Interdisciplinary island metabolism: intersection of flows

and socio-geography approaches to investigate vulnerability, waste colonialism and externalization in the cases of Comoros and New Caledonia. special session: Islands

Islands are closely linked to globalized material flows, with specific constraints and vulnerabilities. Counter-intuitively, they are not examples of metabolic loops of consumption, production and waste. We argue that islands are key territories for better understanding the consequences of flow in terms of vulnerabilities, colonialism and externalization, precisely because of the way they are connected (rather than isolated) to globalized material flows. This research is an interdisciplinary material and geographical analysis of waste realities in two French/formerly French island territories [START_REF] Manglou | Waste colonialism and metabolic flows in island territories[END_REF]Bahers et al., 2022): the Comoros and New Caledonia. It draws on metabolic analysis and waste studies to address the different perspectives that these approaches open up for the study of island territories. An analysis of material flows (EW-MFA method with a spatialization of flows) allows us to sketch out metabolic profiles that show the contribution of the dominant mining and agricultural industries to waste production. This flow approach is combined with a social geography approach. The socio-political context of each island is used to explain contemporary resource and waste management policies and practices.

The comparison of current situations in terms of discourse and flow economics shows how these territories are characterized by the accumulation of waste, which characterizes a colonialism of waste [START_REF] Lepawsky | The changing geography of global trade in electronic discards: Time to rethink the e-waste problem[END_REF][START_REF] Liboiron | Waste colonialism[END_REF][START_REF] Liboiron | Pollution is colonialism[END_REF]. This is not the export of waste from rich countries to poorer ones, but the export of resources extracted from poor countries to rich ones while keeping the extraction waste there. This is the case with nickel waste in Caledonia and oil waste from ylang ylang extraction in the Comoros. This externalization of resources and their circulation thus provokes island vulnerabilities.

In conclusion, there is a need for interdisciplinary research (Bahers et al., 2022) into island metabolism in order to understand the link between the circulation of flows and their sociogeographic consequences. This framework, which combines a territorial ecology and a political ecology of the islands, makes it possible to grasp the intersections between extractive economic sectors and sociometabolic vulnerabilities.


Alberto A Aguilar-Lasserre

Catherine Azzaro-Pantel
email: catherine.azzaropantel@ensiacet.fr

Luc Pibouleau
email: luc.pibouleau@ensiacet.fr

Serge Domenech
email: serge.domenech@ensiacet.fr

Enhanced Genetic Algorithm-Based Fuzzy Multiobjective Strategy to Multiproduct Batch Plant Design

Keywords: Multiobjective Optimization, Genetic Algorithm, Fuzzy Arithmetic, Batch Plant Design

The design of such plants necessary involves how equipment may be utilized, which means that plant scheduling and production must form an integral part of the design problem. This work proposes an alternative treatment of the imprecision (demands) by using fuzzy concepts. In this study, we introduce a new approach to the design problem based on a multiobjective genetic algorithm, taking into account simultaneously maximization of the net present value V P N ~ and two other performance criteria, i.e. the production delay/advance and a flexibility criterion. The methodology provides a set of scenarios that are helpful to the decision's maker and constitutes a very promising framework for taken imprecision into account in new product development stage. Besides, a hybrid selection method Pareto rank-tournament was proposed and showed a better performance than the classical Goldberg's wheel, systematically leading to a higher number of non-dominated solutions.

Introduction

In recent years, there has been an increased interest in the design of batch plant due to the growth of specialty chemical, biochemical, pharmaceutical and food industries. The most common form of batch plant design formulation considered in the literature is a deterministic one, in which fixed production requirements of each product must be fulfilled. However, it is often the case that no precise product demand predictions are available at the design stage [START_REF] Shah | Design of multipurpose batch plants with uncertain production requirements[END_REF].

The market demand for such products is usually changeable, and at the stage of conceptual design of a batch plant, it is almost impossible to obtain the precise information on the future product demand over the lifetime of the plant. Nevertheless, decisions must be taken on the plant capacity. This capacity should be able to balance the product demand satisfaction and extra plant capacity in order to reduce the loss on the excessive investment cost or than on market share due to the varying product demands [START_REF] Cao | Optimal design of batch plants with uncertain demands considering switch over of operating modes of parallel units[END_REF].

The key point in the optimal design of batch plants under imprecision concerns modeling of demand variations. The most common approaches treated in the dedicated literature represent the demand uncertainty with a probabilistic frame by means of Gaussian distributions. Yet, this assumption does not seem to be a reliable representation of the reality, since in practice the parameters are interdependent, leading to very hard computations of conditional probabilities, and do not follow symmetric distribution rules.

In this work, fuzzy concepts and arithmetic constitute an alternative to describe the imprecise nature on product demands. For this purpose, we extended a multiobjective genetic algorithm, developed in previous works [START_REF] Aguilar | Modélisation des imprécisions de la demande en conception optimale multicritère d'ateliers discontinus[END_REF]), taking into account simultaneously the maximization of the net present value V P N ~ and two other performance criteria, i.e. the production delay/advance and a flexibility criterion. The paper is organized as follows. Section 2 is devoted to process description and problem formulation. Section 3 presents a brief overview of fuzzy set theory involved in the fuzzy framework within a multiobjective genetic algorithm. The presentation is then illustrated by some typical results in Section 4.

Process Description and Problem Formulation

Problem Statement

In conventional optimal design of a multiproduct batch chemical plant, a designer specifies the production requirements for each product and total production time for all products. The number, required volume and size of parallel equipment units in each stage are to be determined in order to minimize the investment cost [START_REF] Huang | Fuzzy decision-making design of chemical plant using mixed-integer hybrid differential evolution[END_REF]. The designers must not only satisfy technico-economic criteria, but also respect some due dates. In this framework, this study introduces a new design approach to maximize the net present value V P N ~ and two other performance criteria, i.e. the pro- duction delay/advance and a flexibility criterion. Such an optimal design problem becomes a multi-objective optimization problem (MOOP).

In order to specify the production requirements for each product and total production time for all products, it is almost impossible to obtain some precise information. Indeed, the ability of batch plants to deal with irregular product demand patterns reflecting market uncertainties or seasonal variations is one of the main reasons for the recently renewed interest in batch operations. In this paper, we consider an alternative treatment of the imprecision of the demand by using fuzzy concepts. A genetic algorithm was implemented for solving this problem, since it has demonstrated to be efficient in multi-objective optimization problems.

Assumptions

The model formulation for batch plant design problems adopted in this paper is based on Modi's approach [START_REF] Modi | Design of multiproduct batch processes with finite intermediate storage[END_REF]. It considers not only treatment in batch stages, which usually appears in all types of formulation, but also represents semicontinuous units that are part of the whole process (pumps, heat exchangers,...). Besides, this formulation takes into account mid-term intermediate storage tanks. So, a batch plant is finally represented by series of batch stages (B), semi-continuous stages (SC) and storage tanks (ST). The model is based on the following assumptions:

1. The devices used in a same production line cannot be used twice by one same batch. 2. The production is achieved through a series of single product campaigns. 3. The units of the same batch or semi-continuous stage have the same type and size. 4. All intermediate tank sizes are finite. 5. If a storage tank exists between two stages, the operation mode is "Finite Intermediate Storage". If not, the "Zero-Wait" policy is adopted. 6. There is no limitation for utility. 7. The cleaning time of the batch items is included into the processing time. 8. The item sizes are continuous bounded variables.

Model Formulation

The model considers the synthesis of I products treated in J batch stages and K semicontinuous stages. Each batch stage consists of mj out-of-phase parallel items with same size Vj. Each semi-continuous stage consists of nk out-of-phase parallel items with same processing rate Rk (i.e. treatment capacity, measured in volume unit per time unit). The item sizes (continuous variables) and equipment numbers per stage (discrete variables) are bounded. The S-1 storage tanks, with size Vs*, divide the whole process into S sub-processes. a) Economic criterion evaluation: The net present value method ( V P N ~) of evaluat- ing a major project allows to consider the time value of money [START_REF] Aguilar | Modélisation des imprécisions de la demande en conception optimale multicritère d'ateliers discontinus[END_REF]. Essentially, it helps find the present value in "today's value money" of the future net cash flow of a project. Then, this amount can be compared with the amount of money needed to implement the project. When using this formula, the values of the number of periods (n), discount rate (r) and tax rate (a) take respectively the following classical values 5, 10% and 0 (computation before tax). In order to calculate investment cost (Cost) (2), the working capital (f), revenue ( P V ~), operation cost ( P D ~) and depreciation (A P ) are introduced.

( ) ~= (q 1 ,q 2 ,q 3 ,q 4 ), V P N ~=(NPV 1 ,NPV 2 ,NPV 3 ,NPV 4 )

∑ = + + - - - + - - = n p n p p p p r A a A D V f Cost V P N Max 1 ) 1 ( ) 1 )( ( ~ (1) ∑ ∑ ∑ = = = + + = S s s s s K k k k k k J j j j j j V c R b n V a m Cost 1 1 1 ) ( ) ( ) ( γ β α (2)
The proposed approach involves arithmetic operations on fuzzy numbers and quantifies the imprecision of the demand by means of fuzzy sets (trapezoidal). In this case, the flat line over the interval (q 2 ,q 3 ) represents the precise demands with an interval of confidence at level α=1, while the intervals (q 1 ,q 2 ) and (q 3 ,q 4 ) represent the "more or less possible values" of the demand. The result of the net present value Wang's method (Liou and Wang 1992) was adopted.

( V P N ~) is
The production time necessary to satisfy each product demand must be less than a given time horizon, but due to the nature of the fuzzy numbers, eight different cases for determination of the second criterion may occur. The different cases are reported in figure 2.

The temporal criterion selected is called "common surface", representing the intersection between the sum of the production time (trapezoid) and the horizon of time to respect (rectangle). The calculation of the criterion depends on each case: for example, case1 illustrate the solutions which arrive just in time. The penalization term is equal to an arbitrary value of ω for an advance and 1/ω for a delay in order to penalize more delays than advances. A sensitivity analysis leads to adopt a value of 3 for ω.

c) Flexibility index: Finally, an additional criterion was computed in case of an advance (respectively a delay), representing the additional production (the demand not produced) that the batch plant is able to produce. Without going further in the detailed presentation of the computation procedure, it can be simply said that a flexibility index is computed by dividing the potential capacity of the plant by its actual value.

The problem statement involves four forms of different constraints as reported in literature (Modi and Karimi 1989): Dimension constraints, time constraint, constraint on productivities and the size of intermediate storage tanks.

A Fuzzy Decision-Making Approach for Multiproduct Batch

Plant Design

Overview of Multiobjetive Genetic Algorithm Approach

The GA implemented in this study uses quite common genetic operators. The proposed GA procedure implies the following steps:

1) Encoding of solution. The encoding of the solutions was carried out dividing the chromosome, i.e. a complete set of coded variables, into two parts. The first one deals with the items volumes, which are continuous in the initial formulation. Nevertheless, they were discretized here with a 50 unit range, while their upper and lower bounds were preserved. The second part of the chromosome handles the number of equipment items per stage: the value of these discrete variables is coded directly in the chromosome.

2) Initial population creation. The procedure of creating the initial population corresponds to random sampling of each decision variable within its specific range of variation. This strategy guarantees a population various enough to explore large zones of the search space.

3) Fitness Evaluation. The optimization criterion considered for fitness evaluation involves the net present value (NPV) and two other performance criteria, i.e. the production delay/advance and a flexibility criterion. Traditionally, a GA uses a fitness function, which must be maximized. The fitness for these criteria is equal to their directly calculated values.

4) Selection Procedure. The multi-objective aspects are taken into account in the selection procedure. A hybrid selection method Pareto rank-tournament was proposed and showed a better performance than the classical Goldberg's wheel, systematically leading to a higher number of non-dominated solutions.

The method of tournament preferentially selects the non-dominated individuals of case 1, thus successively and in a consecutive way the procedure selects then if need be cases 2, 3, 4, 5, 6 and 7 (preference for the solutions with greatest common surfaces between the sum of the horizons of time "trapezoid" and the horizon of time to respect "rectangle") until arriving at case 8 (without common surface).

The procedure of selection of the hybrid selection method Pareto rank-tournament into account a Pareto set following the criteria of Pareto dominance are then implemented on the population of individuals and makes it possible to extract the set of the non-dominated Pareto's optimal solutions. 5) Crossover. Two selected parents are submitted to the crossover operator to produce two children. The crossover is carried out with an assigned probability, which is generally rather high. If a randomly generated number is superior to the probability, the crossover is performed. Otherwise, the children are copies of the parents.

6) Mutation.

The genetic mutation introduces diversity in the population by an occasional random replacement of the individuals. The mutation is performed on the basis of an assigned probability. A random number is used to determine if a new individual will be produced to substitute the one generated by crossover. The mutation procedure consists of replacing one of the decision variable values of an individual, while keeping the remaining variables unchanged. The replaced variable is randomly chosen, and its new value is calculated by randomly sampling within its specific range.

7) Registration of all non-dominated individuals in Pareto set.

Pareto's sort procedure is carried out at the end of the algorithm over all the evaluated solutions; at the end of the procedure, the whole set of the non dominated Pareto's optimal solutions, are obtained.

Treatment of an Illustrative Example

We consider an example to illustrate the approach fuzzy-AG based on arithmetic operations on fuzzy numbers and quantifying the imprecision of the demand. The example was initially presented by Ponsich and al. ( 2004): the plant, divided into two sub-processes, consists of six batch stages to manufacture three products.

The GA parameters are the following ones: Population size 200 individuals, number of generations 400 iterations, crossover probability 40%, mutation probability 30% and the stop criterion considered in this study concerns a maximum number of generations to reach.

For the considered example, table 1 shows the values for processing times, size factor for the units, cost data, and the production requirement for each product quantifying the imprecision of the demand by means of fuzzy numbers representing the "more or less possible values".

For the construction of the trapezoid which represents the request for each product, the original values of the demand were used as a reference. To determine the support and the core, one calculated a percentage of opening taking as reference the demand of the original data is computed. 4 Typical Results

Monocriterion Case

For the monocriterion case, the best individuals, that are the surviving ones, are chosen with the Goldberg's roulette. Each individual is represented by a slice of the roulette wheel, proportional to its fitness value. Since the criteria are represented by fuzzy numbers, they were defuzzified (the defuzzified value was calculated with the Liou and Wang's method) in the roulette wheel. GA typical results obtained with V P N ~ as the only criterion to consider are pre- sented in Table 2 (ten runs were performed to guarantee the stochastic nature of the GA). In particular, the value of the best individual of each generation and the average value of the function objective of the population take a traditional form of regular increase, to stabilize itself finally at the end of the research.

Multi-objective Optimization

The multi-objective resolution strategy for multiproduct batch plant design uses two genetic algorithms (bicriteria and tricriteria optimizations are considered).

Bicriteria Case: NPV-The Delays and Advances of the Time Horizon

The first bicriteria analysis takes into account the NPV and the criterion which represents the advances or delays of the time horizon with the hybrid selection method Pareto rank-tournament. To treat the bicriteria optimization, the analysis is performed on the solution of case 2 with the best NPV and on those for case 3 having, on the one hand, larger common surfaces and, on the other hand, the best NPV. The example did not provide any solution of case 1, because the rectangle which represents the horizon of time to respect is larger than the trapezoids obtained for the sum of times of production. Table 3 shows the results of the chosen solutions.

Bicriteria Case: NPV-Flexibility Index

The second bicriteria analysis takes into account the NPV and the criterion which represents the flexibility index of the configuration chosen to produce a possible additional demand. Figure 3 exhibits 277 non dominated solutions of the advance cases "(2, 4, 6 and 8) and of the delay cases (3, 5 and 7). Two solutions of case 2: the first is the solution with the best benefit and the second configuration has the best index of flexibility. The third corresponds to case 3 with the best index of flexibility.

Comparison Between a Hybrid Selection Method (Pareto Rank-Tournament) and the Classical Goldberg's Wheel for the Bicriteria Optimization Case

To evaluate the performance of the selection method of the fuzzy genetic algorithm (combination of Pareto set and tournament), it is proposed to test the algorithm modified with the classical Goldberg's wheel. The same number of surviving individuals is chosen for each criterion (two roulettes). The same parameters for AG with the procedure of Pareto rank-tournament, are used with Goldberg's wheel.

A higher number of non dominated individuals are obtained with the hybrid method because the selection provides compromise solutions between the two criteria and the values of their criteria are better than those obtained with the Goldberg's wheel.

Tricriteria Case: NPV-Delays/Advances-Flexibility Index

Lastly, the fuzzy optimal design of batch plant takes into account simultaneously the three criteria, i.e., NPV, criteria of the advances or delays (common surface) and index of flexibility. The method proposes a sufficiently large range of compromise solutions making it possible to the decision's maker to tackle the problem of the final choice, with relevant information for his final choice.

To analyse the results obtained with the tricriteria case, 6 non dominated solutions are adopted: 3 of case 2, 2 of case 4 and 1 of case 6. These solutions were selected by considering the same policy as for the bicriteria case.

Conclusions

For the most common form of design of a multiproduct batch chemical plant, the designers specify the production requirement of each product and the total production time. However, no precise product demand predictions are generally available. For this reason, an alternative treatment of the imprecision by using fuzzy concepts is introduced in this paper.

In this study, we have introduced a fuzzy-AGs approach to solve the problem of multi-objective optimal design of a multiproduct batch chemical plant. The results obtained on the treated example have shown that three different scenarios were obtained as a fuzzy decision-making approach. The analysis tended to be helpful for decision making.

Its benefits can be summarized as follows:

-Fuzzy concepts allow us to model imprecision in cases where historical data are not readily available, i.e. for demand representation; -The models do not suffer from the combinatorial explosion of scenarios that discrete probabilistic uncertainty representation exhibit;

-Another significant advantage is that heuristic search algorithms, namely genetic algorithms for combinatorial optimization can be easily extended to the fuzzy case;

-The hybrid selection method Pareto rank-tournament was proposed and showed a better performance than the classical Goldberg's wheel, systematically leading to a higher number of non-dominated solutions.

-Multiobjective concepts can also be taken into account.

-The proposed approach thus constitutes an efficient and robust support to assist the mission of the designer, leading to a quite large set of compromise solutions.

Finally, this framework provides an interesting decision-making approach to design multiproduct batch plants under conflicting goals.

Fig. 1 .

 1 Fig. 1. Two trapezoidal fuzzy numbers, Q~= (q 1 ,q 2 ,q 3 ,q 4 ), V P N ~=(NPV 1 ,NPV 2 ,NPV 3 ,NPV 4 )

  treated and analyzed through fuzzy numbers. The demand and the net present value are fuzzy quantities as shown in figure 1. b) Computation of the criterion penalizing the delays and advances of the production time necessary for the synthesis of all the products: for this purpose, we must compare the time horizon H ~ represented by a fuzzy expression (rectangle) and the production time i H ~ (trapezoidal). For the comparison of fuzzy numbers, Liou and

Fig. 2 .

 2 Fig. 2. Eight cases for the minimization of a criterion that penalizes the delays and advances of the time horizon necessary for the synthesis of all the products

Fig. 3 .

 3 Fig. 3. Bicriteria results: NPV-Flexibility Index

Table 1 .

 1 Data used in example

					Processing time τi,j (h)			Size factors (1/kg)
				B1	B2	B3	B4	B5	B6	B1	B2	B3	B4	B5	B6
	Minimum size =250 l	A 1.15	3.98	9.86	5.28	1.2	3.57	8.28	6.92	9.7	2.95	6.57	10.6
	Maximum size = 10 000 l	B 5.95	7.52	7.01	7	1.08	5.78	5.58	8.03	8.09	3.27	6.17	6.57
				C 3.96 γj 0.4	0.29 5.07	0.33 6.01	0.3 5.13	0.2 0.66	0.35 4.37	2.34	9.19	10.3	5.7	5.98	3.14
		Unit price for product i		Coefficients ci,j		Q1=(419520. 428260, 441370, 445740)
			($/Kg)							Q2=(311040, 319140, 330480, 336960)
										Q3=(247680, 258000, 263160,268320)
										H = (5760, 5760, 6240, 6240)
		CP	CO	B1 B2	B3	B4	B5	B6	Cost of mixer=$250V 0.6
	A B C	0.70 0.74 0.80	0.08 0.1 0.07	0.2 0.15 0.34	0.36 0.5 0.64	0.24 0.35 0.5	0.4 0.7 0.85	0.5 0.42 0.3	0.4 0.38 0.22	Cost of reactor=$250V 0.6 Cost of extractor=$250V 0.6 Cost of centrifuge=$250V 0.6
										(Volume V in liter)
						Operating cost factors
				B1		B2	B3	B4	B5		B6
				CE 20		30	15	35	37		18

Table 2 .

 2 Monocriterion (

			V N ~): Fuzzy optimal design of batch plant P
	Product Bis kg TLi h Criterion: The net present value	Information Complémentaire
	A B C	943.3 1145.5 7.3 4.8 899.7 6.6	($) N ~=[722225.3 803765.7 PV 892941.6 969060.6]	V ~=[721977.6 742345.6 764042.2 782142.4] p [$] p D ~= [232095.9 238513.7 245540.9
				251437.2 ] [$]
				I = 698877.8 [$] Ap= 139775.5 [$]
				f= 104831.6 [$] Vs = 1505.2 [l] ∑ i H ~= [5759.9 5925.4 6097.3 6239.9] [h]

Table 3 .

 3 Fuzzy optimal design of batch plant for case 2 (advance) and cases 3 (delays)

		NPV	Common		N	P ~	V	($) and production time Σ H ~	(h)
		(Mean Value)	surface
	Cas 2	860358	653	N	V ~= P	[736120, 817367, 906144, 981801]
				Σ H ~=	[5758, 5916, 6089, 6238]
	Cas 3a	860550	163	N	V ~= P	[736327, 817565, 906332, 981976]
				Σ H ~=	[5772, 5930, 6104, 6253]
	Cas 3b	861823	129	N	V ~= P	[737489, 818728, 907495, 983139]
				Σ H ~=	[5883, 6045, 6222, 6374]


Thibault Clérice
email: thibault.clerice@inria.fr

Anthony Glaise
email: aglaise@laposte.net

Twenty-One* Pseudo-Chrysostoms and More: Authorship Verification in the Patristic World

Keywords: Patristic Studies, Ancient Greek, Stylometry, Siamese Networks, Authorship verification

As the most prolific of the Church Fathers, John Chrysostom (344-407 CE) has a vast textual mass and theological importance that has led to a significant misattribution of texts, resulting in the existence of a second corpus known as the pseudo-Chrysostomian corpus. Like many Greek-language Church Fathers' works, this corpus comprises anonymous texts, which scholars have attempted to reattribute or group together based on factors such as the person's function, biography, ideology, style, etc. One survey conducted by Voicu in 1981 explored potential groupings of such texts and produced a critical list of 21 Pseudo-Chrysostom works identified by scholars, including Montfaucon (1655-1741), one of the first modern editors of Chrysostom's writings. In this paper, we present a novel approach to addressing pseudonymous work in the context of Chrysostomian studies. We propose to employ Siamese networks within an authorship verification framework, following the methodology commonly used in recent computational linguistic competitions. Our embedding model is trained using commonly used features in the digital humanities landscape, such as the most frequent words, affixes, and POS trigrams, utilizing a signal-to-noise ratio distance and pair mining. The results of our model show high AUCROC scores (84.5%). Furthermore, the article concludes with an analysis of the pseudo-Chrysostoms proposed by Voicu. We validate a significant portion of the hypotheses found in Voicu's survey while also providing counter-arguments for two Pseudo-Chrysostoms. This research contributes to shedding light on the attribution of ancient texts and enriches the field of Chrysostomian studies.

Introduction

Late Antiquity literature in Latin and Ancient Greek bears a profound influence from Christian literature, shaping the era's literary landscape. Patristic studies focus on the examination of the earliest Christian authors, spanning from the 1st century CE to the 7th century CE, with some scholars extending the period up to Jean Damascene in the middle of the 8th century CE. Within the realm of Church Fathers, two prominent figures stand out for their enduring impact on the corpora: Augustine (354-430) in Latin and John Chrysostom (344-407) in Ancient Greek. Despite their significance, both Augustine and John Chrysostom encountered a common challenge in Late Antiquity. Numerous works were falsely attributed to them, possibly with the intention of elevating the popularity of these writings, whose actual authors were less renowned. This misattribution has led to the creation of an extensive body of texts with uncertain authorship. According to Voicu [START_REF] Voicu | Une nomenclature pour les anonymes du corpus pseudo-chrysostomien[END_REF], there are "more than a thousand" pseudo-Chrysostomian works whose true originators remain unknown 1 . To address this issue, scholars of patristics have attempted to categorize some of these spurious works, associating them with a single person, who may be anonymous or identified with a specific historical figure.

The task of distinguishing anonymous authors within the pseudo-Chrysostomian corpus is a significant and essential endeavor. By doing so, historians, patristicians, and philologists can work with a more manageable collection of texts. This process offers valuable insights into the construction of Christian theological frameworks, allowing the identification of ideological or thematic clusters within authorship groups. Establishing authorship for texts with at least one proposed date enables scholars to study the evolution of Christian ideology more accurately. Moreover, in rare cases, dating or attributing a specific anonymous text (especially when milestones are present) can provide new dates to other texts, as they might be cited or reused [START_REF] Datema | An unedited homily of Ps. Chrysostom on the birth of John The Baptist (BHG 843k)[END_REF]. This effort holds immense potential for advancing our understanding of the historical and intellectual landscape of Late Antiquity, contributing to a deeper comprehension of the development of the Christian tradition.

In traditional patristics, authorship attribution largely relies on the identification of rare patterns [START_REF] Schatkin | The authenticity of St. John Chrysostom's de Sancto Babyla, Contra Iulianum et gentiles[END_REF], as well as extra-textual clues, such as historical events serving as terminus post quem or ante quem markers, and the analysis of ideological traits or the use of specific quotations. Authors' tendencies to rely on particular parts of the Bible or cite other authors also serve as de facto terminus ante quem. In 1981, Voicu [START_REF] Voicu | Une nomenclature pour les anonymes du corpus pseudo-chrysostomien[END_REF] presented a comprehensive analysis of attributions using such methods in the context of the pseudo-Chrysostomian corpus. He identified 21 text groups for which a common authorship has been proposed by himself or other scholars. However, for some of these pseudo-Chrysostoms, Voicu and the scholars he cites sometimes express hesitancy regarding the attribution or hold differing perspectives on the matter.

In such a context, automatic authorship verification can offer new insights that either support or challenge previous hypotheses. Two other fields that are actively engaged in authorship verification are computational linguistics and digital humanities (DH). In the realm of stylometrical analysis, DH scholars tend to treat texts as bag-of-words and utilize statistically significant features, such as the most frequent words of a corpus [START_REF] Eder | Style-markers in authorship attribution: a cross-language study of the authorial fingerprint[END_REF], part-of-speech 3-grams, and character n-grams [START_REF] Cafiero | Why Molière most likely did write his plays[END_REF]. On the other hand, computational linguistics (CL) has recently shown a preference for treating text as a sequence of words, employing masked language models [START_REF] Stamatatos | Overview of the authorship verification task at PAN[END_REF]. In terms of approach, DH papers lean towards classification (utilizing SVM or similar models) or unsupervised clustering methods. In contrast, CL uses both DL classification methods and siamese neural networks, regardless of the type of input [START_REF] Kestemont | Overview of the cross-domain authorship verification task at PAN[END_REF].

In our research, we propose a novel methodology that combines approaches from both computational linguistics and digital humanities to analyze the 21 potential Pseudo-Chrysostoms identified by Voicu and his predecessors. Specifically, we aim to evaluate the effectiveness of different features, including most frequent words, affixes, and POS 3-grams, within the context of a Siamese network using a linear projection in N dimensions. To maximize the potential of our corpus, we introduce Easy-SemiHard Pair Mining [START_REF] Schroff | FaceNet: A unified embedding for face recognition and clustering[END_REF] to our batch learning process and utilize a signal-to-noise ratio distance [START_REF] Yuan | Signal-To-Noise Ratio: A Robust Distance Metric for Deep Metric Learning[END_REF] to distinguish and separate our texts.

In summary, the contributions of our paper are as follows:

• Introducing a new approach to authorial verification for ancient texts, incorporating Siamese networks and easy-semihard pair mining into the landscape of stylometry within computational humanities. • An analysis of various size for three types of features (POS, MFW, Affixes), which shows consistency in terms with previous studies. • An evaluation of common distances (Manhattan and Euclidian distance [L2]) and a newly introduced one (signal-to-noise ratio distance), which shows that the latter out-performs the best performing Manhattan distance in terms of stability within our settings. • Conducting an in-depth analysis of the results obtained from applying our approach to the 21* pseudo-Chrysostoms' texts identified by Voicu.

The remaining of the paper is organised as follows. Section 2 provides background on the article of Voicu and in general on the issue of the pseudo-Chrysostomian corpus, as well as a deeper background on authorship verification and stylometry in general. Section 3 (Proposed Methods) provides details about the architecture used for the experiments. Section 4 (Experimental Setup) provides insight on the corpus, features selection and metrics. Section 5 provides an evaluation of the results on independent test sets. Section 6 reuses the models built and to provide insight on the PC corpus.

Background and Related Work

Background

Regarding the pseudo-Chrysostomian corpus, a non-specialist might notice the lack of recent discussions concerning various attribution hypotheses. Most of these "old" hypotheses trace their origins back to the 18th century, thanks to the diligent efforts of Bernard de Montfaucon, a Benedictine monk who dedicated his work to publishing the works of Athanasius (1698) and John Chrysostom (1718). Montfaucon's editions and commentary laid the groundwork for later comprehensive editions of numerous patristic texts [START_REF] Quantin | 1588-1613[END_REF], including the renowned Patrologia Graeca compiled by Migne, which is still widely used today as it is now in the public domain. Fortunately, in 1981, Voicu [START_REF] Voicu | Une nomenclature pour les anonymes du corpus pseudo-chrysostomien[END_REF] provided the most recent comprehensive summary of the authorship hypotheses within the pseudo-Chrysostomian corpus. This summary encompassed both refutations and ongoing debates surrounding various authorship attributions. Since then, some new hypotheses have emerged, but there hasn't been a comparable effort to Voicu's in terms of summarizing the existing scholarship.

In his paper, Voicu provides a summary regarding authorship clustering of 88 texts, grouped under the pseudo-identity of 21 different authors by various scholars (PC1 to PC21). On top of the 21 base PC, another group of text is hypothetically attributed to one of them: PC20 is accompanied by PC20b, for which he drafts a possible ensemble without confirmation that they might be of the same authors (see Table 1). The arguments regarding the attributions can vary, and we summarise them in four different categories:

• Theological arguments are based on ideological (in)compatibilities between texts, in the context of a non-unified Christian religion in which subgroups can be found. E.g. PC16 is categorized as a moderate Antiochian, which makes them incompatible with an Alexandrian ideology. • Sequential arguments are based on the obvious continuity between two texts, with established narrative links in both senses (Text A builds on Text B and vice-versa). • Stylistic arguments are based on the style of the authors. They range from the use of what is perceived as "bad Greek" (PC6 and 7) to citation habits regarding the scriptures. • Extra-textual arguments are mostly based on events or texts referenced within the texts, which gives them a common terminus ante quem or post quem, or actually makes them incompatible. It also refers to transmission proofs, such as the constant grouping of the same texts in their transmission history (PC17).

Table 1

List of pseudo-Chrysostoms and the scholars behind the hypotheses. Confidence concerning the grouping of texts under a single pseudonymous author is provided based on Voicu's commentary of each scholar's analysis, including his own. We present a simple typology of the arguments when provided. All the cited scholarship, except for Montfaucon, was published between 1940 (Marx [START_REF] Marx | Procliana: Untersuchung über den homiletischen Nachlass des Patriarchen Proklos von Konstantinopel[END_REF]) and 1981 (Voicu). Most of the cited texts are available in digital formats, specifically in the Thesaurus Lingua Graeca, which is unfortunately closed access 2 . However, one PC could not be tested in our framework: PC19 only refers to one text in Ancient Greek, the other text being a Syriac translation. PC12 has been produced using the OCR available on Google Books, which has been lightly post-corrected.

Related work in stylometry and authorship verification

The present work is related to two fields: computational linguistics (CL) and digital humanities (DH). The existing literature reveals distinct approaches to the problem of authorship verification or authorship identification, primarily influenced by the attributes of each field's corpus and expectations for explainability. We focus on the latest approaches applied in both fields, emphasizing the common technical approaches they have shared in the past.

Computational Linguistics CL offers a clear landscape thanks to PAN, a "series of scientific events and shared texts on digital text forensics and stylometry. " PAN's shared tasks have provided recurring competitions since as early as 2011, focusing on authorship attribution (2011,2012,2018,2019) or verification (2013)(2014)(2015)(2020)(2021)(2022)(2023). As in most other computational linguistics tasks, deep learning has seen a rise in popularity, leading to improved scores but lower explainability. An insight into some of the approaches taken in authorship verification since 2015 reveals the following methods and features used:

• In 2015, Word-based n-grams, sentence length, word frequencies, punctuation frequencies, POS frequencies, and POS n-grams were shared features across different papers [START_REF] Pacheco | Random Forest with Increased Generalization: A Universal Background Approach for Authorship Verification[END_REF][START_REF] Nikolov | Experiments in Author Verification[END_REF]. These features were eventually fed into standard classifiers such as Random Forest or SVMs. • In the 2018 PAN authorship attribution task [START_REF] Kestemont | Overview of the Author Identification Task at PAN-2018: Cross-domain Authorship Attribution and Style Change Detection[END_REF], features continued to focus on characters and word n-grams, with various weighting and normalization methods. While SVMs were commonly used, some early neural network approaches also appeared. • In the 2020 PAN authorship verification task [START_REF] Kestemont | Overview of the cross-domain authorship verification task at PAN[END_REF], methods employing neural networks made an appearance, particularly in the form of Siamese neural networks. Features mainly remained classical stylometric features, such as normalized frequencies of tokens or POS. The winning paper utilized a Siamese network with extracted "linguistic embedding vectors" using an LSTM network with attention to produce document embeddings. • In 2021 [START_REF] Kestemont | Overview of the cross-domain authorship verification task at PAN[END_REF], features and approaches from 2019 were carried over, with a paper using BERT and another one using Siamese networks ranking first and second place, respectively, in the large dataset competition on overall scores, with the first one winning the competition on all provided metrics. • In 2022 [START_REF] Stamatatos | Overview of the authorship verification task at PAN[END_REF], all competitors moved away from SVM and instead used either masked language models' embeddings or previous existing features with Siamese network approaches or fully connected neural networks.

With this summary, we observe that CL is gradually moving away from explainability and increasingly employing text sequences (using RNNs, CNNs, or transformers like BERT or T5) rather than treating texts as bags of words (BoW). Scoring is conducted using various metrics, including AUROC and a modified F1-Score 𝐹 0.5𝑢 [START_REF] Bevendorff | Generalizing Unmasking for Short Texts[END_REF], which "emphasizes correctly-answered same-author cases and rewards non-answers". Siamese networks gained prominence in authorship verification tasks in 2020 and 2021.

In contrast to previous competitions, 2022 departed from 2021 in the type of dataset used. While 2021 and previous shared tasks utilized datasets sharing a common domain (such as fanfictions for 2021), 2022 focused on cross-discourse authorship verification, encompassing "emails, essays, texts messages, and business memos. " This shift resulted in a significant performance drop compared to previous years. Additionally, regarding the dataset, each competition relied on fixed pairings of texts, with limited or no information about the author in the metadata of the documents.

Digital Humanities Unlike CL, DH has largely preferred using feature selection and treats most texts in an authorship attribution framework rather than authorship verification. The three most frequently recognized features are: most frequent words [START_REF] Eder | Rolling stylometry[END_REF][START_REF] Rebora | a war journal, and stylometry: Tackling the issue of short texts in authorship attribution[END_REF], function words, affixes (especially for languages with significant variations in spelling or flexions) [START_REF] Camps | Noisy medieval data, from digitized manuscript to stylometric analysis: Evaluating Paul Meyer's hagiographic hypothesis[END_REF], and POS, with the latter often used as n-grams [START_REF] Cafiero | Why Molière most likely did write his plays[END_REF]. Some other features, such as rhymes [START_REF] Cafiero | Why Molière most likely did write his plays[END_REF], meters [START_REF] Nagy | Metre as a stylometric feature in Latin hexameter poetry[END_REF], and even treebank syntactic tags for Ancient Greek [START_REF] Gorman | Author identification of short texts using dependency treebanks without vocabulary[END_REF], have demonstrated usefulness and stylometrical importance but are less commonly used 3 .

In terms of technologies, tools offering explainability (particularly feature weight) are highly favored, particularly SVMs or, in an unsupervised setting, hierarchical clustering using distances such as Manhattan, Cosine, Burrows' Delta [START_REF] Burrows | Delta': a Measure of Stylistic Difference and a Guide to Likely Authorship[END_REF], or Eder's Delta [START_REF] Eder | Taking stylometry to the limits: Benchmark study on 5,281 texts from Patrologia Latina[END_REF], etc. We hypothesize that the need for explainability arises due to the fundamentally different objectives pursued by computational linguistics (CL) and digital humanities (DH) in the context of authorship verification and identification. While CL is primarily concerned with developing robust models at scale, DH seeks to employ the results as potential evidence in scholarly research. Consequently, the ability to interpret the decisions made by a model becomes invaluable for detecting "invisible biases to the human eye, " such as the presence of unique characters in specific author editions, which might lead the model to overfit and yield erroneous conclusions. This risk is significantly amplified in the context of the frequent diachronic peculiarities prevalent in most DH inquiries or the constraints posed by a limited pool of available historical documents. Conversely, CL typically concentrates on short-term, cohesive, and large corpora, such as social network messages or fan fiction, where interpretability might not be as crucial.

In 2016, General Imposters (GI) methods [START_REF] Koppel | Determining if two documents are written by the same author[END_REF], an authorship verification method, were introduced to DH [START_REF] Kestemont | Authenticating the writings of Julius Caesar[END_REF]. Unlike authorship verification methods of CL, which rely on learning to recognize a true pair of texts from the same author and a pair of texts from different authors, GI introduces impostor authors alongside candidate authors in a classification task. It randomly removes features (50% by default in the stylo package [START_REF] Eder | Stylometry with R: a package for computational text analysis[END_REF]) over 𝑛 experiments (100 by default in stylo) and proposes, for each of these experiments, the author of the closest text as a potential match. If one of the candidates has an overwhelming presence in the closest texts over all experiments, it is proposed as the verified author of the text.

Proposed Method

We propose to bridge the gap between DH and CL practices by introducing supervised authorship verification using text pairing validation through a siamese network. However, Voicu's anonymous PCs should not be used in the context of supervised training and authorship attribution. This is because supervised training relies on having ground truth data to identify the anonymous authors, and for some of Voicu's PCs, they may not even be processed through the general impostors method as they only offer a single pair of texts, which is insufficient for effective training and verification.

Features Following most of the DH literature and PAN approaches until 2020, we select the relative frequency of most frequent words (MFW), POS tri-grams (MFP), and trigram character affixes (MFT) within each text. We reject the use of any punctuation-based features, as punctuation in our texts is the result of the editorial task and can be editor or period dependent (an editor from the 18th century might not punctuate like a contemporaneous one). In order to test these, we tested a range of different values for each features:

1. 0, 100 and 200 most frequent POS-trigrams 2. 0, 250, 500, 750, 1000 most frequent words 43. 0, 250, 500, 750, 1000 most frequent affixes. MFP(100), MFW(1000), MFT(1000) provided the lowest standard deviation and highest AU-CROC with two different distances (cf. Appendix, section B). These numbers, at least for MFW, are in line with current literature, most notably the recent paper from Rebora's [START_REF]Short texts with fewer authors. Revisiting the boundaries of stylometry[END_REF]. We concatenate the relative frequencies of these features into a single vector, denoted as features 𝑇 𝑖 , consisting of 2100 dimensions representing the text sample 𝑖.

Model structure

The model follows a typical architecture of linear layers in the context of Siamese networks (see Figure 1): feature vectors are passed through the same projection layers, paired, and a distance-based decision is proposed. We use linear layers to reduce 𝑇 𝑖 to a given dimension 𝑚, resulting in an embedding representation 𝐸 𝑖 of the text. For the loss and distance metric, we tested Manhattan, Euclidean (L2) and "Signal-to-Noise Ratio" distance [START_REF] Yuan | Signal-To-Noise Ratio: A Robust Distance Metric for Deep Metric Learning[END_REF] (SNR, SNRD for the distance) with contrastive loss. SNR-D considers the noise between an anchor embedding 𝐸 𝑖 and a compared embedding 𝐸 𝑗 , represented as 𝑁 𝑖𝑗 = 𝐸 𝑗 -𝐸 𝑖 , and uses its variance as an informative measure (variance of values across dimensions). The SNR for pair 𝑖, 𝑗 is defined as 𝑆𝑁 𝑅 𝑖,𝑗 =

Learning methods

Unlike PAN tasks, we can build our own corpus without pre-established pairs, allowing us to utilize pair mining methods, which are particularly useful in the context of Siamese networks (see Figure 2). We use Easy-Semi-Hard (ESH) triplet mining [START_REF] Xuan | Improved embeddings with easy positive triplet mining[END_REF][START_REF] Schroff | FaceNet: A unified embedding for face recognition and clustering[END_REF] using the aforementioned distance. Given embeddings 𝐸 and their classes 𝐾 , 𝐸𝑆𝐻 (𝐸, 𝐾 ) computes the distance between all embeddings and provides all positive pairs (𝐾 𝑖 == 𝐾 𝑗 ), as well as semi-hard and hard negative pairs. Negative pairs are considered semi-hard when "they are further away from the anchor than the positive exemplar, but still hard because the squared distance is close to the anchor-positive distance". We use this miner for both the training step and the evaluation step. In their recent paper on the state of authorship verification, Tyo et al. [START_REF] Tyo | On the State of the Art in Authorship Attribution and Authorship Verification[END_REF] showed that ESH mining could help feature-based models be as good as sequential models using transformers or similar layers.

Experimental Setup

Datasets Unlike social networks content or fan-fictions, or even 19th-century literature, Ancient Greek literature spans from 9 BCE to at least 9 CE in the context of our problem. We are dealing with variation in corpus genre, linguistic changes, ideological changes, etc. In order to address this at the training and evaluation step, we design a corpus that is focused on Christian and theological texts using the TLG CD-ROM [START_REF] Berkowitz | Thesaurus Linguae Graecae. Canon of Greek Authors and Works[END_REF] and their XML export through Diogenes [START_REF] Heslin | [END_REF]. Unfortunately, the raw dataset is not shareable, and no current open dataset provides a quantitatively large enough dataset for this period of Ancient Greek5 . For POS tagging, we use Singh et al. [START_REF] Singh | A pilot study for BERT language modelling and morphological analysis for ancient and medieval Greek[END_REF]'s Ancient Greek BERT-based tagger, which is, to our knowledge, the only Ancient Greek tagger trained with both Medieval Greek and classical Greek.

Within the available corpus, we removed:

• any dubia or spuria (de-attributed texts) or anonymous texts;

• any text from John Chrysostom, as the sheer mass of his work and his style seemed to impact the model too much in early experiments, as well as both the Old and New Testaments; • texts marked as commentaries, scholia, fragments, codices transcriptions, or specific recensio;

• any text containing a single line of poetry 6 .

However, all of our most frequent features were extracted from the full Christian corpus. To counter the reduced mass of texts and address the imbalanced state of the dataset, we produced 1000-word samples for each text using the following rules:

• if the text was shorter than 2000 words, we used the 1000 words in the middle of the text, in order to avoid introductions and conclusions. • If the text is larger than 2000 words, we used up to 5 samples of 1000 words, randomly selected within the text except for the first 500 words and the last 500 words.

We then split the corpus in an 80-10-10 ratio along the titles of the various works, so that authors can span in different sets but samples of the same title remain inside the same text (see Figure 3). The final corpus is diverse in genre but is composed of two genres which make up around two-thirds of its texts (Homelia and Treatise) while Oratio, Letters, Hagiography, and other non-frequent genres rapidly decrease in numbers of titles (see Figure 4, full list of authors in appendix, Table D). The PC corpus was not sampled, but relative frequencies were issued using the same process.

Metrics

We chose to evaluate the general performance of the model based on the Area Under the Curve of the Receiver Operating Characteristics (AUCROC), which allows for measuring the relationship between the False Positive Rate (FPR) and the True Positive Rate (TPR). This metric also enables us to select a task-oriented threshold by minimizing the FPR while maintaining a high enough TPR.

Evaluating stability Adding to this metric, we chose to also evaluate the stability of the precision, after seeing in our first experiments important variation between models using the same parameters. We expect the randomness of training neural networks to be responsible for producing latent spaces focusing on different features, and as such wanted to provide a full picture of these potential variation. To evaluate the variation, we evaluate 1. the standard deviation of their AUCROC, independently from the other hyperparameters. Each models was trained three times for each parameters combination, we look at their average standard deviation across distances. 2. using the framework of inter-annotator agreement through the Fleiss Kappa (𝜅, Fleiss and Cohen [START_REF] Fleiss | The Equivalence of Weighted Kappa and the Intraclass Correlation Coefficient as Measures of reliability[END_REF]) -which allows for 𝑁 > 2 annotators -and the test corpus, we look at the 𝑘 of the two best distances, Manhattan and STN (cf. appendix, section B). We train 100 models for each distance. To make binary decision on the test set, we use a threshold based on the development set precision. 3. using the corpus for qualitative analysis, we look at the variation of pairing percentage (described below) within each pseudo-Chrysostomian sub-corpus.

Qualitative analysis method and pairing percentage

To apply our model to the pseudo-Chrysostomian corpus, we need to provide a confidence score to interpret the distance between pairs of texts. We choose to compute the precision of our model at a given distance threshold 𝛿 𝐴𝐵 for each pair A-B (and its reverse B-A) on both the development set and the test set. This threshold produces a "precision threshold" (PT) that allows us to estimate the probability of a false positive at any given distance. For example, if 𝛿 𝐴𝐵 = 0.4 and the precision of the model for distances less than or equal to 0.4 is 0.9, then 𝑃𝑇 𝐴𝐵 = 0.9, indicating a high probability that A and B are from the same author.

In order to have a quick overview for the pseudo-Chrysostomian corpus as a whole, we evaluate the "pairing percentage" at a variety of precision threshold. Pairing percentage indicates how many positive pairing are operated at a given PT.

Hyper-parameters

The hyperparameters used for the experiment were as follows: Adam optimizer, learning rate of 1𝑒 -4 , embedding size of 64, batch size of 64, 30% dropout of features, class sampling of 2, and a minimum of 100 epochs for training. Training was stopped after 20 consecutive bad epochs, using the dev loss as an indicator.

Software implementation

The experiment was implemented using the following software and hardware: it was run on an nVidia RTX 3090 GPU with 24GB of RAM, using torch [START_REF] Paszke | PyTorch: An Imperative Style, High-Performance Deep Learning Library[END_REF], torchmetrics [START_REF] Skafte Detlefsen | TorchMetrics -Measuring Reproducibility in PyTorch[END_REF] for AUROC metrics, pytorch-lightning [START_REF] Falcon | The PyTorch Lightning team, PyTorch Lightning[END_REF] for training, and pytorchmetric-learning [START_REF] Musgrave | PyTorch Metric Learning[END_REF] for the mining operations. The same process was also successfully run on a CPU (AMD Ryzen 5700 with 32GB RAM).

Results

Best distances and models' consistency

On top of the features selection, unlike most traditional DH approaches, training linear models to produce latent spaces can introduce potential randomness due to the initialization of model weights, the choice of optimization algorithms, etc. This possibility is exacerbated by the fact that we are using a changing development pool produced through ESH mining of the full development set.

Looking at the standard deviation of our models across all parameters (Table 2a), STN seems to have the most stable standard deviation moving from dev to test, and the lowest on the test set. This would indicate that STN based model are more stable across runs from a score perspective, and that, at least in our context, that its performances translate in a similar fashion from the dev to the test set. This is further confirmed by our analysis of the variation of AUCROC 𝑑𝑒𝑣 -AUCROC 𝑡𝑒𝑠𝑡 across our parameter search, with a very stable distance between dev and test sets (cf. Table 2b).

Looking at our models randomness further, we can see that the models using STN distance (Table 3) are much more in agreement across 100 different models and training seed than those using Manhattan distance, despite higher AUCROC scores. We can see that STN has more than 2.5 times the 𝜅 of Manhattan, and Manhattan only reaches the .5 𝜅 at 80% of PT. In a philological set-up, this advocates even more for the use of the most stable model.

Looking at the qualitative dataset and pairing percentage, we can see the same phenomenon, with much more unstable pairing using Manhattan rather than STN (Figure 5). At 100 PT, the standard deviation is 50% for more 6 clusters across for the Manhattan distance, a number unseen using Signal-to-noise at the same threshold. Only lower PT achieve such a high standard deviation (See appendix for more figures across 100 models).

Analysis of the results

The results achieved in this study fall within the range of the current state of authorship verification. Tyo et al. [START_REF] Tyo | On the State of the Art in Authorship Attribution and Authorship Verification[END_REF]'s implementation of the N-Grams based model of Weerasinghe et al. [START_REF] Weerasinghe | Feature Vector Difference based Authorship Verification for Open-World Settings[END_REF] and ESH mining achieved between 77% to 91% AUCROC 7 . The proposed model stopped converging after 219 epochs on the ESH mined pairs and provides an 86.75% AUCROC on all pairs of the development set (see Table 4). The model reaches a slightly lower score on the test set with an 83.76% AUCROC. This is inline with the oberserve differences during the parameter sweep. We also would like to note that due to ESH mining, the pairs used in the dev set for loss might change from one iteration to the others, as the embeddings are used to decide which pairs to use, resulting in a different kind of over-fitting than with a traditional fixed development set. The ROC curves (see Figure 6) exhibit a compelling shape, with a significant slope in the initial percentages of the False Positive Rate (FPR), followed by a more gradual increase until reaching 100% True Positive Rate (TPR). This suggests that the model effectively distinguishes between positive and negative pairs, especially in the early stages of classification. UMAP [START_REF] Mcinnes | UMAP: Uniform Manifold Approximation and Projection[END_REF] (Figure 7), while still serving as a proxy for the complexity of the embeddings' dimension, provides a clear depiction of the discrepancy between both sets. In the test set, embeddings of texts from the same authors are notably more distant from each other, indicating that the model's representations are better disentangled in this set. On the other hand, the development set displays more overlap and mixing of embeddings, suggesting that the model's representations are less distinct and might have captured some biases from the data splitting process.

Application on PCs

General results

Using a visualization of pairing percentage using precision threshold (Precision ≥ 0.75, see Figure 8), we see that our overall results are remarkably consistent with the summary proposed by Voicu. Notably, the PC 3, 4, 5, 8, 9, and 13 are not confirmed as single authors by our model and are either indicated as low probability clusters or refuted groupings in Voicu's article. On the other hand, only the 20 and 20b clusters are completely ignored by our model despite having a high chance of being from the same authors according to both Datema [START_REF] Datema | An unedited homily of Ps. Chrysostom on the birth of John The Baptist (BHG 843k)[END_REF] and Voicu. Among the highly connected clusters, PC 1, 2 8 , 12, 16, 17, 18, and 21 are all marked as highly possible or confirmed by Voicu and are found at a precision threshold of 85% or higher, either on dev or test or both.

Closer look at some pseudo-Chrysostoms

We turn our attention to the pseudo-Chrysostoms whose scores are not relating to the philological studies: PC 6-7, 10-11, 14-15, 20, and 20b.

Montfaucon's pseudo-Chrysostoms 6 and 7 Both pseudo-Chrysostom 6 and 7 are hypotheses made by Montfaucon in the 18 th century. PC6 is considered to be written by bad (and stupid) ancient Greek speakers (inepti Graeculi), with ad nauseam use of repetitions and epithets, while ignoring common rules of grammar (PG, 60, 681-682). Regarding PC7 and the De jejunio sermons, the only arguments of Montfaucon (PG, 60, 711-712) were again his absence of knowledge of basic Greek (imperiti Graeculi) and his ability to write mostly nonsensical things (plerumque nugacis).

Looking at the precision threshold matrices in Figure 9, we see that the matrix behaves in a non-reciprocating way, such that some texts are deemed to be of the same author in the sense A-B but not in the sense B-A. Moreover, for PC7, we see that consecutive sermons are unconnected between each other (e.g. De jejuno 3 and De jejuno 4 are not deemed to be of the same author at any precision) but are connected with subsequent or previous sermons (e.g. De jejuno 3 and De jejuno 4 are deemed to be of the same author as De jejuno 5). Given the scores, we might be tempted to confirm a single author for each of the PC6 and PC7 clusters, but we cannot confirm this without reasonable doubt. Our hypothesis, though untested, suggests that if PC6 and PC7 employ such an unconventional form of Ancient Greek compared to the majority of the corpus, they could prove challenging to categorize uniformly.

The case of Pseudo-Chrysostoms 13, 14, and 16 PC13, 14, and 16 are a particular case in Voicu's survey, as PC13 is refuted by Voicu and its texts are dispatched into two other subcorpora: two texts (Contra Iudaeos, Gentiles et Haereticos and In uenerabilem) are thought to go with PC14's De Eleemonsyna, while De Epiphania would go with PC16. Our precision matrix threshold does confirm the unity of PC16 except for the inclusion of PC13's text (see Figure 11a). However, while the pairs connections are low for PC13 and PC14, they seem to be rather high when connected to PC16. Our model seems to recommend the following composition for PC16: The most significant disagreement between traditional scholarship and our model arises with PC20, which is classified as unconfirmed by our model (see Figure 10). Voicu mentions PC20 as the outcome of "on-going research" and comprises 6 base texts (PC20) and 12 potential others (PC20b). Unfortunately, some of these texts are unedited, and we did not have access to them. As of now, none of the texts in PC20 and PC20b form prominent clusters. We can see two pairs that might be interesting to focus on, specifically: The lack of clear clustering and the significant differences in precision threshold values between pairs in opposite directions raise questions about the coherence and authorship attributions within PC20 and PC20b. It is essential to recognize that authorship verification is a complex task, and discrepancies between our model's findings and traditional scholarship can be attributed to various factors, including the linguistic features used for analysis, the dataset's size and quality, and the nature of the texts themselves. We also want to stress that we have a low recall, which can lead to having many false negatives.

• De Eleemosyna (PC14) • In Psalmum Homilia 50 homiliae, 1-2 (PC16) • In Illud: Sufficit Tibi Gratia Mea (PC16) • In Illud: Si Qua Christo Nova Creaturqa (PC16)
Others For other Pseudo-Chrysostom, we propose the following analysis:

• Pseudo-Chrysostom 10 (Figure 11c): This cluster falls into the group of Pseudo-Chrysostoms for which the A-B and B-A distances differ. As it consists of a single pair of texts, and that the confidence is strong in at least one direction, we would be inclined to confirm the assumption of Weyer [START_REF] Weyer | De homiliis quae Joanni Chrysostomo falso attribuuntur[END_REF] regarding their common authorship. • Pseudo-Chrysostom 11 (Figure 11b): This cluster is a subset of Pseudo-Chrysostom 4 and comprises 7 sermons, of which the first three are believed to be of a single author according to Nautin [START_REF] Nautin | Homélies pascales: II. Trois homélies dans la tradition d'Origene[END_REF]. While the unity of the PC4 corpus is not confirmed through our model, our results are more aligned with the PC11 hypothesis, but we would not go as far as confirming it, as the cluster suffers from unmirrored pair distances. • Pseudo-Chrysostom 15 (Figure 11d): In this cluster, we observe a strong reciprocal connection between De Pharisaeo and Ignem Veni Mittere, while In Illud Hominis... shows high variation depending on the pair directions.

Conclusion

Authorship verification is a crucial task in the field of computational humanities and the humanities in general, as it offers a new approach to validate older hypotheses made using traditional philological methods, such as transmission study or stylistic analysis. This is particularly important in patristic studies, where pseudo-author corpora, such as those attributed to Augustine or John Chrysostom, are significant. The challenge lies in dealing with pseudonymous corpora where most authors are unknown.

To address this, we proposed exploring the use of Siamese Networks with embeddings based on known stylometric features, such as the relative frequencies of most frequent words, affixes, and POS 3-grams. To optimize our approach, we leveraged pair mining and signal-to-noise ratio distance, both originally designed for Siamese network architectures. The results we obtained on our development and test sets are very promising, and we argue that, unlike authorship attribution problems, authorship verification problems are less susceptible to overfitting when using such architectures.

Our findings mostly align with the survey conducted by Voicu in 1981, which identified 21 potential unique pseudo-authors in the corpus of the pseudo-Chrysostomian texts (see Table Table 5 Analysis summary. C stands for confirmed, HP high probability, LP low probability, R refuted. On the second line, U stands for cluster unconfirmed, which differs from Refutation as our model is not trained for recall. PC19 could bot been studied because of the Syriac content. 5). Some pseudo-Chrysostoms are highly probable under both the framework of classical philological studies and our approach. However, we encountered discrepancies with two pseudo-Chrysostoms. First, PC1, hypothesized by Montfaucon and refuted by Altendorf [START_REF] Altendorf | Untersuchungen zu Severian von Gabala[END_REF], is mostly confirmed within our framework. Second, we were not able to confirm PC20. It's important to note that our method is designed for precision rather than recall, so this result does not necessarily refute Voicu's hypothesis but still stands out compared to the rest of our results.

In future work, we are interested in extending our approach without relying on a learned embedding space and instead using probabilistic tools for better explainability. The instability of the models we saw with Manhattan distances and the lower -yet existing -instability of STN based models argue for more interpretable and stable models. This kind of work has already been explored by Weerasinghe et al. [START_REF] Weerasinghe | Feature Vector Difference based Authorship Verification for Open-World Settings[END_REF] and would be a valuable addition to the digital humanities landscape. Further analysis on PC20 should be produced, specifically by looking at the research produced by Voicu since 1981 on this particular topic.

Figure 1 :

 1 Figure 1: Architecture of our models: vectors are passed in parallel in the embedding projection layer and a distance vector is then computed.

  𝑣𝑎𝑟(𝐸 𝑖 ) 𝑣𝑎𝑟(𝑁 𝑖𝑗 ) , and the SNR-Distance is SNRD 𝑖𝑗 = 1 SNR 𝑖𝑗 . Unlike distance metrics such as the Euclidean or Manhattan distance, SNRD 𝑖𝑗 ≠ SNRD 𝑗𝑖 , which can provoke unmirrored attributions (cf. our analysis of PC corpora).

Figure 2 :

 2 Figure 2: Learning pipeline using miner to provide pairs within the model.

Figure 3 :

 3 Figure 3: Sample count per author in the training (left), development (centre), and testing set (right).

Figure 4 :

 4 Figure 4: On the left, authors' century of activity, on the right, distribution of genres across works. Each author's century of activity is either provided by the century they are known to live in (uncertain dating) or the century for which they lived most of their adulthood (adulthood being considered as 20 years old and more).

Figure 5 :

 5 Figure 5: Standard deviation of the pairing percentage in clusters depending on the PT. Darker means more variation.

Figure 6 :

 6 Figure 6: ROC curves on both the development set and the test set. Y axis is the True Positive Rate, X axis is the False Positive Rate. Marks on the curve represent the distance threshold used to determine the positives.

Figure 7 :

 7 Figure 7: UMAP 2 dimension projections of the embeddings.

Figure 8 :

 8 Figure 8: Heatmap of the precision thresholds for all pseudo-Chrysostomian corpora. The percentage given in each cell gives the amount of pairs (A-B and B-A) of each PC that are found to be positive given the precision threshold.

(Figure 9 :

 9 Figure 9: Matrix of the precision thresholds for authors whose bad prose is considered a proof of authorship.

• 0 .

 0 91 In Drachmam ...-In Parabolam..., with a .86 PT in the other direction. Both are within a high range of PT and comes from the same PC corpus; • In Illud: Ascendit and In Parabolam de Ficu, with .60 and .74 scores. However, the connection between In Drachmam and In Illud: Ascendit is not strong (.3). Moreover, the precision thresholds for most pairs with high values are not reciprocated when considering the inverse direction: • 0.82 In Rachelem...-De Non Judicando, with a 71-point drop in the other direction; • 0.85 In Rachelem et in Infantes-Quod Mari Similis, with a 75-point drop in the other direction.

Figure 10 :

 10 Figure 10: Matrix of the precision thresholds for the biggest unconfirmed cluster: PC20 and its extension PC20b.

  (a) Precision threshold matrix for Pseudo-Chrysostoms 13, 14, and 16. (b) Pseudo-Chrysostom 4, of which sermos 1-3 are though to be of the same author by Nautin[42]. (c) Pseudo-Chrysostom 10 (d) Pseudo-Chrysostom 15 (e) Pseudo-Chrysostom 17

Figure 11 :

 11 Figure 11: Matrix of the precision thresholds.

Table 2

 2 Variation across the parameters sweep of models according to the distance (and therefore loss) used. Absolute difference between dev and test scores across models depending on the distance used.

	(a) Mean of standard-deviation across the parameters sweep depending on dis-tances. Dev Mean ± Test Mean ± Distance L2 1.04 0.55 1.82 1.02 Manhattan 1.09 0.59 1.84 1.29 STN 1.18 0.68 1.14 0.59	(b) Mean Distance L2 6.63 2.28 0.29 5.26 ± min 25% Median 75% 6.37 7.91 11.66 max Manhattan 6.80 2.68 1.08 5.22 6.87 8.42 12.43 STN 1.87 1.04 0.01 1.10 1.83 2.61 4.91

Table 3

 3 Fleiss Kappa (FK, inter-annotator agreement) using 100 models as annotators on the test set. Predictions on test are thresholded using the development distance found at the given precision.

	Development Precision 100% 99% 95% 90% 85% 80%
			Fleiss Kappa
	Manhattan	0.18	0.18 0.20 0.29 0.40 0.50
	STN	0.50	0.60 0.69 0.72 0.72 0.72

Table 4

 4 Details on the train, dev, and test sets along with their corresponding scores. "First FP Dist. " stands for First False-Positive distance, which indicates the distance at which the first false positive is encountered. The AUCROC is computed on all pairs, such that most text have more negative pairs than positive pairs overall. The metrics are computed using A-B and B-A pairs.

		Authors Texts Samples AucROC 1st FP Dist.	1st FP Classes
	Train	42	384	1294	92.35	0.519	Ephraem Syrus -Cyrillus
	Dev	20	47	155	86.75	0.405	Leontius -Amphilochius
	Test	16	47	132	83.76	0.478 Gregorius Nazian. -Nicetas Chon.

Voicu has been working on a database for these texts, https://www.trismegistos.org/pseudo-chrysostomica/.

All the feature extraction process is shared within the repositories, and the features themselves are made available.

Probably due to the cost of said annotations and the lack of well-performing automatic models for such tasks.

We maxed our value at the size of our sample, 1000 words.

We hope, however, that the Patristic Text Archive[START_REF] Stockhausen | Die Modellierung kritischer Editionen im digitalen Zeitalter[END_REF] will reach one day this kind of milestone for reproducibility purposes. We provide the annotated and processed samples as well as the processing pipeline.

Detected through the presence of TEI-l tags in the XML files.

Note that our corpus are different. Numbers from Tyo et al. are given to provide some form of context, given the lack of application of this method on corpus similar to ours.

The situation is less clear for PC2. It seems that it is well detected by our Manhattan models, see 13.

A. Online Resources

Code and data for the Voicu experiment are available at https://github.com/PonteIneptique/ Chryso-Voicu.

B. Parameters search results

Features and parameters search and their respective AUROC (in %). Each configuration is run three times (seed 42, 128, 256), average and standard deviation are provided. Tested parameters: MFP (0, 100), MFW (0, 250, 500, 1000), MFT (0, 250, 500, 1000), Distance (Manhattan, L2 and STNR). Scores are ranked according to their test AUROC. Only the top 10 are shown, the remaining scores are on the online github repository.

C. Variation of prediction on PC corpora using Manhattan or STN

D. Samples for the corpora


B Le Teurnier

M Boffety

F Goudail

Error mapping in Division of Focal Plane (DoFP) Stokes images and its use for retardance calibration

Keywords:

Integrated Division of Focal plane (DoFP) cameras have recently been developed for polarimetric imaging. These sensors use a grid composed of four different pixels with four different polarizers engraved on them. Four of these different pixels form a superpixel which enables the estimation of the linear Stokes vector with a single acquisition. Estimation of the full Stokes vector can be done by adding a retarder in front of the DoFP camera and performing at least two acquisitions with two different angular positions of the retarder.

As a drawback, DoFP sensors are particularly sensitive to the spatial variations of the scene within a superpixel. Therefore, if these variations are non-negligible compared with the measurement noise, the estimation of the state of polarization is corrupted. We propose a method to map the superpixels in which the estimation can be trusted.

As an example of application, we demonstrate the benefit of using such a mapping to perform dynamic autocalibration of the retardance of the retarder placed in front of the camera for full Stokes imaging. Thanks to measurement redundancy, calibration of the retardance and estimation of the Stokes vector can be jointly performed provided three acquisitions for three different orientations of this retarder are made. However, this method may be perturbed by the spatial variations of a complex scene, and the proposed mapping enables to wisely choose the superpixels that are appropriate for calibration.

INTRODUCTION

Polarization imaging systems lead to contrasts that are invisible in conventional intensity images. [START_REF] Tyo | Review of passive imaging polarimetry for remote sensing applications[END_REF][START_REF] Thilak | Polarization-based index of refraction and reflection angle estimation for remote sensing applications[END_REF][START_REF] Foldyna | Characterization of grating structures by mueller polarimetry in presence of strong depolarization due to finite spot size[END_REF][START_REF] Mu | Optimized design, calibration, and validation of an achromatic snapshot full-stokes imaging polarimeter[END_REF] Such imaging system needs to acquire several analyses of the polarization in order to derive the polarimetric information. DoFP sensors enable to obtain these different analyses in a single acquisition. The existing DoFP are composed of four different pixels with different linear polarizers engraved on them. They allow the estimation of the linear polarization with a single acquisition and thus enable real time polarization imaging. [START_REF] Powell | Calibration methods for division-of-focal-plane polarimeters[END_REF][START_REF] Chen | Calibration method of microgrid polarimeters with image interpolation[END_REF][START_REF] Feng | Polarized-pixel performance model for DoFP polarimeter[END_REF][START_REF] Roussel | Polarimetric precision of micropolarizer grid-based camera in the presence of additive and poisson shot noise[END_REF][START_REF] Lane | Calibration of a polarization image sensor and investigation of influencing factors[END_REF] In order to measure the full Stokes vector, one can add a rotating retarder in front of this DoFP camera and perform at least two acquisitions for different orientations of the retarder. [START_REF] Qi | Real time complete stokes polarimetric imager based on a linear polarizer array camera for tissue polarimetric imaging[END_REF][START_REF] Shibata | Robust full stokes imaging polarimeter with dynamic calibration[END_REF][START_REF] Roussel | On the optimal ways to perform full stokes measurements with a linear division-of-focal-plane polarimetric imager and a retarder[END_REF][START_REF] Li | Theory of autocalibration feasibility and precision in full stokes polarization imagers[END_REF] With three acquisitions with a rotating retarder, the retardance can be calibrated alongside the estimation of the Stokes parameters. 11, [START_REF] Goudail | Precision of retardance autocalibration in full-stokes division-of-focal-plane imaging polarimeters[END_REF][START_REF] Li | Fundamental precision limits of full stokes polarimeters based on DoFP polarization cameras for an arbitrary number of acquisitions[END_REF][START_REF] Teurnier | When is retardance autocalibration of microgrid-based full stokes imagers possible and useful?[END_REF] As a downside, with a DoFP sensor the different pixels used for the estimation of a Stokes vector are not localized at the same place and thus are not seeing the same part of the scene. Therefore, if the scene contains fast spatial variations there will be a strong difference between the polarization seen by each pixel and the estimation will be wrong.

Thus we describe here a method to detect whether or not the variations of the scene within a super-pixel are too important to lead to a reliable estimation. This method uses the redundant information within the measurements of the super-pixel and an estimation of the spatial variations of the intensity in the scene. By combining these two criteria one can obtain an error map which tells where the estimation is not reliable. [START_REF] Teurnier | Definition of an error map for DOFP polarimetric images and its application to retardance calibration[END_REF] We then use this error map to help the calibration of a retarder.

METHODOLOGY

DoFP polarization sensor

DoFP sensors are composed of an array of "super-pixels". Each one contains four pixels with linear polarizers with four different angle at 0°, 45°, 90°and 135°. The four measurements obtained with a super-pixel enable the estimation of the linear Stokes vector. To estimate the full Stokes vector one can use a rotating retarder and perform at least two acquisitions with two different orientations of the retarder. With N acq acquisitions, a super-pixel leads to N mes = 4 × N acq measurements. These measurements are gathered in the vector I and this acquisition is modeled as:

I = WS, (1)
where S = (S 0 , S 1 , S 2 , S 3 ) T is the incident Stokes vector. It is of length K = 4 in the general case, but if we only estimate the linear polarization then we consider only its first 3 component and its length is K = 3. W is a N mes × K matrix called measurement matrix of the super-pixel and describes the acquisition process of the super-pixel. The incident Stokes vector is estimated with:

Ŝ = W + I, (2)
with Ŝ the estimated Stokes vector and the superscript + denotes the Moore-Penrose pseudo-inverse. This estimation model considers a homogeneous incident polarization state on the super-pixel. Unfortunately, the different pixels of the super-pixel usually receive different incident polarizations due to the spatial variations of the scene. If this variations are greater than the measurement noise, it may lead to a significant bias on the estimation of the polarization. We describe here a method to detect the super-pixels which lead to such unreliable estimations.

Redundancy criterion

Within a super-pixel, the number of measurements used for the estimation of the Stokes vector is larger than the number of parameters to estimate. This leads to redundancy between these measurements. The light intensity incident on the super-pixel can be estimated with S 0 a = I 0 + I 90 and S 0 b = I 45 + I 135 . In the ideal case, with no noise and a constant incident Stokes vector on the super-pixel, S 0 a = S 0 b . But if there is inconsistency between the measurements, we may have: R lin = I 0 + I 90 -(I 45 + I 135 ) ̸ = 0. This criterion provides a quantitative evaluation of the inconsistencies between the measurements acquired by the super-pixel. But it does not take into account the calibration of the sensor and it is not applicable to imaging systems composed of a sensor and a rotating retarder used to measure the full Stokes vector.

We thus propose a generalization of this criterion. Using the singular value decomposition (SVD) of the measurement matrix W, one has:

W = UDV T , (3)
where U and V are unitary matrices of dimensions N mes × N mes and K × K. D is a N mes × K matrix with a K × K first block which is diagonal and contains the singular values of the measurement matrix W. The last N mes -K rows of D contains only zeros. The superscript T denotes the transposition. By substituting Eq. 3 in Eq. ( 2) we have:

U T I = DV T S. (4)
The last N mes -K elements of the vector U T I are null, due to the properties of D, in the ideal case when considering a homogeneous incident Stokes vector and in absence of noise. We then define the matrix U R composed of the last N mes -K columns of the matrix U and compute:

R = U T R I. (5)
The vector R is null in the ideal case with a homogeneous incident Stokes vector and without noise. Furthermore, in this conditions and in the case of the sensor used alone for linear polarization estimation, the criterion R is a scalar and is proportional to R lin .

In practice, we mostly have Poisson noise, but with a mean value high enough to consider it as an additive Gaussian random variable which variance is equal to the mean value of the signal. We can derive from this consideration that the elements of the vector R are centered Gaussian random variables. One can thus obtain a vector of standard Gaussian random variable T whose elements are defined with:

T(i) = R(i) VAR[R(i)] . (6)
Then, if the noise is the only perturbation affecting the measurements, the Euclidean norm of the vector T follows a χ 2 law and a threshold can be applied with a constant false alarm rate. [START_REF] Teurnier | Definition of an error map for DOFP polarimetric images and its application to retardance calibration[END_REF] This criterion based on the redundant information is then able to detect whether the inconsistencies between the measurements are significant in front of the noise.

This detector suffers a flaw. It does not detect the intensity variations within the super-pixel. This is highlighted on Figure 1 where are displayed the estimated intensity S 0 with the detected inconsistencies displayed in red and the estimated degree of linear polarization (DoLP). The scene acquired is a blank cardboard with large black letters printed on it. We used the DoFP sensor alone to estimate only the linear polarization state. We observe, on the estimated DoLP, estimation artifacts located on the intensity edges of the scene: estimated DoLP is significantly higher on these edges without physical explanation. However the detector defined in Equation ( 6) does not detect these edges as problematic even though there is clearly an error of estimation. Thus, we need to use a separate intensity variation detector to help this redundancy based criterion.

Intensity Variation detection

We designed an intensity variation detector with constant false alarm rate. Doing so, we only need to set one parameter, the false alarm rate, to apply thresholds to both the redundancy based criterion and the intensity variation criterion. This intensity variation detector uses the intensity estimations on neighboring super-pixels, as shown Figure 2, so that it is independent from the polarization states. This detector uses the log-likelihood ratio between the two following hypotheses:

• There is no intensity variation and the signal is mostly Poisson noise of constant mean over the super-pixels.

• There is an intensity variation within this array and we consider four regions within the 4 × 4 pixels array selected. The signal within the four regions follows four different Poisson laws of parameters λ a , λ b , λ c and λ d as shown on Figure 2. Then the log-likelihood ratio can be computed as:

17 ln(R) = 4 1 4 k ( λk ln( λk )) -k λk 4 ln k λk 4 . ( 7
)
where the λk are the estimation of the parameters λ k for the four different super-pixels surrounding the super-pixel of interest. We demonstrated 17 that this log-likelihood ratio is proportional to a χ 2 law when the measurement noise is the only perturbation. Thus a threshold can be applied with a constant false alarm rate to detect an intensity variation.

The error map obtained with this intensity variation detector is then combined with the error map based on the redundancy based criterion. An example is shown on Figure 3 where we use the same cardboard scene as for 1. In green is displayed the error map obtained with the redundancy criterion and in red the error map given by the intensity variation detector for a false alarm rate P fa = 10 -3 . In yellow we display the pixels for which both criteria detect anomalies. As we can see the intensity variation detector detects well the edges of the black letters where we observed estimation artifacts. Thus it is a good complement to the redundancy based criterion.

APPLICATION

We demonstrate the utility of such error map for the dynamic calibration of a retarder placed in front of the DoFP camera for the estimation of the full Stokes vector. Such calibration is possible when at least three acquisitions are performed for different orientations of the retarder.

Autocalibration

Dynamic calibration (we usually call it autocalibration) is performed by jointly estimating the retardance δ of the retarder and the Stokes vector S from the measurements of a super-pixel by using the redundancy between the intensity measurements within a super-pixel. To do so, we optimize the criterion F(δ):

δ = arg min δ [F(δ)], (8)
with where I is the vector of the measured intensities, I d the identity matrix and W(δ) the measurement matrix of the super-pixel considering the several acquisitions for the several orientations of the retarder of retardance δ. [START_REF] Teurnier | When is retardance autocalibration of microgrid-based full stokes imagers possible and useful?[END_REF] Then the estimated retardance δ is introduced in the measurement matrix W(δ) which is substituted in Equation 2:

F(δ) = [I d -W(δ)W(δ) + ]I 2 , (9)
Ŝ = W(δ) + I, (10)
Only one super-pixel is needed to perform this autocalibration. However, using a set of several super-pixels reduces the uncertainty on this estimation. In order to wisely choose the super-pixels used for this autocalibration, it has been shown [START_REF] Teurnier | When is retardance autocalibration of microgrid-based full stokes imagers possible and useful?[END_REF] that, in the presence of additive Gaussian noise, we must consider the super-pixels which maximize the value of the reduced signal to noise ratio SNR δ = SNR×DoLP. Furthermore, there is the constraint that SNR δ > 8, otherwise the autocalibration is not possible. In practice we thus choose the super-pixels which maximize the estimated ŜNR δ and respect ŜNR δ > 8.

Error mapping

However as seen on Figure 1, it often happens that the estimated DoLP, and then ŜNR δ , is over estimated due to artifacts of estimation where there is significant spatial variations in the scene. This may lead to select superpixels for the autocalibration which are not reliable. This scenario is shown on Figure 4 where are displayed the intensity image of the scene used with the set of super-pixels selected by maximizing the estimated ŜNR δ displayed in red (left), and the histogram of the estimated retardances for this set of super-pixels plotted in red (right). We are using a quarter wave-plate (QWP) so we expect a retardance close to 90°. The super-pixels selected are mainly located on intensity edges for which we have seen that the estimated DoLP is biased. These super-pixels lead to huge errors on the estimated retardances which are not gathered around the expected value.

Then we compute the error map for this scene which is displayed on Figure 5 in yellow on the gray level intensity image. Then we use this error map to select the super-pixels which maximize the value of ŜNR δ outside of the set of super-pixels which have been detected as unreliable. This lead to the set of super-pixels displayed in green on Figure 4 (left). They are located mostly on a smooth area of the scene where the estimation is more reliable. The histogram of the values of estimated retardances using this set of super-pixels is plotted in green (right). We see that this histogram is centered around the expected value for the retardance with a mean value of 87.6 • and a standard deviation of 4.6 • . This set of super-pixels thus gives a suitable estimation of the retardance for the used wave-plate (90°).

CONCLUSION

In conclusion, the generalized redundancy criterion R provides information about the inconsistencies between the measurements within a super-pixel and can take into account the calibration of the sensor as well as being able to work with full Stokes imaging system. We derived a CFAR detector from this criterion and as it is not able to detect errors which are due to intensity edges we proposed another CFAR detector to detect these intensity variations. The error map obtained by combining these two detectors enables to have a map of the super-pixels for which we can not trust the estimation. We demonstrated that such mapping makes it possible to perform dynamic calibration of the retardance of a wave-plate used within a full Stokes imaging system even with a complex scene. Such error mapping may be useful for other applications such as demosaicing strategies.

Figure 1 .

 1 Figure 1. An example of error map and estimated DoLP on a scene composed of a blank cardboard with large black letters printed on it. The intensity estimated in gray level with the error map displayed in red (left) and the estimated DoLP (right). The error map used here uses only the redundancy criterion to detect the inconsistencies between the measurements within a super-pixel.

Figure 2 .

 2 Figure 2. A 4 × 4 pixels array. The four pixels at the center form the super-pixel of interest. The four super-pixels highlighted by the color squares are used to measure the four parameters λa, λ b , λc and λ d corresponding to the sum of the measured intensities within these super-pixels.

Figure 3 .

 3 Figure3. The error map obtained by combining the redundancy criterion and the intensity variation detector. The intensity is displayed in gray level, the redundancy criterion based detector in green and the intensity variation detection in red. In yellow appear the pixels detected by both detectors as problematic.

Figure 4 .Figure 5 .

 45 Figure 4. The histogram of the estimated retardance value for the N chosen super-pixels. Without using the error map (in red) and with excluding the pixels detected by the error map (in green)