Olivier Cavadenti

Victor Codocedo

Jean-Franc ¸ois Boulicaut

Mehdi Kaytoue

When Cyberathletes Conceal Their Game: Clustering Confusion Matrices to Identify Avatar Aliases

Video game is a very lucrative industry, unleashed by the ubiquity of gaming devices, multi-player networks and live broadcasting platforms. Games generate large amounts of behavioural data which are valuable to face the new challenges of video game analytics such as detecting balance issues, bugs and cheaters. In electronic sports (e-sports), cyberathletes conceal their online training using different aliases or avatars (virtual identities), which allow them not being recognized by the opponents they may face in future competitions (with cash prices challenging already most of the traditional sports). It was recently suggested that behavioural data generated by the games allows predicting the avatar associated to a game play with high accuracy. However, when a player uses several avatars, accuracy drastically drops as prediction models cannot easily differentiate the player's different avatar aliases. Since mappings between players and avatars do not exist, we introduce the avatar aliases identification problem and propose an original approach for alias resolution based on supervised classification and Formal Concept Analysis. We thoroughly evaluate our method with the video game Starcraft 2 which has a very wide and active community with players from diverse cultures and nations. We show that under some circumstances, the avatars of a given player can easily be recognized as such. These results are valuable for e-sport structures (to help preparing tournaments), and game editors (detecting cheaters or usurpers).

I. INTRODUCTION

Currently, video games are a popular and lucrative industry generating more revenue than both Cinema and Music. This recent and fast development has been catalysed by several factors. First, an increasing part of the population has access to connected electronic devices (computers, smart-phones, etc.), and to the Web including a myriad of single and/or multiplayer games. Video games are now a popular leisure activity. Furthermore, the recent development of competitive gaming and live video game streaming platforms [START_REF] Kaytoue | Raïssi. Watch me playing, I am a professional: a first study on video game live streaming[END_REF] makes electronic sports (e-sports) a reality [START_REF] Taylor | Raising the Stakes : E-Sports and the Professionalization of Computer Gaming[END_REF]. Professional players on contract with teams and sponsors are taking part in competitions with cash prices challenging already most of the traditional sports. Cyberathletes are widely followed in so-called off-line events and daily supported on the Web through their own live broadcasts [START_REF] Kaytoue | Raïssi. Watch me playing, I am a professional: a first study on video game live streaming[END_REF] generating the fourth largest stream of data in 2015 for the platform, Twitch.tv. People not only enjoy playing, but also enjoy watching the others for a variety reasons [START_REF] Cheung | Starcraft from the stands: understanding the game spectator[END_REF], [START_REF] Kaytoue | Raïssi. Watch me playing, I am a professional: a first study on video game live streaming[END_REF]. Games generate tremendous amounts of behavioural data, valuable to face the many new industrial challenges such as detecting balance issues [START_REF] Bosc | Mining balanced sequential patterns in RTS games[END_REF], bugs and cheaters [START_REF] Von | Machine learning and data mining in call of duty (invited industrial talk)[END_REF], but also valuable This research has been partially funded by the French National Project FUI AAP 14 Tracaverre 2012-2016. for several academic fields such as artificial intelligence [START_REF] Ontanon | A Survey of Real-Time Strategy Game AI Research and Competition in StarCraft[END_REF], computer human interactions [START_REF] Yan | Masters of control: Behavioral patterns of simultaneous unit group manipulation in starcraft 2[END_REF] and cognitive sciences [START_REF] Thompson | Video game telemetry as a critical tool in the study of complex skill learning[END_REF].

In this paper, we introduce a new problem: the avatar aliases identification problem. In most of online competitive games, players need an "avatar" (i.e., an online identity) to log in the game network. Nothing forbids a player to have several avatars and actually, it is a very common practice for professional players, or cyberathletes, of several video game franchises (e.g. League of Legends, Starcraft 2, Dota 2, Counter Strike). Players generally have one official avatar for official tournaments, and several others to conceal their tactics while playing without being recognized by other players they may meet online: global rankings and leagues are public just as in chess and tennis, while game logs (called replays) are available and prone to analysis by means of visualization 1 , machine learning and data mining techniques [START_REF] Low-Kam | Mining statistically significant sequential patterns[END_REF], [START_REF] Synnaeve | A bayesian model for opening prediction in RTS games with application to starcraft[END_REF] just as in standard sport analytics. Accordingly, we are facing a set of players, generating behavioural data (game logs or traces), in a one-to-many relationship with avatars (the many-to-many relationship will be discussed later). Our goal is not to discover this mapping, since players privacy is protected. Instead, the avatar aliases identification problem aims at discovering groups of avatars belonging to the same player. Solving this problem is motivated by the growing need of e-sport structures to study the games and strategies of the opponents (tournament preparation), and the security challenges of game editors (detecting usurpers).

Recently, it was suggested that behavioural data hide individual identifying patterns [START_REF] Thompson | Video game telemetry as a critical tool in the study of complex skill learning[END_REF], [START_REF] Yan | Masters of control: Behavioral patterns of simultaneous unit group manipulation in starcraft 2[END_REF], (tested on the real time strategy game Starcraft 2 ( c Blizzard)). After choosing features related to keyboard usage, Yan et al. showed that a classifier can be trained to predict with high accuracy the avatars involved in a game play [START_REF] Yan | Masters of control: Behavioral patterns of simultaneous unit group manipulation in starcraft 2[END_REF]. Nevertheless, they purposely considered datasets without players having several avatars (what we call avatar aliases). Indeed, in presence of such avatar aliases, the prediction accuracy drastically degrades, since prediction models fail at differentiating two avatars of the same player. We build on this work by proposing an approach called D ÉBROUILLE for answering the avatar aliases identification problem: it relies on mining confusion matrices yielded by a supervised classifier using Formal Concept Analysis [START_REF] Ganter | Formal Concept Analysis[END_REF], and exploits the confusion a classifier has in presence of avatar aliases when they belong to the same player. Experimental evaluation shows very good results under certain restrictions imposed to the set of players involved in a dataset. The remainder of this paper is organized as follows. Section II introduces the problem settings, our claims and goals. We formally present an original methodology in Section III for discovering avatar aliases and the evaluation strategy in Section IV. Our results are assessed through an extensive evaluation in Section V. Related work is discussed in Section VI before we conclude.

II. PROBLEM SETTINGS

A. Starcraft 2 in competitive gaming

We study the real-time strategy game (RTS) Starcraft 2, a franchise released in 2010 which met success in competitive gaming and e-sports [START_REF] Taylor | Raising the Stakes : E-Sports and the Professionalization of Computer Gaming[END_REF]. A standard game involves two players and each chooses a faction (Zerg, Protoss or Terran) with different weaknesses and strengths (following a rock/paper/scissors principle). Players control buildings and units through an aerial view of the battleground map, collect resources to build an army and achieve victory by destroying the opponent's forces. A player needs an avatar to enter the dedicated multi-player system called Battle.net. In the context of e-sport, our careful attention is given to the avatar aliases identification problem that affects the game 2 . Nothing forbids a user from having multiple avatars, which is actually a common practice among professional players for a variety of reasons. For example, as they compete in international tournaments, they may need an avatar in each different Battle.net server (USA, Europe, Korea, etc.). Another important reason is that professional users may need to hide their tactics while training before competitions. For this reason, cyberathletes may create different avatars with names such as lIlIllIlIlll called bar code avatars (as in Table III) for practising on public servers without being recognized. Practise on public networks corresponds to a major time of their activity [START_REF] Taylor | Raising the Stakes : E-Sports and the Professionalization of Computer Gaming[END_REF]. Currently, among the best 50 avatars in the global Starcraft 2 ranking system, 46 avatars are bar code 3 . As argued in the introduction, multi-aliases prompt several problems. Our goal is to formalize and solve the avatar aliases identification problem which consists in finding avatars belonging to the same player without any information about the individual but his game behaviour.

B. Behavioural data

During a game, a player performs a series of actions which generates a trace. Basic actions rely on mouse usage, provided with semantics: "selection of a unit", "attack with selected unit", "collect resources with selected unit", etc. Many (if not all) of these actions can be performed with the mouse by selecting a series of menus on the screen. However, to improve the amount of actions a user can perform per minute (a critical issue in this kind of video games), expert players prefer to bind these actions directly to the keyboard through the use of customized hotkeys, also called control groups [START_REF] Yan | Masters of control: Behavioral patterns of simultaneous unit group manipulation in starcraft 2[END_REF]. As such, any action, done through mouse or keyboard, is stored in replay files that are available on the Web and usable for observing and studying other players strategies. Replays constitute extremely rich behavioural data, and we use them for answering our problem. Table I shows two traces associated to a single game: Player TAiLS starts by selecting with the mouse a building 2 Multi-aliases are known as "smurfs" players within the gamer community. 3 http://www.sc2ranks.com visited on 2015, May 28 th called Base, assigns it to hotkey 1 (a), selects some units with the mouse (s), select units attached to hotkey 2 (s), etc. Let us consider the following scenario. Given a trace, is it possible to find its associated avatar? In Table I, given the trace in the first row, is it possible to say that the avatar associated to it is RorO? We can approach this question from a classification point of view. Given a training set of traces labelled with their associated avatars, we would like to classify new "instances" (new traces) in the space of classes represented by the set of avatars. As such, we can apply any supervised classification technique in this direction. Recently, Yan et al. [START_REF] Yan | Masters of control: Behavioral patterns of simultaneous unit group manipulation in starcraft 2[END_REF] suggested that behavioural data, as presented in the previous subsection, offer predictions with high accuracy, when there are no aliases in the dataset. In presence of aliases however, the accuracy drastically drops, as trained models are not able to differentiate properly different avatars of a same individual.

The reason why game traces allow good avatar predictions, when there are no aliases, is related to the cognitive process involved in the activity of playing [START_REF] Thompson | Video game telemetry as a critical tool in the study of complex skill learning[END_REF]. A game of Starcraft 2 is short (between 15 and 20 minutes in average), in real time, and several tasks and strategies are repeated thousands of time to be achieved instinctively: mastering the game requires an intense multi-tasking activity (not sequential) and daily practise. Novice's traces will seem messy during the learning process, but will converge with practise towards distinctive patterns, or individual signatures.

D. Towards confusion clustering to identify avatars aliases

In general, players use several avatars that generate traces. This is modelled in Figure 1. The mapping owns : P layer → Avatar is however not known (or very partially by the game editor that keeps it private). The only information available is the mapping generates : Avatar → T race. Notice that we make the assumption that an avatar belongs to a single player (individual). This may not hold: nothing forbids two persons from sharing an avatar. However, as an avatar is ranked according to all its games (wins and loses) in a world ranking system, it is fairly safe to assume in general that professional players (our focus) do not share their avatars to protect their reputations but also privacy when playing in secrecy.

The general intuition of our approach to tackle the avatar aliases identification problem is now introduced. Our model can be split in two sub-tasks, namely "finding Trace patterns associated to Avatars" and "finding Avatar clusters associated to Players". For the first task, a classifier is built as explained in the previous subsection, for predicting the avatar involved in a game. In presence of avatar aliases, its accuracy drastically drops. However, it is a fair hypothesis that the classifier confusion shall be spread locally among avatar aliases. As such, the second step involves a particular clustering of the confusion matrix, that outputs avatar aliases candidates. We formalize this step with FCA: a fuzzy set operator on class memberships enables the finding of candidate patterns.

III. METHOD FORMALIZATION

Let A be a set of avatars and T be a set of traces such that for a given avatar a ∈ A, the set T a ⊆ T is the set of all traces generated by a. Consider a classifier ρ where labels are the avatars to predict. Our method consists in using the confusion matrix generated by a classifier ρ and analyse how confusion between labels is spread to extract candidates of avatar aliases. This process has two steps: firstly, specific patterns are extracted from the confusion matrix, secondly, they are scored, ranked and post processed. The corresponding pseudo-code of our approach, called D ÉBROUILLE for unscramble in French, is given in Algorithm 1 and is detailed hereafter.

A. Classifying Traces

A classifier is a function ρ : T → A that assigns the avatar ρ(t) ∈ A to a given trace t ∈ T . Let n = |A| be the number of avatars in A, from any classifier ρ, one can derive a confusion matrix

C ρ n×n = (c i,j ) where c i,j = |{t ∈ T ai s.t. ρ(t) = a j }|
Each row and column of C ρ correspond to an avatar, while the value c ij is the number of traces of avatar a i that are classified by ρ as of avatar a j . The normalized confusion matrix is given by Cρ

= [c i,j /|T ai |]
where Cρ i,i = 1 for any i ∈ [1, |A|] means all the traces of avatar a i are correctly classified by ρ. Table II gives an example of normalized confusion matrix between five avatars. It is worth mentioning that we have generalized the classifier to a given function ρ and thus, we will treat it as a "black box" with an input (in the form of a set of traces T and avatars A) and an output which corresponds to the confusion matrix. Internally, features built from traces can be chosen independently, and the classifier can split the set of traces into train and test sets using different strategies.

Algorithm 1 D ÉBROUILLE pseudo code

Require: Cρ : normalized confusion matrix, λ cluster score threshold, s score threshold. Ensure: P list of pairs of avatar ranked by cluster score. for all p ∈ pairs do 7:

if p ∈ P and cluster score(p) > λ then 8:

P ← P ∪ {p} 9: P ← rank P according to the cluster score 10: return P

B. Clustering Avatars

Our goal is to discover groups of avatars that belong to the same player. Our intuition is that a classifier will hardly differentiate these avatar aliases, hence the confusion matrix values should be high and concentrated around them. This is exemplified in Table II: avatars {a 1 , a 2 } are candidates to belong to the same player, {a 4 , a 5 } shall belong to another user, while a 3 stays as singleton with a diagonal high value. Hence, a reasonable clustering of avatars would be given by the partition {{a 1 , a 2 }, {a 3 }, {a 4 , a 5 }}. 2. These conditions come from the fact that, if a i , a j correspond to the same player, traces in T ai have the same probability of being classified as a i or a j (the same for traces in T aj ). Furthermore, for a trace of avatar a i , it is required that the probability of classification is spread between a i and a j only, meaning that

Cρ ij + Cρ ii 1 (similarly for a j ).
Using this rationale, in what follows we propose (i) to extract patterns from the confusion matrix, and (ii) to post process them to provide groups of candidate avatar pairs. The first step is achieved thanks to Formal Concept analysis (FCA [START_REF] Ganter | Formal Concept Analysis[END_REF], [START_REF] Ganter | Pattern structures and their projections[END_REF]), while we define scoring functions and ranking for the second step.

1) Fuzzy pattern structure: Let us define the fuzzy set of membership degrees L A where L = [0, 1], such as the mapping function δ : A → L A assigns membership values for the avatar a i in the fuzzy set L A based on the normalized confusion matrix. Simply, this is a mapping that assigns to a i its corresponding row in Cρ which we denote Cρ i . We model accordingly a confusion matrix Cρ as a pattern structure (A, (L A , ), δ). The operator is a meet operator in a semi-lattice (idempotent, commutative and associative), and is defined as follows, given two avatars a i , a j ∈ A:

δ(a i ) δ(a j ) = min( Cρ ik , Cρ jk ) , k ∈ [1, |A|] δ(a i ) δ(a j ) ⇐⇒ δ(a i ) δ(a j ) = δ(a i )
Actually, corresponds to the fuzzy set intersection and (L A , ) is a partial order over the elements of L A which can be represented as a semi-lattice.

The pattern structure (A, (L A , ), δ) is provided with two derivation operators, forming a Galois connection [START_REF] Ganter | Pattern structures and their projections[END_REF]. Formally, we have, for a subset of avatars A ⊆ A and a fuzzy set d ∈ L A such as:

A = a∈A δ(a) d = {a ∈ A | d δ(a)} A pair (A, d) is a pattern concept iff A = d and d = A.
Pattern concepts are ordered by extent inclusion such that for (A 1 , d 1 ) and (A 2 , d 2 ) we have:

(A 1 , d 1 ) ≤ (A 2 , d 2 ) ⇐⇒ A 1 ⊆ A 2 (or d 1 d 2 )
Intuitively, a pattern concept (A, d) contains a fuzzy set d which can be represented as a vector d = d j with length |A| where each value d j is the minimum for all rows i in column j of matrix Cρ such that a i ∈ A.

Example. The Table II illustrates a confusion matrix involving five avatars. It has been obtained from a classifier ρ. We illustrate first how pattern concepts are generated:

δ(a 1 ) = {a 0.6 1 , a 0.4 2 , a 0 3 , a 0 4 , a 0 5 } δ(a 2 ) = {a 0.4 1 , a 0.55 2 , a 0.05 3 , a 0 4 , a 0 5 } δ(a 1 ) δ(a 2 ) = {a 0.4 1 , a 0.4 2 , a 0 3 , a 0 4 , a 0 5 }
2) Scoring concepts and extracting aliases: The scoring function s : L A → [0, 1] is given as follows: for a pattern d,

s(d) = |A| j=1 d j
It is clear that function s is decreasing w.r.t. the order of pattern concepts, i.e.

(A 1 , d 1 ) ≤ (A 2 , d 2 ) =⇒ s(d 1 ) ≤ s(d 2 )
. Thus, pattern concepts can be mined up to a given score threshold analogously as formal concepts can be mined up to a given minimal support, as it is done in pattern mining [START_REF] Giacometti | 20 Years of Pattern Mining: a Bibliometric Survey[END_REF]. We can appreciate that the higher the score of a given pattern, the more confused is the classification of traces of avatars a ∈ A by ρ in Cρ and thus, they become candidates for merging. This property directly follows from the choice of our similarity operator as a fuzzy set intersection, which behave as a pessimistic operator (returning minimum values).

The pattern mining step is executed as follows and corresponding to the M ineF uzzyConcepts step (Line 2) in Algorithm 1. From the confusion matrix we compute all possible pattern concepts using the addIntent algorithm [START_REF] Van Der Merwe | AddIntent: A New Incremental Algorithm for Constructing Concept Lattices[END_REF]. Pattern concepts are then ranked according to their score (Line 3) and converted into a list of pairs (Line 5). For example, if a pattern concept extent contains three avatars a 1 , a 2 and a 3 , we convert this concept into pairs (a 1 , a 2 ), (a 1 , a 3 ) and (a 2 , a 3 ). The order among pairs of the same pattern is disregarded.

The addIntent algorithm is known to have a linear complexity w.r.t. the number of possible pattern concepts [START_REF] Van Der Merwe | AddIntent: A New Incremental Algorithm for Constructing Concept Lattices[END_REF] which can grow exponentially w.r.t. the number of avatars in A. In our case, experimental evidences suggest that the number of pattern concepts in this setting is much lower than A 2 , given the empty intersection between most pairs of avatars in A. Finally, given that the scoring function is monotonous w.r.t. the order , it can be used as a filter to stop the calculation of meaningless patterns.

Example. Continuing the previous example, we have:

s({a 1 , a 2 } ) = 0.8 (1)
s({a 4 , a 5 } ) = 0.75 (2)
s({a 1 , a 2 , a 4 } ) = 0.05 (3)
ii + Cρ ij + Cρ ji + Cρ jj 2.
Consider that the pair of avatars (a i , a j ) respects these conditions. It is easy to see that (a i , a j ) will necessarily be a candidate pair highly ranked from the previous step.

Cρ ij Cρ jj min( Cρ ij , Cρ jj ) Cρ ii Cρ ji min( Cρ ii , Cρ ji ) =⇒ min( Cρ ij , Cρ jj ) + min( Cρ ii , Cρ ji ) 1
Thus, the set of avatar clusters we are looking for are contained within the set of candidate pairs and moreover, they are highly ranked. In order to filter the list of candidates from pairs that do not hold the avatar cluster definition, we propose a cosine similarity measure between a couple of vectors calculated for each avatar as follows. Let (a i , a j ) be a candidate pair, the cluster score is defined as:

cluster score(a i , a j ) = cosine( Cρ ii , Cρ ij , Cρ jj , Cρ ji )
The cluster score establishes a measure of how close is a candidate pair from being an avatar cluster. The logic of this follows from the following scenario. Consider that the traces of avatar a i were all correctly classified meaning that Cρ ii = 1 and that the traces of avatar a j were all incorrectly classified as a i , meaning that Cρ ji = 1, thus we have the following section of the normalized confusion matrix:

ai aj ai 1 0 aj 1 0
We can observe that the pair (a i , a j ) will be contained in the set of candidate pairs and will be highly ranked, even though it is not an avatar cluster since it violates the first condition. The cluster score for this particular case can be calculated as:

cluster score(a i , a j ) = cosine( 1, 0 , 0, 1 ) = 0 meaning that this candidate pair is not an avatar cluster. Notice that for the pair of avatars such that a ii = 1 and a jj = 1, the cluster score is 1 (cosine between parallel vectors) while the pair is not an avatar cluster. Indeed, this is true, however this pair would have a score s equal to 0 and would be at the bottom of the ranked candidate pairs. A third kind of pair occurs when the traces of a i and a j are all incorrectly classified as a third avatar a k . In such a case, the cluster score is 0.

The post processing step is executed as follows as depicted in Algorithm 1. Given a ranked list of candidate pairs yielded from the previous step (Line 2 and 3), each pair is evaluated using the cluster score. Given an arbitrary threshold λ, if the cluster score of the candidate pair is below this threshold, then it is rejected (Line 7 and 8). Candidate pairs are re-ranked into a final list of avatar clusters (Line 9).

IV. EVALUATION

In this section, we provide a detailed evaluation procedure used in the experiments for assessing the ability of our approach at finding avatar aliases. Table III: An example of five Battle.net accounts and their respective avatars

A. Avatars matching

As we do not have information about the users behind the avatars, it is not possible to actually evaluate the candidate pairs for merging using a "ground truth". Instead, we perform an indirect evaluation of our approach using three different strategies one being specific to the game Starcraft 2. Indeed, as illustrated in Figure 2, the avatar system of Starcraft 2 is more elaborated than our general model given in Figure 1. The Table III 1) Nicknames: Each avatar is associated with a non-unique (nick-)name. Names are chosen by users and can be changed at any time. To change the name, users have to pay a fee to the videogame company. This means that, even when changing the name is possible, users do not change their name often.

We can identify three situations for the change of name. Firstly, users want anonymity and thus, they change their name to avoid being recognized by other users. We can consider this as a "cheap" way to achieve anonymity, since it is actually cheaper than creating a new account, and the user does not have to re-classify her new account into a top-league (actually, quite expensive in terms of play-hours). Since changing the name does not require a change in the account, even with a new name users are actually easily recognizable. We will discuss this in the following section. Secondly, users may join a team. Teams are groups of avatars that frequently play together collaboratively against other teams. Teams also have associated names which users usually add to their names as a suffix. Thirdly, users may change their name by any other reason.

It is clear that, when finding two candidate avatars for merging, we cannot rely simply in comparing their names. On the one hand, very popular names such as "Batman" or "Superman" may be used by several users. On the other hand, even if a user has different accounts, and we successfully identify them for merging, nothing forbids the user to use different avatar names for those accounts. Thus, names will be used as weak indicators for avatar merging.

2) URL: In Starcraft 2, as described in Table III, an avatar is associated with a unique account. This account can be identified by the URL associated to the avatar which contains information about the location of the avatar (European Union, USA, Korea, etc.), the ID number and the avatar's name. As we have discussed, users are free to change the name of their avatars by paying a fee. When this is done, the URL changes by removing the old name and including the new one. However, since it is the same account, the location and the ID number remain the same.

The URL is a strong indicator for avatar merging since it is quite obvious that, given two avatars with the same associated account, they correspond to the same user. Usually, we would integrate the traces of these avatars into a single one before the trace classification step (and actually, we do this for the first of our experiments). Instead, we will leave them as they are since they provide us with a sort of "ground truth". That is, if our system is able to merge avatars of different accounts, it should be able to merge avatars of the same account.

3) Surrogate Avatars: Given the set of traces T and the set of avatars A, we generate a partition of A in two different subsets A γ and A θ (a partition means that A = A γ ∪ A θ and A γ ∩ A θ = ∅). For each a ∈ A θ , we generate a partition in T a with components T a 1 and T a 2 where a 1 , a 2 are called the surrogate avatars of a. Let Ãγ be the set of all surrogate avatars, we build the set à = Ãγ ∪ A θ .

Intuitively, surrogate avatars are known to belong to the same user. Thus, they provide a "ground truth" to evaluate our approach.

In Figure 4 the chart at left shows the long-tail distribution of the number of games played by avatar. We assume that professional players (those we are looking to disambiguate) are those that belong to the head of the curve, i.e. those that play the most. We consider this assumption fair since, in order to become professionals, players have to practice and perform in several competitions yielding a high number of games played. Thus, the set A γ is built from a fraction γ of the avatars with the highest number of games played.

Similarly, we consider a minimal number of games (traces) to actually include the avatar a in A θ . We assume that a user that has played a few games with an Avatar has no reasons to create a different one. We are well aware that this may not be the case for users that already have a different avatar and are just starting with their second. However, as we will discuss next, this induces an "imbalance" issue that may affect the classifiers' ability to group avatars of the same user. For an avatar with a few traces, the classifier will fail to provide a good prediction and the confusion matrix will present values which are explained by randomness rather than by the fact that two avatars belong to the same user. For example, consider an avatar a such as |T a | = 2. Its row in Cρ will contain two nonzero values 0.5 in two different columns. It is easy to see that this avatar would likely conform a pattern concept with the avatars corresponding to those columns. In order to avoid this, we use a threshold θ such as for all a ∈ T θ we have |T a | ≥ θ.

The problem of balance refers to the fact that we are not certain about the distribution of time spent by a user among her different avatars. Put more simply, given that a user has two avatars, the question is if she prefers one over the other (meaning that one of her avatars has more traces associated than the other) or if she plays equally with one or the other (meaning that both of her traces have a similar number of traces associated). This is an important issue since it affects directly the efficacy of our approach. If in general, users play many more games with one of their avatars than the others, the classifier applied to the traces will be less effective and will tend to classify the traces of the avatars with fewer games on the avatar with more games. To study this issue in a deeper way, we introduce the parameter β as a balance between the traces distributed over the surrogate avatars. Consider that for an avatar a we have that |T a | = 100, this is the avatar has 100 associated traces. When creating the surrogate users a 1 and a 2 a β = 0.5 yields that both surrogate users will have 50 associated traces, i.e.

B. Evaluation Metrics

To evaluate our approach we will measure the precision, recall and f-measure of the first 100 ranked avatar clusters. Given the ranking r (after cluster score filtering using λ), we have:

precision(r) = T P T P + F P recall(r) = T P T P + F N F -measure(r) = 2 • precision(r) • recall(r) precision(r) + recall(r)
Where T P, F P and F N stand for true positives, false positives and false negative, respectively. We will consider true positives as combinations of avatar names (N AM ES), U RL and surrogate avatars (SU G). False positives will be any candidate pair which does not belong to the true positive set in a given ranking. It is worth noticing that a pair considered as a false positive under this definition may not actually be one. We consider them false positives since we do not have enough information to consider them as true positives, meaning that their avatar names do not match, their URL is different and they are not part of our own set of surrogate avatars. They are in fact the kind of pairs we are looking for. False negatives are those candidate pairs that should have been considered as true positives, but that do not appear in the ranking.

The figure at left in Figure 3 shows the initial candidate pairs extracted from a confusion matrix generated by a Sequential Minimization Optimization (SMO) classification algorithm implemented in Weka. The classifier parameters were left as default, while the parameters of our approach for this particular figure are γ = 0.05 (top 5% of users were converted into surrogates) and θ = 5 (users with less than 5 games were extracted from the dataset). Within the figure, a point represents a pair of avatars. If the avatars are surrogates, the point is represented with a red circle. If they have the same account, the point is represented with a green triangle and, in the case they have the same name, the point is a yellow star. In any other case (false positive), the point is a blue cross. False positives are annotated with the nick-names of the avatars. Figure 3 shows the top 20 without cluster score filtering (λ = 0) and presents very bad results. Only 8 out 20 points are not false positives (40% of precision). The figure at right in Figure 3 shows the top 20 after cluster score filtering (λ = 0.9) with very good results. Actually, only 1 out of 20 points is a false positive and it represents a couple of avatars that belong to the player known as aLive 4 . It is clear that in this particular case, our system is able to provide a precision of 100%, even though we just report 95% (19/20).

We also report on other three measures, namely P@10 (precision in the first 10 elements of the ranking), mean average precision (MAP), the receiver operating characteristic (ROC) (and the ROC area under the curve -AUC). For the sake of brevity, we do not provide a description of them. For further information on these metrics we refer the reader to [START_REF] Christopher | Introduction to Information Retrieval[END_REF].

V. EXPERIMENTS

This section reports a thorough evaluation of our approach, for answering the avatar aliases identification problem. We begin by introducing our datasets and by highlighting the prediction ability of game traces when there are no aliases in the dataset.

A. Rough replay collections

Any game of Starcraft 2 is recorded into a file called replay which contains all data necessary for the game engine to replay the game. Replays are shared on dedicated websites 5 . Along with replays, a set of parsing tools which allows extracting information from the replays are openly available 6 . Using these tools we have created two collections for our study.

COLLECTION 1 -Replays without avatar aliases: This collection has been chosen for studying the efficacy of classifiers to recognize avatars of traces. Thus, we have purposely selected a collection of game replays which cannot contain avatar aliases. This is the case for the 2014 World Championship Series (second season 7 ) in which users are forced to register their real names. The collection contains a total of 955 one-versus-one high level games and 171 unique players.

COLLECTION 2 -Replays with possible avatar aliases:

We gathered all the replays available on the website Spawning Tool8 on the month of July 2014, for a total 10,108 one-versusone games and 3,805 players. This collection corresponds to a real world situation, and is used for evaluating our avatar alias resolution approach according to Section IV. Figure 4 shows the distribution of the number of games by avatar for both replay collections. The distribution for Collection 2 corresponds to a long tail where 10% of players participate in more than 67% of the games. The distribution for Collection 1 is explained by the elimination process of the WCS qualifiers, meaning that the distribution of game played by user gradually increments as they go up the classification ladder. Respectively, in average each player participates in 5 and 9 games in Collections 1 and 2. Charts at center and right in Figure 4 illustrate the proportion of each type of action used as feature for avatar classification in our approach. Particularly, these figures show the actions for the first ten and thousand seconds Figure 3: Candidate pairs ranking with λ = 0 (left) and with λ = 0.9 (right) Figure 4: Distribution of the number of games by avatar, proportion of executed actions for first ten (resp. thousand) seconds of the game, respectively. The object selection is the most prominent event in the first seconds of the game (the warm up phase), totalling 80% of the actions, after which the use of hotkeys becomes the most important action. This is the main reason why we will use object selection frequencies as features (in previous work, only hotkeys were used as features [START_REF] Yan | Masters of control: Behavioral patterns of simultaneous unit group manipulation in starcraft 2[END_REF]). The chart at right in Figure 4 can be explained as follows. In the first minutes of the game there are not many options for the player to execute leading to a high proportion of clicking and selection events. After the 2 or 3 minutes, the user has built up a wider variety of options to execute which leads to hotkey bindings. The major part of bindings are made only once in the game, thus the highest proportion of hotkey events around 200 seconds. In the rest of the game, these proportions stabilize.

B. Experimental set up

Two main experiments were conducted. In Collection 1, we apply a set of classifiers to test the efficacy of predicting the avatar of a trace. In Collection 2, we apply classification and clustering to perform avatar alias resolution. Both experiments share the steps of (i) parameter selection, (ii) dataset creation and (iii) classification. The second experiment also considers a fourth step (iv) clustering, scoring and post processing.

1) Parameter selection: Throughout this document we have described a range of parameters which compensate for the fact that we know very little of the users we are looking for. For example, the parameter γ compensates for the fact that we do not know which is the proportion of avatars in a dataset that correspond to aliases of the same user. The parameter β compensates for the fact that we do not know in which proportion aliases are used. For example, β = 0.5 represents the fact that users may use their aliases in an equal proportion, while β = 0.8 represents the fact that users may use one alias much more than the other. The parameters in our approach are a manner of discovering under which conditions the notions of avatar alias resolution holds. For this reason, we have selected a wide spectrum of parameter combinations for our experiments. The following list present the names and meaning of each parameter.

τ : a time threshold for traces. Only the actions of the first τ seconds are considered in the dataset for classification. We also use a threshold for the number of actions, thus only the first τ actions can be considered γ: proportion of users converted into surrogates aliases θ: minimum number of games played by an avatar to be retained in the dataset β: surrogates' balance is the proportion of games attributed to one of the two surrogates derived from an avatar λ: cluster score threshold (see Section III)

2) Dataset creation: Having defined a set of parameter value, we generate datasets for classification from Collections 1 and 2. Each dataset contains traces as instances and avatars as class labels. Features of traces are a vector of numerical attributes and a couple of categorical attributes. Vector dimensions are associated with a canonical order over the repertory of events in the game. The value of each dimension for a given trace is the number of times the event was executed in the trace. A final dimension considers the average actions per minute (APM) associated with that trace. Categorical attributes correspond to the race used by the player (possible values: Protoss, Terran or Zerg) and the final status of the game (possible values: Win or Lose). Datasets are stored in the attribute-relation file format (ARFF) for the Weka system.

A single dataset is built for each different provided selection of attributes. For collection 1 we created 92 datasets, while for collection 2 we created 64 datasets.

3) Classification: Each dataset is classified using the Weka machine learning software and evaluated using 10-fold cross validation from which we obtain a confusion matrix. To represent the generality of our approach, we chose four different classifiers, namely K Nearest Neighbours (KNN), Naive Bayes (NBAYES), J48 decision tree (J48) and Sequential Minimization Optimization (SMO). Parameters for each of the classifiers were left as default.

4) Clustering, scoring and post processing: Each confusion matrix was processed by the Sephirot addIntent implementation 9 to obtain a set of pattern concepts. Scoring and post processing were implemented in ad-hoc python scripts.

C. Experimental results

1) Classifying avatars: Figure 5 shows the precision and the ROC area obtained for 92 datasets created for Collection 1. The parameter τ ranged over 23 values in an exponential scale, initially from 10 to 90 seconds then from 100 to 900 and finally from 1000 to 5000 seconds (the longest game in this collection has around 5300 seconds) and thus, the x axis of each figure is in logarithmic scale. For each measure, four figures corresponding to four different settings of θ are presented. Each line corresponds to a different classifier.

The figures present an empirical evaluation that the initial assumption, that avatars are very easily recognizable based in the signatures left in the traces they generate while playing, is true. For each different setting, ROC area is always around 100% showing the robustness of the approach under different parametrizations. Precision is always maintained over 60%, achieving it minimal value for the SMO classifier with θ = 5 and τ > 1000. Actually, this also confirms two of our previous assumptions. Firstly, it is hard to recognize users that have played a few games, meaning that the larger the value of the θ threshold, more discriminative power has the classifier. Secondly, users are recognizable in the first few minutes of the game. The precision curves show a slight concave behaviour hinting a maximum of the precision w.r.t. the time cut used for traces. Indeed, this agrees with the hotkey use description given for Figure 4 in the chart at right. Users can be efficiently discovered by their hotkeys binding settings. As the game progress, traces may differ given that the number of options in the game greatly increase and vary in execution regarding different opponents.

2) Identifying multiple aliases: The goal of experiment 2 is evaluating our approach for finding avatar aliases. We have generated 48 datasets considering different parametrizations. As already discussed in the previous experiment, the efficacy of the classifiers achieves its best in the first few minutes of the game. Thus, we have selected three different τ values, namely 30, 60 and 90 seconds. We have picked the same values for θ as in the previous experiments. Surrogates were generated for the first 5, 10, 15 and 20 percent of the most active users in the dataset (γ). For this particular experiment, we have set the balance β = 0.5. All 48 datasets were processed using the four classifiers previously mentioned yielding a total of 192 confusion matrices. The top part of the Table IV shows a summary for the evaluation results using the top 100 pairs of avatar clusters found with parameters τ = 90, θ = 20 (248 avatars including surrogates), γ = 0.2 (41 surrogates) and λ = 0.9. The top, medium and bottom parts of the table contain the evaluations when looking for surrogates only, surragates and URLs, surrogates, URLs and names, respectively.

Results indicate that our approach is very efficient at identifying surrogate avatars under this parametrization. This is particularly true for KNN and the J48 classifiers achieving very high recall values. In the upper table, while precision is low it is worth noticing that in the top 100, there are only 41 surrogates meaning that the maximum achievable precision is 0.41. The classifier KNN is particularly good in this measure achieving an almost perfect value (0.4 of 0.41). All four classifiers achieve a very high precision in the first 10 results (P@10) while two of them get a perfect score. Indeed, one of the main characteristics of our approach is the good ranking it generates over the avatar pairs. This fact is confirmed by the good MAP and ROC area under the curve (AUC) values achieved by all four classifiers. Both these measures slightly degrade when including in the set of true positives URLs and names. In the case of the latter, this can be understood since not all avatars with the same name necessarily belong to the same user (as we have previously stated, same nick-names is a weak indicator). Thus, pairs of avatars with the same name will be more evenly distributed over the ranking or can even be found at the bottom indicating that they do not belong to the same user. This fact is reflected in the gap between the high grow of precision and low degradation of recall, i.e. avatars with the same name are evenly distribute between the pairs retrieved and those that were not.

A special mention deserve the URL true positives. As we have discussed, avatars with the same URL necessarily belong to the same user. Hence, we would have expected that in the first 10 pairs retrieved we could find an even distribution of surrogates and URLs. Instead, for all classifiers, P@10 is more than 80% surrogates (while the rest is always URLs -P@10 in the medium part of the table). The reason behind this is that we have purposely selected a balance of 0.5 for the surrogate distribution of traces, while we do not have control over this value for the URL pairs. The lower part of Table IV shows a summary of results when looking for just surrogates while varying the balance in the distribution of traces between them. We can clearly observe that the performance of the approach quickly degrades as more imbalanced gets the distribution (the higher the β value). Actually, for some classifiers it is not possible to obtain a single good result, even when we have lowered the λ threshold to 0.8. As URLs are not necessarily balanced, classifiers tend to predict the label of a trace belonging to an avatar with less traces to one with more traces. Issues related to learning from imbalanced datasets are reviewed in [START_REF] He | Learning from imbalanced data[END_REF] and need to be considered when selecting a proper classifier for our particular application.

VI. RELATED WORK

Recently, Yan et. al [START_REF] Yan | Masters of control: Behavioral patterns of simultaneous unit group manipulation in starcraft 2[END_REF] suggested that behavioural patterns discriminating skills, but also player themselves, can be Parameters:: γ = 0.2, θ = 20, λ = 0. Table IV: Summary of evaluation measures over the resulting avatar cluster list yielded by the alias resolution approach (at top), and avatar clustering when varying the balance (β) (at bottom). Each entry represents a confidence matrix yielded by the respective classifier discovered in the way they use their keyboard when playing Starcraft 2. They gathered 3,316 replays and took as features the frequencies of each control group key were used (30 features) for the whole game. They showed that these features allow predicting with high accuracy the league in which an avatar is playing 10 . A second result tells that a basic SVM classifier can predict the avatars involved in a game with high accuracy (≥ 0.95 accuracy with a leave-one-out validation), even when the avatar has few numbers of samples (between 2 and 20). Yan et. al showed that hotkeys (control groups) yield unique behavioural patterns of a user, but they did not present a way to discover avatar aliases: they even removed avatars with high probability of being aliases (e.g., bar code names).

Using control groups as features is actually inspired by several works in software and security applications. Indeed, typing patterns allow identifying users by their typing characteristics [START_REF] Peacock | Typing patterns: A key to user identification[END_REF]. The tedious task is to determine the appropriate behavioural metrics and features [START_REF] Yampolskiy | Behavioural biometrics: a survey and classification[END_REF]. For example, keystroke dynamics and typing rhythm are crucial for authenticating a user based on habitual patterns [START_REF] Monrose | Keystroke dynamics as a biometric for authentication[END_REF]. Recently, an investigation around Starcraft 2 [START_REF] Thompson | Video game telemetry as a critical tool in the study of complex skill learning[END_REF] highlighted that the predictive impor-tance of features is not constant across levels of expertise, while Yan et al. [START_REF] Yan | Masters of control: Behavioral patterns of simultaneous unit group manipulation in starcraft 2[END_REF] ensured that complex features, even spatio-temporal, are not important: only the frequency of hotkeys is enough to output highly accurate classifiers; we emphasized this fact by showing that only the first few minutes of game play are enough to recognize the player. Starcraft 2 and other real time strategy games (RTS) in general, face several research challenges in artificial intelligence [START_REF] Ontanon | A Survey of Real-Time Strategy Game AI Research and Competition in StarCraft[END_REF] including opponent modelling and learning. Game traces/logs from replay files tend to be more and more used to tackle these problems since this information is easy and free to gather. We can notice several works focusing especially on tactical and strategic aspects, such as predicting army locations and opponents actions [START_REF] George | A data mining approach to strategy prediction[END_REF], [START_REF] Synnaeve | A bayesian model for opening prediction in RTS games with application to starcraft[END_REF] and automatically discovering build orders [START_REF] Low-Kam | Mining statistically significant sequential patterns[END_REF], [START_REF] Bosc | Mining balanced sequential patterns in RTS games[END_REF]. All these works focus on effective actions made by players (build orders, micro/macro management) while we use here only the very first few actions of the warm up phase that one could consider as noise.

VII. CONCLUSION AND FUTURE WORK

Video game analytics is a growing field of data science, crucial, if not vital, for the biggest game producers and editors. It comes with many challenges, the holy grail being to find all the ingredients that could assure an indisputable success of a game directly at its release. Pragmatically however, behavioural big data is gathered and analysed to answer several problems, including the design of better artificial agents, game balancing, bugs and cheaters and usurpers detection, etc. Games are then patched, cheaters are banned, and this cycle restarts. Behavioural data is also a gold mine in the context of electronic sports and competitive gaming, for reaching the same goals as in standard sport analytics 11 .

We introduced the problem of avatar aliases identification, when there exists no mapping between individuals and their avatars. This is an important problem for game editors, but also for e-sport structures. Our method relies on the fact that behavioural data hide individual characteristic patterns, which allows making predictive approaches very accurate. Nevertheless, this good performance quickly degrades when data hides avatar aliases, which is why we based our analysis on confusion matrices.

Our results are encouraging, and suggest several perspectives of further development. Firstly, we assumed the mapping between players and avatars to be one-to-many. While this assumption is reasonable in the case of expert players in Starcraft 2, a general model suggests a many-to-many mapping in general. Secondly, we focused on expert players (having many historical samples), and it may be that predictive models are less powerful for novice players, since they use hotkeys in a smaller proportion. New features could be introduced in this case. For example, one could consider sequential features with sequential classifiers like in [START_REF] Dafé | Learning sequential classifiers from long and noisy discrete-event sequences efficiently[END_REF]. However, we believe that novice players do not hide behind aliases. How the method can be adapted for other video games is also challenging: the competitive games Dota 2 and League of Legends do not propose customizable hotkey systems as in Starcraft 2, thus one needs to find good features from behavioural data. Finally, we proposed an approach in the context of formal concept

Figure 1 :

 1 Figure 1: A simple model of game traces generations

1 : P ← ∅ 2 :

 12 C ← M ineF uzzyConcepts( Cρ , s) 3: C ← rank C according to s 4: for all c ∈ C do 5: pairs ← pairs from the pattern concept extent of c 6:

3 )

 3 Post-processing candidates: Consider the clustering condition previously formalized as Cρ ij Cρ ji Cρ ii Cρ jj and Cρ

Figure 2 :

 2 Figure 2: The trace generation model in Starcraft 2

|T a 1 |

 1 = |T a 2 | = 50. With β = 0.7 we will have |T a 1 | = 70 and |T a 2 | = 30 and so on.

Figure 5 :

 5 Figure 5: Classification results for Collection 1: precision and ROC area under the curve (AUC) distribution on 23 points of τ for four θ values. τ points were varied exponentially (10-90, 100-900, 1000-5000).

Table I :

 I Traces of a match for two players C. Predicting avatars from behavioural traces

	Avatar	Game trace	Outcome
	RorO	s,s,hotkey4a,s,hotkey3a,s,hotkey3s, ...	Lose
	TAiLS	Base,hotkey1a,s,hotkey1s,s,hotkey1s, ...	Win

Table II :

 II Confusion Matrix exampleMore formally, given a normalized confusion matrix Cρ , we would like to find pairs of avatars a i , a j ∈ U such that Cρ

ij Cρ ji Cρ ii Cρ jj and Cρ ij + Cρ ji + Cρ ii + Cρ jj

  exemplifies this model.

	User	owns	Account	contains	Avatar	generates	Trace

For example, https://sites.google.com/site/sc2gears/ for Starcraft

http://wiki.teamliquid.net/starcraft2/ALive

http://wiki.teamliquid.net/starcraft2/Replay Websites

http://sc2reader.readthedocs.org/

http://wcs.battle.net/sc2/en/articles/wcs-2014-season-2-replays

http://spawningtool.com/

Leagues regroup players by level, following an ELO-like ranking system, from bronze, silver, gold, platinum, diamond, master, to grand master, the later involving the best 200 players of each continent.

http://www.sloansportsconference.com analysis, but it is clear that other methods can be used and compared, relying on spectral clustering or biclustering.


Select interferograms -Study phase values

2.

Map the sliding compartment(s)

4.

Analyse kinematic 5.1. Extract profiles and topography 5.2. Profiles analysis Kinematic evolution

6.

Recalculate the interferograms -Subtract by median 3.


Nicolas Verrier

Corinne Fournier

Anthony Cazier

Thierry Fournel

Co-design of an in-line holographic microscope with enhanced axial resolution: selective filtering digital holography

Keywords: numbers: (090.1995) Holography: Digital holography, (100.3190) Image processing: Inverse problems, (100.6640) Image processing: Superresolution; (100.3010) Image processing: Image reconstruction techniques, (120.3940) Instrumentation, measurement, and metrology: Metrology

Common-path digital in-line holography is considered as a valuable 3D diagnostic techniques for a wide range of applications. This configuration is cost effective and relatively immune to variation in the experimental environment. Nevertheless, due to its common-path geometry, the signal to noise-ratio of the acquired hologram is weak as most of the detector (i.e. CCD/CMOS sensor) dynamics is occupied by the reference field signal, whose energy is orders of magnitude higher than the field scattered by the imaged object. As it is intrinsically impossible to modify the ratio of energy of reference to the object field, we propose a co-design approach (Optics/Data Processing) to tackle this issue. The reference to object field ratio is adjusted by adding a 4-f device to a conventional in-line holographic set-up, making it possible to reduce the weight of the reference field while keeping the object field almost constant. Theoretical analysis of the Cràmer-Rao lower bounds of the corresponding imaging model illustrate the advantages of this approach. These lower bounds can be asymptotically reached using a parametric inverse problems reconstruction. This implementation results in a 60 % gain in axial localization accuracy (for of 100 µm diameter spherical objects) compared to a classical in-line holography set-up.

I. INTRODUCTION

Digital in-line holography is a widely used tool for 3D imaging in severe experimental environments [START_REF] Jones | Application of in-line holography to drop size measurement in dense fuel sprays[END_REF][START_REF] Jericho | Submersible digital in-line holographic microscope[END_REF][START_REF] Jericho | In-line digital holographic microscopy for terrestrial and exobiological research[END_REF][START_REF] Berg | Digital holographic imaging of aerosol particles in flight[END_REF][START_REF] Verrier | Inline particle holography with an astigmatic beam: setup self-calibration using an "inverse problems" approach[END_REF]. The common-path geometry of a classical in-line holography set-up makes it possible to perform quantitative 3D imaging with a limited number of optical accesses as well as relative immunity to variations in the experimental conditions (e.g. variations in temperature, vibrations in the imaging path). These features have increased interest in the use of digital in-line holography for application in a wide range of domains including fluid dynamics [START_REF] Berg | Digital holographic imaging of aerosol particles in flight[END_REF][START_REF] Katz | Applications of holography in fluid mechanics and particle dynamics[END_REF], and bio-medical imaging [START_REF] Bishara | Lensfree on-chip microscopy over a wide field-of-view using pixel super-resolution[END_REF][START_REF] Mudanyali | Wide-field optical detection of nanoparticles using on-chip microscopy and self-assembled nanolenses[END_REF]. Nevertheless, as both the reference field and the object field have a common optical path, it is impossible to act on one or the other independently. This issue can be tackled with an off-axis digital holography configuration [START_REF] Leith | Reconstructed wavefronts and communication theory[END_REF], which, for instance, makes it possible to characterize both amplitude and phase objects [START_REF] Cuche | Digital holography for quantitative phase-contrast imaging[END_REF][START_REF] Charrière | Cell refractive index tomography by digital holographic microscopy[END_REF] down to shot noise limits [START_REF] Lesaffre | Noise and signal scaling factors in digital holography in weak illumination: relationship with shot noise[END_REF][START_REF] Verrier | Vibration of low amplitude imaged in amplitude and phase by sideband versus carrier correlation digital holography[END_REF][START_REF] Lopes | Singular-value demodulation of phase-shifted holograms[END_REF]. However, the complexity of the experimental set-up is not suitable for severe experimental environments.

Several strategies have been proposed to get round this problem. Signal processing based approaches have been demonstrated based on a priori knowledge of the object [START_REF] Yang | Depth-offocus reduction for digital in-line holography of particle fields[END_REF], deconvolution strategies [START_REF] Latychevskaia | Depth-resolved holographic reconstructions by three-dimensional deconvolution[END_REF], or compressive imaging [START_REF] Rivenson | Compressive fresnel holography[END_REF][START_REF] Rivenson | Improved depth resolution by single-exposure in-line compressive holography[END_REF]. Experimental schemes, based on the use of two orthogonal view points, have also been considered at the cost of a more complex experimental set-up and a mandatory view registration [START_REF] Buchmann | Ultra-highspeed tomographic digital holographic velocimetry in supersonic particle-laden jet flows[END_REF]. However, all * nicolas.verrier@univ-st-etienne.fr these approaches rely on hologram back-propagation to retrieve information about the recorded objects, therefore introducing artifacts such as aliasing ghosts or twinimage noise [START_REF]Sampling of the diffraction field[END_REF][START_REF] Fournier | Digital in-line holography: influence of the reconstruction function on the axial profile of a reconstructed particle image[END_REF][START_REF]Exact analysis of the effects of sampling of the scalar diffraction field[END_REF].

Instead of transforming the data through light backpropagation calculations, inverse problems (IP) approaches have been successfully applied to the reconstruction of in-line and off-axis holograms [START_REF] Soulez | Inverse-problem approach for particle digital holography: accurate location based on local optimization[END_REF][START_REF] Soulez | Inverse problem approach in particle digital holography: out-of-field particle detection made possible[END_REF][START_REF] Denis | Inline hologram reconstruction with sparsity constraints[END_REF][START_REF] Bourquard | A practical inverse-problem approach to digital holographic reconstruction[END_REF][START_REF] Fournier | Digital Hologram Processing in On-Axis Holography[END_REF]. The aim of these approaches is to find, in the least-square sense, imaging models that best match the acquired data, and allow accurate estimation of the parameters of the objects under investigation (e.g. 3D position and size of a particle in in-line particle holography). It should be noted that a maximum a posteriori (MAP) approach can be used to regularize the maximum likelihood estimation. Despite good accuracy, this device is limited by the signal to noise ratio of the diffraction pattern (i.e. the object signal), which prevents the detection of particles less than 5 µm in diameter with a unitary magnification configuration. To improve accuracy, while keeping ease of use of the experimental set-up, one can consider optimizing both the experimental and the data processing design.

Here, we consider an optics/data-processing "codesign" scheme to improve the axial accuracy of a conventional in-line holographic set-up, while preserving its experimental advantages (i.e. limited amount of optical accesses, space bandwidth product, and relative robustness to experimental environment variation). The proposed approach is applied, to the particular case of spherical opaque particle holograms. To this end, we propose a modification of in-line holography involving a selective 4-f filtering stage. Associated parametric imaging model is discussed, and the optimal filter design is discussed through a Cràmer-Rao lower bounds analysis. The advantages of the approach are demonstrated through numerical and bench-top experiments resulting in an axial localization accuracy gain of 60 %. gram content is performed by a 4-f stage, made of two f = 150 mm achromatic doublets (Thorlabs AC508-150-A-ML R ), which makes it possible to perform all optical Fourier filtering operations. The test object -a chrome deposited opaque disk whose diameter is 2r = 100 µm (diameter ±1 µm, roundness ± 0.25 µm Optimask R ) -is positioned in the object focal plane of the 4-f device, and illuminated by a collimated field. Therefore, in the Fourier space of the 4-f arrangement, the reference field is focalized, whereas the wave diffracted by the object spreads in the whole spatial frequency space. Note that this configuration is similar to the one proposed in Ref. [START_REF] Micó | Phaseshifting Gabor holography[END_REF], where phase shifting in-line holography was considered. This configuration makes it possible to act on the reference field while keeping the object field highfrequency content. The filtering mask, whose optimal design is discussed in the following sub-sections, consists of a semi-transparent disk deposited on a glass substrate. This filtering mask acts on the amplitude of the optical Fourier transform of the interference field: the central part of the Fourier spectrum is attenuated, while the remainder of the spectrum remains unchanged, which allows adjustment of the reference to object field ratio. The filtered Fourier transform is then collected by the second lens, which performs another Fourier transform, leading to a reference attenuated version of our original interference field, which is propagated over a distance z to the recording plane. The interference is finally recorded on a 12 bit 4872 × 3248 pixel with 9 µm pitch CCD sensor (Prosilica GE4900 R ).

B. Imaging model

To be able to optimally design the filtering mask, we had to build the imaging model associated with our experimental set-up. Considering a conventional in-line holography set-up, the recorded intensity is

I z (x, y) ∝ |1 -ϑ (x, y) * h z (x, y)| 2 , ( 1
)
where ϑ is the 2D aperture function of the particle under investigation, which is unity within the aperture and zero elsewhere. The symbol * denotes the 2D spatial convolution product, and the impulse response of free-space propagation h z (x, y) is

h z (x, y) = 1 iλz exp i π λz x 2 + y 2 . ( 2
)
In the remainder of this article, z is set so that the assumption z ≫ r 2 /λ holds. Under this assumption, the Huygens-Fresnel diffraction integral finds an analytic expression, often called the Tyler/Thompson model [START_REF] Tyler | Fraunhofer holography applied to particle size analysis a reassessment[END_REF].

The intensity I z (x, y) recorded for a spherical opaque object located at a distance z from the imaging sensor, and without filtering operations, is thus given by

I z (x, y) ∝ 1 - 1 λz F x λz , y λz {ϑ (ξ, η)} sin π λz x 2 + y 2 + 1 λz F x λz , y λz {ϑ (ξ, η)} 2 , (3)
Considering an opaque diffraction particle, the Fourier transform of ϑ is

F x λz , y λz {ϑ (ξ, η)} = 2πr 2 λz 2πr x 2 + y 2 × J 1 2πr x 2 + y 2 λz , (4)
with J 1 the Bessel function of the first kind. Introducing the cardinal Bessel function of the first kind J 1c , Eq. ( 4) becomes

F x λz , y λz {ϑ (ξ, η)} = 2πr 2 J 1c 2πr x 2 + y 2 λz . (5)
It should be noted that, under the assumption of a diluted particle sample, an additive intensity model can be built from Eq. (3). In this situation, the interference between object waves will not been taken into account. However, a more complete non-linear imaging model can be considered if this situation is encountered.

Coming back to Eq. ( 1), it is possible to build an imaging model that accounts for the selective filtering of the field. Considering that both lenses perform an optical Fourier transform of the field distribution in their object focal plane, we can modify Eq. (1) to

I z (x, y) ∝ F x λf , y λf F x 0 λf , y 0 λf {1 -ϑ (ξ, η)} ×M α x 0 λf , y 0 λf * h z (x, y) 2 , (6)
with M α x0 λf , y0 λf the opacity distribution of the filtering mask defined as

M α x 0 λf , y 0 λf =    α if x0 λf 2 + y0 λf 2 ≤ ρM λf 1 otherwise ,( 7
)
where ρ M is the physical size of the filtering mask, and (x 0 , y 0 ) are the spatial coordinates in the Fourier space. The expression given in Eq. ( 6) can be further expanded leading to

I z (x, y) ∝ α 2 -α 2πr 2 λz J 1c 2πr x 2 + y 2 λz sin π λz x 2 + y 2 + 2πr 2 λz α (1 -α) J 1c 2πr x 2 + y 2 λz sin π λz x 2 + y 2 * J 1c 2πρ M x 2 + y 2 λf + second order terms, ( 8
)
where α is the amplitude transmittance of the filtering mask. Details of the calculations leading to Eq. ( 8) are provided in Appendix A. As J 1c 2πρM √

x 2 +y 2 λf can be considered to have a compact support (most of its energy is in the low frequency components), Eq. ( 8) can be further simplified leading to our imaging model

I z (x, y) ∝ α 2 -α 2πr 2 λz J 1c 2πr x 2 + y 2 λz sin π λz x 2 + y 2 + 2πr 2 λz α (1 -α) J 1c 2πr x 2 + y 2 λz sin π λz x 2 + y 2 × ϑ M (x, y) + second order terms. ( 9
)
It can be seen that Eq. ( 9) is a high pass-filtered version of Eq. ( 3) (further shown in Fig. 4). Considering Eqs. ( 7) and ( 9), it is possible to simulate filtered holograms.

In the remainder of this article, second order terms of Eq. ( 9) will be neglected. As a matter of fact, as it is discussed in Appendix B, its maximal value (considering our experimental set-up parameter) is at least one order of magnitude lower than the other terms in Eq. [START_REF] Leith | Reconstructed wavefronts and communication theory[END_REF]. To be able to acquire holograms in optimal conditions, we will focus on the co-design of the filter.

C. Filter co-design

Two parameters need to be accounted for in order to filter our reference field. First, the filter radius ρ M can be chosen so that the imaging model remains simple, while keeping an analytic expression. This aspect will be convenient to analyze the influence of the filtering mask opacity. Then, the filter transmittance is chosen in order to have optimal performance of the imaging system.

Determination of the filter radius

The role of the filtering mask is to act on the Fourier space low frequency components, mainly associated with the reference field, while keeping the high frequency content almost constant. Considering Eq. ( 6), it can be seen that the filtering is performed on the low frequency content of the Fourier transform of the particle transmittance 1 -ϑ (x 0 , y 0 ). Its analytic expression is straightforwardly derived from Eq. ( 4) and is

F x 0 λf , y 0 λf {1 -ϑ (x 0 , y 0 )} = δ 0,0 x 0 λf , y 0 λf -F x 0 λf , y 0 λf {ϑ (x 0 , y 0 )} , (10)
where δ 0,0 is the Dirac distribution. In order to link the filter radius ρ M to observable quantities such as the particle radius, while keeping an analytic image formation model, the filter radius ρ M is chosen so that it extends over the first cancellation of the modulation function given Eqs. ( 4) and [START_REF] Cuche | Digital holography for quantitative phase-contrast imaging[END_REF]. It should be noted that the choice of the first cancellation is arbitrary: considering that the object field is not disturbed by the filtering mask, every cancellation of the modulation function can be selected to design ρ M . Therefore the mask radius ρ M is given by

ρ M = 1.22λf 2r . (11)
Thus, according to our experimental configuration, λ = 660 nm, f = 150 mm, and r = 50 µm, we obtain ρ M = 1.2 mm. The size of the mask is fixed in the remainder of the article. It should be noted that Eq. ( 11) can be used to adjust either f or r, while keeping ρ M constant. Illustration of the proposed scheme is depicted Fig. 2. Here, the effect of the selective filtering is underlined by simulating the hologram, using Eq. ( 9), corresponding to a r = 50 µm particle positioned at z = 130 mm from a 1024 × 1024 sensor, whose pixel pitch is 9 µm. For the purpose of qualitative comparison, the hologram without filtering, i.e. considering α = 1, is shown in Fig. 2(a). The field in the filtering plane, which corresponds to the interference light field Fourier transform modulus (Fig. 2(b)) is filtered considering a mask with a radius ρ M = 1.2 mm as predicted by Eq. ( 11), and a transmittance α amp = 0.13, as illustrated in Fig. 2(c). Comparing the unfiltered field (Fig. 2(a)) with its filtered equivalent (Fig. 2(d)) makes it possible to qualitatively assess the advantages of the proposed scheme. As can be seen by comparing the simulations, higher frequencies of the interference signal are more highly contrasted throughout imaging field.

Influence of the opacity of the filtering mask on the accuracy of parameter estimation

In estimation theory [START_REF] Kay | Fundamentals of statistical signal processing: estimation theory[END_REF], minimal achievable variance of an imaging model parameter can be assessed through Cramer-Rao Lower Bounds (CRLB) computation [START_REF] Cramér | Mathematical methods of statistics[END_REF][START_REF] Rao | Information and the accuracy attainable in the estimation of statistical parameters[END_REF][START_REF] Réfrégier | Noise theory and application to physics: from fluctuations to information[END_REF][START_REF] Fournier | On the single point resolution of on-axis digital holography[END_REF]. These bounds can be estimated by inverting the Fisher information matrix associated with the imaging model concerned. Under the assumption of Gaussian white noise, and considering an imaging model relying on four parameters θ i=1,4 , the Fisher information matrix is a 4 × 4 matrix whose elements are proportional to the imaging model gradients along the parameter direction. Considering f , ρ M and α as fixed parameters, the Fisher information matrix associated to the imaging model of Eq. ( 9) is given by [START_REF] Fournier | On the single point resolution of on-axis digital holography[END_REF]

F Mi,j = SNR 2 × x y ∂I n (x, y) ∂θ i ∂I n (x, y) ∂θ j , (12)
where SNR stands for the hologram signal to noise ra-tio, and θ ∈ {x i , y i , z i , r i } is the set of parameters of the imaging model. The centered and normalized imaging model I n is given by either Eq. ( 3) or Eq. ( 9). In the case of uncorrelated imaging model parameters, the nondiagonal terms vanish. The CRLBs on the imaging model can finally be obtained by inverting the Fisher information matrix (Eq. ( 12)). These bound, denoted σ x,y,z,r , correspond to the diagonal elements of the inverted matrix. Under the assumption of non-correlated imaging model parameter, the CRLB are given by

F -1 M =     σ 2 x ε yx ε zx ε rx ε xy σ 2 y ε zy ε ry ε xz ε yz σ 2 z ε rz ε xr ε yr ε zr σ 2 r     , (13)
with ε ij ≪ σ x,y,z,r . Under this assumption, the diagonal terms of F -1

M can be written as

σ 2 i ≈ 1 F Mi,i = 1 SNR 2 x y ∂I n (x, y) ∂θ i 2 -1 . ( 14
)
To compare the accuracy of the conventional digital holography set-up (relying on the imaging model given in Eq. ( 3)) with that of our proposed experimental arrangement (associated with the imaging model of Eq. ( 9)), we study the ratio between the standard deviation on the parameter estimation for both configurations

σ filt i σ Th i = SNR Th SNR filt    x y ∂I Th ∂θi 2 x y ∂I filt ∂θi 2    1/2 , (15)
where the superscipts Th, and filt are respectively associated with the image formation models given in Eqs.

(3) and [START_REF] Leith | Reconstructed wavefronts and communication theory[END_REF]. It should be noted that the image of the model gradients differs only in the center. In the case of an attenuation of the first diffraction lobe, the change in the gradient energy impacts less than 4 % of the pixels. Furthermore, the change occurs in the low frequency region where energy of the gradient is low. Thus, the ratio of the gradient energies is close to unity, and Eq. ( 15) becomes

σ filt i σ Th i ≈ SNR Th SNR filt (16)
Therefore, to characterize the effect of filtering parameters estimation on the accuracy gain, Eq. ( 16) is computed for α ∈ [0, 1]. The filtering procedure will mainly affect the hologram SNR. Therefore, considering Eq. ( 14), the estimation accuracy will be improved accordingly. Fig. 3 shows the results obtained with the evolution of σ filt x,y /σ Th x,y (dashed blue line), σ filt z /σ Th z (blue circles), and σ filt r /σ Th r (blue diamonds). The hologram SNR was estimated considering calculations derived in Sec. II D. As expected, the greater the opacity of the filtering mask, the higher the hologram SNR (see dashed orange curve in Fig. 3), and hence the better the parameter estimation. It should be noted that this theoretical analysis does not account for experimental parameters such as maximal light source energy, imaging sensor dynamics and quantization, which are likely to prevent from the use of a filtering mask with α → 0. Nevertheless, it emphasizes the fact that filtering most of the reference beam contribution improves the accuracy of imaging model parameters.

D. Improving the signal to noise ratio

As stated earlier, the proposed filtering improves the contrast of high frequency features of the recorded hologram. This effect can be proved experimentally by recording holograms with and without filtering. Considering the imaging model without filtering (see Eq. ( 3), the maximal intensity I max z in the recording plane can be written as

I max z = I 0 + ∆I 0 , (17)
where I 0 is the reference wave offset and ∆I 0 is the amplitude of the first diffraction lobe given by

∆I 0 = πr 2 λz I 0 . (18)
The introduced notations are highlighted in Fig. 4. In the same way, the maximal recorded intensity with filtering I max filt can be described as the sum of an offset I 1 and an amplitude ∆I 1 so that

I max z filt = I 1 + ∆I 1 , (19)
with I 1 = α 2 I ′ 0 , and ∆I 1 given by The ratio µ between the second and the first maxima of a cardinal Bessel function of the first kind has been added to account for the reduction of the offset, as well as the evolution of the hologram SNR according to α. In practice, to compensate for the decrease in the signal attributed to the filtering process, the intensity of the laser is adjusted to cover the whole camera dynamic range

∆I 1 = πr 2 λz αµI ′ 0 . (20)
I max z filt = I max z . (21)
According to the value of the ratio µ, two scenarios, scenario and scenario in Fig. 5, can be considered:

1. if µ ≤ 1 the first diffraction lobe, corresponding to the first maximum of the cardinal Bessel function of the first kind, remains preponderant, the filtering process will only slightly improve the high frequency content of the recorded hologram.

2. if µ > 1 the second diffraction lobe becomes predominant, there will be a marked improvement in the high frequency features in the recorded hologram.

illustrated in Fig. 5, the switch between the scenarios takes place when the opacity of the mask is such that 1-α = 0.83. This value was estimated by calculating the ratio µ between the two first diffraction lobes according to Eq. ( 9). Considering these aspects, and using Eqs. [START_REF] Rivenson | Compressive fresnel holography[END_REF] to [START_REF] Fournier | Digital in-line holography: influence of the reconstruction function on the axial profile of a reconstructed particle image[END_REF] it is possible to link both I 0 and I ′ 0 to fulfill Eq. ( 21) leading to

I ′ 0 = 1 + πr 2 λz α α + µπr 2 λz I 0 . (22)
The same can be done to obtain the amplitude ratio between the filtered and unfiltered hologram

∆I 1 ∆I 0 = µ 1 + πr 2 λz α + µ πr 2 λz (23)
Fig. 6 gives an example of an intensity profile recorded with α so that the intensity of the second diffraction is close to that of the first diffraction lobe. The unfiltered intensity profile corresponds to the light gray curve. The second to first amplitude ratio is A nd 2 /A st 1 ≈ 0.13 and the SNR of the recorded pattern is driven by both the offset I 0 and the maximum amplitude of the signal I max z . In this configuration, the SNR is fixed. In the filtering configuration (blue curve in Fig. 6), the ratio A nd 2 /A st 1 can be adjusted according to α. When α is chosen such that A nd 2 /A st 1 ≥ 1, the SNR of the recorded hologram is driven by the amplitude of the second diffraction lobe, thus enhancing the high frequency content of the hologram, as can be seen in Figs. 2 and7. In this configuration 1 -α = 0.85, and the ratio µ = A nd 2 /A st 1 ≈ 1, which is in good agreement with the imaging model analysis proposed in Fig. 5. Moreover, the SNR of the filtered hologram (estimated has the ratio between the second diffraction lobe signal and the noise variance) has been estimated to be 5.19 times higher than that of the unfiltered hologram. This is in accordance with the results obtained in Fig. 3. Adjusting the hologram SNR makes it possible to act on the estimation accuracy of our imaging model parameters as discussed through our CRLB analysis: the higher the SNR, the better the parameter estimation accuracy. It should however be noted that a sufficient part of the reference beam has to be kept in order to record interference and therefore to function in a holographic regime.

III. EXPERIMENTAL RESULTS

The qualitative improvement of the filtering scheme presented above was confirmed by the simulation results. For experimental validation, a custom made filtering mask (transmittance in intensity α = 0.1 ± 15% Optimask R ) with radius ρ M = 1.2 mm was positioned within the 4-f filtering stage. The corresponding mask amplitude transmittance was α = 0.32. A hologram of a 2r = 100 µm opaque circular disk was recorded with and without the filtering mask in the 4-f stage. The image of the object through the 4-f arrangement was positioned at z = 130 mm from the sensor whose area was cropped to 1024 × 1024 pixels for the purpose of illustration.

Example of holograms are shown in Fig. 7 without (a), and with (b) filtering. The addition of the filtering mask improves the high frequency content contrast. This aspect is made clearer by calculating radial means around the interference pattern. These are shown in red, in Figs.

7(a) and (b).

To better point out the improvement of the proposed scheme on the accuracy of z and r estimation, a statistical experiment was performed. Statistical series of holograms were acquired with the experimental arrangement illustrated in Fig. 1. Series of 100 holograms, randomly shifted using a mechanical (x, y) translation stage, were acquired without and with the filtering mask (α = 0.32 in amplitude) in the 4-f filtering stage. Holograms were then reconstructed using an Inverse Problems (IP) reconstruction scheme [START_REF] Soulez | Inverse-problem approach for particle digital holography: accurate location based on local optimization[END_REF][START_REF] Soulez | Inverse problem approach in particle digital holography: out-of-field particle detection made possible[END_REF]. This approach aims at finding, in the least-square sense, the imaging model that best matches the recorded data. For parametric imaging models relying on a few parameters (i.e. object 3D positions and size), these approaches lead to almost unsupervised algorithms, and optimal signal reconstruction [START_REF] Fournier | On the single point resolution of on-axis digital holography[END_REF][START_REF] Mortensen | Optimized localization analysis for singlemolecule tracking and super-resolution microscopy[END_REF].

The IP algorithm used in this study is derived from that of [START_REF] Soulez | Inverse-problem approach for particle digital holography: accurate location based on local optimization[END_REF][START_REF] Soulez | Inverse problem approach in particle digital holography: out-of-field particle detection made possible[END_REF] and consists of two steps:

1. A coarse estimation step, aimed at finding the best-matching element in a discrete model dictionary, built by varying the imaging model parameters {x i , y i , z i , r i }. Discrete model dictionaries were built for non-filtered and filtered holograms using Eqs. ( 3) and ( 9) respectively.

2. A local optimization step, which, with a sub-pixel accuracy, fits the imaging model using, as first guess, the coarsely estimated parameters (see step 1).

Unfiltered holograms were processed using the imaging model proposed in Eq. ( 3), whereas the filtered holograms relied on the imaging model proposed in Eq. ( 9). For each hologram, we thus obtained the object's lateral positions {x, y}, axial location z, as well as its radius r. Doing this for all the filtered and holograms makes it possible to extract the experimental statistics concerning parameter estimation accuracy. Due to the random nature of the imposed lateral shift, improvement in lateral accuracy is not discussed here rather we focus our attention on the improvement in accuracy of the z and r parameters, which are assumed to remain constant over the whole hologram sequence. Note that joint opti- 3) as an imaging model, and red circles correspond to the filtered holograms (associated imaging model corresponding to Eq. ( 9)).

mization of the hologram sequence, discussed in [START_REF] Verrier | Digital holography superresolution for accurate three-dimensional reconstruction of particle holograms[END_REF][START_REF] Verrier | 3d tracking the brownian motion of colloidal particles using digital holographic microscopy and joint reconstruction[END_REF] but not implemented here, can be used to further improve parameter estimation accuracy.

The results obtained are summarized in Fig. 8 for the evolution of z (Fig. 8(a)), and r (Fig. 8(b)). Reconstruction of the filtered hologram is represented by red circles (the corresponding imaging model is given in Eq. ( 9)), and blue crosses represent unfiltered holograms (see. Eq. ( 3) for the reconstruction imaging model). It can be seen that both z and r reconstructed values are less dispersed when selective filtering is performed (see data circled in red in Fig. 8). This can be confirmed by estimating the standard deviation σ z,r and the standard error of the mean σ z,r of z and r data-sets. Results, for in-line holog- raphy (corresponding to the set-up proposed Fig. 1 without a filtering mask) and filtered in-line holography are summarized in Table I. The filtering mask parameters were chosen based on the CRLB analysis. Within this configuration, the average SNR of the hologram sequence was estimated to be 3.3 times higher for the filtered holograms, revealing a good accordance with the theoretical results depicted in Fig. 3. As can be seen from the standard deviation and from the standard error of the mean on the estimation of both z and r, the proposed experimental modification allows an improvement in the accuracy of both parameters, namely σ Th z /σ filt z = 1.58 and σ Th r /σ filt r = 1.7, thus illustrating the ability of the proposed filtering scheme to enhance the axial localization accuracy of a conventional in-line holography set-up. It should be noted that this enhancement is well bellow the theoretical prediction (rarely reached) in Fig. 3. This can be explained by the Gaussian white noise hypothesis, not experimentally fulfilled, made in our IP reconstruction scheme that may lead to overestimation of the CRLB values of the imaging model parameters.

IV. CONCLUSION

An optics/data-processing "co-design" scheme is proposed as a way to simultaneously improve the object sizing, and axial localization accuracy of a conventional inline holographic configuration. This approach, based on a close interaction between the experimental set-up and the data reconstruction scheme, is based on a slight modification of a conventional Gabor holographic set-up allowing selective filtering of the reference field while keeping the object field almost intact. This approach has several advantages in terms of the sensor's dynamic range allocation, detection signal to noise ratio, and pattern matching accuracy.

For optimal functioning, the design of the filtering mask has been discussed, through CRLB analysis of the imaging model, leading to a clever choice of the mask opacity. The concept of selective filtering in-line digital holography was qualitatively validated by both digital and bench-top experiments confirming the interest of the method for SNR, and high frequency content improvement of recorded holograms. Quantitative assessment was made through the statistical analysis of 100 holograms with and without selective filtering showing a 60 % improvement in the accuracy of the axial localization.

Other experimental studies are now needed to identify the limits of the improvement in accuracy with higher mask opacity. Complementary experiments should also be performed on more realistic 3D samples (e.g. droplet jets, colloidal suspension . . . ). Thanks to improvement in SNR, our configuration should make it possible to reconstruct smaller objects.

It should be noted that such an optics/data-processing "co-design" methodology is not limited to the particular case of digital holography but can be generalized to any imaging technique that relies on a known parametric imaging model. For more complex objects (non spherical, transparent, phase objects) the herein discussed method can generalized considering a MAP reconstruction [START_REF] Denis | Inline hologram reconstruction with sparsity constraints[END_REF]. This approach thus paves the way for the development of cost effective imaging devices with optimal performance. The second lens of the 4-f filtering device again performs a Fourier transform of the field given in Eq. (A4). The field in the image focal plane of the second lens is therefore given by (omitting the constant phase terms)

U (x, y, 4f ) = iλf F -kxλf,-kyλf ÛF (k x , k y ) , (A7) which simplifies as U (x, y, 4f ) = α-2πr 2 F -kxλf,-kyλf J 1c 2πr k 2 x + k 2 y ×M α (k x , k y )] . (A8)
The minus sign in the spatial coordinates (x = -k x λf, y = -k y λf ) makes it possible to account for the Gouy phase shift that occurs in the 4-f filtering device [START_REF] Gouy | Sur une propriété nouvelle des ondes lumineuses[END_REF].

Let us now introduce the geometrical aperture of the filtering mask ϑ M (k x , k y ), which can be defined as

ϑ M (k x , k y ) = 1 if k 2 x + k 2 y ≤ ρM λf 0 otherwise , (A9)
allowing Eq. (A5) to be modified as

M α (k x , k y ) = 1 -(1 -α) ϑ M (k x , k y ) . (A10)
Introducing M α (k x , k y ) in Eq. (A8) makes it possible to derive the amplitude in the 4-f device image plane

U (x, y, 4f ) = α -ϑ (x, y) + (1 -α) × ϑ (x, y) * kx,ky J 1c 2πρ M k 2 x + k 2 y . (A11)
Propagation of Eq. (A11) to the sensor plane can be achieved through spatial convolution according to

U (x, y, z) = U (x, y, 4f ) * x,y h z (x, y) , (A12)
with h z (x, y) the free space impulse response defined in Eq. ( 2). The amplitude in the sensor plane can be written as

U (x, y, z) = α -ϑ (x, y) * x,y h z (x, y) +(1 -α) ϑ (x, y) * x,y h z (x, y) * kx,ky J 1c 2πρ M k 2 x + k 2 y . (A13)
Finally, the intensity in the sensor plane can be estimated considering and will be considered as our imaging model for the selective filtering digital in-line holography set-up.

I z (x, y) = |U (x, y, z)| 2 = |R (x, y)| 2 -2ℜ {R * (x, y) O (x, y)} + |O (x, y)| 2 , ( A14
Appendix B: Neglecting the second order term

As discussed in the article main matter, our imaging model relies on a modification of the classical particle hologram formation model proposed in Ref. [START_REF] Tyler | Fraunhofer holography applied to particle size analysis a reassessment[END_REF] and Eq. ( 3). In this model, the interference between objects are neglected assuming a diluted investigated medium. Furthermore, considering spherical objects of radius r i located at a distance z i so that

z i ≫ πr 2 i λ (B1)
the second order term of Eq. ( 3) can be neglected, thus leading to a simple and linear imaging model. In our proposed imaging model, the same assumption holds. It should be noted that, in our experimental configuration, the distance z corresponds to the distance between the filtered image of our object through the 4f-stage to the imaging sensor.

To illustrate this aspect, both first and second order terms of Eq. ( 9) can be simulated according to our experimental parameters. As it can be noticed from Fig. 9, both the first order (a) and second order (b,c) terms span over the whole frequency space and cannot be simply canceled by our high-pass filtering strategy. However, having a closer look to the maximal values of each term, one can verify that the second order term can be neglected compared to the contribution of the reference wave and the first order term. Using Eq. ( 9) with the experimental parameters used for the experiment depicted As can be noticed from Eq. (B2), the second order term is 40 times as low as the first order term and can therefore be legitimately neglected. Moreover, both reference and first order terms are of the same order of magnitude, which confirms the result of Fig. 6.

  FIG. 1. (Color online) Experimental configuration for hologram acquisition.

FIG. 2 .

 2 FIG. 2. Illustration of the selective filtering. (a) Hologram simulated without filtering. (b) Fourier transform modulus of the particle hologram: F x 0 λf , y 0 λf {1ϑ (ξ, η)}. (c) Filtered

20 FIG. 3 .

 203 FIG. 3. (Color online) Evolution of the hologram SNR (orange dotted line), and the standard deviation of x, y (dashed blue line), z (blue circles), and r (blue diamonds) as a function of the filtering mask opacity 1α. All the parameters are normalized to the values of the classical imaging model (see Eq. (3)).

FIG. 4 .FIG. 5 .

 45 FIG. 4. (Color online) Schematics of the recorded intensity profiles (a) without, and (b) with filtering. These schematics are not to scale, as the aim is simply to better highlight the notations introduced in the main matter.

FIG. 6 .

 6 FIG. 6. (Color online) Radial intensity profile of the recorded hologram without (gray solid curve), and with (blue solid curve) fitering for 1α = 0.85 (data average has been removed from the intensity profile curves).

FIG. 7 .

 7 FIG. 7. (Color online) Experimental illustration of the filtering effect. (a) Hologram recorded without the filtering mask. (b) Hologram recorded with the filtering mask. Intensity profiles, in white, were obtained by radially averaging the intensity profiles along the circumference of the interference pattern.

  FIG. 8. (Color online) Evolution of the estimated object to sensor distance (a) and object radius (b) as a function of the fram index. Blue crosses correspond to the non-filtered filtered holograms that are processed using Eq. (3) as an imaging model, and red circles correspond to the filtered holograms (associated imaging model corresponding to Eq. (9)).

Fig. 6 FIG. 9 .

 69 FIG. 9. (Color online) (a) Representation of the first order term of Eq. (9). (b) Representation of the second order term of Eq. (9). (c) Close up view of the white square in (b).

TABLE I .

 I Standard deviation and standard error of the mean (estimated over 100 samples) of the axial position z and radius r estimation for both classical and filtered in-line holography

		σz (µm) σz (µm) σr (nm) σr (nm)
	In-line holgraphy 46.7	0.47	127.8	1.29
	Filtering	29.6	0.29	75.2	0.75
	Ratio	1.58	1.58	1.7	1.7

The authors acknowledge financial support provided by Université de Lyon through its Programs "Investissements d'Avenir" (ANR-1 1-IDEX-0007), and LABEX PRIMES (ANR-11-LABX-0063).

Appendix A: Calculation of the imaging model Details of the calculations leading to the imaging model given in Eq. ( 9) are given here. The amplitude of the light-field in the object plane U (x, y, 0) (see Fig. 1 for details) can be expressed as

where A 0 is the constant reference beam amplitude. In the remainder of this Appendix, A 0 will be A 0 = 1. The object is positioned in the object focal plane of the 4f filtering device. The first lens therefore performs an optical Fourier of the amplitude distribution given in Eq. (A1)

where (k x , k y ) = (x/ (λf ) , y/ (λf )) are the coordinates in the filtering plane. The Fourier transform of the aperture function F kx,ky [ϑ (x, y)] is given by [START_REF] Goodman | Introduction to Fourier optics[END_REF]

As we are only interested in the recorded intensity, the constant phase terms of Eq. (A3) have been omitted. Introducing Eq. (A3) in Eq. (A2) leads to

δ 0,0 (k x , k y ) being a Dirac distribution. A filtering mask M α (k x , k y ) is applied to the amplitude distribution given Eq. (A4). This amplitude mask is designed so that