Nardine Osman

Mark D'inverno

Carles Sierra

Leila Amgoud

Henri Prade

Matthew Yee-King

Roberto Confalonieri

Dave De Jonge

Katina Hazelden

An Experience-Based BDI Logic: Motivating Shared Experiences and Intentionality

This paper proposes the notion of experience to help situate agents in their environment, providing a link on how the continually evolving environment impacts the evolution of an agent's BDI model and vice versa. Then, using the notion of shared experience as a primitive construct, we develop a novel formal model of shared intention which we believe more adequately describes social behaviour than traditional BDI logics that focus on individual agents. Whilst many philosophers have argued that collective intentionality cannot always be equated to the collection of the individual agents' intentions, there has been no AI model that addresses this issue. We believe this is the first attempt to develop an explicit notion of shared experience from an AI perspective.

I. INTRODUCTION

Existing models of BDI agents typically assume the agent has its belief base already built [START_REF] Inverno | Engineering AgentSpeak(L): A formal computational model[END_REF]- [START_REF] Winikoff | Jack intelligent agents: An industrial strength platform[END_REF]. This paper extends such work by presenting an agent model that proposes the notion of experience, which supports situating agents in their environments and closes the BDI circle: an agent's experience in its environment impacts its beliefs, which in turn affect its desires, then intentions, leading to new actions, and back round again to new experiences. In other words, comparing to existing BDI approaches, experience is proposed as part of the missing link between the state of the environment and an agent's mental state.

After presenting the basic agent model, we then introduce a logical formalism (X-BDI) to describe the agent reasoning and whose semantics is grounded in the model. This formalism delivers the main novelly of our proposal: defining the notion of shared experience based upon the notion of individual experience, and using this definition to address the issue of collective intentionality. Following the intuitive assumption that shared experiences and shared intentions provide the basis of social behaviour, 1 which is the foundation of multiagent system research, we say we propose a model that provides the necessary tools for reasoning over such social behaviour.

The remainder of this paper is organised as follows: we start with a specification of the agent model, then we introduce the logic for the experience-based BDI model, and follow it with an example that illustrates the impact of this work. We then close with some literature review and concluding remarks.

II. THE AGENT MODEL

We argue that the physical world is populated with objects (some of which may be labelled as agents) and changes through events (where some subset may be labelled as actions performed by agents). Here, our focus is not on the physical world itself, but on how it is perceived by agents. It is the perception of the agent situated in a given environment that affects the agent's beliefs, desires, and intentions. More precisely, we introduce the notion of experience, which is the result of the agent perceiving its environment and forming some conceptions about it. We argue that experiences shape beliefs, 2 beliefs influence desires, and desires drive intentions. Intentions in turn lead the agent to act in its environment (where the action could be as simple as to observe), which results in the agent having new experiences. In other words, we say that experience is the missing link that completes the BDI cycle as it helps illustrate how the environment impacts the evolution of an agent's BDI and vice versa. This section provides the basic definitions of experiences, beliefs, desires, and intentions. We note that our proposal is based on the assumption that an agent can perceive its environment, resulting in having a set of percepts. Although we note that percepts may sometimes be imagined (following the cognitive neuroscience view that considers imagination as a second order perception [START_REF] Schütz-Bosbach | Prospective coding in event representation[END_REF], [START_REF] Perrett | Seeing the future: Natural image sequences produce "anticipatory" neuronal activity and bias perceptual report[END_REF]). As such, the set of agent α's percepts is referred to as P α = R α ∪ I α , where percepts are either real (R) or imagined (I). We also note that percepts, both real and imagined, could describe various types of elements, amongst these are percepts of agents (AG) and percepts of actions (AC).

A. Experiences

An experience is simply viewed here as a number of percepts that the agent decides to group together. Experiences may be basic (composed of percepts) or compound (composed of a combination of percepts and other experiences). An example of the former would be the sunset one witnessed. An example of the latter would be going out, which contains nested experiences such as having a drink, seeing a show, etc.

We enrich the definition of an experience (although these may be considered complementary) by adopting Kant's argument that the basic cognitive hard wired relationships are time ordering (≺), causality (❀), and spatial (⊲⊳) relationships. We also adopt Kant's view that human beings follow the categorical imperative method that makes them classify things as good or bad. As such, we introduce the evaluation function (♥) for evaluating experiences.

We state that the set of all experiences of agent α is E α (where the set of all experiences is referred to as E). A single experience of agent α, E α ∈ E α , is defined as the tuple:

E α = P ′ α ∪ E ′ α , ≺, ❀, ⊲⊳, ♥ • P ′ α ∪E ′
α describes the experience's content (referenced as content(E α )), which may be composed of percepts (P ′ α ⊆ P α ), sub experiences (E ′ α ⊆ E α ), or a combination of both.

• ≺⊆ (P ′ α ∪ E ′ α ) × (P ′ α ∪ E ′ α )
defines a temporal partial order over the elements of content(E α ).

• ❀⊆ (P ′ α ∪E ′ α )×(P ′ α ∪E ′ α ) defines a causal relationship over the elements of content(E α ), and it should satisfy the property ❀⊆≺.

• ⊲⊳⊆ (P ′ α ∪ E ′ α ) × (P ′ α ∪ E ′ α )
× S defines a spacial relationship between the elements of content(E α ), and it should satisfy the property (a, b, s) ∈⊲⊳⇒ (a, b) ∈≺ ∧(b, a) ∈≺. An example of the set of spacial relationships could be S = {above, below, left, right, behind, infront}.

• ♥ : L → V defines the experience's evaluation, which maps a label to an evaluation space. An example of an evaluation space is V = {positive, negative}. The label is intended to represent the evaluation criteria, such as L = {impact, usefulness, contentment, surprise}.

The evaluation criteria may either represent rational criteria, such as whether "the experience helps one achieve their goals", or emotional ones.

a) Basic Experience: We distinguish a special type of experience, a basic experience E α , in which all the contents of the experience are single percepts: E α = P ′ α , ≺, ❀, ⊲⊳, ♥ . Agent α's set of all basic experiences is referred to as E α . b) Basic Action: An interesting type of basic experience is the basic action, which we define as a basic experience, whose content contains only the percepts describing the pre-conditions of the action, the action percept describing the action itself (AC ∈ AC), and the percepts describing the post-conditions of the action. The temporal relation then states that all pre-condition percepts should precede the action percept and that the action percept should precede all the post-condition percepts. Additionally, the causal relation states that the action percept leads to the post-condition percepts. A basic action is then defined as a basic experience E α that satisfies the following properties:

1-∀ x ∈ content(E α ) • (∃ y ∈ content(E α ) • x ≺ y ∨ y ≺ x) 2-∀ x, y, z ∈ content(E α ) • x ≺ y ∧ y ≺ z ⇒ y ∈ AC α 3-| AC α ∩ content(E α ) | = 1 4-∀ x, y ∈ content(E α ) • x ❀ y ⇒ x ∈ AC α ∧ y ∈ AC α

B. Beliefs

SImilar to existing BDI models, we say an agent holds a set of beliefs B about itself, its environment, its feelings (if any), and so on.

C. Desires

We say desires D are experiences that one desires to be realised in the environment. In other words, the percepts composing these experiences are imagined ones, as they have not yet happened or may never happen. For example, one can desire to win the Nobel prize. A desire is then a basic experience E such that content(E) ⊂ I, where I describes the set of imagined percepts. We note that desires may or may not be feasible. Furthermore, agents may or may not commit to realising their desires; they usually attempt to realise a subset of their desires. Desires that the agent commits to realising through a concrete plan are called intentions, which we introduce next.

D. Intentions

We say intentions are desires with plans: T = D × P, where a plan is defined as a partial order of imagined basic experiences: P ⊆ E ×E. We note that basic experiences include basic actions, and an imagined basic action is an action that the agent has committed to realise but has not realised yet.

E. Agent Model

An agent model is then defined as the tuple:

P, E, B, D, T

where P is the agent's set of all percepts, E its set of experiences, B its set of beliefs, D its set of desires, and T its set of intentions.

III. EXPERIENCE-BASED BDI LOGIC

We now define a BDI logic that is grounded on the notion of experience which we will call X-BDI. We first define its syntax, then we give its semantics based on the formal definitions of the previous section, and finally present a sample of the logic's inference rules.

A. Syntax

• If ϕ is a propositional well formed formula then ϕ ∈ X-BDI

• If ϕ is a propositional well formed formula, α ∈ A is an agent, and P is the set of plans, then Action(α, ϕ) ∈ P

• If ϕ ∈ X-BDI, α ∈ A is an agent, and p ∈ 2 P is a plan then E(α, ϕ), B(α, ϕ), D(α, ϕ), I(α, ϕ, p) ∈ X-BDI • If ϕ ∈ X-BDI, G = {α, β, . . . } ⊆ A, and p ∈ 2 P then CE(G, ϕ), JE(G, ϕ), SE(G, ϕ, p), CD(G, ϕ), JD(G, ϕ), SD(G, ϕ, p), CI(G, ϕ, p), JI(G, ϕ, p), SI(G, ϕ, p) ∈ X-BDI • If ϕ, ψ ∈ X-BDI then ¬ϕ, (ϕ ∨ ψ) ∈ X-BDI
The meaning of the symbols are as follows: G is a set of agents describing a group (which we also refer to as an agent community), Action has the obvious meaning, B, D, and I are the classical BDI symbols, and E stands for Experience. We use the letter C as a prefix to the E, B, D, and I symbols to mean Common (i.e. what is common to all agents in a group); J to mean Joint (i.e. what is common to all agents in a group and all those agents are aware that it is common to all agents in the group); and S to mean Shared (i.e. what is common to all agents in a group, and all those agents are aware that it is common to all agents in the group, and the shared belief literal is related to the agents willingly acting together a shared plan). Thus, the expression:

SI({α, β}, seeBCNplay, {Action(α, tv), Action(β, stadium)})
is a literal in X-BDI expressing (according to the following semantics) that α and β have the shared intention to see Barcelona play a football match by α planning to see the match on tv and β going to the stadium.

B. Semantics

Formally, we understand the model of the world (which is populated by agents) as a Kripke structure where the state of the world and the mind state of agents evolve due to the actions of agents. For instance, if ϕ is perceived by α (and it becomes part of α's experiences), then α may decide to believe ϕ and we can say that the formula B(α, ϕ) is generated in α's belief base. Similarly, we can define the semantics of formulae like D(α, ϕ). We say agents may have beliefs about other agents, more concretely about the intentions, desires, beliefs, and experiences of other agents. This means that we give semantics to nested expressions. For instance, α may have seen an agent β in the environment watching a football match and jumping with joy when Barcelona scores a goal and thus α deduces from that experience that β desires that Barcelona wins, and that can be represented in the logic as B(α, D(β, winBCN)). Thus, given a set of agents {α, β, . . . }, we define an X-BDI semantic model as a pair ω = W, A where W is a classical logic model (i.e. interpretations for propositions) and A is a vector of agent models A = α, β, . . . : a) Propositions: The interpretation of classical formulae is straightforward. We will use the symbol |= PL to refer to classical satisfaction.

W, A

|= ϕ iff ϕ ∈ PL and W |=PL ϕ b) Actions: We say it is sufficient for one agent in the community to have a percept of the action (even if the percept is not shared by others) and that the world is consistent with the changes in the environment caused by the action, for the action to be considered true, or satisfied (recall that AC α is α's set of perceived actions). The satisfaction of a plan consists on the satisfaction of each individual action in the plan.

W, A |= Action(α, ϕ) iff ∃ β ∈ A • (α, ϕ) ∈ AC β and W, A |= ϕ W, A |= p iff p ∈ 2 P and ∀ a ∈ p • W, A |= a c
) Experiences: A model satisfies that α had an experience with content ϕ if it exists in α's set of experiences E α . A group of agents had a common experience tagged as λ if all had an experience that semantically entails λ. Similarly, a group of agents had a joint experience tagged as λ if all had an experience that semantically entails λ and they recognise each other as part of the experience. Finally, a group of agents had a shared experience tagged as λ if it was a joint experience, they all actively participated in it, and the plan of action is said to be satisfied (in other words, each action has been perceived by at least one agent).3

W, A |= E(α, ϕ) iff ϕ ∈ Eα W, A |= CE(G, λ) iff ∀ α ∈ G • ∃ ϕ ∈ X-BDI W, A |= E(α, ϕ) ∧ ϕ → λ W, A |= JE(G, λ) iff ∀ α, β ∈ G • ∃ ϕ ∈ X-BDI • W, A |= E(α, ϕ) ∧ ϕ → λ ∧ B(β, E(α, ϕ)) W, A |= SE(G, λ, p) iff W, A |= JE(G, λ) and W, A |= p and ∀ α ∈ G • ∃ Action(α, ϕ) ∈ p • W, A |= ϕ → λ d) Beliefs:
The semantics for beliefs is similar to experiences. However, the notion of shared belief is defined to exist when the agents in the group recognise one another in a shared experience that led all of them to hold the belief.

W, A |= B(α, ϕ) iff ϕ ∈ Bα W, A |= CB(G, ϕ) iff ∀ α ∈ G • W, A |= B(α, ϕ) W, A |= JB(G, ϕ) iff ∀ α, β ∈ G • W, A |= B(α, ϕ) ∧ B(β, B(α, ϕ)) W, A |= SB(G, ϕ, p) iff W, A |= JB(G, ϕ) and ∃ λ ∈ X-BDI • W, A |= SE(G, λ, p) ∧ λ → ϕ e) Desires:
We use the same approach used for beliefs.

W, A |= D(α, ϕ) iff ϕ ∈ Dα W, A |= CD(G, ϕ) iff ∀ α ∈ G • W, A |= D(α, ϕ) W, A |= JD(G, ϕ) iff ∀ α, β ∈ G • W, A |= D(α, ϕ) ∧ B(β, D(α, ϕ)) W, A |= SD(G, ϕ, p) iff W, A |= JD(G, ϕ) and ∃ λ ∈ X-BDI • W, A |= SE(G, λ, p) ∧ λ → ϕ f) Intentions:
The intention of an agent to reach a goal by following a plan is true if it is internalised in the intentions of the agents. Common intentions and joint intentions follow the same patterns as in the case of beliefs. However, a shared intention requires each agent to intend part of the plan to achieve the goal. That is, there is no shared intention if any agent is not involved in fulfiling (at least) part of the plan for achieving the goal.

W, A |= I(α, ϕ, p) iff (ϕ, p) ∈ Iα W, A |= CI(G, ϕ, p) iff ∀ α ∈ G • W, A |= I(α, ϕ, p) W, A |= JI(G, ϕ, p) iff ∀ α, β ∈ G • W, A |= I(α, ϕ, p) ∧ B(β, I(α, ϕ, p)) W, A |= SI(G, ϕ, p) iff ∀ α, β ∈ G • ∃ pα ⊂ p • pα = ∅ ∧ W, A |= I(α, ϕ, pα) ∧ B(β, I(α, ϕ, pα))
We assume classical propositional deduction for propositional fragments of X-BDI and modus ponens on the X-BDI formulae. X-BDI axioms are those of propositional logic plus specific axioms for X-BDI expressions. Due to space limitations, here we include axioms for Desires only:

⊢ D(α, ϕ) ∧ D(β, ϕ) ↔ CD({α, β}, ϕ) (1) ⊢ CD(G, ϕ) ∧ D(β, ϕ) ↔ CD(G ∪ {β}, ϕ) (2) ⊢ JD(G, ϕ) → CD(G, ϕ) (3) ⊢ CD({α, β}, ϕ) ∧ B(α, D(β, ϕ)) ∧ B(β, D(α, ϕ)) ↔ JD({α, β}, ϕ) (4) ⊢ JD(G ∪ G ′ , ϕ) → JD(G, ϕ) ∧ JD(G ′ , ϕ) (5) ⊢ SD(G, ϕ, p) → JD(G, ϕ, p) (6) ⊢ JD(G, ϕ, p) ∧ SE(G, λ, p) ∧ (λ → ϕ) → SD(G, ϕ, p) (7)
IV. EXAMPLE

With the basic definitions we can now specify the exact steps needed for intentional agents to achieve shared intentions (and hence, shared experiences).

Motivating the Interest in a Shared Experience.: Agents may have different motives for engaging in shared experiences, for various reasons based on their own personal beliefs and desires. In BDI models, agents' actions are the result of their intentions, which are a subset of desires that the agent has chosen to realise. As such, the action of approaching other agents could in many cases be the result of a desire of sharing an experience with that agent.

Investigating & Realising a Joint Desire: When one agent desires a shared experience with other agents, then it will contact those agents to convince them to share its desire, and eventually fulfil that desire. In multiagent systems, this could be achieved through argumentation [START_REF] Maudet | Argumentation in multi-agent systems: Context and recent developments[END_REF], or collaborative planning [START_REF] Grosz | Planning and acting together[END_REF]. This step describes the agent's investigation of whether a joint desire for achieving the shared experience may be realised. If an agent believes that all other agents hold the same desire and that they are aware of each one of them holding that desire, then the agent will believe that there is a joint desire to achieve the shared experience.

Investigating & Realising a Shared Intention: After a joint desire is born, agents are then committed to finding the plan of action for fulfilling the desired shared experience. Argumentation (or argumentation-based negotiation) is usually used here to make sure that all parties agree to who does what and under what conditions. If an agent believes that all other agents intend to execute their role in a given plan and that they are all aware of each others' intentions, then the agent will believe that there is a shared intention for executing that given plan.

Realising a Shared Experience: Agents are now committed to carrying out the actions of the plan that they are responsible for. When the plan has been executed, the shared experience is realised.

In what follows, we take the scenario of two agents that are interested in the shared experience of buying a gift together. Let us say the agents adopt the names of their human owners, c for Carla and b for Bill, and they are interested in buying a gift for Mary together (the desire to buy a gift for Mary is referred to as g). In what follows, we describe what actions the agents could perform based on their X-BDI, and the changes resulting in their X-BDI for each action perceived occurring in their environment.

1) Motivating the Interest in a Shared Experience. Say

Carla is interested in buying Mary a gift, but because she cannot afford a gift on her own, she desires to buy the gift with one of Mary's friends (referred to as MaryFriend). As such, her belief base includes:

D(c, SE({c, MaryFriend}, g, inanyway)) (i)
Note that inanyway refers to some plan of action that has not been decided upon yet. Bill desires to share any experience with Carla (referred to as anything), possibly because he is dependent on her, or maybe he is secretly in love with her.

As such, Bill's belief base includes:

D(b, SE({c, b}, anything, inanyway)) (ii)
2) Investigating & Realising a Joint Desire. In order for Carla to fulfil her desire of finding a partner for buying Mary a gift with (literal (i) in Carla's belief base), she contacts Mary's friends, asking each "Would you like to buy a gift for Mary with me?" Bill replies with a "Yes" to fulfil his desire of sharing any experience with Carla (literal (ii) in Bill's belief base). This exchange of information leads to Carla adding the following to her belief base:

B(c, D(b, SE({c, b}, g, inanyway))) (iii) B(c, B(b, D(c, SE({c, b}, g, inanyway))))

And deducing that:

B(c, JD({c, b}, SE({c, b}, g, inanyway)))

Similarly, Bill modifies his belief base by adding:

B(b, D(c, SE({c, b}, g, inanyway))) (iv) B(b, B(c, D(b, SE({c, b}, g, inanyway))))
And deducing that:

B(b, JD({c, b}, SE({c, b}, g, inanyway)))

As such, the following now holds:

JD({c, b}, SE({c, b}, g, inanyway)) (v)
Note that the joint desire for a shared experience may be the result of differing individual desires. In our example, Carla just needs someone to share her expenses, whereas Bill just wants to do anything jointly with Carla. Also note that (v) is deduced from axiom (4), the belief literals (iii) and (iv), and formula CD({c, b}, SE({c, b}, g, inanyway)), which in turn is deduced from axiom (1) and belief literals (i) and (ii).

All the deduced formulae that we mention in this section are deduced in a similar straightforward manner.

3) Investigating & Realising a Shared Intention. After both agents agree that there is a joint desire for buying the gift together, they go on to argue on the details of the plan they will follow. For example, where to buy the gift? What to buy? In this paper, we do not dwell on the details of how a plan is agreed upon. Assuming both agents agree on a plan p, and they both communicate their agreement to each other, then Carla's belief base is updated to contain the following:

I(c, SE({c, b}, g, p), pc) B(c, I(b, SE({c, b}, g, p), pb)) B(c, pc ⊂ p ∧ pc = ∅ ∧ pb ⊂ p ∧ pb = ∅)
And Carla deduces:

B(c, SI({c, b}, SE({c, b}, g, p), p))
Note that p c is Carla's part of the plan and p b is Bill's part of the plan. Similarly, Bill's belief base is updated by adding:

I(b, SE({c, b}, g, p), pb) B(b, I(c, SE({c, b}, g, p), pc)) B(b, pc ⊂ p ∧ pc = ∅ ∧ pb ⊂ p ∧ pb = ∅)
And Bill deduces:

B(b, SI({c, b}, SE({c, b}, g, p), p))

As such, the following now holds:

SI({c, b}, SE({c, b}, g, p), p)

Note that each agent commits to performing its own part of the plan (in this case, p c and p b ) to make sure the plan is realised, and both agents believe that this plan will realise their shared experience. 4) Realising a Shared Experience. After executing their actions and the plan is fulfilled, if the agents can perceive that the plan has been fulfilled, then Carla's belief base is updated to contain the following:

B(c, SE({c, b}, g, p))
Similarly, Bill's belief base is updated by adding:

B(b, SE({c, b}, g, p))
And the shared experience is said to have been realised:

SE({c, b}, g, p)

V. BACKGROUND The literature on BDI is vast. However, since the original proposal by Bratman [START_REF] Bratman | Intentions, Plans, and Practical Reason[END_REF], most work has concentrated on providing agent-oriented programming languages based on BDI concepts such as Jason [START_REF] Bordini | Programming Multi-Agent Systems in AgentSpeak using Jason[END_REF], Jack [START_REF] Winikoff | Jack intelligent agents: An industrial strength platform[END_REF], AgentSpeak [START_REF] Inverno | Engineering AgentSpeak(L): A formal computational model[END_REF], [START_REF] Rao | AgentSpeak(L): BDI agents speak out in a logical computable language[END_REF], 3APL [START_REF] Hindriks | Architecture for agent programming languages[END_REF] or 2APL [START_REF] Dastani | 2APL: a practical agent programming language[END_REF]; and on logical approaches to the BDI model such as modal logic [START_REF] Rao | Modeling rational agents within a BDIarchitecture[END_REF], first-order logic [START_REF] Rao | AgentSpeak(L): BDI agents speak out in a logical computable language[END_REF], or degrees [START_REF] Parsons | An approach to using degrees of belief in BDI agents[END_REF], [START_REF] Casali | A graded BDI agent model to represent and reason about preferences[END_REF]. Differently from existing work, we extend the BDI model with the notion of experience, and use that to introduce the notions of shared experiences and collective intentionality.

Another related line of work with respect to the notion of experience is Case-Based Reasoning (CBR) [START_REF] Aamodt | Case-based reasoning: Foundational issues, methodological variations, and system approaches[END_REF]. In CBR certain types of (problem-solving) experiences are kept in memory and recalled in order to be reused in the solution of new problems. More importantly, the notion of a shared experience is also not central in CBR; although federated CBR [START_REF] Ontañón | Learning when to collaborate among learning agents[END_REF] discusses a remotely related issue, which is the communicability of knowledge and the coordination amongst agents for acquiring new problem solving capabilities and improving performance.

The main contribution of our work is on experiences and shared intentionality. Individual actions are dictated by individual intentions, which are the result of individual beliefs, and desires. But how do collective actions come about? Philosophers, especially those interested in action theory, have been more and more interested in the notion of collective actions, intentionality, and belief. Since these issues have not been addressed by current BDI approaches, we now discuss existing philosophical views that have addressed these issues. Some argue that collective attitude simply refers to the fact that the majority of the group's members share that attitude [START_REF] Quinton | The presidential address: Social objects[END_REF]. These accounts were labelled as summative accounts by Gilbert [START_REF] Gilbert | On social facts, ser. International library of philosophy[END_REF]. He argued the insufficiency of simple summative accounts (SSA) and presented the complex summative accounts (CSA) as one approach to address some of their problems by introducing the notion of common knowledge. For example, a group G intends X if and only if most of the members of G intend X and it is common knowledge in G that most of the members of G intend X.

In our model, simple summative accounts are described by common modalities. For example, a common experience is an experience that has been experienced by the individuals of the group. Complex summative accounts are described by joint modalities. For example, a joint experience is an experience that is common to a group, and the members of the group are aware that they have all experienced the same experience.

Gilbert further argues that both the SSA and CSA accounts are not the right approaches since a group's attitude cannot always be described in terms of the individual ones, even if it was enhanced with common knowledge. For instance, a group of people who are jogging in the park are aware of each other's intentions to jog; yet, there is something different between this group and a group of friends who decide to jog together.

While our model does define common and joint modalities, we note that these collective modalities are simply used to describe the collection of individual modalities. They are not used to describe the groups' intentions as a whole. For that, the shared modality is introduced. A shared experience is a joint experience in which the agents are actively involved. A shared intention is an intention shared by the group members to carry out a predefined plan for fulfilling a joint desire.

Searle [START_REF] Searle | The construction of social reality[END_REF] states that the we-intend cannot be reduced to a set of I-intend, even if it was supplemented with mutual beliefs. He argues that collective intentions should combine the sense of acting with the sense of willing something together. In our model, the notion of shared experience does not necessarily imply that the different agents sharing an experience were willing to perform the actions they did together. However, the shared intention is defined as having a shared plan, where each agent intends to fulfil its part of the plan. The very definition of an agent's intention is its willingness and commitment to execute the corresponding plan. As such, a shared intention is realised only if the agents are willing and committing to execute their actions as detailed by the plan agreed upon.

Bratman [START_REF] Bratman | Faces of Intention, ser. Cambridge Studies in Philosophy[END_REF], like Searle, does not believe in a plural agent (representing a group as a whole) that could hold shared intentions, nor does he believe that shared intentions can be reduced to individual intentional states. In his definition, an intention is shared if and only if our intentional actions are coordinated by making sure our personal plans of action meld together. As such, shared intentions give rise to argumentation and negotiation for agreeing on the coordinated plans of actions. Bratman further argues, like Searle, that a single agent can have a we-intention, which has attracted a great deal of criticism [START_REF] Stoutland | Why are philosophers of action so anti-social?[END_REF], [START_REF] Velleman | How to share an intention[END_REF], since one cannot intend what he/she cannot fulfil (in other words, one cannot intend actions that may only be executed by others).

Similar to Bratman, we say an intention is shared only if there is a shared plan of action that meshes and coordinates individual actions, although the level of coordination remains loose. For example, the plan may state who does what, without going into the order or the pre and post conditions of actions. Unlike Bratman and Searle, we say a shared intention cannot be held by a single agent (although single agents may form beliefs about shared intentions), but by the group of agents whom the plan of action cannot be carried out without. Agents can only intend to perform their part of a shared plan.

VI. CONCLUSION

This paper presented a formal agent model that introduces the notion of experience, which situates agents in their environment, providing a clear and tangible link that was missing in past BDI approaches and that help describe how the environment (and its changes) impact the evolution of an agent's BDI and vice versa.

We also built a logical formalism, the X-BDI, to describe the agent's reasoning and whose semantics are based on our proposed agent model. The main novel contribution of this work is defining, through the X-BDI logic, the notion of shared experience and using it as a fundamental construct to underpin collective intentionality, which we believe are the basis of social behaviour and the foundation of multiagent systems in general. While BDI was used as a model for individual agent reasoning, X-BDI can be used (as our example illustrates) as a model for collective agents' reasoning by illustrating when, why, and how do social interactions take place.

We argue that if social behaviour was not based on shared experiences and intentionality, then it would be the result of peers coincidentally acting in a similar way, which is not the type of social behaviour that interests our research work in multiagent systems.

The rich interplay between experiences and beliefs and how experiences impact the formation of beliefs has been discussed in[START_REF] Amgoud | Experiences -a forgotten component of epistemic states[END_REF].[START_REF] Amgoud | Experiences -a forgotten component of epistemic states[END_REF] argues why the classical knowledge representation approaches that equate the set of beliefs to the set of experiences is not expressive enough as it cannot describe what is "not possible" or what is "possible for sure", and it illustrates how experiences may be used to modify beliefs, such as reassessing one's generic beliefs when an experience contradicting this generic belief is observed.

For simplification, we abuse notation and write a ∈ p to mean that action a is 'part' of plan p and p ′ ⊆ p to mean that plan p ′ is 'part' of plan p.

ACKNOWLEDGMENT This work is supported by: the ACE ERA-Net project; the CBIT project (funded by the Spanish Ministry of Science & Innovation TIN2010-16306); and the Agreement Technologies project (CONSOLIDER CSD 2007-0022, INGENIO 2010).


Nelly Pustelnik

IEEE Member Herwig Wendt
email: herwig.wendt@irit.fr

Patrice Abry

Ieee Fellow

LOCAL REGULARITY FOR TEXTURE SEGMENTATION: COMBINING WAVELET LEADERS AND PROXIMAL MINIMIZATION

ABSTRACT

Texture segmentation constitutes a classical yet crucial task in image processing. In many applications of very different natures (biomedical, geophysics,...) textures are naturally defined in terms of their local regularity fluctuations, which can be quantified as the variations of local Hölder exponents. Furthermore, such images are often naturally embedded in the class of piece-wise constant local regularity functions. The present contribution aims at proposing and assessing a segmentation procedure for this class of images. Its originality is twofold: First, local regularity is estimated using wavelet leaders, a novel multiresolution quantity recently introduced for multifractal analysis but barely used in local regularity measurement, comparisons against wavelet coefficient based estimation are conducted; Second, the challenging minimal partition problem underlying segmentation is convexified and conducted within a customized proximal framework. The estimation of the number of regions and their target regularity is obtained from a total-variation estimate that enables the actual use of proximal minimization for texture segmentation. Performance is assessed and illustrated on synthetic textures. Index Terms-Texture segmentation, piece-wise constant, local regularity, wavelet leaders, proximal minimization.

MOTIVATION, RELATED WORKS, CONTRIBUTIONS

Texture segmentation from local regularity fluctuations. Texture characterization and segmentation constitutes a challenging task in image processing (see e.g., [START_REF] Haralick | Statistical and structural approaches to texture[END_REF][START_REF] Laine | Representations for texture segmentation[END_REF][START_REF] Jung | Texture segmentation via non-local non-parametric active contours[END_REF][START_REF] Unser | Texture classification and segmentation using wavelet frames[END_REF][START_REF] Kaplan | Extended fractal analysis for texture classification and segmentation[END_REF] and references therein). In a variety of applications of possibly very different natures, the relevant information characterizing textures, and thus conveying the information to be analyzed, consists of the fluctuations across space of their local regularity. This is notably the case for biomedical textures such as bone [START_REF] Benhamou | Fractal analysis of radiographic trabecular bone texture and bone mineral density: two complementary parameters related to osteoporotic fractures[END_REF] or breast [START_REF] Kestener | Waveletbased multifractal formalism to assist in diagnosis in digitized mammograms[END_REF] (see also [START_REF] Lopes | Fractal and multifractal analysis: A review[END_REF] for a review), for surface imagery [START_REF] Mallick | Discrepancy between subcritical and fast rupture roughness: A cumulant analysis[END_REF], for satellite imagery [START_REF] Roux | A wavelet-based method for multifractal image analysis. III. Applications to high-resolution satellite images of cloud structure[END_REF], or for image registration [START_REF] Véhel | Multifractals, texture and image analysis[END_REF][START_REF] Pesquet-Popescu | Stochastic fractal models for image processing[END_REF].

Often, local regularity is assessed in terms of the so-called Hölder exponent h(x) [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF]. In essence, h(x) consists of the power law exponent quantifying the decrease of the powers of local fluctuations, measured via multiresolution quantities TX (a, x) (e.g., wavelet coefficients) around a given space position x when the analysis scales a goes towards fine scales, i.e., ETX(a, x) 2 ≃ C(x)a 2h(x) when a → 0.

Related works: Hölder exponent estimation and segmentation.

Local regularity based analysis of textures can naturally be organized into two sub-questions: i) Accurately estimating the Hölder exponents for each location; ii) Splitting the image into regions where estimates can be considered constant, so as to achieve texture segmentation. Though these issues have received significant efforts over the past two decades for 1D signals (cf. e.g., [START_REF] Benassi | Gaussian processes and pseudo-differential elliptic operators[END_REF][START_REF] Gonc | Multiple window and local scaling exponent estimation[END_REF]), much less contributions were dedicated to 2D fields or images (see a contrario [START_REF] Pesquet-Popescu | Stochastic fractal models for image processing[END_REF] and references therein).

Estimation relies on two key choices, with crucial impact on performance: Multiresolution quantities TX (a, x) from which the h(x) are measured; Range of analysis scales a to be practically involved in the estimation. For multiresolution quantities, most works conducted so far relied on either increments, oscillations or wavelet coefficients. Bilinear time-scale representations were also used [START_REF] Gonc | Multiple window and local scaling exponent estimation[END_REF]. The choice of the range of scales is naturally framed into a classical bias-variance trade-off. While theoretically, estimation should be conducted in the limit of fine scales, practically a finite range of scales am ≤ a ≤ aM has to be used. Choosing a large range of scales yields smaller estimation variance at the price of increasing the bias because the TX(a, x) at large scales are less well localized (cf. e.g., [START_REF] Benassi | On roughness indices for fractional fields[END_REF][START_REF] Gonc | Multiple window and local scaling exponent estimation[END_REF]).

For segmentation, i.e., to detect changes amongst Hölder exponents measured at different locations, this trade-off also turns crucial. This is why both estimation and change detection have mostly been conducted in model dependent frameworks. Fractional Brownian motion (fBm) [START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF], the paradigm model for scale invariance, implies a constant and unique Hölder exponent across sample fields. Multifractional Brownian motion extends fBm by allowing piece-wise smooth variations of h(x) across space (while preserving joint Gaussianity) [START_REF] Ayache | The covariance structure of multifractional brownian motion, with application to long range dependence[END_REF] and constitutes essentially the only model with spacedependent h that has been studied and used. The problem has mostly been addressed for 1D signals and barely for images (see a contrario [START_REF] Pesquet-Popescu | Stochastic fractal models for image processing[END_REF][START_REF] Kestener | Waveletbased multifractal formalism to assist in diagnosis in digitized mammograms[END_REF] which aimed at identifying particular points in the image defined by specific values of h(x)). Furthermore, to our knowledge, little work has been published aiming at detecting changes in h(x) in a context where an exact model is not a priori assumed (see a contrario [START_REF] Gonc | Multiple window and local scaling exponent estimation[END_REF] for 1D signals).

With respect to the generic issue of image segmentation, efficient variational methods were proposed [START_REF] Kass | Snakes: Active contour models[END_REF][START_REF] Mumford | Optimal approximations by piecewise smooth functions and associated variational problems[END_REF][START_REF] Caselles | Geodesic active contours[END_REF][START_REF] Kichenassamy | Gradient flows and geometric active contour models[END_REF][START_REF] Chan | Active contours without edges[END_REF][START_REF] Couprie | Power watershed: A unifying graph-based optimization framework[END_REF], however none of them guarantees convergence towards a global minimum, and thus, obtained solutions strongly depend on initialization. Instead, a convex relaxation of the minimal partition techniques were proposed in [START_REF] Chan | Algorithms for finding global minimizers of image segmentation and denoising models[END_REF][START_REF] Pock | A convex relaxation approach for computing minimal partitions[END_REF]. In [START_REF] Pock | A convex relaxation approach for computing minimal partitions[END_REF], the algorithmic solution is based on a Arrow-Hurwicz type primal-dual algorithm but requires inner iterations and upper boundedness of the primal energy in order to improve convergence speed. Also, a proximal solution in a context of disparity estimation was proposed in [START_REF] Hiltunen | Comparison of two proximal splitting algorithms for solving multilabel disparity estimation problems[END_REF]. Alternatively, recent techniques are framed into variational principles, making use of comparisons between pairs of neighboring patches [START_REF] Ni | Local histogram based segmentation using thewasserstein distance[END_REF][START_REF] Jung | Texture segmentation via non-local non-parametric active contours[END_REF]. Notably, in [START_REF] Ni | Local histogram based segmentation using thewasserstein distance[END_REF], a convex relaxation of the original problem was envisaged so as to ensure to find the global minimizer, while in [START_REF] Jung | Texture segmentation via non-local non-parametric active contours[END_REF] a more general but non-convex framework was proposed, which yields good results at the price of a well-chosen initialization. However, the computational cost is a limitation of these methods.

Goals and contributions.

In this context, the first originality of the present contribution consists in performing region-segmentation of images, that excludes the recourse to a fully parametric model for the data, but instead only assumes that images belong to the class of piecewise constant Hölder exponent functions. Its second originality lies in a twofold departure from earlier works: For multiresolution quantities, wavelet leaders are chosen and shown to yield improved statistical performance for Hölder exponent estimation, compared to the classical wavelet coefficients (cf. Section 2); For segmentation, using estimated Hölder exponents as inputs, a proximal algorithm is devised. It essentially relies on the customization of the formulation proposed in [START_REF] Hiltunen | Comparison of two proximal splitting algorithms for solving multilabel disparity estimation problems[END_REF], which expands on that proposed in [START_REF] Pock | A convex relaxation approach for computing minimal partitions[END_REF] to efficiently handle the resulting large size problem. Further, we propose an original method to a priori estimate the number of regions as well as their mean regularity, an initialization step that turns compulsory and crucial for the actual use of the proximal approach in that context. These are detailed in Section 3.

This combination (use of wavelet leaders and of proximal algorithm with original initialization) enables us to propose an efficient data-model free piecewise constant Hölder exponent minimal partition image segmentation procedure, that furthermore depends on very few parameters that need to be tuned manually.

WAVELET COEFFICIENTS AND LEADERS FOR H ÖLDER EXPONENT ESTIMATION

Local regularity and Hölder exponent. Let X denote the bounded 2D function (image) to be analyzed. Local regularity around position x 0 ∈ R 2 is measured by the so-called Hölder exponent h(x 0 ), defined as the largest α > 0, such that there exists a constant C > 0 and a polynomial Px 0 of degree less than α, such that

|X(x) -Px 0 (x)| ≤ C|x -x 0 | α in a neighborhood x of x 0 .
When h(x 0 ) is close to 0, the image is locally very irregular and close to discontinuous. Conversely, a large h(x 0 ) corresponds to a locally smooth field. For example, when h(x 0 ) increases towards 2 then the fields becomes smoother and smoother up to being differentiable. Fig. 1-b below displays a Gaussian texture, with piecewise constant Hölder exponents, the region with the lowest h (outer ring) is more irregular then the one with the largest h (inner ring).

Wavelet coefficients. Let φ(x) and ψ(x) denote the scaling function and mother wavelet defining a 1D multiresolution analysis. Let the 2D wavelets be defined as:

ψ (0) (x) = φ(x1)φ(x2), ψ (1) (x) = ψ(x1)φ(x2), ψ (2) (x) = φ(x1)ψ(x2), ψ (3) (x) = ψ(x1)ψ(x2) with x = (x1, x2). The collections ψ (m) j,k (x) = 2 -j ψ (m) (2 -j x -k)
of dilated (to scales a = 2 j ) and translated (to space positions x = 2 j k) templates of ψ0 form a basis of L 2 (R 2 ) for well chosen functions ψ. Let d

(m) X (j, k) = X, ψ (m) j,k
denote the (L 1normalized) coefficients of X of the so-called discrete wavelet transform (DWT) of X. Readers may refer to [START_REF] Mallat | A wavelet tour of signal processing[END_REF] for further details.

Wavelet leaders. Wavelet leaders were recently introduced in the context of multifractal analysis [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF][START_REF] Wendt | Bootstrap for empirical multifractal analysis[END_REF], their use in the context of local regularity measurement has however been barely considered. The wavelet leader LX (j, k), located around position x 0 = 2 j k, is defined as the local supremum of all wavelet coefficients taken within a spatial neighborhood across all finer scales 2 j ′ ≤ 2 j :

LX (j, k) = sup m=1,2,3, λ j ′ ,k ′ ⊂Λ j,k |d (m) X (j ′ , k ′ )|, (1)
λj,k = [k2 j , (k + 1)2 j ), Λj,k = p∈{-1,0,1} 2 λj,k+p [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF][START_REF] Wendt | Bootstrap for empirical multifractal analysis[END_REF].

Local regularity estimation.

It has long been known that, for large (but not all) classes of images [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF], wavelet coefficients reproduce the Hölder exponent h(x 0 ), in so far as, for all k 0 such that 2 j k 0 ≈ x 0 :

1 3 3 m=1 |d (m) X (j, k 0 )| ≃ Cd(x 0 )2 jh(x 0 ) when 2 j → 0.
More recently, it has been shown [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF][START_REF] Wendt | Bootstrap for empirical multifractal analysis[END_REF] that wavelet leaders systematically (that is: for all classes of images) reproduce the Hölder exponent h(x 0 ): LX (j, k 0 ) ≃ CL(x 0 )2 jh(x 0 ) when 2 j → 0. This naturally leads to the estimation of h(x 0 ) by means of linear regressions:

hd(x 0 ) = j 2 j=j 1 wj ln 1 3 3 m=1 |d (m) X (j, k 0 )|, (2)
hL(x 0 ) = j 2 j=j 1 wj ln LX (j, k 0 ), (3)
with weights wj chosen to perform non weighted regression [START_REF] Gonc | Multiple window and local scaling exponent estimation[END_REF]. Section 4 compares estimates obtained from either wavelet coefficients or leaders and shows that the latter yield a substantial decrease in variance and are thus to be preferred.

MINIMAL PARTITION SEGMENTATION USING PROXIMAL TOOLS

Let Ω ⊂ R 2 denote a spatially continuous domain and f ∈ L 2 (Ω) the image that we intend to label. Note that, in Section 4, f will successively model the estimated Hölder exponents obtained from wavelet coefficients or leaders, i.e., f = hd or f = hL. The Mumford-Shah approach in [START_REF] Mumford | Optimal approximations by piecewise smooth functions and associated variational problems[END_REF] consists in computing a piecewise constant approximation of data f . Assuming that f can be approximated by Q labels such that (Ωq)1≤q≤Q models the Q distinct regions, this problem can be written as min

Ω 1 ,...,Ω Q Q q=1 Ωq (f -vq) 2 dx + 1 2 Q q=1 Per(Ωq) s.t. Q q=1 Ωq = Ω, (∀q = p), Ωq ∩ Ωp = ∅, (4)
where vq stands for the value of f in region Ωq (with, by convention, vq ≤ vq+1), where the left-hand-side term consists of the sum of the variances of f over all regions, and where Per(Ωq) measures the perimeter of region Ωq, and where the constraints imposed on Ωq ensure to obtain a non-overlapping partition of the image. The discrete analogue of Model ( 4) is the Potts model which is known to be NP-hard to solve. A solution to circumvent this difficulty consists in relaxing this non convex formulation into a convex approximation [START_REF] Pock | A convex relaxation approach for computing minimal partitions[END_REF]. From now on, the sampled version of the image f is considered and denoted as a vector of pixels f = (f (n) )1≤n≤N ∈ R N . The Q regions Ωq are labeled through an auxiliary variable u = (u (n) )1≤n≤N , such that u (n) = vq if and only if the pixel n belongs to the region Ωq. Further, one can introduce Q binary functions θ = (θ1, . . . , θQ) such that, for every q ∈ {1, . . . , Q},

(∀n ∈ {1, . . . , N }), θ (n) q = 1 if u (n) ≥ vq, 0 otherwise. ( 5
)
It results that the labeling function u can be recovered from θ = (θ1, . . . , θQ) through the relation

(∀n ∈ {1, . . . , N }), u (n) = Q q=1 (vq -vq-1)θ (n) q , (6)
with v0 = 0. A bijection between u and θ is guaranteed by constraining, for every n ∈ {1, . . . , N }:

Bn = {θ (n) ∈ {0, 1} × . . . × {0, 1}, 1 ≥ θ (n) 1 ≥ . . . ≥ θ (n) Q ≥ 0},
whose convex relaxation is

Bn = {θ (n) ∈ [0, 1] × . . . × [0, 1], 1 ≥ θ (n) 1 ≥ . . . ≥ θ (n) Q ≥ 0}.
Moreover, for every n ∈ {1, . . . , N }, θ

(n) 1 = 1.

Convex criterion

The convexification of Model (4) leads to the following minimization problem involving the Q binary functions θ = (θ1, . . . , θQ) [START_REF] Pock | A convex relaxation approach for computing minimal partitions[END_REF]:

minimize θ 1 ,...,θ Q Q-1 q=1 N n=1 (θ (n) q -θ (n) q+1 )(f (n) -vq) 2 + N n=1 θ (n) Q (f (n) -vQ) 2 + 1 2 Q q=1
ρTV(Hθq, V θq)

s.t. (∀n ∈ {1, . . . , N }), θ (n) 1 = 1, (∀n ∈ {1, . . . , N }), 1 ≥ θ (n) 2 ≥ . . . ≥ θ (n) Q ≥ 0, ( 7
)
where H ∈ R N×N and V ∈ R N×N are matrix representations of, respectively, the horizontal and vertical first-order discrete differences, and where the total variation reads, for every

η = η (n) 1≤n≤N ∈ R N and ζ = ζ (n) 1≤n≤N ∈ R N , ρT V : η (n) 1≤n≤N , (ζ (n) 1≤n≤N → N n=1 |η (n) | 2 + |ζ (n) | 2 .
The functions involved in [START_REF] Kestener | Waveletbased multifractal formalism to assist in diagnosis in digitized mammograms[END_REF] are convex, lower semi-continuous, and proper. H and V are diagonalizable in the Fourier domain and the proximity operator of each function has a closed form. It results that the proximal algorithms PPXA+ [START_REF] Pesquet | A parallel inertial proximal optimization method[END_REF] can be used to efficiently find the minimum of [START_REF] Kestener | Waveletbased multifractal formalism to assist in diagnosis in digitized mammograms[END_REF].

Estimation of (vq)1≤q≤Q

The a priori choice of (vq)1≤q≤Q is likely to strongly impact the estimated (θq)1≤q≤Q. Here, we propose to extract this values from a denoised estimator of f such that

f * = arg min g f -g 2 + λ ρTV(Hg, V g) (8)
where λ > 0 denotes a regularization parameter, that impacts the quality of the denoised estimate. Section 4 further details the estimation of the (vq)1≤q≤Q from f * .

RESULTS

Numerical simulations.

The potential and performance of the combined estimation and segmentation procedures described in Sections 2 and 3 above are illustrated and assessed by application to independent realizations of synthetic images. They are produced numerically according to a 2D multifractional Brownian field model [START_REF] Benassi | Gaussian processes and pseudo-differential elliptic operators[END_REF][START_REF] Ayache | The covariance structure of multifractional brownian motion, with application to long range dependence[END_REF], whose definition has been slightly modified here to ensure an homogeneous variance across the image:

X(x) = C(x) R 2 e ıxξ-1 |ξ| h(x)+ 1 2 dW (ξ), (9)
where dW (ξ) is 2D Gaussian white noise and h(x) denotes the prescribed Hölder exponent function. The normalizing factor C(x) ensures that the local variance of X does not depend on the location x. This model is chosen here for convenience in synthesis. Note that the proposed estimation and segmentation procedures do not rely at all on any knowledge of the model. A sample field of such processes is shown in Fig. 1-b. Analysis is conducted using a standard 2D DWT with orthonomal tensor product Daubechies mother wavelets with Nψ = 2 vanishing moments. Regularity is estimated using the scaling range (j1, j2) = (1, 3).

Estimation.

To compare estimation performance obtained with wavelet leaders against those produced with wavelet coefficients, the estimation procedure described in Section 2 is applied to 100 independent copies, of size N × N , with N = 1024 and with uniform Hölder exponent: h(x) ≡ H.

Segmentation.

Synthetic data are produced according to the model in [START_REF] Mallick | Discrepancy between subcritical and fast rupture roughness: A cumulant analysis[END_REF] with piecewise constant Hölder exponent function h(x) = Hq for Region q. Each region is thus regarded as a different texture and the goal is to achieve texture segmentation of the image. As an instructive example, we use images consisting of Q = 3 different regions with piecewise constant Hölder exponents on concentric disks, as shown in Fig. 1-a. The three different textures can clearly not be identified from the histogram of the image pixel values shown in Fig. 1-c. In simulations, N × N = 512 × 512.

The estimation procedures of Section 2 are applied to such images, yielding local estimates of the Hölder exponents which are plotted as images and histograms in Fig. 2-a (wavelet coefficients) and in Fig. 3-a (wavelet leaders). Despite a visual perception of 3 different regions for the wavelet coefficient based estimates hd, the corresponding histogram (Fig. 2-b) is unimodal and fails to reveal the existence of three constant-h regions in the image, a direct The proximal tool based segmentation procedure described in Section 3 is now applied to both hd(x) and hL(x), initialized as described above. Results are compared in Fig. 2-f and Fig. 3-f, (for one single image) and quantified in Table 2 (averaged across 100 independent copies). This shows that the number of mis-classified pixels is systematically smaller when segmentation is based on hL(x). Differences between wavelet coefficient and leader based performance may appear small, this is however only a consequence of the chosen simple geometry for the piece-wise constant regions. Indeed, quantifying the number of mis-classified pixels within narrow strips comprising the borders between the piecewise constant-h regions shows that segmentation based on hL(x) significantly outperforms that based on hd(x), a direct consequence of the better localization and lower variance of the hL(x). Wavelet leader based segmentation thus provides far better border delimitation, a crucial issue in biomedical texture segmentation, for instance. Our codes were developed in MATLAB R2011b and the segmentation of a 512 × 512 image with Q = 3 labels requires only 5 minutes on an 2.

CONCLUSIONS AND PERSPECTIVES

We have proposed, to the best of our knowledge, the first fully operational texture segmentation procedure based on texture local regularity. The segmentation procedure is designed for the class of piece-wise constant regularity images. It relies on combining improved regularity estimates based on wavelet leaders with the proximal solution to the minimization of the convex criterion underlying the segmentation problem. The proximal solution further relies on a total variation and proximal based initialization. The procedure is illustrated here using realizations of stochastic Gaussian model process with prescribed region-wise constant local regularity. The segmentation procedure does, however, not rely on any a priori model knowledge besides that of data belonging to the class of piece-wise constant regularity images. The performance of the proposed procedure for different region geometry, selection of scaling range, and sensitivity to local regularity differences are currently being systematically quantified. Comparisons against alternative segmentation features, such as local entropy, will be conducted. Comparisons of the robust convex optimization based segmentation achieved here against recent classification techniques, such as the one proposed in [START_REF] Ni | Local histogram based segmentation using thewasserstein distance[END_REF][START_REF] Jung | Texture segmentation via non-local non-parametric active contours[END_REF] will also be made. Moreover, extensions of the procedure to the class of piecewise smooth local regularity images are currently under study. Preliminary analysis of real biomedical textures yields promising results and will be further investigated.

Fig. 1 .

 1 Fig. 1. Synthetic data. (a) Piecewise constant Hölder exponent mask (H1, H2, H3) = (0.25, 0.50, 0.75), (b) Sample field, and (c) Corresponding histogram of the pixel values.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Wavelet coefficient segmentation. (a) Estimate hd, (b) Histogram of hd, (c) h * d , (d) Histogram of h * d , (e) Segmentation based on thresholding of h * d , and (f) Labeling function u obtained from proximal segmentation with f = hd.

Table 1 . Estimation Performance. Coefficients

 1 Table 1 reports estimation performance for different values of H and clearly indicates that leader-based estimates of h systematically outperform wavelet coefficient based ones, with significant decrease in variances at the price of increased biases, overall yielding substantially decreased mean-square errors.

	H =	0.25		0.50		0.75
		WavC WavL WavC WavL WavC WavL
	Bias	-0.36	0.03	-0.19	0.08	-0.08	0.11
	Std	0.52	0.19	0.55	0.37	0.60	0.50
	RMSE	0.63	0.19	0.58	0.38	0.60	0.52
	min	-0.14	0.27	0.27	0.47	0.56	0.69
	max	-0.09	0.31	0.37	0.70	0.77	1.10

vs leaders.

Table 2 .

 2 4 GHz Quad-Core Intel Xeon and 6 GB of RAM. Segmentation Performance. Coefficients vs. leaders

	% of miss-	entire	around borders
	classified pixels	image	(∼ 2.10 4 pixels)
		hd	hL	hd	hL
	BasicThresh 33.3 22.1 93.6	62.2
	ProxSeg	3.4	3.0	9.4	8.6