for ICA use centering and whitening based on eigenvector decomposition and reduction of dimension as main processing steps.Whitening ensures that the input observations are equally treated before dimension reduction.

Many algorithms of different complexities have been developed for ICA [START_REF] Stone | Independent Component Analysis: A Tutorial Introduction[END_REF]. For separating satellite gravity signals, we consider the FastICA algorithm, which is a computationally highly efficient method for performing the estimation of ICA [START_REF] Hyvärinen | Independent component analysis: Algorithms and applications[END_REF]. It uses a fixed-point iteration scheme that has been found to be 10-100 times faster than conventional gradient methods for ICA [START_REF] Chambers | Evaluation of new GRACE time-variable gravity data over the ocean[END_REF].

Assuming that an observation vector y collected from N sensors is the combination of P (N ≥ P) independent sources represented by the source vector x, considering a linear statistical model, it can be written as:

y = Mx (1)
where M is the mixing matrix whose elements m ij 's (1 ≤ i ≤ N, 1 ≤ j ≤ P) indicate to what extent the j th source contributes to the i th observation. The columns {m j } are the mixing vectors.

ICA aimed at estimating the mixing matrix M and/or the corresponding realizations of the source vector x, only knowing the realizations of the observation vector y, under the following assumptions [START_REF] Lathauwer | An introduction to independent component analysis[END_REF].

1) The mixing vectors are linearly independent.

2) The sources are statistically independent.

The original sources x's can be simply recovered by multiplying the observed signals y's with the inverse of the mixing matrix also known as the "unmixing" matrix

x = M -1 y (2)
ICA consists of three numerical steps. The first step of ICA is to center the observed vector, i.e., to remove its mean m = E{y}. The second step consists in whitening the vector y to remove any correlation between the components of the observed vector. In other words, the components of the white vector y have to be uncorrelated, and their variances equal unity.

Letting C = E{yy t } be the correlation matrix of the input data, we define a linear transform B that verifies the following two conditions:

y By   (3)   t P E yy I   (4)
where I P is the identity matrix of dimension P × P.

This is easily accomplished by considering

B = C -1/2
(5)

A reduction of the dimension of the data to the number of independent components P, discarding the too small eigenvalues obtained by principal component analysis, is performed simultaneously as the whitening.

For the third step, an orthogonal transformation of the whitened signals is used to find the separated sources by rotation of the joint density. The appropriate rotation is obtained by maximizing the nonnormality of the marginal densities, since a linear mixture of independent random variables is necessarily more Gaussian than the original components.

We apply a linear ICA through the FastICA algorithm made available at http://www.cis.hut.fi/projects/ica/fastica/ to unravel the independent components of the monthly gravity field anomaly in the level-2 GRACE products. We assumed that the GRACE level-2 products from CSR, GFZ, and JPL are the different observations of the same monthly gravity anomaly, the land hydrology and ocean mass anomalies, and the north-south stripes, the independent sources.

III. DATA SET

A. Model Outputs Used to Generate Synthetic Signal 1) Land Hydrology Signal From WGHM Model: TheWater-GAP Global Hydrology Model (WGHM) was specifically designed to estimate river discharge for water resource assessments [START_REF] Döll | A global hydrological model for deriving water availability indicators: Model tuning and validation[END_REF]. It computes 0.5° × 0.5° gridded time series of monthly runoff, snow depth, soil water within the root zone, ground water and surface water storage in rivers, lakes, and wetlands, and river discharge. Here, we used the total water storage (TWS) grid for

March 2006 resampled at 1° to represent the continental part of the water cycle.

2) OBP From ECCO Model: Ocean bottom pressure (OBP) data represent the vertically integrated mass of the atmosphere and ocean above a peculiar point of the seafloor. In this

B. GRACE Products

The GRACE mission, sponsored by NASA and DLR, has been collecting data since mid-2002.

1) Raw level-2 products: The level-2 raw data consist of monthly estimates of geopotential coefficients adjusted for each 30-day period from raw along-track GRACE measurements by different research groups (i.e., CSR, GFZ, and JPL). These coefficients are developed up to 60• (or a spatial resolution of 333 km) and corrected for oceanic and atmospheric effects [START_REF] Tapley | GRACE measurements of mass variability in the Earth system[END_REF], [START_REF] Bettadpur | CSR level-2 processing standards document for level-2 product release 0004[END_REF] to obtain residual global grids of ocean and land signals corrupted by a strong noise.

These data are available at ftp://podaac.jpl.nasa.gov/grace/.

2) The monthly raw solutions (RL04) from CSR, GFZ, and JPL were destriped and smoothed by Chambers [START_REF] Chambers | Evaluation of new GRACE time-variable gravity data over the ocean[END_REF] for hydrological purposes. These three data sets are available for several averaging radii (0, 300, and 500 km on the continents and 300, 500, and 750 km on the oceans) at ftp://podaac.jpl.nasa.gov/tellus/grace/monthly.

IV. RESULTS

A. Simulated GRACE Data

In order to validate the ICA method for separation of signals in GRACE products, we make a synthetic data set by adding different levels of noise to reference observations and inverting the results. March 2006 was chosen because continental water mass amplitudes were particularly strong and thus recognizable (i.e., ∼600 mm of equivalent water thickness (EWT) over the Amazon basin).

We generated a simulated GRACE observation by adding three complementary components: land hydrology, ocean, and noise signals. The land hydrology component comes from the TWS, expressed in terms of EWT, of the WGHM model. We used OBP from the ECCO model to simulate the ocean mass anomaly. To be consistent with the spatial resolution of GRACE, TWS and OBP signals were filtered with a 200-300-km Gaussian low-pass filter.

The noise added in our simulations was obtained as the difference between level-1 and level-2 land water solutions from CNES/GRGS [START_REF] Lemoine | Temporal gravity field models inferred from GRACE data[END_REF] over the same monthly time period and normalized from 10% to 100% of the maximum of the synthetic ocean and land signals.

To generate three synthetic sets of observations, similar to the GRACE products from CSR, GFZ, and JPL, for instance, as input for the ICA method, we multiply the noise signal by 0.9, 1, and 1.1 before adding the land hydrology and ocean components. The simulated GRACE observations present a clearly non-Gaussian histogram. Each input grid is converted in a vector of 64 800 (= 360 × 180 grid points) values sorted according to their geographical coordinates for both land and oceans. Hence, we can apply the ICA method to these simulated GRACE observations.

We performed the ICA for levels of noise from 10% to 100%. These levels are defined as the ratio between the maximum of amplitude of the noise and the maximum of amplitude of the simulated GRACE signal. We present the results for a level of noise of 100% (Fig. 1, top).

The FastICA method allows for successfully isolating the hydrological signal over land and ocean from the noise whatever the level of noise. The results of the ICA can be seen in the middle and bottom panels of Fig. 1. The synthetic GRACE input is corrupted by a strong noise whose maxima are of the same intensity as those of the hydrological signal. After applying the ICA method, two clearly separated modes appear: the hydrological signal and noise. The synthetic data correspond to an overdetermined case, that is to say, a case where the number of observations y is greater than the number of sources x. Both spatial patterns and maximum intensity are well separated. Numerical comparisons between the synthetic hydrological signal and its estimate by the ICA method are presented in Table I for increasing levels of noise. These differences logically increase with the noise. In all cases, the mean bias amplitude is lower than 0.5 mm of EWT and slightly negative. The maximum rms is lower than 3 mm of EWT, and the maximum difference reaches 35.5 mm, that is to say, 0.5% (6%, respectively) for a reference hydrological signal amplitude of 600 mm.

This preliminary test of using ICA for separating synthetic signals is successful, as the structures of the hydrological models are fully recovered with realistic amplitudes.

B. Real GRACE Data

We now apply the ICA method to the real GRACE data for separating useful hydrological signals from uncorrelated noise for a given monthly period. We consider three 1° global grids corresponding to the CSR, GFZ, and JPL level-2 GRACE solutions for March 2006 as three independent observations of the Earth's gravity field variations. Our starting assumption is that the level-2 GRACE solution is a linear combination of different spatially independent gravity sources, as long as this resulting mixing does not have a pure Gaussian distribution.

Unfortunately, the histogram of the raw GRACE solutions is purely Gaussian as noisy stripes dominate the observation.

We preprocessed the level-2 GRACE solutions using Gaussian filters with different radius from 300 to 900 km. When the GRACE solutions are Gaussian filtered with averaging radii ranging from 300 to 500 km, the distributions of the observations are sufficiently non-Gaussian to allow the use of the ICA method. For averaging radii lower than 300 km (or greater than 500 km), the signal is dominated by the north-south stripes (respectively, the smoothing of the GRACE solutions is too important); thus, the corresponding histogram is Gaussian. As a consequence, the ICA method is unable to isolate the hydrological signal over land and oceans from noise in these cases.

We applied FastICA to Gaussian prefiltered GRACE solutions for averaging radii of 300, 400, and 500 km. As ICA is linear, the ICA-based CSR, GFZ, and JPL solutions only differ from a scaling factor for each specific component. The ICA-filtered CSR, GFZ, and JPL solutions are obtained by multiplying the j th independent component with the j th mixing vector [START_REF] Tapley | GRACE measurements of mass variability in the Earth system[END_REF]. As the last two modes exhibit similar histograms corresponding to the north-south stripes, we present their sum in Fig. 2. The distribution of each independent component is clearly separated.We chose to present in Fig. 3 the independent the Gaussian-filtered using a radius of 400 km level-2 GFZ solution and the independent components corresponding to ICAfiltered GFZ solutions. The first component is clearly related to continental hydrology with variations in the range of +450/-300 mm (350/ -200 mm, 300/ -150 mm) of EWT for an averaging radius of 300 km (respectively, 400 and 500 km) and to ocean signals with variations in the range of ±230 mm (±180 mm, ±150 mm) of EWT for an averaging radius of 300 km (respectively, 400 and 500 km; see Table II). The larger water mass anomalies are observed in the Amazon, Congo, and Ganges basins over land, and in the Mediterranean Sea and the Gulf of Carpentaria by northern Australia over ocean (Fig. 3, middle). Components 2 and 3 (we present their sum in Fig. 3, right bottom) correspond to the north-south stripes, i.e., resonances in the satellite's orbits. They are smaller by a factor of three or four (Table II).

C. Comparisons With Destriped and Smoothed GRACE Data

The first component has been compared to the destriped and smoothed level-2 GRACE data processed by Chambers [START_REF] Chambers | Evaluation of new GRACE time-variable gravity data over the ocean[END_REF] for each data center. The results of this comparison are presented in Table III over land and oceans for radii of 300 and 500 km. A good agreement between the ICA estimates and the destriped and smoothed GRACE data is observed over the continents and increases with the averaging radius.

This result is consistent with the fact that the dynamics of the noise in the prefiltered observations logically decreases as the averaging radius increases. Table II presents the main characteristics of the ICA estimates for 300-, 400-, and 500-km cases and compares them to the characteristics of the destriped and smoothed solutions. For a 300-km averaging radius, the ICA estimates present a stronger signal and the same variability as destriped and smoothed solutions, whereas for an averaging radius of 500 km, signal amplitudes are similar, but the ICAbased solutions exhibit a lower variability. For an averaging radius of 400 km, the ICA estimates are very similar to the destriped and smoothed products at 500 km (same level of variability, correlation of 0.9) but exhibit a stronger signal.

Over the oceans, the agreement between the ICA estimates and the destriped and smoothed products is lower (Table III). This is mainly due to the fact that the GRACE signal over the oceans is generally weaker than the land signals by factors of two or three, both on seasonal and interannual time scales and also because the mean monthly ocean model was not added back. Moreover, as the signal over the oceans is very different from one GRACE raw product to another, the ICA approach is unable to well separate the geophysical signal from the northsouth stripes.

V. CONCLUSION

Our study is the first attempt to destripe satellite gravity information using an ICA-based strategy, only taking the statistical properties of the observations into account and without any extra information. After having been successfully tested on synthetic noisy data (rms error lower than 0.5% for a maximum amplitude of noise equal to the maximum of the continental hydrology signal), ICA has been applied to separate the statistically independent components of the Earth's gravity field by combining monthly level-2 land water solutions processed by the three official providers of GRACE products (CSR, GFZ, and JPL). The main component corresponds to the water mass anomaly on continents and oceans. The second and third ICA estimates are pure noise of less amplitude, to be cancelled from the GRACE signals for geophysical interpretation. On real GRACE data, this approach requires the use of a preprocessing filter to verify the assumption of non-Gaussianity of the observations. This method gives equivalent results as the destriping method on the continents and even less smoothed results. On the oceans, as the signal is weaker and the input is quite different, ICA is unable to extract correctly the geophysical signal from the noise. Our study demonstrates the ability of ICA to discriminate efficiently geophysical components from the total measured gravity field, as continental hydrology change, and even in the presence of important uncorrelated noise. In the future, this efficient linear technique offers an interesting alternative to classical filtering, for example, it would be considered for cleaning an entire time series of GRACE geoids from polluting noise.

FIGURESFig. 1 .

 1 FIGURES

Fig. 2 .

 2 Fig. 2. Normalized histograms of the GRACE water storage from filtered with a Gaussian filter of 400 km of radius for March 2006 and of the corresponding ICA-estimated independent components.

Fig. 3 .

 3 Fig. 3. GRACE water storage from GFZ filtered with a Gaussian filter of 400 km of radius for March 2006 (top); 1 st ICA component corresponding to land hydrology and ocean mass (middle); the sum of the 2 nd and 3 rd corresponding to the North-South stripes (bottom). Units are mm of EWT.

  climate of the ocean (ECCO) model with data assimilation of sea surface topography from altimetry (ECCO-JPL). The OBP data are available at http://ecco.jpl.nasa.gov/external/.We remove from the TWS (OBP, respectively) grid the time average TWS (OBP, respectively) map between January 2003 and December 2007 to derive anomaly grids, thus being consistent with the processing of GRACE data over land and oceans.

study, we use the 1° × 1° OBP grid for March 2006 from the estimating the circulation &

ACKNOWLEDGEMENTS

This work was supported in part by the fondation Sciences et Techniques pour l'Aéronautique et l'Espace in the framework of the CYMENT Project.

TABLES


published or not. The documents may come

The curved interface with variable thickness

Francesco Dell'Isola, W.