R Fablet
email: rfablet@ifremer.fr

Statistical learning applied to computer-assisted fish age and growth estimation from otolith images

Keywords: statistical learning, otolith interpretation, otolith image analysis, computer-assisted fish age and growth analysis

Statistical learning applied to computer-assisted fish age and growth estimation from otolith images

Problem statement and related work

The acquisition of fish age and growth data from the analysis of calcified structures is at the core of stock assessment and marine Ecology issues. Fish otoliths are among the most commonly used structures for age determination. As illustrated in Fig. 1 for a plaice otolith, otoliths are formed through an accretionary process, which may result in an alternation of translucent and opaque rings. For many species, seasonal rings can be observed. Hence, fish age can be estimated by counting these rings, while the estimation of otolith growth resorts to measuring the distance between the otolith center and the successive seasonal growth rings.

One should stress that the analysis of otolith microstructures may also be considered to analyze daily growth increments. In this paper, the emphasis is given on seasonal growth rings, though the presented methodology is generic and cold be applied to the analysis of otolith microstructures.

[Figure 1 about here.] Assessment or ecological issues typically require the interpretation of several thousand of calcified structures a year per stock or species. This tedious task is achieved by expert readers. Precision and accuracy issues are of key interest since age and growth data serves as the basis for further analysis: for instance, this data is used to estimate biological parameters (abundance, mortality,...) [START_REF] Mesnil | An integrated system for production fish aging: image analysis and quality assurance[END_REF], and they also provide the time reference to calibrate other measures (e.g., chemical signatures) extracted from the calcified structures [START_REF] Pontual | Manual of Fish Sclerochronology, chapter Otolith microchemistry[END_REF].

Therefore, there is a need for defining protocols and tools targeted at quality assurance and quality control [START_REF] Robertson | Development of an artificial neural network for automated age estimation[END_REF].

Computer-assisted tools provide means for the standardization of objective age reading techniques with a view to improving accuracy and precision levels. A va-riety of solutions can be designed to help in the acquisition of age and growth data (for instance, the automation of the acquisition of series of otolith images [START_REF] Ogor | Tnpc software: digital processing of calcified structures[END_REF] or the storage and management of bases of images of interpreted calcified structures (CS) [START_REF] Robertson | Development of an artificial neural network for automated age estimation[END_REF][START_REF] Ogor | Tnpc software: digital processing of calcified structures[END_REF])) or to automate this task thansk to the estimation of fish age and growth estimation from CS images [START_REF] Fablet | Automatic fish ageing from otolith images using statistical learning[END_REF][START_REF] Guillaud | Autonomous agents for edge detection and continuity perception on otolith images[END_REF][START_REF] Robertson | Development of an artificial neural network for automated age estimation[END_REF][START_REF] Troadec | Use of deformable templates for otolith 2D growth ring detection by digital image processing[END_REF]. The research effort has been mainly focused on the automation of the estimation of fish age and growth. Two broad categories of approaches can be distinguished: 1D methods [START_REF] Lagardère | Age estimation in common sole solea solea larvae: validation of daily increments and evaluation of a pattern recognition technique[END_REF][START_REF] Welleman | Automatic ageing of Plaice otoliths by means of image analysis[END_REF] restrict their analysis to an intensity profile extracted along a given reading axis, while 2D techniques [START_REF] Guillaud | Autonomous agents for edge detection and continuity perception on otolith images[END_REF][START_REF] Troadec | Use of deformable templates for otolith 2D growth ring detection by digital image processing[END_REF] are aimed at providing a 2D segmentation of growth rings. While being far more robust, proposed 2D techniques are still limited to the detection of likely growth rings, and do not cover the interpretation task which requires a priori information to cope with false rings ro checks. The study by Robertson and Morrison (Robertson and Morison, 1998) was the first attempt to benefit from readers' experience to design a better system based on neural networks. Though promising, the reported results highlight the need for a better extraction of the information conveyed by otolith images. Though promising, the reported results highlight that the exploited image features (Fourier coefficients of the intensity signal along a given axis) do not to bring significant improvement for age estimation compared to other features (eg, otolith weight, fish size and sex). This somewhat contradicts the interpretation protocols followed by experts, who mainly rely on the interpretation of the internal otolith macrostructures to estimate the age. Hence, there is need for an improved automated extraction of the information conveyed by otolith images in terms of alternation of translucent and opaque rings.

In this paper, a statistical learning strategy is also adopted and it is assumed that one can benefit from experts' experience by means of bases of interpreted otolith images. Whereas Robertson and Morrison only focus on fish age estimation from otolith images, two other applications are investigated: the estimation of individual growth patterns from otolith images and the definition of indicators of the reliability of readers' interpretation. To our knowledge, our work is the first attempt to provide such tools for computer-assisted aging. The key point of this work is to state the considered within a pattern classification framework, so that it benefits from the recent advances in statistical learning [START_REF] Bishop | Neural Networks for Pattern Recognition[END_REF][START_REF] Scholkopf | Learning With Kernels: Support Vector Machines, Regularization, Optimization and Beyond[END_REF]Zhu and Hastie, 2001). They provide us with the methodological background to infer interpretation rules from training examples. This paper is organized as follows. Section 2 points out the key ideas of kernel-based statistical learning. Section 3 presents the information extraction stage within otolith images and the application to fish age estimation. The analysis and the estimation of individual otolith growth patterns is discused in Section 4. In Section 5, the design of a confidence measure of readers' interpretation is detailed. Results for plaice otolith samples are reported in Section 6 and Section 7 provides concluding remarks and an outline for future work.

Statistical learning using kernel methods

Age and growth analysis from otolith images is regarded as a pattern classification issue. More precisely, two main issues are considered: mapping otolith images to age groups and evaluating whether or not a given otolith growth pattern is realistic.

In the first case, the classes are given by the age group and numerical descriptors will be extracted from otolith images to define a feature space. In the second case, it will resort to a binary (realistic vs. non-realistic) classification issue, where the feature space will be defined from otolith growth patterns. Thus, in both cases, a generic classification issue is considered: assigning a class to observed feature vectors.

To solve for this issue, kernel approaches are among the most efficient techniques [START_REF] Scholkopf | Learning With Kernels: Support Vector Machines, Regularization, Optimization and Beyond[END_REF]. The kernel principle is illustrated in Fig. 2.

The use of a kernel function can be viewed as a means to map the original feature space to a higher dimensional space, within which the considered classification issue should be easier solved for. Given a feature space X , kernel approaches rely on the introduction of the so-called kernel function K. Its key feature is to be defined as a dot product within a mapped space:

∀(x, x ) ∈ X 2 , K(x, x ) =< Φ(x), Φ(x ) > (1)
where Φ defines a mapping from the initial feature space X to the transformed one Φ(X ). A typical example of kernel function is given by the radial basis function:

K rbf (x, x ) = exp(-(x -x ) 2 /2σ 2 ).
It should be stressed that the mapping function Φ is never actually used. All the computations only involve its product.

This property is often referred to as the kernel trick.

[Figure 2 about here.]

In the following, the focus is given to two categories of kernel approaches:

Support Vector Machines (SVM) and Kernel Logistic Regression (KLR).

Support vector Machines

SVMs have initially been developed by Vapnik [START_REF] Vapnik | Statistical learning theory[END_REF] for character recognition. Given a binary classification issue within the feature space X and a training set {(x i , y i ) ∈ X × {-1, 1}}, SVM techniques resort to the estimation of the hyperplane best separating as illustrated in 2. In the transformed feature space Φ(X ), the hyperplane equation is given by: < Φ(w), Φ(x) > +b = 0, where w is the vector normal to the hyperplane, and b a translation parameter. Each half-space separated by this hyperplane refers to the two classes assigned according to the 10 decision function y = sign (< Φ(w), Φ(x) > +b).

More formally, SVMs are maximum margin classifiers and resort to the following minimization [START_REF] Scholkopf | Learning With Kernels: Support Vector Machines, Regularization, Optimization and Beyond[END_REF]:

( ŵ, b) = arg min (w,b) 1 2 Φ(w) 2 + C i η i (2)
under the constraints ∀i, y i (< w, x i > +b) ≥ 1η i and η i ≥ 0. Variables η i are measures of the errors of the classification of the data (x i , y i ): Thus, the term i η i is a complexity-related penalty term, and the constant C a regularization parameter used to balance the maximization of the margin against this complexity term.

Solving for this minimization issue comes to select support vectors {x SV j } within the training data {x i } and to estimate their weights {α j } such that the hyperplane vector w is given by Φ(w) = j α j Φ(x SV j ). Hence, the binary decision function over X is defined as:

d SV M (x) = sign   j α j K(x SV j , x) + b   (3)
The parameters of a SVM model are the supports vectors {x SV j }, the weights {α j },

5
the bias b and the kernel function K.

Kernel Logistic regression

Contrarily to SVMs, kernel logistic regression (KLR) is aimed at estimating a probabilistic output P KLR (y(x) = 1) of the classification function. Using a logistic model, P KLR (y(x) = 1) is set as (Zhu and Hastie, 2001):

P KLR (y(x)) = 1 + exp i α i K(x i , x) + b -1 (

Parameter estimation

In practice, SVMs and KLR models require the selection of the appropriate kernel function K. In this paper the radial basis function is used:

K rbf (x, x ) = exp(-(x -x ) 2 /2σ 2 ) parametrized

Automated fish age estimation

In a first step, the aim is to infer fish age from otolith images. To state this issue as a classification issue, it is first required to define an appropriate feature space X . Therefore the computation a vector of numerical descriptors of the 2D image content is needed. Consequently, the extraction of relevant features from otolith images (in terms of alternation of translucent and opaque rings) is first detailed.

Then, the application of a statistical learning scheme is described to automate fish age estimation from the defined feature space.

From 2D image content to 1D representation

Since otolith interpretation is usually performed by experts along a predefined reading axis. For instance, for plaice otoliths, as shown by Fig. 1, experts mainly consider the zone corresponding to the greatest growth. Consequently, our automated analysis the analysis of the content of otolith images can then be restricted to a region around the main reading axis. Let us stress that the proposed approach can be easily extended to several reading axes if required.

[Figure 3 about here.]

A template-based approach inspired from [START_REF] Troadec | Use of deformable templates for otolith 2D growth ring detection by digital image processing[END_REF] is then exploited to extract a one-dimensional signal depicting the 2D image content within an angular sector around the considered reading axis. More precisely, as illustrated in Fig. 3 given the otolith center 0 and an angular sector S, the following 1D signal s S is computed:

s S (α) = med(I(O + α * T S )) (5)
where med() is the median operator, T S is the template model used within S, I the image intensity function, and α a scaling factor between 0 and 1. As in [START_REF] Troadec | Use of deformable templates for otolith 2D growth ring detection by digital image processing[END_REF], the template model T S is given by the sampled external otolith shape within S. s S (α) is then the median intensity value along the template T S scaled by α w.r.t. the otolith center O. Since the scaling of the external otolith shape w.r.t.

the otolith center provides a good approximation of the shape of the seasonal rings [START_REF] Troadec | Use of deformable templates for otolith 2D growth ring detection by digital image processing[END_REF], s S depicts oscillations corresponding to the alternation of translucent and opaque rings within S.

Peak-based representation

Expert interpretations rely on the detection of ridge and valley structures corresponding to translucent and opaque rings. In order to detect ridge and valley structures within S, one needs to extract the meaningful extrema of the signal s S . This is achieved by determining the zero-crossings of derivative of s S . Since otolith growth is not linear but rather exponential, s S is not stationary in terms of frequency content. It is then needed to adapt the scales at which the derivative of s S is computed. This is done by applying a convolution to a Gaussian kernel whose variance is deduced from an a priori exponential otolith growth model. This is similar in spirit to the demodulation scheme applied in [START_REF] Lagardère | Age estimation in common sole solea solea larvae: validation of daily increments and evaluation of a pattern recognition technique[END_REF].

Given the positions of the detected maxima (or equivalently the minima) of s S {t 1 , ....., t n }, a new normalized peak-based signal s P B S is defined by:

s P B S (t) = min {ρ(t -t k /t n )} k∈{1,...,n} (6)
where ρ is a distance kernel. The computation of {t 1 , ....., t n } is based on the detection of the zero crossings of the first derivative of s G S , estimated by finite differences. In practice, ρ(u) = 2atan(u)/π. The resulting signal s P B S only depends on the extrema position as shown in Fig. 4. In particular, compared to s G S , s P B S is invariant to the variability of intensity ranges over images.

[Figure 4 about here.]

Application to fish age estimation

The feature space defined by the normalized peak-based representation of the content of the otolith is rather high-dimensional (typically about 1000 dimensions).

Hence, the straightforward application of SVM and KLR techniques is not feasible. A Principal Component Analysis (PCA) is therefore applied to retain only the first N P CA principal components (typically, N P CA < 100).

Given this reduced feature space, fish age estimation comes to map an image n to an age group given its feature vector x n . To cope with this multi-class issue, binary SVM and KLR models are extended to multi-class classification. A one vs. all strategy [START_REF] Scholkopf | Learning With Kernels: Support Vector Machines, Regularization, Optimization and Beyond[END_REF] The reader is referred to [START_REF] Fablet | Automatic fish ageing from otolith images using statistical learning[END_REF] for a more detailed description of the training issues, especially in terms of improvement of the generalization properties of the learned model.

Analysis and estimation of individual growth patterns

In addition to inferring fish age from otolith images, a key application is to estimate individual otolith growth patterns and at evaluating the relevance of otolith interpretations w.r.t. previously interpreted samples. Both tasks require the development of an appropriate otolith growth model.

Otolith growth is usually modeled using some a priori mean model, such as exponential and polynomial models, and the associated statistical distribution characterizing the variability of the studied growth phenomena. This category of statistical models is not suited for the issues considered in this paper. It rather comes to evaluate whether or not a given growth pattern is likely to be an actual one.

Due to the huge natural variability as illustrated in the experiments (see Section 6), the distance to a simple mean a priori model is not sufficient to tackle this binary classification issue. Therefore, a statistical learning scheme is exploited in order to train a system from labeled (realistic vs. non-realistic) examples of individual growth patterns.

Statistical modeling of individual growth patterns

Let us denote by g an otolith growth pattern defined by the sequence of the distances {g 1 , ..., g A } from the otolith center to the translucent (or equivalently opaque) rings. Our aim is to evaluate the likelihood P Gr (g) = P (g is realistic).

The probabilistic otolith growth model P Gr (g) is estimated by statistical learning from a dataset of growth patterns labeled as realistic or non-realistic. Let us denote by G this dataset. Kernel logistic regression is applied using the feature vector X g = {g, ∆g} with ∆g = {g ig i-1 } for any sample g ∈ G. The computation of ∆g to define this feature vector is motivated by the key information brought by growth increments to discriminate actual growth patterns from erroneous ones. Let us stress that the length of the feature vector X g depends on the number of growth rings. Therefore, one model P Gr (.|A = a) is learned for each age class a.

The relevance of the learned models P Gr (.|A) depends on the ability of the samples in G to convey enough information to discriminate realistic growth patterns from erroneous ones. Given a set of otolith images interpreted by an expert, it is assumed that training samples for the "realistic" class are provided, but it is also needed to determine examples for the "non-realistic" class. The following procedure is exploited. For all the training images, the signal s S (α) as defined in Eq.5 is extracted, and the positions of its local maxima {t i } i∈ [1,Nt] are computed.

From this position set {t i }, the 2 Nt subsets obtained by selecting from 1 to N t ring positions in {t i } are generated and the associated growth configurations are computed. From this set of 2 Nt growth configurations, all those different from the expert interpretation are kept as samples for the class "non-ralistic". This scheme is illustrated in Fig. 5 [Figure 5 about here.]

For each age class a, the probabilistic model P Gr (.|A = a) is estimated from the resulting training set using a cross-validation scheme as explained in Section 2.

Automated estimation of individual growth patterns

The definition of the probabilistic otolith growth models {P Gr (.|A = a)} is first applied to the estimation of individual growth patterns from otolith images.

Given an otolith image I, the first step comes to extract the s S (α) as defined in Eq.5 and to compute the positions {t i } of its local maxima. The age estimator introduced in Section 3 infers an age class a(I) from s S . Then, one generates the set G(a(I)) of all the growth configurations corresponding to the age class a(I) which can be generated from the set of the candidate ring positions {t i }.

Growth patterns within G(a(I)) are sorted according to their likelihood w.r.t. the growth model P Gr (.|A = a(I)). The one with the greatest likelihood provides the estimated otolith growth pattern ĝ: ĝ = arg max g∈G(a(I))

P Gr (g|A = a(I)) (7)
Let us point out that this scheme could also be used as an interactive tool for the acquisition of otolith growth data for previously interpreted samples: given an expert estimation of age, the positions of the growth rings can be automatically recovered.

Confidence measure of otolith interpretations

Beyond the automated estimation of fish age and growth from otolith images, the computation of a confidence measure is investigated to evaluate the reliability of otlith interpretations. As illustrated by Fig. 5 for two different interpretations of the same otolith, the aim is to define a measure pointing out that the top interpretation is unlikely, while the bottom one seems correct.

Formally, the definition of this confidence measure is viewed as the evaluation of the likelihood P Conf (g is relevant|I) that a given interpretation (or growth pattern) g is relevant given the observed otolith image I. Exploiting Bayes rule, P Conf (g is relevant|I) is written as :

P Conf (g is relevant|I) ∝ P (I|g is relevant)

• P (g is realistic)

(8) P (I|g is relevant) is aimed at determining how likely the image I is correctly explained by the interpretation g in terms of age and growth. As a first step, this likelihood is computed from the statistical model P Age (a(g)|.) introduced in Section 3, with a(g) the age estimate associated with the growth pattern g. Therefore, the peak-based representation s P B S is computed within an angular sector specified by the expert as detailed in Section 3, and the likelihood P (I|g is relevant) is evaluated as:

P (I|g is relevant) = P Age (s P B S |a(g)). (9)
Similarly, P (I|g is not relevant) is defined as 1 -P Age (s P B S |a(g)). P (g is realistic) conveys a priori information on growth patterns. It is evaluated from the probabilistic growth model P Gr (g|A = a(g)), as introduced in Section 4. Besides, P (g is not realistic) is given by 1 -P Gr (g|A = a(g)).

P Conf (g is relevant|I) is then expressed as follows:

P Conf (g is relevant|I) = P Age (s P B S |a(g))

• P Gr (g|a(g)) • Z -1 [START_REF] Scholkopf | Learning With Kernels: Support Vector Machines, Regularization, Optimization and Beyond[END_REF] with the normalization factor Z given by:

Z = P Age (s P B S |a(g)) • P Gr (g|a(g)) + 1 -P Age (s P B S |a(g)) (1 -P Gr (g|a(g))) (11)
Given an image I and an interpretation g, the closest P Conf (g is relevant) to one, the more relevant the interpretation. Conversely, the closest P Conf (g is relevant)

to zero, the less relevant the interpretation. Let us point out that this relevance measure needs to be interpreted with respect to the variability encountered within the training sets exploited to learn the probabilistic models P Age and P Gr .

Results

Otolith database

[Figure 6 about here.]

[Figure 7 about here.]

Experimental results are reported for a database of 320 images of plaice otoliths from age group 1 to 6 caught during the fourth quarter in 1994 and 2000. The ground truth for fish age estimation is given by the interpretation of an expert.

Due to the lower number of samples in age groups 5 and 6, these two groups are merged for aging issues to end up with a five-class issue: age groups 1 to 4 and 5+.

This grouping can also be motivated by the fact that age groups 4 and less usually represent more than 85% of the commercial landings. Similarly, too few samples from age group 0 were available to add this class in our experiments. The age distribution of this otolith set is given in Fig. 6.a. As an illustration, Fig. 7 depicts sample images for the different age groups. Fig. 6.b also stresses the biological variability encountered when dealing with fish age and growth analysis. More precisely, this plot proves that there is a high degree of overlap in the distance from the otolith center to the first three translucent rings. This feature makes it infeasible to develop an accurate a priori prediction of the ring positions and leads to a complex multi-class classification issue.

For different issues considered in this paper, the training set is formed by selecting the two third of the images of each age class, while the remaining images are used for the test set. The training set and the test set then depict the same age distribution displayed for the whole image databse in Fig. 6.

Automated ageing

In Fig. 8 is reported the classification results for the images of the test set (i.e., one third of the considered image dataset for each age group), the mean rate of correct classification (w.r.t. the expert ground truth) with SVMs and KLR is of 88%. Not surprisingly, classification errors are more numerous for the older age groups. it should also be stressed that for 96% of the samples the absolute ageing error is below or equal to one. This mean rate of correct age estimation is close to inter-expert agreement rates. This highlights the interest of the proposed scheme and the need for a carefully designed pattern recognition scheme to achieve good classification performance.

[Figure 8 about here.]

These results can be favorably compared to those obtained by previous approaches [START_REF] Guillaud | Autonomous agents for edge detection and continuity perception on otolith images[END_REF]. In addition, the application to routine aging seems promising since these results are in the range of the levels of agreement observed between expert readers. The reader is referred to [START_REF] Fablet | Automatic fish ageing from otolith images using statistical learning[END_REF] for a more detailed evaluation of the proposed approach for automated ageing and a comparison between different feature types (frequency vs peak-based features) and statistical classifiers (neural nets vs. SVMs). This evaluation stresses that kernel-based learning outperforms neural networks, and that the proposed feature set leads to significant improvements (greater than 10% in terms of mean correct classification rate) compared to previous work.

Estimation of individual growth patterns

[Figure 9 about here.]

As an illustration of the proposed approach for the estimation of individual growth patterns, Fig. 9 depicts an otolith image belonging to the age class 4. It reports the candidate ring positions which are provided by the maxima of the template-signal s S , the expert interpretation. From the set of growth configurations of age class 4, the configuration with the greatest likelihood value P Gr (g|a = 4) is depicted. As shown by Fig. 9, the actual growth pattern is successfully estimated. Similarly to the age estimation application, the probabilistic growth model is trained from the two third of the considered plaice datasets.

A quantitative evaluation of the proposed approach has also been performed.

Given the test set of otolith images, the statistical learning approach has been applied to the estimation of the growth pattern, given the age estimate provided by the expert. For 95% of the samples, one retrieves the same growth interpretation as the one provided by the expert.

Confidence measure of otolith interpretations

[Figure 10 about here.]

In Fig. 10 several examples of the computation of a confidence measure of the interpretation of otolith growth rings for plaice otoliths are displayed. For each image, the confidence measure is compputed for two different interpretations the expert one and an erroneous one. As expected, the confidence measure is greater than 0.5 in the first case and closer to zero in the latter. Besides, Fig. 10 shows that the computation of these confidence measures may assist the reader in ambiguous cases. For instance, for the second and third examples reported in Fig. 10, it helps in interpreting the first growth ring. This case illustrates the potential interest of this confidence measure for routine ageing, where the interpretation of the first ring is known to be one of the key issue.

These different examples stress that the computation of these confidence measures of otolith interpretation could be used for routine ageing issues: either as an interactive tool to assist the expert for ambiguous cases, or as an automated verification tool to detect erroneous or unlikely interpretations within a set of interpreted otolith sets with a view to asking for a second reading.

Conclusion

This paper has discussed the use of statistical learning techniques, namely kernel approaches, to benefit from a training set of interpreted data in order to cope with computer-assisted fish age and growth analysis. Information extraction within otolith images has been regarded as the computation of a template-based 1D signal summing up the image content within a predefined area of interest (typically defined as an angular sector around the main reading axis to age determinations for rather "easy-to-read" species such as flat fish otoliths, more complex cases need to be investigated. For instance, similar types of probabilistic otolith growth models should be considered for situations where the growth analysis needs to be performed along polylines or curved axis. Otherwise, whereas the proposed peak-based representation of the content of otolith images and the exploited learning strategies can be regarded as generic, the emphasis should be given on the information extraction stage (for instance, for the detection of the growth rings) to cope with samples like cod or hale otoliths. Besides, the issue of combining the proposed approaches to schemes dedicated to 2D ring segmentation [START_REF] Guillaud | Autonomous agents for edge detection and continuity perception on otolith images[END_REF][START_REF] Troadec | Use of deformable templates for otolith 2D growth ring detection by digital image processing[END_REF] is also of particular interest.

List of Figures

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  4) where {x i } are the training data, {α i } their estimated weights, and b the estimated bias. One should point out that KLR does not share the important property of sparsity of SVMs: whereas in SVMs, only a few training samples are kept as the support vectors to actually define the classification function, all training samples contribute to the probabilistic output of the KLR model. However, the estimation of their parameters are issued from a slight adaptation of the SVM algorithms (Zhu and Hastie, 2001).

  by the scale parameter σ. In addition, in both cases, it is required to estimate the optimal value of the regularization parameter C. Since no theoretical result exists to set σ and C, one generally uses a crossvalidation strategy to determine the best parameter setting within a grid of predefined values of the regularization parameters C and of the kernel parameter σ. The set of training samples T is first split in a number n (typically, n is set to 5) of subsets {T 1 , ..., T n } of equal number. The cross-validation procedure is then applied as follows. At iteration k between 1 and n, a SVM (respectively KLR) model is estimated using the {T j , j = k} as the training samples. The performance of the estimated classifier is evaluated according to a loss function (for instance, the mean misclassification rate) for the samples of the subset T k . The final stage consists in computing the mean loss value for the different values of the regularization and kernel parameters, and in retaining the parameter setting with the best classification performance.

  is used. It comes to train several binary classifiers, using a given age group as the first class and merging all the other age groups to define the second class. The outputs of these binary classifiers are combined according to a simple voting rule to lead to am ulti-class classification. When using KLR models, let us denote by P Age (.|A = a) the resulting probabilistic model for the age class a.

69Figure 1 :

 1 Figure 1: Illustration of plaice otolith interpretation for a 4 year old individual. The otolith image has been acquired using a binocular microscope under transmitted light. The interpretation of the winter translucent rings is displayed by the markers set on the radial drawn on the main reading axis.

Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 2345 Figure2: Key principle of kernel-based classification: map the original feature to a higher-dimensional feature space via the kernel function Φ and determine the separating hyperplane with maximum margin (from[START_REF] Scholkopf | Learning With Kernels: Support Vector Machines, Regularization, Optimization and Beyond[END_REF]).

Figure 8 :Figure 9 :

 89 Figure 8: Plot of the ageing errors from age group 1 to age group 5+ for the plaice test set.

  ). Using statistical learning, this signal can be used to infer fish age. New probabilistic models of otolith growth, learned from otolith samples interpreted by an expert, have been introduced to discriminate realistic growth patterns from erroneous one. These models are exploited to estimate individual otolith growth patterns and to compute confidence measures of otolith interpretations.The evaluation of the proposed approach on a set of several interpreted plaice otoliths from the fourth quarter validates the proposed models and demonstrates their key interest for routine age and growth estimation, especially in the context of quality assurance and quality control in age determinations. There are a number of potential applications such as the complete automation of the acquisition of age and growth data for otolith subsets, the automation of the creation of bases of CS images and of the associated interpretations from aged CS datasets, or computeraided CS interpretation. At the operational level (for instance, for the determination of age structures used for stock assessment), the key benefits expected from these tools are to reduce the cost of the acquisition of age data from CS, while improving data quality (mainly, in terms of precision) and data traceability (both in terms of age determinations and in terms of growth ring interpretations through the automation of the creation of databases of images of interpreted materials). Some potential limits of the application of statistical learning strategies should also be pointed out. Since they rely on the sets of interpreted CS structures with a view to automatically inferring the associated interpretation rules and to applying these rules to new samples, the capability of the trained systems to efficiently extrapolate from the training set obviously depends on the features and the associated spread within the training set w.r.t. to those within the test set. For instance, if significant changes occur in otolith growth for a given stock, new training sets might be needed to apply these techniques from one year to another. This is of key importance and should be evaluated in future work. Similarly, such techniques are not able to cope with age classes not present within the training sets and their performance in terms of precision and accuracy rely on the quality of the training data. In most cases, the latter provides upper bounds for the performances of the trained system. One might point out that it should be preferred to build training sets from known age materials, if available. Consequently, the transfer to the operational level of the proposed methodology should be evaluated on a species-and stock-specific basis. While it should first provide a relevant alternative and support

1

  Illustration of plaice otolith interpretation for a 4 year old individual. The otolith image has been acquired using a binocular microscope under10 transmitted light. The interpretation of the winter translucent rings is displayed by the markers set on the radial drawn on the main reading axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	2	Key principle of kernel-based classification: map the original fea-
	15	ture to a higher-dimensional feature space via the kernel function
		Φ and determine the separating hyperplane with maximum margin
		(from (Scholkopf and Smola, 2002)). . . . . . . . . . . . . . . . .
	3	Template-based approach aimed at extracting a 1D representation
		of the image content within a region of interest. . . . . . . . . . .
	4	Illustration of the peak-based representation for the otolith image

 Zhu, J. and Hastie, T. (2001). Kernel logistic regression and the import vector machine. In Int.Conf. on Neural Information Processing Systems, Vancouver.

Acknowledgements:

The author wishes to thank A. Benzinou for fruitful discussion on the templatebased model and N. Le Josse for his input to this work. Besides, the author is thankful to A. Ogor for his help in the acquisition of the image dataset.


Ronan Fablet
email: rfablet@ifremer.fr

Sylvain Pujolle

Anatole Chessel
email: achessel@ifremer.fr

Abdesslam Benzinou
email: benzinou@enib.fr

F Cao
email: fcao@irisa.fr

VARIATIONAL LEVEL-SET RECONSTRUCTION OF ACCRETIONARY MORPHOGENESIS FROM IMAGES

come

Variational level-set reconstruction of accretionary morphogenesis from images.

INTRODUCTION AND PROBLEM STATEMENT

Shape deformation has been widely studied in the field of computer vision mainly for applications to shape matching or deformable object analysis [START_REF] Caselles | An axiomatic approach to image interpolation[END_REF]16]. In this paper, we focus on the deformation of structures which grow according to an accretionary process. In the biological domain, examples of such structures include tree trunks, the shells of shellfish, fish otoliths, corals, bones, vertebrae.... Other examples related to crystallization process could also be cited. The characteristics of the accretionary growth generally vary over time, generally with some given periodicity (mainly daily and annual), in terms of crystalline organization or composition. This leads to the presence of successive dated layers with different optical densities and provides the basis for exploiting such structures as biological archives: for instance, for the reconstruction of the daily temperatures of the environment [START_REF] Daverat | Reconstruction of life histories chronologies from measures sampled on the otolith using mixture models and hidden Markov models: case study of the eel (anguilla anguilla) otolith Sr:Ca ratios[END_REF] or for the analysis of individual migration paths [START_REF] Durand | Reconstruction of wavelet coefficients using total variation minimization[END_REF].

Concerning shape characteristics, the presence of these successive layers potentially permit to back-track the evolution of the shape of the observed structure from the primary core to the outline. The morphogenesis of such structures could then be studied at the individual level, and not only from a statistical point of view at a population level. However, this issue does not reduce to matching successive shapes [START_REF] Caselles | An axiomatic approach to image interpolation[END_REF]16], since the automated segmentation of the growth rings is itself a complex problem due to the presence of blind areas and so-called subjective contours [START_REF] Gonzalez | Digital Image Processing[END_REF]. But, it can be stated as an inverse problem for which the goal is to reconstruct the series of the 2D shapes of the observed structure within some observation plane given an image of this structure. Image orientations (i.e., the orientations of image gradients) locally convey relevant information on the shape of the growth rings, since they can be viewed as estimations of the growth directions. Consequently, the reconstruction of the series of shapes of the processed biological structure is solved for such that these shapes are locally normal to image orientations. Formally, a level-set representation of the accretionary growth permits to derive a variational formulation from purely geometric constraints including both orientation-based and shape regularity constraints. The reconstruction of the morphogenesis then re-y Now t=t0 t t=0 Otolith image x Fig. 2. Level-set representation of the accretionary growth process: the evolution of the shape is described by a potential function U , such that the growth shape at time t is its levelline Γ t (U ) = {p ∈ R 2 such that U (p) = t}. sorts to a variational interpolation which is efficiently solved for within a multiresolution framework.

PROPOSED APPROACH

Joint level-set representation and variational setting

The accretionary growth process is modeled as a potential function U defined over R 2 with values in [O, T ] such that the level set Γ t (U ) of U for level t is the shape of the considered biological structure within a given observation plane at time t, that is to say

Γ t (U ) = {p ∈ R 2 such that U (p) = t} (1)
Intrinsically, such a representation guarantees that the growth is normal to the shape, as commonly assumed for accretionary growth. Note that, in practice, the actual tile information might not be available, so that the values of potential function U are not actual time values, but are homogeneous to time.

Based on this level-set representation, we aim at estimating the potential function U associated with a given biological structure. Formally, some boundary conditions are assumed to be provided. These conditions include the shapes at times 0 and T, which may be automatically extracted [START_REF] Caselles | Geodesic active contours[END_REF]. solving for the reconstruction of U is then stated as the computation of the extension of the boundary values G B associated with boundary conditions B, to the whole domain R 2 according to some energy criterion E:

U = arg min U∈U (GB) E(U ), (2)
where U(G B ) is the set of the functions from R 2 to R which are equal to G B on boundary set B. Energy E(U ) is split into two terms: a regularization term issued from a geometric shape prior and a data-driven term setting orientation-based constraints:

E(U ) = E R (U ) + γE O (U ), (3)
where γ is a weighting factor balancing the relative influence of each term. The regularization term is defined as the sum of the perimeter of the growth shapes computed as

E R (U ) = t∈[0,T ] p∈Γt(U) 1. (4)
The second term relies on orientation-based constraints. More precisely, the gradients of the image depicting the observed biological structure permit to locally infer the local growth orientations. Let us denote by ω the orientation field computed as the normalized image gradient ∇I/|∇I|, where ∇I is the gradient of image I. For low gradient values, the computation of the orientation is highly noisy. The norm of the gradient is however known as a relevant measure of the reliability of this gradient information. Thus, a confidence map α is defined as g(|∇I|), where g is a continuous stepwise function rescaled between 0 and 1. As the normal to shape Γ t (U ) is expected to be tangent to orientation ω given the confidence weight α, energy term E O (U ) is computed as follows

E O (U ) = t∈[0,T ] p∈Γt(U) α(p) • ∇F (p) |∇U (p)| , ω(p) ⊥ 2
(5) Minimizing explicitly criterion E(U ) is however infeasible in practice, since it would require to extract all the levellines of U . From the co-area formula [START_REF] Monasse | Fast computation of a contrast invariant representation[END_REF], an equivalent formulation can be derived such that the resulting criterion only involves the spatial derivatives of U . More precisely, the minimization of E(U ) resorts to:

U = arg min U∈U (GB) p∈R 2 |∇U |+γ•α•|∇U |•ρ ∇U |∇U | , ω ⊥ (6)
This variational setting involves two classical terms: the first one is regularization term set as a the norm of the gradient of the potential function and the second is the data-driven term, which aims at aligning the orientation field of U to orientation constraints ω. In this second term, norm |∇U |, inherited from the co-area transform, is a weighting factor such that more influence is given to points where the gradient of the level-set representation is high (conversely, where the growth is slow).

Criterion ( 6) is derived from purely geometric constraints. In addition, level-set representations are contrast invariant (i.e., a contrast change does not affect the geometry of the levelsets but only leads to different indexes for each level-set) [START_REF] Odobez | Separation of moving regions from background in an image sequence acquired with a mobile camera[END_REF]. Consequently, additional constraints are required to prevent from retrieving unstable solutions. As constraints on the surface between successive level-lines are equivalent to constraints on first-order statistics p(U ) of U , minimization (6) will be carried out subject to p(U ) is uniform. The choice of a uniform prior is arbitrary. Other priors might be used, if available, for instance priors related to a priori growth laws.

Minimization issue

To carry out the above constrained minimization, a two-step iterative approach is proposed: the first step solves for unconstrained minimization [START_REF] Daverat | Reconstruction of life histories chronologies from measures sampled on the otolith using mixture models and hidden Markov models: case study of the eel (anguilla anguilla) otolith Sr:Ca ratios[END_REF], and the second step carries out an histogram equalization [START_REF] Huber | Robust Statistics[END_REF] to map the current solution onto the constraint that p(U ) is uniform. We detail below the unconstrained minimization. Besides, to improve the convergence of this procedure and reduce the dependence with respect to the initialization, we adopt a multiresolution framework.

Unconstrained minimization ( 6) is solved for using a discrete/Markovian setting of the associated variational criterion. Hence, energy E(U = is computed as:

p∈R 2 |∇U p | + γ • α p • |∇U p | • ρ ∇U p |∇U p | , ω ⊥ p , (7)
where ∇U p is approximated as (U p1 -U p3 , U p2 -U p4 ) with p 1 , p 2 , p 3 , p 4 the four western, northern, eastern and southern neighbors of pixel p in the image. Besides, as the absolute norm | | is not derivable in 0, its use may lead to numerical instabilities. Therefore, it is replaced by ρ(x) = √ 2 + x 2 . An iterated weighted least square (IRLS) scheme is then exploited to minimize the resulting criterion [START_REF] Lin | Geometric Measure Theory: An Introduction[END_REF]. This schemes iterates the computation of weights β(p) = φ (∇U p ), where φ(x) = ρ (x)/x, and the computation of the solution of the weighted least-square issue

min U∈U (GB) p∈R 2 β(p)|∇U p | 2 +γ •α p •• ∇U p |∇U p | , ω ⊥ p 2 (8)
given weight maps β. Using an incremental Gauss-Seidel scheme, we recursively solve for the optimal incremental update of potential U p at pixel p given the potential function elsewhere. This step relies on a linearization of the cost function with respect to the increment. This incremental refinement is applied to all pixels except those belonging to boundary set B.

From the updated solution Ũ of the unconstrained criterion [START_REF] Daverat | Reconstruction of life histories chronologies from measures sampled on the otolith using mixture models and hidden Markov models: case study of the eel (anguilla anguilla) otolith Sr:Ca ratios[END_REF], a new solution is derived such that the constraint that p(U ) is uniform applies. This step simply comes to implementing an equalization of the histogram of Ũ. As stressed previously, this equalization step only changes the contrast between level-lines, but the geometry of the level-set representation remains unchanged. For computational efficiency, this equalization step is not applied after each incremental update, but typically every one hundred iterations. This parameter setting was experimentally proven to ensure the convergence to non-degenerate estimations.

Multiresolution minimization

Minimization (6) is obviously not convex. To prevent from being trapped in local minima and to ensure a better robustness to the initialization, a multiresolution scheme is used. Given a Gaussian pyramid of the orientation field ω, the multiresolution approach comes to solve for the estimation of the potential function U at successive resolutions, from the coarsest resolution to the finest one. The final estimate at a given resolution serves as the initialization at the next resolution. In practice, Gaussian pyramids with four levels are exploited.

Initialization

The computation of a relevant initialization to the proposed gradient-based multiresolution scheme is important to ensure a fast and pertinent convergence. The initialization is delivered by the AMLE (Absolutely Minimizing Lipschitz Extension) of the boundary conditions to the whole domain [START_REF] Chauvaud | The shell of the Great Scallop Pecten maximus as a tool for high frequency paleoenvironmental reconstruction[END_REF]. Since the AMLE resorts to the minimization of the L ∞ norm, the AMLE can be viewed as the less smooth interpolation operator among all interpolation operator. The AMLE is implemented as a diffusion operator ∂U /dt = U ∇∇ where U ∇∇ is the second-order derivative in the direction of the gradient of U . The key properties of the AMLE is that it is an artifactfree and oscillation-free interpolant actually capable of taking into account isolated boundary points as well as curves.

RESULTS

Results of the reconstruction of the morphogenesis of different real structures are reported to demonstrate the efficiency of the proposed method. In all these results the boundary conditions are simply issued from the extraction of the outline and the position of the manually marked growth center. The parameter setting was defined as follows: potential values of U ranging from O (growth center) to 1 (outline), γ = 10 and = 1. Concerning computational time, the proposed method runs in about one minute for 1000x1000 images under a Pentium IV at 2.5GHz.

Fig. 3 reports a first example with a pollock otolith. Animated versions of these results can be viewed at webpage www.ifremer.fr/lasaa/rfablet/mottolith. The observed rings are seasonal rings (dark zones are winter rings whereas white zones correspond to summer rings). It should however be noted that none of the annual rings is actually closed. They all involve subjective contours to be closed (especially the first dark ring). Consequently, extracting closed rings appear as complex issue if no additional a priori information is available. Conversely, the proposed framework successfully re- cover the deformations of the otolith from the core to the outline. The proposed level-set representation intrinsically encodes that closed shapes have to be reconstructed. Besides, the constraints issued from local image orientations convey sufficient information so that the estimated level-set representation approximately fits to the observed growth rings. Note, for instance, that local shape deformations such as the one occurring towards the top of the otolith is correctly taken into account. Besides, it can stressed that the left and right otolith sides are correctly matched. Results reported in Fig. further illustrates the relevance of the proposed framework for a tree trunk involving rather circular concentric ring shapes, and a hake otolith involving complex ring structures. In both cases, relevant shape series are reconstructed regarding the complexity of the task.

The proposed solution for the reconstruction of the accretionary morphogenesis of biological structures open new research directions with a view to exploiting the invaluable potential of these biological archives in terms of information extraction (growth axis extraction, ring segmentation), morphogenesis analysis and modeling as well as 2D information calibration and decoding (for instance, to analyze information along different growth axis).

Fig. 1 .

 1 Fig. 1. Illustration of the goal of the paper: image of a structure growing according to an accretionary process (a pollock otolith) in a given observation plane containing the growth center (top); reconstructed series of the evolved shapes (down).

Fig. 3 .

 3 Fig. 3. Reconstruction of the series of growth shapes for the pollock (Pollachius pollachius) otolith image depicted above based on the gradient-based orientation field: image of the pollock otolith (left), reconstructed potential function U (middle), series of shape superposed to the otolith image (right).

Fig. 4 .

 4 Fig. 4. Examples of reconstruction of the series of growth shapes: image of the tree trunk section (left), reconstructed potential function U (middle), series of shape superposed to the processed image (middle left), image of a hake otolith section (left), reconstructed potential function U (middle right), series of shape superposed to the processed image (right).


Nicolas Courbin

Ronan Fablet
email: ronan.fablet@ifremer.fr

Capucine Mellon

Hélène De Pontual

Are hake otolith macrostructures randomly deposited? Insights from an unsupervised statistical and quantitative approach applied to Mediterranean hake otoliths

Keywords: age, age estimation method, quality control, growth, Merluccius merluccius, otolith

Are hake otolith macrostructures randomly deposited? Insights from an unsupervised statistical and quantitative approach applied to Mediterranean hake otoliths

Introduction

Catch proportions at age are key inputs to stock assessment models and are routinely used for fisheries management. The time and the duration of life history events are also required information for many ecological studies. To provide information of the age structure of exploited populations, the ages of almost a million fish are estimated annually using otoliths [START_REF] Campana | Otoliths, increments, and elements: keys to a comprehensive understanding of fish populations?[END_REF]. However, the interpretation of fish otoliths is far from a trivial task and data quality remains crucial. For instance, a review of age data for ICES stocks shows that data quality is modest or poor for 75% of the stocks (H. Sparholt pers. comm.). There is therefore a clear need to assess the accuracy and precision of age estimation procedures (see [START_REF] Campana | Accuracy, precision and quality control in age determination, including a review of the use and abuse of age validation methods[END_REF] as well to develop quality control mechanisms [START_REF] Morison | Quality issues in the use of otoliths for fish age estimation[END_REF]. Besides these aspects, the interpretation of otolith growth marks is often a complex task in which subjectivity increases with the complexity of the structural pattern of the otolith. Although the complexity of the interpretation depends on various parameters, it is above all species-dependent. Improved understanding of the biological meaning of growth marks and the development of a quantitative framework aimed at defining an objective procedure for otolith interpretation are complementary solutions to reducing interpretation subjectivity, and are especially needed for complex species.

We seek to develop the quantitative framework here, for European hake (Merluccius merluccius). Recent advances in signal processing and pattern recognition provide appropriate tools to achieve this goal, as exemplified in other applications to otolith research [START_REF] Fablet | Automated fish age estimation from otolith images using statistical learning[END_REF][START_REF] Fablet | Unsupervised Bayesian reconstruction of individual life histories from otolith signatures: case study of Sr : Ca transects of eel (Anguilla anguilla) otoliths[END_REF]. The development of a measure of otolith similarity in terms of structural information is viewed as a first step towards extracting relevant patterns in complex arrangements. The European hake is a good candidate for such developmental work. For Atlantic hake, the precision of age estimation has recently improved as a consequence of international otolith reading workshops being convened (e.g. [START_REF] Anon | Report on improving sampling of western and southern European Atlantic Fisheries[END_REF]. However, the results of recent tagging experiments [START_REF] De Pontual | A pilot tagging experiment on European hake (Merluccius merluccius): methodology and preliminary results[END_REF][START_REF] De Pontual | Evidence of underestimation of European hake growth in the Bay of Biscay, and its relationship with bias in the agreed method of age estimation[END_REF] appear to invalidate the interpretation criteria agreed at one such workshop and described by Pineiro and Sainza (2003). The tagging work suggested age overestimation of the recovered fish and therefore an underestimation of somatic growth [START_REF] De Pontual | Evidence of underestimation of European hake growth in the Bay of Biscay, and its relationship with bias in the agreed method of age estimation[END_REF], which clearly impacts stock assessment and management advice [START_REF] Bertignac | Consequences of bias in age estimation on assessment of the northern stock of European hake (Merluccius merluccius) and on management advice[END_REF]. Uncertainties in age and growth information on Mediterranean hake are well documented [START_REF] Oliver | Age et croissance du merlu (Merluccius merluccius L.) des îles Baléares[END_REF][START_REF] Morales-Nin | Growth of juvenile Merluccius merluccius in the Gulf of Lions (NW Mediterranean) based on otolith microstructure and length-frequency analysis[END_REF] and have been discussed in several workshops [START_REF] Gfcm | Report of the Technical Consultation of Methodologies used for fish age-reading[END_REF][START_REF] Oliver | Report on the age-reading Workshop on Mediterranean European hake and Sardine, Palma de Mallorca, Spain, 10-15[END_REF]. Knowledge on growth has been improved for the early life stages from an analysis of otolith microstructures in different areas of the Mediterranean [START_REF] Arneri | Aspects of the early life history of European hake from the central Adriatic[END_REF][START_REF] Morales-Nin | Recruitment and post-settlement growth of juvenile Merluccius merluccius on the western Mediterranean shelf[END_REF][START_REF] Belcari | Age determination and growth of juveniles of the European hake, Merluccius merluccius (L., 1758), in the northern Tyrrhenian Sea (NW Mediterranean)[END_REF]. However, to date, structural patterns of otoliths (in terms of macrostructures) have been suggested to be unusable for age estimation [START_REF] Morales-Nin | Otolith growth and age estimation in the European hake[END_REF].

To investigate these issues, a quantitative method has been developed. From an analysis of large sets of otoliths, it performs an unsupervised extraction of a typology of otolith macrostructures and the statistical analysis of associated otolith macrostructure patterns. Here, we describe the proposed approach and demonstrate how it can be applied to discussing relevant questions regarding structural patterns of Mediterranean hake otoliths.

Material and methods

Biological material

We used a set of Mediterranean hake otoliths (sagittae) previously prepared for routine age estimation in transverse sections. The fish were collected from both commercial landings and scientific surveys, the international bottom trawl survey (MEDITS), the small pelagic survey (PELMED), and the fishing technology survey (TECPEC), carried out in the Gulf of Lions from June to November of 2002 and 2003. The size range of the fish was 8-50 cm. Sampling design requires 10 fish per 1-cm size class, but in 2003, there were very few fish from 19 to 25 cm. Total length (cm), total weight (g), sex (male, female, undetermined), and maturity were recorded for all fish. The length distribution of the processed sample set is shown by sex and sampling year in Figure 1.

Suggested approach

With a view to analysing the structural patterns of otoliths, a key step is to define a similarity measure between otoliths in terms of structural information, i.e. a numerical measure quantifying how similar two otoliths are, based on a numerical representation of structural patterns. This is viewed as a key component of the framework needed to determine objective and validated protocols for otolith interpretation. Such a similarity measure provides the basis of standard analytical tools, for instance statistical tests or classification tools, for analysing the structural patterns of otoliths. Formally, the key feature of the proposed approach is to map a sequence of macrostructures observed on a section of an otolith to a one-dimensional structural signal.

Such a representation of the structural information of otoliths is required to encode the presence of different types of macrostructure [START_REF] Campana | Otoliths, increments, and elements: keys to a comprehensive understanding of fish populations?[END_REF][START_REF] Panfili | Manual of Fish Sclerochronology[END_REF]. Macrostructures that are visible on otoliths may have different origins. Seasonal growth structures can be differentiated from checks [START_REF] Panfili | Manual of Fish Sclerochronology[END_REF], but it needs to be stressed that checks might be associated with major ontogenetic change as well as with fish-specific responses to environmental or endogenous factors. Although the former type of checks can be used as relevant time references at a group or population level, the latter type can be regarded as structures formed at non-specific, i.e. random, times during the life of a fish1 . As far as ageing and temporal otolith calibration are concerned, differentiating random checks from other age-related macrostructures is required to evaluate structural similarity. As a first step, the determination of such a typology of otolith macrostructures can be derived from an unsupervised statistical analysis of characteristic features of the macrostructures observed in a set of otoliths. Exploiting the resulting macrostructure typology, any sequence of otolith macrostructure can be converted to a sequence of labels, in terms of macrostructure type, before the computation of a onedimensional structural signal and the introduction of a structural similarity measure.

Macrostructure descriptors and parameters

The transverse sections were observed with a stereomicroscope connected to a high resolution video camera and a computer. Observations were made under reflected light at constant magnification in order to avoid bias that may be induced by different scales of observation. A clearing medium (a mixture of glycerine and alcohol) was used to enhance the contrast of observations. Information acquisition consisted of specifying a radial from the nucleus to the edge of the ventral side of the otolith, following the maximum growth axis (Figure 2). Each translucent macrostructure identified and delimited by markers was automatically characterized by three criteria (Figure 3): width (W), distance from the nucleus to the middle of the zone (D), and translucency intensity (TI). The last of these is computed as a normalized measure of image intensity2 . Width and distance to nucleus are determined in mm. During the observations, three types of translucent zone were observed: large translucent zones close to the nucleus (WTZC), large translucent zones distant from the nucleus (WTZD), and thin translucent zones or checks (TTZ) (Figure 4). Two observations per otolith were carried out by the same reader to assess the precision of interpretation. Although the operator had no previous experience of otolith analysis, interpretation was subsequently modified for just 17% of the sample. Measurements (W, D, TI) were automatically acquired on digital images using TNPC software [START_REF] Fablet | TNPC (digital processing of calcified structures): user manual[END_REF].

Unsupervised typology extraction of otolith macrostructure

Let us use {M i } to denote the specific set of otolith macrostructures where each macrostructure M i is characterized by its feature vector

{ } i i i i M M M M TI D W y , , =
. The unsupervised determination of macrostructure typology is stated as the extraction of the relevant clusters (or classes) in the feature space . To this end, the distribution of the macrostructures within this feature space is modelled as a Gaussian mixture. Each macrostructure type T is assumed to be described by a Gaussian distribution g(.|Θ

} {

TI D W , ,

T ,Σ T ), where Θ T is the vector of mean characteristics (W T , D T, TI T ) and Σ T the covariance matrix:

( ) ( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Θ - Σ Θ - - Σ = Σ Θ - T T T T T T T y y y g 1 2 2 1 exp 2 1 , π ,
The mixture model evaluates the likelihood of a macrostructure characterized by its feature vector y as follows:

{ } ( ) ∑ = Σ Θ = Σ Θ K k T T T T k T T T k k k k k k k y g y p 1 ) , ( , , π π
, where K is the number of macrostructure types and { } k T k π the prior probabilities of the different macrostructure types. Following a priori analysis of structural patterns of otoliths, a mixture model with three components is considered. The estimation of model parameters is carried out according to the maximum lilkelihood criterion (see Appendix).

This methodology permits the identification within a quantitative and objective framework of a typology of otolith macrostructures. Of all macrostructures types, those corresponding to thin structures with clear spatial variability are unlikely to be relevant for age estimation or associated with checks (TTZ). The other types of macrostructure are considered to be major structures (WTZC and WTZD), potentially age-related and meaningful at a population or group level. It must be stressed that they may include both actual annual growth structures and checks associated with major ontogenetic change.

Representation of the structural patterns of otoliths

Given the extracted macrostructure typology, any particular macrostructure can be assigned a macrostructure type from its descriptors. Formally, given a macrostructure M described by its feature vector M y , the most likely macrostructure type T M is derived from the estimated mixture model (see Appendix).

Applying this probabilistic rule of labelling to all macrostructures, the structural information conveyed by an otolith can be encoded by its associated sequence of macrostructure labels. Two types of representation are investigated. First, coding WTZC and WTZD labels as code 1 and TTZ labels as code 2, an otolith is described by a series of 1 and 2 according to the nature, number, and positions of the translucent zones along the radial. Such representation is compact and straightforwardly defines a categorization of otolith patterns (Figure 5). Statistical analysis of the distribution of the observed otolith pattern categories permits investigation of whether or not specific structural patterns can be identified for otolith subsets. We stress, however, that this representation of structural information does not take into account relative distances between macrostructures.

A second representation of the structural pattern of otoliths can then be introduced. Given the sequence of macrostructure labels { }

{ } i M M i i D W ,
, we define a normalized structure-based otolith signal s (Fablet, 2006a) as a function of the distance to the nucleus. Keeping only the main macrostructures (i.e. macrostructures labelled as WTZC and WTZD), signal s is set to -1 within macrostructure zones and as a sinusoidal function rescaled between -1 and 1 elsewhere (Figure 5). As this structural signal is defined as a function of the distance to the nucleus, it intrinsically encodes both the number and the relative positions of the macrostructure zones. Issues arising from differences in the number of otolith macrostructures as well as from comparing otoliths of different age or length can then be dealt with.

Analysis of structural otolith similarity

The proposed representations of the structural information conveyed by otoliths provide the basis for evaluating the similarity between otoliths in terms of structural pattern. A first approach relies on the statistical analysis of the relative frequencies of the otolith pattern categories observed for a given otolith set. These statistics permit investigation of meaningful differences among otolith subsets (for instance, otolith groups formed according to fish sex or year of catch) using standard non-parametric statistical tests [START_REF] Legendre | Numerical Ecology[END_REF].

A second approach uses the computed structural signals to define explicitly the similarity measure between two otoliths as a function of the difference between these signals. Rather than straightforwardly exploiting the Euclidean distance, this similarity measure is defined as the minimum distance between the structural signals of two otoliths with respect to admissible growth-warping functions [START_REF] Myers | A comparative study of several dynamic time-warping algorithms for connected word recognition[END_REF]. Such a definition is aimed at coping with the variability of the structural pattern derived from individual otolith growth variability. Similar approaches are exploited in medical imaging to analyse differences among individuals, for instance for brain mapping [START_REF] Hellier | Retrospective evaluation of inter-subject brain registration[END_REF]. Formally, the similarity measure D(s 1 ,s 2 ) between two signals s 1 and s 2 is defined as follows:

( )

2 2 1 2 1 ) ( ) ( min ) , ( ∫ Φ - = Ξ ∈ Φ D D s D s s s S , ( 1
)
where Ξ is the set of admissible warping functions, defined in our case according to regularity constraints, i.e. the set of smooth monotonous functions with bounded variations3 [START_REF] Hellier | Retrospective evaluation of inter-subject brain registration[END_REF].

For any otolith set, this similarity measure is used to compute a similarity matrix that stores the similarity measures computed for all otolith pairs. From this similarity matrix, the organization of the similarities of the structural patterns of otoliths within the considered sample sets can be visualized using linear or non-linear mapping [START_REF] Borg | Modern Multidimensional Scaling[END_REF], and statistical differences between otolith subsets can be evaluated using standard distance-based statistical tests, for instance Mantel's test [START_REF] Legendre | Numerical Ecology[END_REF]. For this study, this numerical framework was applied to test for sexbased and cohort-related differences in structural patterns of otoliths.

Results

Macrostructure typology

A database of 628 otoliths was considered, 80% of the original available otolith set (fish length range 8-50 cm). In ~10% of the samples, sections prepared according to the routine otolith preparation protocol had not passed through the nucleus, and poorly contrasted otoliths accounted for another 10%. In both cases, precise determination of the features of the translucent zones (width, distance to the nucleus, and opacity) was not possible, so to preclude bias in the analysis, those samples were not considered. As shown in Figure 6, omitting those otoliths does not lead to any particular sampling bias.

The unsupervised extraction of the typology of otolith macrostructures derived from the Gaussian mixture model led to three standard modes for the whole dataset (Figure 7). Three macrostructure types were associated with those modes. Translucency intensity was not a discriminant feature in any case.

Of the three modes, the biggest prior was at 0.5127 and corresponded to the thinnest marks (0.02<W<0.08 mm) along the radial (0.4<d<3.9 mm): they are referred to as checks (TTZ). The mean position was d = 1.78 mm and W = 0.04 mm. The second Gaussian mode (prior 0.1781) was associated with thicker marks (0.08<W<1.2 mm) close to the nucleus (0.5<d<1.6 mm): they are referred to as large translucent zones close to the nucleus (WTZC). Their mean position was d = 0.91 mm, and mean width W = 0.37 mm. The third mode (prior 0.30919) involved rather thicker marks (0.08<W<0.84 mm), far from the nucleus and widely distributed (1.16<d<3.8 mm): they are referred to as large translucent zones distant from the nucleus (WTZD). Their mean position was d = 2.20 mm and their mean width W = 0.19 mm.

Correlation hypotheses between W and D features were tested for each mode using a Student's t correlation test for the estimated covariance matrix. Although there was no correlation for the TTZ mode (p > 0.1), correlations between the distance to the nucleus and the width of the translucent zones for the two other WTZ modes were significant (p < 0.0001).

Assignment of each macrostructure to each of the three types is illustrated in Figure 8.

Otolith pattern categorization

Global analysis of 628 otoliths led to the creation of 157 different otolith pattern categories based on encoding by 1 WTZC, 1 WTZD, and 2 TTZ. The number of occurrences of these categories ranged from 1 to 53 (Figure 9a). Note that just 18 categories were found nine times or more and that they included 63% of the samples (Figure 9b). In that case, one-third of the macrostructure sequence begins with a TTZ and two-thirds with a WTZ.

Statistical analysis of wide translucent macrostructures

We note empirically that the number of WTZs increased with fish size, as illustrated for samples of indeterminate sex, females, and males (Figures 10a, b andc). More precisely, for hakes of indeterminate sex and length 8-20 cm, most of the observed structural patterns (65%) did not contain any WTZ, whereas above 20 cm the observed structural patterns involved at least one WTZ (Figure 10a).

Figures 10b andc indicate a succession of peaks corresponding to an increase of WTZ number in the structural sequences with fish length. Notably, WTZ deposition takes place on average at greater length in females than in males. The otoliths with one WTZ were mainly 17-26 cm for males and 18-32 cm for females. Those with two WTZs dominated between 27 and 33 cm for males and between 33 and 38 cm for females. The structural sequences with three WTZs formed a peak between 35 and 37 cm for males and were in the majority between 39 and 46 cm for females. Thereafter, for both sexes, otoliths with four WTZs dominated. The most complex structural sequences, with six and seven WTZs, have been found only for females, and they appeared at a relatively large size, at 42 and 49 cm, respectively (Figure 10c).

Statistical analysis of the computed structure-based otolith signals

Structural patterns of otoliths were further analysed using the similarity measure introduced in the previous section, to test for significant similarities or difference among sample subsets. More precisely, we asked two questions: does the structural pattern of otoliths remain stable over time for young fish, and do males and females from a given length range differ in their otolith pattern?

For hake of 8-18 cm, the typological analysis was similar for 2002 and 2003, with the presence of checks and translucent zones close to the nucleus not very broad (<0.4 mm) (Figures 11a andb). Analysis of the structural otolith similarity matrix between otoliths from these two years revealed three groups of patterns, but no group specific to a given year (Figure 11c). There was no significant difference based on structural similarity defined by Equation (1) (p = 0.468). Otoliths of young hake shared similar structural patterns in 2002 and 2003. Different otolith patterns between males and females have been recorded before. The structure-based otolith similarity measure was also used to test for this quantitatively. Focusing on the length range 24-34 cm, for which there were significant numbers of otoliths, it was clear that the otolith patterns of both males and females involved the three types of macrostructure, with visually similar distributions of checks and translucent rings close to the nucleus (Figures 11d ande). However, analysis of structural otolith similarities with respect to sex led to a significant difference in terms of structural patterns (p = 0.049), illustrated by mapping the similarity matrix, which involves several clusters, some specific to just one sex (Figure 11f).

Discussion

Workshops devoted to improving European hake age determination have been conducted for at least ten years, in order to agree age-estimation criteria and to standardize methodology. They resulted in increased precision of ages [START_REF] Piñeiro | Report of the 3rd Workshop on European hake age reading[END_REF]), but could not address the issue of accuracy, emphasizing that the criteria needed to be validated. Chemically tagging fish as a basis of markrecapture analysis, thought to be the best method for validating the periodicity of growth increment formation [START_REF] Campana | Accuracy, precision and quality control in age determination, including a review of the use and abuse of age validation methods[END_REF], has recently been used for Atlantic hake [START_REF] De Pontual | A pilot tagging experiment on European hake (Merluccius merluccius): methodology and preliminary results[END_REF], and the first results show that growth is underestimated as a consequence of age-overestimation (de [START_REF] De Pontual | Evidence of underestimation of European hake growth in the Bay of Biscay, and its relationship with bias in the agreed method of age estimation[END_REF]. Underestimation of growth during the first year of life has also been demonstrated recently by [START_REF] Arneri | Aspects of the early life history of European hake from the central Adriatic[END_REF], Morales-Nin and Moranta ( 2004), [START_REF] Kacher | Distribution and growth of 0-group European hake in the Bay of Biscay and Celtic Sea: a spatial and inter-annual analysis[END_REF], and [START_REF] Belcari | Age determination and growth of juveniles of the European hake, Merluccius merluccius (L., 1758), in the northern Tyrrhenian Sea (NW Mediterranean)[END_REF], based on an assumption of daily deposition of micro-increments. Therefore, various studies support the hypothesis that hake is a fast-growing species. The initial results of a haketagging programme started in 2005 in the Gulf of Lions confirmed the fast-growing hypothesis (C. Mellon, unpublished data). The lack of a methodology for validating age estimates despite the many years of age-reading experience in Europe underscores the need for a method to be developed that reduces the subjectivity of expert interpretation and is based on determination of a typology of otolith macrostructures.

Structural patterns of otoliths

Hake otolith rings are visible at low magnification. On a seasonal scale, the accretion process alternatively produces opaque and translucent zones, which differ in their organic matter/mineral ratio, crystal size, and the width of their primary increments [START_REF] Panfili | Manual of Fish Sclerochronology[END_REF]. Some species, such as plaice (Pleuronectes platessa), lay down simple patterns in their otolths, and a pair of translucent ring/opaque zones is considered to be deposited annually. For other species, such as hake, otolith interpretation is complex because of the presence of numerous macrostructures. There are many thin translucent zones that probably correspond to short environmental and/or physiological events, and the difficulties in interpreting such otoliths often increases with the size of the fish.

Classification of the rings is a major problem in European hake. Here, a systematic delimitation of all types of translucent zone, thin and large, on the main radial with markers, and a statistical treatment of a large set of radials was carried out. The unsupervised statistical analysis of the distribution of the characteristics of these macrostructures led to definition of three main macrostructure types, two corresponding to wide translucent zones, and one to thin translucent zones. The last mode is associated with a large variance in the distance to the nucleus. This mode is therefore likely to be associated mainly with specific fish responses to environmental or endogenous factors. By contrast, the spatial extent of the other two modes is more clearly defined, suggesting that they are relevant at a population level, for instance in response to seasonal variations or major ontogenetic events. To our knowledge, this is the first time that quantitative discrimination of meaningful translucent zones has been attempted for otolith checks. The application to Mediterranean hake otoliths, whose interpretation is difficult, stresses the potential of the proposed framework to be applied to other fish species with complex otolith patterns From the extracted macrostructure typology, a probabilistic labelling scheme can be derived to assign any particular macrostructure to a macrostructure type. This scheme provides the means of investigating similarities in terms of structural information among otoliths. The number of WTZs evolves coherently with fish length, i.e. their distribution against fish length is clearly modal and the mean length associated these modes increases with the number of WTZs. In addition, an evaluation of statistical differences based on structural otolith similarity reveals no significant difference between the structural patterns of otoliths of young fish sampled in 2002 and in 2003, supporting the assumption that otolith patterns may be stable over time. We stress that the proposed similarity measure allows for gradual changes in growth patterns over time. Hence, the similarity in terms of structural information has to be considered in terms of the relative positions of the relevant translucent zones. The analysis of hake 24-34 cm long requires the detection of a significant difference between male and female otoliths in terms of structural patterns. Such a result is coherent with known differences in growth patterns of male and female hake. As otolith structures relevant at a population level can be assumed to be deposited in a similar way for males and females, it is to be expected that they will be found at a greater length for females than for males, as revealed by our analysis.

The results show that our approach allows us to differentiate, using quantitative features, random macrostructures (likely to be associated with individual physiological responses) from otolith macrostructures that are clearly not random and that can be considered at a population level. This finding contradicts that of [START_REF] Morales-Nin | Otolith growth and age estimation in the European hake[END_REF], who concluded from analysis of sulcal ring patterns of Mediterranean hake that ring formation in the otolith showed no synchronicity at a population level and was dependent mainly on physiological events. Our results therefore provide a basis for future work aimed at defining a well-founded and objective age estimation protocol for European hake. To this end, our proposed methodological framework should be applied at a broader scale, specifically to otolith subsets sampled in different seasons and years, with a view to determining the period of deposition of the macrostructures and to constructing models of otolith patterns. Also, given that our analysis was limited to fish <50 cm, there would likely be merit in applying the proposed methodology to older fish. We anticipate that the unsupervised extraction of associated otolith macrostructure typology will lead to the identification of more macrostructure types potentially meaningful for age determination. As correlations between the distance to the nucleus and the width of the translucent zones were significant for the two WTZ modes, application to otolith sets involving greater fish lengths might require consideration of other features to characterize translucent zones. In particular, it might be important to consider a relative width measure (for instance, relative to the width of neighbouring translucent zones) to account for the fact that the width of the translucent zones decreases as the fish become older.

Fast growth hypothesis

Assuming that the number of WTZs is correlated to fish age, it is of interest to compare the distribution in the number of WTZ, in terms of fish length, with the current hypotheses on hake growth. Two growth hypotheses, fast (FGH) and slow (SGH), have been given in the literature for the first year of life. A slow growth rate (1.15 cm month -1 ) was estimated for the Gulf of Lions by [START_REF] Aldebert | Estimation de la croissance du merlu dans le golfe du Lion par l'analyse des fréquences de taille[END_REF] and [START_REF] Morales-Nin | Growth of juvenile Merluccius merluccius in the Gulf of Lions (NW Mediterranean) based on otolith microstructure and length-frequency analysis[END_REF], and another of between 0.7 and 1.2 cm month -1 for the Ligurian Sea by [START_REF] Orsi-Relini | Spatial-temporal distribution and growth of Merluccius merluccius recruits in the Ligurian Sea, observations on the 0 group[END_REF]. Those values (Table 1) were calculated using a method based on length frequency analysis (Fishparm/Bhattacharya). Fast growth (defined as 1.2-2.5 cm month -1 ) was estimated for the Catalan coast at >1.6 cm month -1 by [START_REF] Morales-Nin | Recruitment and post-settlement growth of juvenile Merluccius merluccius on the western Mediterranean shelf[END_REF], in the Gulf of Alicante at 1.7 cm month -1 by [START_REF] Garcia-Rodriguez | How fast does hake grow? A study on the Mediterranean hake (Merluccius merluccius L.) comparing whole otoliths readings and length frequency distributions data[END_REF], and in the northern Tyrrhenian Sea also at 1.7 cm month -1 by [START_REF] Belcari | Age determination and growth of juveniles of the European hake, Merluccius merluccius (L., 1758), in the northern Tyrrhenian Sea (NW Mediterranean)[END_REF]. Those values of growth rate were obtained from the Fisat program for length frequency analysis.

To compare the evolution of the number of otolith macrostructures labelled as meaningful with respect to these two growth hypotheses (slow and fast), we analysed the length distribution of the fish whose otoliths were assigned numbers (from zero to seven) of WTZ otolith structures (Figure 10). These length distributions are compared with the predicted length ranges for ages 1-7 for each growth hypothesis. The outcome is that the fast growth hypothesis applies to our results better than the slow growth hypothesis, for both sexes (Figure 10). For FGH, structural sequences consist of zero and one WTZ during the 1 st year, predominantly one and two WTZs during the 2 nd year, and mainly two and three WTZs during the 3 rd year (Figure 10a). For the 2 nd and 3 rd years, the appearance of an additional translucent ring seems to correspond to a 2 nd half of that year of life. Conversely, there is no consistency with SGH. This analysis should be considered with respect to growth variability, which may be high for hake. Further, it is likely that not all macrostructures are seasonal structures. Among the structures identified as potentially age-related, some are likely to be associated with major ontogenetic events. Hence, more than one WTZ structure could be observed for some years of the fish life. It is not straightforward to generate a perfect match between the length distribution of a given number of observed otolith WTZs and the length range derived from FGH at a specific age. The important result regarding hake otolith interpretation is that the otolith patterns involve some population-related determining factors.

The establishment of a validated ageing protocol requires further understanding and characterization of the conditions and processes leading to formation of the WTZ macrostructures. To this end, future work should rely on experimental approaches based on both tank and field experiments.

Methodological aspects and future work

The methodological developments described in this paper are aimed at characterizing in an unsupervised and objective way the deposition of otolith macrostructures from a statistical analysis of the structural information in sets of otoliths. To this end, an unsupervised typology of otolith macrostructures has been derived. This typology then permitted definition of a similarity measure between otoliths in terms of structural information, with a view to quantifying similarities or differences among otolith subsets with respect to macrostructure patterns.

The potential of the proposed methodological framework for analysing structural patterns of otoliths has been demonstrated for a set of several hundred otoliths of Mediterranean hake, which is defined as a complex species in terms of otolith interpretation. The proposed framework could be applied similarly to other complex (in terms of age) or new species for which no validated interpretation protocol is available. Moreover, such unsupervised analysis can also be applied to collections of marked otoliths derived from mark-recapture experiments. In those cases, extraction of the macrostructure typology can also rely on a time-calibration of the macrostructures associated with tagging and recapture date(s) and should contribute to the development of validated and objective interpretation schemes.

Another area of application for the proposed methodology is the comparison, in quantitative terms, of the interpretation of otolith patterns by experts. Evaluation of intra-and inter-expert agreement in otolith interpretation serves as a basis for standardizing age estimation protocols and implementing quality assurance and control in age estimation [START_REF] Morison | Quality issues in the use of otoliths for fish age estimation[END_REF]. Such associated quantitative analysis is restricted to computing agreement rates or the coefficients of variation of age structures, because quantitative evaluation of the similarity of the interpretation of otolith macrostructures is not possible because of the absence of metrics. The proposed measure of similarity of otolith patterns can be applied to the patterns interpreted by experts with a view to quantifying interpretation differences, though such application will rely on a predetermined macrostructure typology agreed among experts. The analysis could be performed repeatedly for a single expert to evaluate potential drift in interpretation, between experts to check for consistency of interpretation protocol among experts and/or institutes, and for training to evaluate the relevance of the interpretation with respect to annotated reference collections.

The main limitation of the proposed framework lies in its need for manual detection of otolith macrostructures, i.e. selection of the positions and widths of the macrostructures. This acquisition step is time-consuming and could induce some subjectivity. However, the required degree of expertise for the task is low because it is limited to selecting the positions of the macrostructures and does not involve their interpretation in terms of growth structures or checks. This statement is supported by the low variability observed between two acquisition sessions for a reader with no previous experience in otolith interpretation. Future work will include coupling of the proposed framework with automated detection of otolith macrostructures (Fablet, 2006b) in order to reach fully automated and unsupervised characterization of otolith patterns. Additionally, we will investigate the use of complementary features of otolith macrostructures, especially in terms of oxygen and carbon isotope descriptors (e.g. Hoie et al., 2004), with a view to enriching the macrostructure typology. Additional descriptors will be performed within the methodologies described here. Finally, future work will pursue the development of measures of structural otolith similarity accounting for typology and the relative positions of macrostructures and variability in otolith growth, especially for applications aimed at checking protocol coherency. [START_REF] Borg | Modern Multidimensional Scaling[END_REF]. n is the number of otoliths analysed, and p the value of the p-statistic in the permutation test of Mantel [START_REF] Legendre | Numerical Ecology[END_REF]. n and p need to be italicized, some lettering sizes need to be adjusted down, SEX is mis-spelled, and I don't like the solid black line down the middle

Running headings

∏ Σ Θ = Σ Θ Σ Θ i k T T T M k T T T k k k i k k T k T k T k k k y p , , max arg , , ˆ, , π π π .
To solve this maximization issue, we use the EM (Expectation-Maximization) algorithm. Refer to [START_REF] Bishop | Neural Networks for Pattern Recognition[END_REF] for further detail on implementation of this iterative procedure.

Assigning a macrostructure type to a given macrostructure

Assignment of a macrostructure type T M to a given macrostructure M, characterized by the associated features y M , is derived from maximizing the posterior classification likelihood: [START_REF] Aldebert | Estimation de la croissance du merlu dans le golfe du Lion par l'analyse des fréquences de taille[END_REF], and to the fast growth hypothesis (FGH) of [START_REF] Garcia-Rodriguez | How fast does hake grow? A study on the Mediterranean hake (Merluccius merluccius L.) comparing whole otoliths readings and length frequency distributions data[END_REF].

Hypothesis

FigureFigure 1 .

 1 Figure legends do not embolden lettering in any Figure Figure 1. Length distribution of the processed sample by sex and sampling year.Figure 2. Example of the analysis of translucent zones of hake otoliths from a transverse section of the sagitta, with a radial ( ) and markers ( ).Figure 3. Illustration of the features used for characterizing the translucent zones (appearing dark when observed under reflected light) of hake otoliths: width (W), distance to nucleus (D), and translucency

Figure 2 .

 2 Figure 1. Length distribution of the processed sample by sex and sampling year.Figure 2. Example of the analysis of translucent zones of hake otoliths from a transverse section of the sagitta, with a radial ( ) and markers ( ).Figure 3. Illustration of the features used for characterizing the translucent zones (appearing dark when observed under reflected light) of hake otoliths: width (W), distance to nucleus (D), and translucency

Figure 3 .

 3 Figure 1. Length distribution of the processed sample by sex and sampling year.Figure 2. Example of the analysis of translucent zones of hake otoliths from a transverse section of the sagitta, with a radial ( ) and markers ( ).Figure 3. Illustration of the features used for characterizing the translucent zones (appearing dark when observed under reflected light) of hake otoliths: width (W), distance to nucleus (D), and translucency intensity (TI), the last computed as the mean image intensity. Italicize W and D, as in text and equations Figure 4. Examples of the three types of translucent structures derived from the proposed unsupervised statistical analysis: a wide translucent zone close to the nucleus (WTZC), a wide translucent zone distant from the nucleus (WTZD), and a thin translucent zone or check (TTZ).Figure 5. Representations of structural patterns of otoliths: definition of otolith pattern categories corresponding to specific sequences of labelled macrostructures in terms of macrostructure typology (bottom left), and the associated structural signal accounting for both type and relative position of the

Figure 5 .

 5 Representations of structural patterns of otoliths: definition of otolith pattern categories corresponding to specific sequences of labelled macrostructures in terms of macrostructure typology (bottom left), and the associated structural signal accounting for both type and relative position of the translucent macrostructures (bottom right). Make some of the lettering smaller -it is too large Figure6. Length distribution of the set of discarded otoliths, by sex.

Figure 7 .Figure 10 .Figure 11 .

 71011 Figure 7. Parameters of the Gaussian mixture model showing three modes for the distribution of the features of translucent structures. Ellipses are drawn to indicate the standard deviation of each Gaussian mode (we display the ellipses corresponding to a distance of one standard deviation to the Gaussian

  the parameters of the mixture model is carried out according to the Maximum Likelihood (ML) criterion. Model parameters are determined

Fig

  Fig. 1.

Fig

  Fig. 7.

Table 1 .

 1 Von Bertalanffy growth parameters of Merluccius merluccius according to the slow growth hypothesis (SGH) demonstrated by

We subsequently refer to this second type of check as a random check, given that they cannot be associated with specific age-related fish life history events meaningful at a population or group level. By contrast, major ontogenetic checks and growth macrostructures form a macrostructure category referred to as potentially agerelated macrostructures.

 2 The normalized image-intensity measure is normalized with respect to the local mean and standard deviation of image intensities along the analysed reference growth axis to remove bias that may be induced by varying lighting and otolith reflection conditions. Local mean and standard deviation of image intensities are estimated using a Gaussian window whose scale parameter is set with respect to the macrostructure width.

Considering the set of smooth monotonous functions with bounded variations leads to an additional regularity term in the minimization defined by Equation (1). This regularity term is evaluated as the norm of the gradient of the warping function Φ. For further detail of the implementation of this minimization, the reader is referred to[START_REF] Hellier | Retrospective evaluation of inter-subject brain registration[END_REF], a study that uses a similar formulation in the field of medical imaging.

Acknowledgements

We thank J. L. Dufour for preparing otolith sections, A. Ogor for help in monitoring the TNPC imageanalysis system, and A. Payne, V. Parisi and an anonymous reviewer for their valued comments on the submitted manuscript.