Jean B Lasserre

VOLUME OF SLICES AND SECTIONS OF THE SIMPLEX IN CLOSED FORM

Given a vector a ∈ R n , we provide an alternative and direct proof for the formula of the volume of sections ∆ ∩ {x : a T x <= t} and slices ∆ ∩ {x : a T x = t}, t ∈ R, of the simplex ∆. For slices the formula has already been derived but as a byproduct of the construction of univariate B-Splines. One goal of the paper is to also show how simple and powerful can be the Laplace transform technique to derive closed form expression for some multivariate integrals. It also complements some previous results obtained for the hypercube [0, 1] n .

Introduction

In Marichal and Mossinghof [START_REF] Marichal | Slices, slabs, and sections of the unit hypercube[END_REF] the authors have provided a closedform expression of slices and slabs of the unit hypercube cube [0, 1] n . In the interesting discussion on the history and applications (e.g. in probability and statistical mechanics) of this problem, they mention how similar but earlier results had been already proved, notably by Pólya in his PhD dissertation. In [START_REF] Marichal | Slices, slabs, and sections of the unit hypercube[END_REF] the authors' proof relies on a signed simplicial decomposition of the unit cube and the inclusionexclusion principle whereas Pólya's approach was different and related the volume to some sinc integrals as also did Borwein et al. [START_REF] Borwein | Multi-variable sinc integrals and volumes of polyhedra[END_REF] much later. For more details the interested reader is referred to [START_REF] Marichal | Slices, slabs, and sections of the unit hypercube[END_REF] and the references therein.

For the simplex one can find several contributions in the literature for integrating polynomials and defining cubatures formula; see for instance the recent work of Baldoni et al. [START_REF] Baldoni | How to integrate a polynomial over a simplex[END_REF] and the many references therein. But concerning the slice of a simplex it turns out that a formula for the volume of the slice has already been derived ... as a by-product in the construction of univariate B-splines! Indeed as explained in Micchelli [8, pp. 150-153], in their construction of univariate B-splines of degree n -1, Curry and Schoenberg showed that they are interpreted as volumes of slices of a n-simplex! In the description1 of Chapter 4 in [START_REF] Micchelli | Mathematical Aspects of Geometric Modeling[END_REF] one may even read "This chapter explores the powerful idea of generating multivariate smooth piecewise polynomials as the volume of slices of polyhedra."

Contribution. The goal of this note is to provide a relatively simple and direct proof for volumes of sections and slices of the simplex without taking a detour via the theory of univariate B-splines. It also shows how powerful and "easy" can be Laplace techniques for such a purpose. We consider sections and slices of the canonical simplex ∆ := {x : e T x ≤ 1} (where e ∈ R n is the vector of ones). That is, given a vector a of the unit sphere S n-1 and some t ∈ R, we want to compute the n-dimensional (resp. (n -1)-dimensional) Lebesgue volume of the sets

Θ(a, t) := ∆ ∩ { x ∈ R n : a T x ≤ t } (1.1) S(a, t) := ∆ ∩ { x ∈ R n : a T x = t } (1.2)
Notice that in contrast to the unit hypercube [0, 1] n , the simplex ∆ is not a cartesian product and the "canonical" signed simplicial decomposition of the sliced hypercube used in [START_REF] Marichal | Slices, slabs, and sections of the unit hypercube[END_REF] has no analogue for the simplex ∆. So for the simplex a different approach is needed and ours is based on the Laplace transform technique already used in our previous work in [START_REF] Lasserre | Solving a class of multivariate integration problems via Laplace techniques[END_REF] for computing a certain class of multivariate integrals.

In particular and in contrast to the case of the sliced hypercube, some special care is needed when some weights a i , i ∈ I (for some subset I ⊂ {1, . . . , n}), are identical, in which case the generic formula for the volume of the section of the simplex degenerates.

Finally, the result of [START_REF] Marichal | Slices, slabs, and sections of the unit hypercube[END_REF] for the unit hypercube can be retrieved from our results either (a) directly by applying the same Laplace technique to the context of the hypercube or (b) in two steps by first decomposing the unit hypercube into the union of 2 n simplices and then applying our "sliced-simplex formula" to each of the simplices.

Main result

Given a scalar x ∈ R, the notation (x) + stands for max[0, x]. Let us first recall the following result already proved in [START_REF] Barvinok | Computing the volume, counting integral points, and exponential sums[END_REF] and [START_REF] Lasserre | Solving a class of multivariate integration problems via Laplace techniques[END_REF] for the canonical simplex ∆. Lemma 2.1 ([1], [START_REF] Lasserre | Solving a class of multivariate integration problems via Laplace techniques[END_REF]). Let c := (c 1 , . . . , c n ) ∈ R n , c 0 := 0, with c i = c j for every distinct pair (i, j). Then

(2.1) ∆ exp(-c T x) dx = n i=0 exp(-c i ) j =i (c i -c j ) .
We will use Lemma 2.1 to prove the following result for the sliced simplex and a section of the simplex as well.

Theorem 2.2. Let a = (a 1 , . . . , a n ) ∈ S n-1 , a 0 := 0, and assume that a i = a j for any pair (i, j) with i = j. Then

vol (Θ(a, t)) = 1 n! n i=0 (t -a i ) n + j =i (a j -a i ) . (2.2) vol (S(a, t)) = 1 (n -1)! n i=0 (t -a i ) n-1 + j =i (a j -a i ) . (2.3)
Proof. We first assume that a ≥ 0. Let f : R + → R be the Lebesgue volume of Θ(a, t) so that f (t) = 0 for every t < 0. Let

F : C → C its Laplace transform L[f ] λ → F (λ) := ∞ 0 exp(-λt) f (t) dt,
whose domain is D := {λ ∈ C : ℜ(λ) > 0}. Hence with λ real, λ > 0,

F (λ) = ∞ 0 exp(-λt) f (t) dt = ∞ 0 ∆∩{x:a T x≤t} dx exp(-λt) dt = ∆ ∞ a T x exp(-λt) dt dx [by Fubini-Tonelli] = 1 λ ∆ exp(-λ a T x) dx = 1 λ n+1 n i=0 exp(-λ a i ) j =i (a j -a i ) = n i=0 1 n! j =i (a j -a i ) • Γ(n + 1) λ n+1 L[t n ] • exp(-λ a i )
where we have used (2.1). Notice that the function

λ → h(λ) := n i=0 1 n! j =i (a j -a i ) • Γ(n + 1) λ n+1 • exp(-λ a i )
is analytic on D and coincides with F (λ) on (0, +∞) ⊂ D. Therefore by the Identity Theorem for analytic functions (see e.g. Freitag and Busam [START_REF] Freitag | Complex Analysis, Second Edition[END_REF]) F (λ) = h(λ) on D; Next recall that λ → exp(-λc)L[t n ](λ) is the Laplace transform of u c (t)(t -c) n with u c (t) being the Heavyside function t → u c (t) = 1 if t ≥ c and u c (t) = 0 otherwise; see e.g. [START_REF] Widder | The Laplace Transform[END_REF]. Therefore,

f (t) = 1 n!   { i:a i ≤t } (t -a i ) n j =i (a j -a i )   = 1 n! n i=0 (t -a i ) n + j =i (a j -a i )
,

which is the desired result (2.2). Next, notice that if n is even then f is differentiable and if n is odd f is differentiable for all t ∈ ∪ i {a i }, with f ′ (t) = 1 (n -1)! n i=0 (t -a i ) n-1 + j =i (a j -a i ) , t ∈ {a 0 , a 1 , . . . , a n }.
Moreover, for all t ∈ {a 0 , a 1 , . . . , a n }, the volume of the section {x : a T x = t} ∩ ∆ is given by: vol(∆ ∩ {x :

a T x = t}) = a f ′ (t), t ∈ {a 0 , a 1 , . . . , a n } = 1 (n -1)! n i=0 (t -a i ) n-1 + j =i (a j -a i )
, since a ∈ S n-1 and where the first equality follows from a formula provided in [START_REF] Lasserre | An analytical expression and an algorithm for the volume of a convex polytope in R n[END_REF] for the (n -1)-volume of the (n -1)-dimensional facet of an arbitrary polytope. Finally, by using a simple continuity argument, formula (2.2) remains valid at those points t = a i , i = 0, . . . , n.

Let us now consider the case where a i < 0, i ∈ I, for some nonempty subset I ⊂ {1, . . . , n}. Hence we may and will suppose that a 1 < a 2 < . . . < a n with a 1 < 0. Let

t * = min { a T x : x ∈ ∆ } = min {a i : i ∈ I } = a 1 < 0,
and observe that f (t) = 0 whenever t ≤ t * . Therefore let t → g(t) := f (t + t * ) so that g(t) = 0 if t ≤ 0. Its Laplace transform reads

∞ 0 exp(-λt) g(t) dt = ∞ 0 exp(-λt)f (t + t * ) dt = ∆ ∞ a T x-t * exp(-λt) dt dx = exp(λt * ) F (λ) = n i=0 exp(-λ (a i -t * )) n! j =i (a j -a i ) Γ(n + 1) λ n+1 L[t n ]
,

provided that a i = a j for all distinct pairs (i, j); and so

g(t) = 1 n!   { i:a i ≤t+t * } (t -a i + t * ) n j =i (a j -a i )   = 1 n! n i=0 (t -(a i -a 1 )) n + j =i (a j -a i )
, provided that a i = a j for all distinct pairs (i, j). But then as f (t) = g(t -a 1 ) we obtain the desired result (2.2). Finally (2.3) is obtained as before.

So if the a i 's are now ordered with increasing values

a 1 < a 2 < • • • < 0 < • • • a n then from (2.
2) one can see that the volume function t → Θ(a, t) of the sliced simplex is a piecewise polynomial of degree n on [a 1 , a 2 ] ∪ [a 2 , a 3 ] ∪ • • • ∪ [a n-1 , a n ] and constant equal to zero on (-∞, a 1 ] and equal to 1/n! on [a n , +∞).

As explained in Micchelli's book [8, p. 50-54], formula (2.3) for the volume of slices of ∆ was already known to Curry and Schoenberg in their construction of (piecewise polynomials) B-Splines. Starting from the formula of the B-Splines of degree n -1 their proved that it can be interpreted as volumes of slices of ∆. (k,ℓ). An obvious way to see what happens is to perform a perturbation analysis. So suppose that a i = a j for all distinct pairs (i, j) = (k, ℓ) and let a ℓ := a k + ǫ with ǫ > 0 sufficiently small so that now a i = a j for all pairs (i, j) (including the pair (k, ℓ)). Then formula (2.2) reads

1 n! i =k;i =ℓ (t -a i ) n + j =i (a i -a j ) + 1 ǫ n! (t -a k ) n + j =k;j =ℓ (a j -a k ) - (t -a k -ǫ) n + j =k;j =ℓ (a j -a k -ǫ) and therefore when ǫ → 0 one obtains vol (Θ(a, t)) = 1 n! i =k;i =ℓ (t -a i ) n + j =i (a i -a j ) - 1 (n -1)! (t -a k ) n-1 + j =k;j =ℓ (a j -a k ) for all t = a k .
A similar analysis can be done if several coefficients a i 1 , a i 2 , a i k are identical to some value s, in which case a term of the form 1

(n-k)! (t-s) n-k + j =i 1 ,...,i k (a j -s)
appears.

2.2.

For an arbitrary simplex. For an arbitrary simplex Ω with set of vertices V := {v 1 , . . . , v n+1 } and a ∈ S n-1 such that a T v = a T w for any pair of distinct vertices (v, w) ∈ V 2 , formula (2.2) becomes

f (t) = 1 n!     v∈V (t -a T v) n + w∈V ;w =v a T (v -w)     .
Similarly, at every point t ∈ {a T v : v ∈ V }, the volume of the section {x : a T x = t} ∩ Ω is given by: vol(Ω ∩ {x :

a T x = t}) = 1 (n -1)!     v∈V (t -a T v) n-1 + w∈V ;w =v a T (v -w)     .
2.3. The unit hypercube [0, 1] n . We now use the same Laplace technique to retrieve a formula for the unit hypercube [0, 1] n already provided in Marichal and Mossinghof [START_REF] Marichal | Slices, slabs, and sections of the unit hypercube[END_REF]. So define the sets

B := { x ∈ R n + : x ≤ 1 } B a (t) := { x ∈ R n + : x ≤ 1; a T x ≤ t}, t ∈ R.

2. 1 .

 1 The case of identical weights. Let us indicate what happens to formula (2.2) if e.g. a k = a ℓ for some pair

  First assume that 0 < a ∈ S n-1 and let f : R + → R be the Lebesgue volume of B a (t) andF : C → C its Laplace transform L[f ] λ → F (λ) := ∞ 0 exp(-λt) f (t) dt,

http://epubs.siam.org/doi/abs/10.1137/1.9781611970067.ch4

This work was supported by a grant of the PGMO program of the Fondation Mathématique Jacques Hadamard (FMJH), Paris).

where here its domain is D := {λ ∈ C : ℜ(λ) > 0}. With λ real, λ > 0,

where 1 U ∈ {0, 1} n is the indicator of the set U. Therefore proceeding as in the proof of Theorem 2.2, Theorem 1]. Similarly, the volume for a slice of the hypercube can be deduced by taking the derivative of f (t).

Finally the case where a ∈ S n-1 has some negative entries can be recovered as we did for the simplex.


Nawel Takouachet
email: n.takouachet@estia.fr

Jérémy Legardeur
email: j.legardeur@estia.fr

The Role of the Facilitator during Digital Creative Sessions

Keywords: Digital creativity session, Software for collaborative creativity, Creativity facilitator ACM Classification H.5.2: User Interfaces, H.5.3: Group and Organization

RESUME

L'animation des sessions de créativité permet d'améliorer l'efficacité d'un groupe pour la résolution de problèmes, la recherche de nouveaux concepts ou de solutions innovantes. Ce papier aborde le rôle de l'animateur pendant une session de créativité dans un environnement numérique. Nous évaluons les logiciels collaboratifs existants qui utilisent la métaphore du Post-it® et nous montrons que ces applications ont été conçues pour être utilisées sans forcément tenir compte de la présence d'un animateur. Ils offrent cependant un ensemble de fonctions qui sont pertinentes pour le groupe et ses individus. En nous basant sur ces constatations, nous proposons un ensemble de fonctionnalités principales pour intégrer pleinement le rôle de l'animateur aux applications digitales lors des sessions de créativité.

INTRODUCTION

Creativity engages the mind ability to bring new relevant insights to solve a challenge. This process is indeed a prerequisite driver of the innovation, which is the action of turning the creative ideas into real products and services.

Increasingly, companies invest and place value on creative thinking and innovation to face rapid advances and increasing competition in new business, strategic, technological and human resources development.

Companies aim to boost their employee's creativity for reaching successful outcomes and ensuring sustainability of their activities.

There are numerous creative techniques and most of them are based on the brainstorming principle developed by Alex Osborn [START_REF] Osborn | Applied imagination; principles and procedures of creative problem-solving[END_REF]. Osborn with his colleagues developed brainstorming based methods such as SCAMPER [START_REF] Eberle | Scamper: Games for Imagination Development[END_REF], Creative Solving Problem [START_REF] Parnes | A source book for creative thinking[END_REF]. These techniques can be applied during creative session to help generate order and prioritize information, combine and chunk ideas, propose solutions and find alternative. One of the wide popular tools that can be easily adapted and used with different creative techniques is sticky notes. Ideas are recorded on small colored paper that can be easily moved and then used for other tools such as theming and linking.

Today, and thanks to technological advances, the principle of sticky note is transposed to graphical interfaces with multiple software applications, allowing low costs solutions with new useful features such as portability, collaborative work on distant sites, flexible organization, backups and restores steps. Nevertheless, despite of these multitude of available digital applications, the use of paper-based sticky notes during creative session, especially professional consulting services continued to be dominant and far from being dwindled.

In this paper, we attempt to understand why digital applications for creative session have difficulty to assert as an alternative to paper-based conventional methods? We present a review of existing software for creative session based on sticky notes and similar component. We highlight the fact that these applications do not take into account the specific role of the facilitator. Indeed, the facilitator is an important active member of the creative meeting, able to understand the group dynamics and specialized in leading and pushing creativity from a group. As a matter of fact, if the facilitator is successful, the objective of the creativity session will be achieved: solutions will be produced, understood and agreed by all participants.

Therefore, it is essential that digital applications for creative session integrate facilitations steps and functions. To this end, we propose useful guidelines that help to design embedded facilitation-oriented features in digital creativity sessions.

FACILITATOR ROLE

Generally, group members are not trained in working group processes. The facilitator is a group process trainer, he is charged to help the group to solve problems and take process-related decisions [START_REF] Smith | The t-group trainer: Group facilitator or prisoner of circumstance[END_REF][START_REF] Schwarz | The skilled facilitator: practical wisdom for developing effective groups[END_REF][START_REF] Ackermann | Participants' perceptions on the role of facilitators using group decision support systems[END_REF][START_REF] Phillips | Faciliated work groups: Theory and practice[END_REF].

The facilitator understands and involves the group dynamics to guide and stimulate interaction between the participants. He seeks to keep the creativity process moving forward and ensure reaching the session outcomes with clear thinking and good global participation. Blumberg and Golembiewski [START_REF] Blumberg | Learning and change in groups[END_REF] described a list of the facilitator' functions, as follows:

1. Creates situations conducive to learning. 2. Establishes a model of behavior. 3. Provides new values in process. 4. Facilitates the flow of information. 5. Participates as an expert. 6. Protects participants from unnecessary stress and attack. 7. Confronts the group regarding its processes.

During the creative process, the facilitator practices different creativity techniques that are appropriate to the context, the group nature and the problem to resolve.

COLLABORATIVE TAKING NOTES SOFTWARE

We present in this section an overview of existing software that support collaborative creativity thinking and offered the possibility to generating and structuring ideas using sticky notes or similar component. We include in this study digital applications that cover colocated and remote collaborative work. We are interested in how these applications take into account the role of the facilitator during the creative process. Indeed, we focus on useful features and functions during creativity sessions, in particular those requested in the facilitation mode.

Evernote [Erreur ! Source du renvoi introuvable.] is a powerful application widely used by the general public. It allows taking notes, photos, create to-do lists, record audio reminders and search notes. It synchronizes all notes on all the computers, phones and tablets. Nevertheless, Evernote does not directly support realtime collaboration; it has to be used with LiveMinutes [START_REF] Liveminutes | LiveMinutes Inc[END_REF], a web-based application that allows users to collaborate on their notes in real time and simultaneously synchronized with Evernote. Microsoft Office OnNote [START_REF] Onnote | [END_REF] offers the same services as its competitor Evernote. It has to be linked with Outlook 2013 to let a user create a shared workspace and made edits in real-time by all the invited participants.

However, even by integrating the collaborative aspect, Evernote and OneNote are principally note taking software and do not implement specific functions for creativity sessions. Kalemmer et al. [START_REF] Klemmer | The designers' outpost: A tangible interface for collaborative web site design[END_REF] developed Designers' Outpost, a tangible user interface that combines physical and electronic representations of sticky notes. Users can write on a paper sticky note with a standard pen and place it on the electronic board. The interface integrates a computer vision component for capturing and digitalizing the physical sticky notes. Users can link sticky notes by drawing a line from one note to another with a stylus; when a sticky note is moved on the board, all its associated links follow. Everitt et al. [START_REF] Everitt | Two worlds apart: Bridging the gap between physical and virtual media for distributed design collaboration[END_REF] extended the Designers' Outpost to support remote collaboration on distributed media spaces.

Note that authors have observed two styles of interacting with the board: the facilitator and the open board style. In the facilitator style, one person chairs the board and engages discussion between participants. Progressively, as concepts arise, the facilitator links notes together and creates notes that synthesize the discussion content. In the open board style, however, everyone gets to actively create notes and place it on the board.

Hilliges and al. [START_REF] Hilliges | Designing for collaborative creative problem solving[END_REF] have designed a brainstorming application that uses an interactive table and a large wall display. Authors investigated how collaborative creativity can be supported in digital environment while maintaining face-to-face communication. They developed pen-based user interfaces and focused design on handwriting and fluid gestures interaction. Participants can create a sticky note by simply drawing a square on the table surface. During divergent thinking step, the participants switch on the board and they can move and reorganize sticky notes. The application implements a clustering function which is largely used by the facilitator during the creative session. It allows creating clusters by simply drawing a circle around some sticky notes. Clusters are merged by dragging them close together and can be connected to each other or to a single sticky note by drawing a line connecting borders.

Tele-Board [START_REF] Gericke | Tele-board: Follow the traces of your design process history[END_REF][START_REF] Gumienny | Supporting creative collaboration in globally distributed companies[END_REF] is a remote collaboration web that supports idea generating and structuring. The system integrates a video conference feature to synchronous transparent whiteboard overlay. Two whiteboards are installed on two remote sites. Gestures and facial expression of each participant acting on one of the two whiteboards are displayed as transparency on the other whiteboard. In this way, the group member are immersed with a co-presence feeling and take thus advantage of the benefits of face to face communication.

Tele-board implements a large number of features that are highly requested in the facilitator mode during the session of creativity. It allows forming clusters or even hierarchical structures that can be exported in mind map format. In addition, creativity templates can be loaded as background pictures and used to group and classify the notes generated by participants. Furthermore, the Tele-Board system offers the possibility to save states of the whiteboard. Users can eventually navigate through different whiteboard states and eventually load a state to continue working at any previous step. This property is also useful for analyzing the progress of the creativity process and helps debriefing.

Stormboard [4] is a web brainstorming and real-time collaboration application. It allows generating ideas using the mouse and the keyboard in form of text, image, documents and sketches on different devices. Once a storm created, other individuals can be easily invited to participate. All ideas can be commented allowing the team to discuss and refine ideas. Each participant has a number of votes that can be fully or partially used to designate his favorite ideas. has the advantage to provide great models of templates. It also allows bringing out advanced statistical reports taking into account the position of the sticky notes on the templates. However, in the current version of Stormboard, sticky note cannot be linked or clustered.

EFFECTIVE FACILITATION-ORIENTED FUNCTIONS

Digitalization of creativity session give groups creativity several advantages over traditional (Figure 1), paperbased meeting such as:

 Easy generation and capture of ideas,  Collaboration intra-distributed groups across space and time,  Collaboration in real time,  Flexible structuration and organization, Automatic recording of the different stages of the session,  Conducting a focus: selection, zoom, grouping,  Voting ideas and export statistical reports.

On the other hand, previous comparative studies have shown that providing facilitation advice to group members can enhance the group interaction and improve derived results [START_REF] Jessup | Group facilitation and group support systems[END_REF]. Furthermore, researchers suggest two possible complimentary manners to improve the group interactions: the effectiveness of the group facilitation and the use of appropriate technology to support groups [START_REF] Hamilton | The facilitation of organizational change: An empirical study of factors predicting change agents' effectiveness[END_REF][START_REF] Clawson | The importance of facilitator role behaviors in different face to face group support systems environments[END_REF].

Nevertheless, already existing software we presented in the previous section do not distinguish explicitly the role of the facilitator of the creativity session from others participants. Indeed, all participants get to actively participate and have all access rights for creating, editing, organizing and evaluating notes during the session. This explains the difficulty of the professional facilitators to use this kind of software; it is clearly difficult for them to manage the creativity session, in the same manner as they do in the paper-based sessions, in several stages by progressively activating the participants' accessibility to certain functions.

According to these findings and drawing on concrete experiences of practitioners, we have identified a set of potential features that have to be taken into account to build an embedded facilitation in digital creativity sessions.

Explicitly separate the role of the facilitator from other participants

Two kinds of users have to be distinguished from each other in the interface:

 The facilitator: he has access to all functions and can enable or disable at any time users access to any function.  The other participants: they can access to the enabled functions and interact according to the directives of the facilitator.

Configurable and flexible session settings

The facilitator has to be able to configure the session settings according to creative process that will adopt. Indeed, he should be able to choose the:  number subgroups of participants: the facilitator can eventually divide the participants in subgroups.  number of the involved whiteboards: the facilitator enters the groups' participants and can load a background template.  number of the processing steps: for each step, the facilitator chooses the options/functions accessible by the participants.

Configurable and flexible processing steps settings

For each step, the facilitator chooses, through a checkbox, functions to be activated, principally:

Mode of interaction

 Individual: participants can only create ideas on their dispositive but they cannot see ideas the group members had contributed. The animator sees ideas happen on his workspace and can

Sticky-notes options

CONCLUSION

In this work we remind the importance of the facilitator role for assisting the group to be effective and reproduce relevant results. Despite the potential benefits of the group-facilitation, we were noted that current collaborative applications for creativity do not taking into account the specific role of the facilitator. We believe that implementing the facilitator style into digital applications for creativity session has a great potential for improving group interaction and outcomes. It may furthermore encouraging practitioners and professional consultants to adopt these applications and moving beyond classical paper-based approaches.

Building on this outcome, we introduced some guidelines to build effective facilitation-oriented functions in digital creativity sessions. The key element is to allow the facilitator to plan, organize and guide the creativity session as he usually does.

Our future work will focus on the design of a facilitatorparticipants model of interaction for creativity process in digital environment, specifying each mode of interaction and their relationship during the different stage of the creativity process.

Figure 1 .

 1 Figure 1. Paper-based creativity sessions


Usha Rashmi Bhaskara

M M C G Warmoeskerken

Attachment of β-Cyclodextrins on Cotton and In uence of β-Cyclodextrin on Ester Formation with BTCA on Cotton

Keywords: β-Cyclodextrin, Cotton, BTCA, FTIR-ATR spectroscopy

Cotton was treated with β-cyclodextrin (BCD) and two derivatives of β-cyclodextrin (2-hydroxypropyl-β-cyclodextrin and monochlorotriazinyl-β-cyclodextrin) to assess the optimal type for xation with cotton. e experimental results showed that treatment of cotton with BCD using the crosslinker BTCA resulted in higher xation than the treatments with the other two derivatives. e concentration of BTCA used did not signi cantly in uence the amount of xation of BCD on cotton. FTIR-ATR spectroscopic analysis showed that the amount of ester formed on the fabric was in uenced by the addition of BCD on cotton with BTCA in comparison to crosslinking of only BTCA with cotton. e laundering tests showed relatively poor washfastness of the β-cyclodextrins on the fabrics.

Introduction

In this work, three di erent types of β-cyclodextrins (β-cyclodextrin (BCD), 2-hydroxypropyl-β-cyclodextrin (HP BCD), and monochlorotriazinyl-β-cyclodextrin (MCT BCD) were attached to cotton. e aim of the work was to determine the optimum type of β-cyclodextrin for xation on cotton and to check the in uence of crosslinker concentrations on xation of BCD on cotton. Additionally, the amount of ester formed on the fabric due to crosslinked BCD cotton cellulose was assessed. BCD in this work refers speci cally to unmodi ed β-cyclodextrin, while β-cyclodextrins refers to modi ed and unmodi ed β-cyclodextrins in general and the three cyclodextrin types used in this work as a whole.

β-Cyclodextrins are cyclic oligosaccharides composed of glucose units linked by α-1,4-glycosidic bonds. Each β-cyclodextrin unit has a hydrophobic cavity that can act as a host for a hydrophobic guest molecule depending on the size, geometry, and molecular weight of the guest molecule. ese types of host-guest complexes nd application in control release of various drugs in the pharmaceutical industry and in other applications such as slow release of fragrances, aromatic oils, additives in the food industry, and so forth. Normally, a 1:1 complexation ratio is found between the cyclodextrin molecule and the guest molecule, while sometimes 1:2 or 2:1 ratios are also found. 1 e inclusion of antimicrobial activity on textiles by incorporating antimicrobials as guest molecules in textilebound β-cyclodextrins are of particular interest.

Derivatives of β-cyclodextrin are formed by substituting hydroxyl groups with alkyl or hydroxyalkyl groups on the glucose units of the β-cyclodextrin molecule, leading to changes in solubility and complex-forming selectivity. 2 HP BCD is one such derivative of β-cyclodextrin. e molecular weight of HP BCD is 1399 g/mol as compared to 1135 g/mol for BCD. [START_REF] Roquette | Technical Sheet, Kleptose Betacyclodextrins and Hydroxypropyl Betacyclodextrins[END_REF] HP BCD is interesting due to its increased solubility in water as a result of the substituted hydroxyls with 2-hydroxypropyl groups in the anhydroglucopyranose units, due to which the inclusion complex forming ability is said to increase. Another popular derivative for xation on textile is MCT BCD. MCT BCD, with an average molecular weight about 1560 g/mol, is particularly interesting because of the substitution of hydroxyls in the anhydroglucopyranose units with monochlorotriazinyl groups, which results in a compound that can be xed to cotton without a crosslinker. 4 In this investigation, BCD and HP BCD were xed to cotton via 1,2,3,4, butanetetracarboxylic acid (BTCA) crosslinker. BTCA is a non-formaldehyde based crosslinker that has proven e ective in terms of durable press performance in comparison with other polycarboxylic acids such as citric acid, malic acid, or succinic acid. BTCA has four carboxylic acid groups, one to three of which are said to be involved in crosslinking with cotton during the esteri cation reaction.

Cotton is known to esterify with BTCA during the curing process in two steps: rst, by formation of a ve-membered cyclic anhydride through the dehydration of adjacent carboxyl groups, and second, by the formation of ester links between the anhydride intermediate and cellulose. Fig. 1 shows the mechanism of BCD xation to cotton via BTCA through the esteri cation reaction. is has been veri ed by researchers with Fourier transfer infrared (FTIR) spectroscopy. [START_REF] Medronho | [END_REF] For bifunctional polycarboxylic acids such as malic acid or succinic acid, only one carboxyl group is said to be involved in esteri cation with cotton. 6 Citric acid is cheaper than BTCA and also more eco-friendly, however, it is known to cause fabric yellowing. 7 BTCA used at certain application concentrations is also known to give ame resistant properties to blends 8 and to cotton 9 (i.e., Class 1 ame retardant e ect according to test method ASTM D1230). BTCA is also said to be more resistant to hydrolysis, and therefore, more wash durable as well. 9,10 Sodium hypophosphite (SHPI) is a widely-used catalyst with BTCA since it gives almost no yellowing and good wrinkle recovery properties. It speeds up the process of BTCA esteri cation with cotton by weakening hydrogen bonds between the carboxylic groups. Use of SHPI reduces the temperature required for cyclic anhydride formation when compared to the use of BTCA alone. 11 In general, SHPI is also considered an optimal BTCA catalyst in comparison with other catalysts, especially in terms of the durability of durable press performance. 12,13 e amount of BCD, HP BCD, and MCT BCD xation were assessed by simple gravimetric measurements and the well-known phenolphthalein test. Cotton xed to BCD was characterized using FTIR-ATR spectroscopy and di erential scanning calorimetry (DSC). ese two techniques were only used for the BCD-treated cotton. Finally, durability of the attached β-cyclodextrins was analyzed by repeated laundering tests.

Experimental Materials

e plain woven cotton (fabric density) 170 g/m 2 was obtained from w Testgewebe. BCD and HP BCD were obtained from Roquette. BTCA, sodium hydroxide, sodium carbonate, tris(hydroxymethyl)aminomethane (Tris), phenolphthalein, and SHPI were obtained from Sigma Aldrich. MCT BCD was obtained from CTD Inc. Demineralized water was used in all experiments.

Various β-Cyclodextrin Treatments

Cotton fabrics were treated using the exhaustion method at room temperature (RT) with the liquor to cloth ratio (LCR) of 20:1. BCD treatment was done in a beaker where the required amounts of BTCA (10, 30, or 60 g/L) and SHPI were added to the calculated liquor volume and stirred until they were dissolved. e required amount of BCD was added (0, 40, 60, 80, or 100 g/L) and dissolved. Fabric samples were placed inside the liquor and stirred for 5 min. A er treatment, the fabric samples were pre-dried at 110 °C for 10 min, cured at 160 °C for 5 min, rinsed in water (60 °C, 5 min), rinsed with tap water, and nally air dried. e curing temperature was kept at 160 °C, since it is known that anhydride formation with SHPI occurs at this temperature. 14 e use of higher temperatures may lead to higher esteri cation (i.e., more crosslinking of BTCA), however, this would also lead to more tensile strength loss. 15 e BTCA to SHPI molar ratio used was 1:0.3. 16 In the literature, the BTCA to SHPI mass ratios of 3:1,11 2:1, 17 and 1:1 [START_REF] Šauperl | [END_REF] were also reported. SHPI catalyst can be e ective starting from 0.3% (3 g/L) to 11% concentrations. 19 To check the in uence of curing conditions and the in uence of the BTCA to SHPI ratio on the amount of BCD xed on cotton, a series of trials with increased curing temperature Fig. 1. Attachment of BCD to cotton with BTCA. [START_REF] Medronho | [END_REF] Source reproduced with permission from Elsevier.

(180 °C) and increased BTCA to SHPI molar ratio (1:1) were performed and the xation amount obtained was compared to the amount achieved under standard treatment conditions (160 °C, BTCA to SHPI molar ratio of 1:0.3).

For treatment with HP BCD (0, 40, 60, 80, or 100 g/L), the same procedure was followed, with the exception that the amount of BTCA used was only 30 g/L.

For treatment with MCT BCD (0, 40, 60, 80, or 100 g/L), the procedure was the same as above with the exception that no BTCA/SHPI was added and the treatment was performed in alkaline medium (pH 11, 125 mM sodium carbonate solution). e treatment time, pre-drying, curing at 160 °C, and sample rinsing a er the treatment were the same as in the two previously mentioned cases.

Analysis and Measurement

Gravimetric Measurements

To assess the amount of β-cyclodextrin xed, gravimetric measurements were made before and a er sample treatment (Eq. 1). Before each measurement, samples were conditioned at 20 °C and 65% relative humidity for 24 h. Eq. 1 W b is the sample weight before treatment and W f is the sample weight a er treatment.

Quanti cation with Phenolphthalein

e amount of β-cyclodextrin xed onto a fabric can be estimated using an alkaline phenolphthalein solution. Phenolphthalein can complex within the β-cyclodextrin cavity. e absorbance of the alkaline phenolphthalein can be measured with UV-vis spectrophotometer at 550 nm.

A calibration curve was made to plot the decrease in absorbance of phenolphthalein solution with the increase in concentration of β-cyclodextrin. For the calibration curve, a series of test tubes were set up, each containing a 5 mL solution (V in L). Each test tube contained 1 mL of varying concentrations of β-cyclodextrins (C in g/L) made in the bu er solution (Tris bu er solution, 0.05 M, pH 7), and 4 mL of phenolphthalein solution at a xed concentration (in g/L). From the curve t equation, a relationship between absorbance and concentration of β-cyclodextrins was determined. e concentration of phenolphthalein and sodium carbonate used were taken from the literature. 20,21 e phenolphthalein working solution was prepared by preparing a 4 mM stock solution of phenolphthalein in ethanol, and taking 1 mL of this stock and mixing it with 4 mL of ethanol and 95 mL of 125 mM sodium carbonate solution. e pH of this 0.04 mM phenolphthalein solution was 11. For the xed β-cyclodextrin quanti cation, 4 mL of this phenolphthalein working solution was added to 1 mL of Tris bu er solution (0.05 M, pH 7), and samples (2.5 cm × 2.5 cm) of β-cyclodextrin treated fabric was put in this 5 mL solution. e nal concentration of the 5 mL phenolphthalein in the test tubes was 0.032 mM. Since phenolphthalein can undergo photo-oxidation (sodium carbonate at the concentration used was supposed to stabilize it), the test tubes were covered with aluminum foil and kept for 2 h in a dark cabinet. e absorbance of the remaining solution was then measured at 550 nm using Cary 100 UV spectrophotometer (Agilent Technologies).

From the absorbances noted from the test tubes and the curve t equation, the mass of β-cyclodextrin in the 5 mL of phenolphthalein solution and nally on the fabric could be estimated. e mass of the di erent β-cyclodextrins (BCD, HP BCD, and MCT BCD) xed on cotton (F in mg/g) was calculated using Eq. 2.

Eq. 2 C is the β-cyclodextrin concentration measured in g/L, V is volume in L, and M is the dry weight of fabric sample in g.

To check for the in uence of BTCA on phenolphthalein quanti cation of BCD, homogenous absorbance tests were done with phenolphthalein and BCD solutions containing BTCA. A 0.04 mM solution of phenolphthalein was made and divided into two beakers. In one beaker, 1 g/L of BTCA was added. en two series of test tubes were arranged and one was lled with 4 mL of phenolphthalein solution with BTCA and the other without the BTCA. BCD solutions (1 mL) of various concentrations made in 0.05 M Tris bu er were then added to all the test tubes and the absorbances were then measured at 550 nm with the UV spectrophotometer.

FTIR-ATR Spectroscopy

BTCA crosslinks cotton through esteri cation reactions. e e ectiveness of crosslinking various amounts of BTCA on cotton can be indirectly assessed using FTIR spectroscopy. Crosslinking e ectiveness is expressed in terms of degree of esteri cation. e relative amount of ester formed on di erent BTCA-treated cotton fabrics is obtained from the ester peak intensity, while acid xed to the fabric can be derived by the carboxylate anion peak intensity in FTIR spectroscopic measurements of these fabrics. e ratio between the two peaks, called the carbonyl band intensity ratio, gives the degree of esteri cation on the crosslinked cotton. is method is based on the semi-quantitative assessment of degree of esteri cation by the measurement of the carbonyl band intensities. 6,22,23 e resulting ester carbonyl band intensity ratio is correlated to the wrinkle recovery angle and tensile strength of fabric treated with BTCA. 15 Samples are pre-treated with sodium hydroxide (NaOH) or ammonia vapor before analysis. It was reported that the NaOH method of sample preparation is suitable only for the assessment of degree of esteri cation on BTCAtreated cotton and not for cotton crosslinked with other polycarboxylic acids. 24 Pre-treatment of BTCA-treated cotton with hydrogen chloride (HCl) vapor leads to the protonation of ionized carboxylate groups (-COO -to -COOH) on cotton. is consequently shows up as a new peak at 1720 cm -1 in FTIR spectroscopic analysis. is can be reversed with exposure of BTCA-treated cotton to ammonia vapor. 25 In this work, FTIR-ATR spectroscopic measurements were performed to assess the in uence of BCD xation with BTCA on the formation of esters on cotton. BCD concentrations were varied along with BTCA concentrations. e ester intensity of the fabrics were then checked at standard curing conditions (i.e., cotton samples were treated with 30 g/L and 60 g/L of BTCA and with 0 g/L to 100 g/L of BCD), and the ester intensity on cotton was measured. e in uence of curing conditions on formation of ester on BCD-treated cotton was also assessed (i.e., cotton samples were treated with 0 g/L and 100 g/L of BCD and 30 g/L and 60 g/L of BTCA at altered curing conditions (such as changed curing temperature and BTCA to SHPI ratio), and the ester intensity was measured again.

Preparation for FTIR-ATR analysis involved treating cotton fabric samples with 0.1 M NaOH for a minimum of 3 min, washing o NaOH from the fabric, and then drying at 90°C for 10 min before FTIR-ATR measurements. Values given by the ammonia vapor method were not comparable to the NaOH method. Ammonia vapor can complex with the BCD cavity resulting in inconsistent measurements. erefore, the NaOH method of pre-preparation was used for FTIR-ATR measurements in this paper. FTIR-ATR spectra were obtained on a Bruker IFS 66/S spectrophotometer, equipped with Silver gate ATR cell employing a cylindrical Ge crystal (n = 4.0) with a diameter of 7 mm and single re ection (SpectraTech). e samples were tightly pressed against a Ge crystal. Spectra were recorded over the range 4000-600 cm -1 , with a resolution of 4 cm -1 and averaged over 128 spectra.

Di erential Scanning Calorimetry

Di erential scanning calorimetry (DSC) measurements were used to check the in uence of BCD and BTCA treatment on thermal decomposition of cotton. Cotton samples were treated with 10 g/L, 30 g/L, and 60 g/L of BTCA, and 0 g/L and 100 g/L of BCD at various curing conditions. ese measurements were made under a nitrogen ow of 20 mL/ min. Samples were rst held isothermally for 1 min at 25 °C, and then heated to 550 °C at the rate of 10 °C/min. Cooling was done at the rate of 60 °C/min. Sample weights for these measurements were around 9.3 mg to 9.8 mg.

Laundering Tests

Cotton was treated with BCD, HP BCD and MCT BCD at 100 g/L application concentration. BCD and HP BCD crosslinking with cotton was done with 30 g/L of BTCA (and the earlier mentioned BTCA to catalyst ratio). Curing was done at 160 °C for 5 min for all samples as previously. ese samples were laundered with 4 g/L of AATCC Reference Detergent (without optical brighteners). Laundering was done at LCR of 50:1 at 60 °C for 30 min. A er laundering, the samples were rinsed in tap water and then dried in the oven for 15 min at 110 °C. Six repeated laundering cycles were performed and the amount of xation was assessed a er each cycle.

Results and Discussion

Gravimetric Measurements

Weight gain (W%) measurements vs. concentration of β-cyclodextrins for various treatments are shown in Figs. 2a andb. Fig. 2a shows weight gain results for treatments with BCD at various BTCA concentrations, while Fig. 2b shows results obtained for the treatments with all three types of β-cyclodextrins (BCD, HP BCD, and MCT BCD) used. Even when no BCD was added, W% values on cotton increased with increased BTCA concentration (Fig. 2a). Use of 10 g/L BTCA gave a steady increase in W% values with increased BCD concentration. However, when greater amounts of BTCA (30 g/L and 60 g/L) were used, there was no steady corresponding increase in W% values with increased BCD concentration. In the presence of BCD, W% values obtained for the treatments with 30 g/L BTCA were lower than than for samples treated with 10 g/L BTCA. All these facts imply that BTCA concentration predominantly in uences W% values.

Quanti cation of β-Cyclodextrins

From the amount of β-cyclodextrin xed on the cotton, shown in Fig. 3a as F (mg/g) against applied concentrations of β-cyclodextrin (g/L) with various concentrations of BTCA, there appeared to be no signi cant in uence of crosslinker concentration on xation amount. It was also noted that 30 g/L of BTCA resulted in lower standard deviation in xation values. e xation of the di erent β-cyclodextrins on cotton is also given in moles/g of cotton in Table I. e in uence of curing conditions on the xation of BCD on cotton was also noted for 30 g/L and 60 g/L of BTCA with 100 g/L of BCD. On changing the curing conditions to a higher curing temperature and with an increased BTCA to SHPI ratio, signi cantly increased xation was not seen in the case of 30 g/L of BTCA (Table II). When curing conditions were changed in the case of 60 g/L of BTCA, no signi cant change in the amount of BCD xation was noted with an increase in curing temperature to 180 °C. However, with the additional increase in the BTCA to SHPI ratio from Fig. 2b compares W% values for cotton treated with various types of β-cyclodextrin. Comparison was made between BCD, HP BCD (both xed with 30 g/L BTCA), and MCT BCD treated fabrics. e in uence of increased β-cyclodextrin concentration on W% values was obvious only for cotton treated with BCD (Fig. 2a). HP BCD showed a slight loss in W% values with increased concentration, while MCT BCD did not show any in uence of increased concentration on the amount attached to cotton (almost constant W% values). ese results indicate that W% measurements, due to their high standard deviation and inclusion of the crosslinker weight (for BCD and HP BCD), did not accurately correspond to the actual xation amount. 1:0.3 to 1:1 mole ratio at this higher curing temperature, the amount of BCD xation was in fact reduced (Table II). From this data, even though a greater amount of BTCA was used, increased curing temperature and catalyst concentration may mean greater esteri cation, it may not mean increased xation of BCD. e amount of xation was greatest for the BCD, followed by the other two derivatives of β-cyclodextrin (Fig. 3b and Table I). For HP BCD, although the W% measurements showed decreased W% values with increased concentration of HP BCD applied, the measurement with phenolphthalein showed a linear increase in HP BCD xed on cotton. e reason for the lower xation of HP BCD as compared to BCD on cotton under the same given xation conditions could be due to the long chain of the substituted 2-hydroxypropyl groups of HP BCD in place of hydroxyl groups in the anhydroglucopyranose units, [START_REF] Roquette | Technical Sheet, Kleptose Betacyclodextrins and Hydroxypropyl Betacyclodextrins[END_REF] which are less accessible for the required reaction. ese substituted groups allow increased solubility of HP BCD in aqueous solution (solubility at RT was 100 g/L). However, these groups may not guarantee increased xation to cotton since xation is determined by the number of free -OH groups on the C6 position of the glucopyranose unit of β-cyclodextrin molecule available for esteri cation. 26 An instability of the phenolphthalein measurements was noted as predicted during the quanti cation of BCD xed on the fabrics in the presence and absence of BTCA. e results of the homogenous tests done with phenolphthalein and BCD in the presence and absence of BTCA can be seen in Fig. 4, where absorbance A of phenolphthalein is shown against concentration of BCD (g/L) in the presence and absence of BTCA. In the presence of BTCA, the absorbance (A) values of the phenolphthalein solutions increased slightly, however, the readings stabilized with increased BCD concentration. e slight phenolphthalein measurement instability noted above was therefore ignored since this instability appeared to resolve when the amount of BCD xed on the fabrics increased. this peak intensity. erefore when BCD was added to the system, the relative degree of ester formation was reduced as compared to when only BTCA was present.

Anhydride intensities at 1782 cm -1 and 1852 cm -1 showed that with increasing BCD application concentration, these intensities were reduced when 30 g/L of BTCA was used, with a mixed trend when 60 g/L of BTCA was used (Table IV). If anhydride intensities had increased while ester intensities decreased, it could have been said that lower esteri cation was due to lower mobility of the anhydrides intermediate formed to access the cellulosic hydroxyl groups. is is known to happen if the molecular weight of a polycarboxylic acid is quite large. 28 Since this was not the case, this e ect occurred unrelated to formation of the anhydride intermediate.

Reactivity of carboxylic acids and alcohols towards esteri cation are limited by steric hindrances. 29 Steric hindrances in BCD can have an in uence on cellulosic surface gra ing. 30 erefore, steric hindrances could have led to the reduction in ester formation between BTCA, BCD, and cellulose when compared to ester formation with only BTCA and cellulose, resulting in less BCD crosslinking.

e degree of esteri cation increased with BTCA amount, therefore in general, the ester carbonyl band intensity ratio increased with increased BTCA concentration. 23,24 e ester linkages were homogenously spread within the fabric, while the carboxylates had higher concentrations on the outer surface rather than in the fabric interior. 22 erefore, FTIR-

FTIR-ATR

FTIR-ATR spectroscopic analysis was performed to determine if BTCA esteri cation of cotton fabric was in uenced by the addition of BCD. For esteri ed cotton, the 1720 cm -1 peak represents the overlap of the carbonyls (C=O) from the acid and the ester of BTCA. Treatment with NaOH converts the acid of BTCA on the fabric to carboxylate anion resulting in a new increased peak at 1570 cm -1 . A er complete conversion of acid to carboxylate anion with NaOH, intensities of the 1720 cm -1 peak re ect the total ester on the fabric. e esteri cation degree of the fabric was calculated by dividing the two mentioned intensities. e resulting number is the average number of ester groups formed per BTCA molecule. is result indicates the e ectiveness of BTCA as a crosslinking agent or degree of esteri cation. For all FTIR-ATR data, the spectra was normalized for the 1313 cm -1 peak. Present experiments only dealt with data from the 1720 cm -1 (ester intensity) peak.

Figs. 5a andb shows the FTIR-ATR spectra of cotton fabrics treated with di erent concentrations of BCD and with 60 g/L and 30 g/L of BTCA. e di erences in the ester peak intensity can be seen among the various fabrics. is could not be observed in the FTIR-ATR spectra of cotton treated with 10 g/L BTCA (spectra not shown here). Fig. 5c shows the ester carbonyl band (1720 cm -1 ) intensity normalized at 1313 cm -1 of cotton fabrics treated with BTCA and BCD under various curing conditions. e ester band intensity represents the total amount of ester in the fabricester groups between cellulose and single-bonded acid molecules and ester groups crosslinking the cellulose and BCD molecules. Crosslinking occured quite randomly on cotton in the presence of BTCA due to the presence of the di erent mono-or di-esters of BTCA. With BCD included, additional permutations and combinations, such as BCD-BTCA, BCD-BTCA-BCD, cotton-BTCA, cotton-BTCAcotton, cotton-BTCA-BCD, BCD-BTCA-cotton-BTCA-BCD, cotton-BTCA-BCD-BTCA-cotton, were expected. [START_REF] Medronho | [END_REF]27 e rst two combinations listed were washed away during the rinsing step. e ester intensity with increased BCD application concentration with 30 g/L and 60 g/L of BTCA is given in Table III. e ester intensity was slightly more when 60 g/L was used as compared to 30 g/L BTCA use. A trend of reduced ester intensity with increased BCD application concentration was noted. e total amount of ester formed on the fabric increased with increased curing temperature or increased BTCA to SHPI ratio as expected (Fig. 5c). However, the addition of BCD in each case seemed to reduce ATR spectroscopy for calculation of the ester carbonyl band intensity ratio was not used since FTIR-ATR spectroscopy is a near surface analysis technique. Use of the carboxylate intensity for degree of esteri cation calculations would lead to erroneous results.

Di erential Scanning Calorimetry (DSC)

According to mechanism put forth by Wu, 31 the increased amount of ester on cotton can lead to ame resistanceesteri cation of the C-6 hydroxyl of cotton can prevent the formation of levoglucosan by reducing cotton depolymerization. Tetracarboxylic acid esteri cation leads to increased thermal activation energy and thermal stability, and greater char content (due to the increased carboncarbon network).

DSC spectra can show temperature and heat release changes in materials during pyrolysis. In the current work, the untreated cotton sample showed an endothermic peak around 370 °C due to char formation. For ame retardant treated cotton, thermal decomposition temperatures measured by DSC were lowered by a signi cant degree indicating the in uence of ame retardants. 32 e heat ow curve of all treated samples showed a small endothermic peak shi (Figs. 6a andb). However, no signi cant decomposition temperature di erences were noted between the DSC curves of cotton samples treated with BTCA and with BCD when compared with native cotton. A slight, unrepeatable shi was noted when curing conditions were increased in some cases. Additional tests using a microscale combustion calorimeter (not shown here) showed no signi cant di erences between the various treatments on cotton.

Laundering Tests

Rigourous laundering tests were performed showing that the amount of β-cyclodextrin xed on the cotton reduced signi cantly with each wash cycle. Laundering conditions were taken from the list prescribed in the ISO 105C06 standard. MCT BCD was entirely removed a er only one wash, while BCD and HP BCD lasted for a few washes (Fig. 7). Lower washfastness of ester linkages can be expected in the presence of BCD and HP BCD in comparison with crosslinking of only BTCA with cotton. With only BTCA attached, all the ester bonds (one to three formed) crosslink only with cotton, while attached BCD or HP BCD have at least one ester linkage remote from the cotton portion of the molecule. Apart from this, the addition of BCD into the system shows that the ester formation is also lowered (seen by the reduced ester intensity) in the system as compared to crosslinking with only BTCA, probably due to steric factors. ese two factors result in the lower washfastness of the β-cyclodextrins xed with BTCA to cotton. However, the reason for low wash durability of MCT BCD was not clear since the monochlorotriazinyl groups were expected to be covalently bound to cotton. e phenolphthalein method of quanti cation may lead to under estimation of β-cyclodextrins xed onto cotton. is would be due to steric hindrance of surface β-cyclodextrin molecules preventing phenolphthalein molecules from entering β-cyclodextrins xed inside the fabric. 30 is direction of research needs further investigation. Some preliminary results with washing at lower temperatures and without detergent show better washfastness than observed here.

Conclusions

e results of this investigation show that β-cyclodextrin (BCD) and two β-cyclodextrin derivatives (HP BCD and MCT BCD) were successfully attached to cotton. Attachment of BCD with BTCA on cotton showed a greater amount of xation than when MCT BCD and HP BCD were used. Varying the amount of the BTCA crosslinker did not have a signi cant in uence on the amount of xation of β-cyclodextrins on cotton under the curing conditions used. FTIR-ATR spectroscopic measurements showed that the total amount of ester formed on the fabric was reduced by addition of BCD in comparison to the only BTCAtreated cotton. DSC measurements showed no signi cant di erences in the thermal decomposition peak temperatures due to BTCA treatment on cotton as compared to nontreated cotton, however all treated fabrics showed a small endothermic shi in the heat ow curve.

Laundering tests showed that, under severe washing conditions, the amount of β-cyclodextrins on cotton was drastically reduced with each wash cycle indicating poor washfastness. Despite poor wash durability, this technology can be used in applications where washing is infrequent, such as in wound dressings, where a BCD-treated textile could host antibacterial molecules for slow and controlled release.

Fig. 2 .

 2 Fig. 2. a) W% values of cotton treated with varying concentrations of BCD and 10, 30, and 60 g/L BTCA. b) W% values of cotton treated with varying concentrations and types of β-cyclodextrins. BCD and HP BCD were xed to cotton with 30 g/L BTCA, while MCT BCD was attached directly to cotton. For both a) and b), all values were an average of three measurements per value.

Fig. 3 .

 3 Fig. 3. a) Amount of BCD xed on cotton with varying applied concentrations of BCD and 10, 30, and 60 g/L of BTCA. b) Amount of β-cyclodextrins xed on cotton treated with varying concentrations and types of β-cyclodextrins. BCD and HP BCD were xed with 30 g/L of BTCA, while MCT BCD was attached directly. For both a) and b), all values are an average of three measurements per value.

Fig. 4 .

 4 Fig. 4. In uence of BTCA on phenolphthalein in homogenous solution containing BCD. All values are an average of three measurements per value.

Fig. 5 .

 5 Fig. 5. a) FTIR-ATR spectra of cotton samples treated with 60 g/L BTCA; (a) untreated cotton and cotton treated with (b) 0, (c) 40, (d) 60, (e) 80, and (f) 100 g/L BCD respectively. All samples pretreated with 0.1 M NaOH prior to measurement. b) FTIR-ATR spectra of cotton samples treated with 30 g/L BTCA and BCD. e legend shows the BCD application concentration used. c) Normalized ester peak intensity with change in curing temperature and BTCA:SHPI mole ratio shown in brackets. BCD applied was 100 g/L, and 30 g/L and 60 g/L of BTCA were used. All readings were an average of three samples. a)

Fig. 6 .Fig. 7 .

 67 Fig. 6. a) DSC gures of treated and untreated cotton. b) DSC gures of cotton treated with and without BCD at various curing conditions. Legends for a) and b) include BTCA concentration (g/L), BCD concentration (g/L), BTCA: SHPI mole ratio, and curing temperature (°C).a)

Table II . Fixation F of BCD (100 g/L) at Di erent Curing Conditions and BTCA Concentrations

 II

	10 g/L BTCA
	30 g/L BTCA
	60 g/L BTCA
	Treatment Conditions		F (mg/g)	SD
	BTCA	Curing	BTCA:SHPI
	Concentration	Temperature	Molar Ratio
	(g/L)	(°C)
	60	160	1:0.3	7.82	2.05
	60	180	1:0.3	7.45	0.29
	60	180	1:1	5.95	0.57
	30	160	1:0.3	8.83	0.36
	30	180	1:0.3	9.10	0.81
	30	180	1:1	8.91	0.58
	BCD + 30 g/L BTCA
	+ 30 g/L BTCA

Table III . FTIR Ester Peak Intensity with Di erent Concentrations of BTCA and BCD

 III

	BTCA	BCD Application	Ester Peak	SD
	Concentration	Concentration	Intensity (cm -1 )
	(g/L)	(g/L)
	30	0	0.372	0.029
	30	40	0.346	0.021
	30	60	0.334	0.022
	30	80	0.296	0.016
	30	100	0.291	0.013
	60	0	0.390	0.007
	60	40	0.393	0.058
	60	60	0.357	0.023
	60	80	0.351	0.057
	60	100	0.343	0.007

Table IV . FTIR Anhydride Intensities with Di erent Concentration of BTCA and BCD

 IV

	BTCA	BCD	Anhydride	SD	Anhydride	SD
	Concentration	Application	Peak		Peak
	(g/L)	Concentration	Intensity		Intensity
		(g/L)	(1782 cm -1 )		(1852 cm -1 )
	30	0	0.341	0.018	0.346	0.022
	30	40	0.337	0.016	0.339	0.015
	30	60	0.323	0.018	0.332	0.023
	30	80	0.300	0.017	0.304	0.020
	30	100	0.295	0.022	0.296	0.023
	60	0	0.302	0.024	0.315	0.020
	60	40	0.334	0.016	0.341	0.021
	60	60	0.300	0.038	0.305	0.028
	60	80	0.311	0.041	0.315	0.038
	60	100	0.309	0.012	0.311	0.013

Acknowledgements e authors would like to thank Clemens Padberg (MTP, University of Twente), Bert Vos (Production Technology, University of Twente), and Jacob S. Lopulissa, Salina Sarwari, and Ernest Cichomski (ETE group, University of Twente).

Author


Marco Pizzolato
email: marco.pizzolato@inria.fr

Aurobrata Ghosh
email: aurobrata.ghosh@gmail.com

Timothé Boutelier
email: timothe.boutelier@olea-medical.com

Rachid Deriche
email: rachid.deriche@inria.fr

Magnitude and complex based diffusion signal reconstruction

come

Introduction

Diffusion Weighted magnetic resonance imaging (DW-MRI) measures the signal attenuation due to the loss of spin phase coherence caused by particles subject to Brownian motion. DW-MRI thus is inherently a low Signal to Noise Ratio (SNR) technique. Indeed, increasing the diffusion-weighting measured by the b-value or decreasing the voxel size can further reduce the SNR, causing the signal to be close to the background noise level [START_REF] Jones | Squashing peanuts and smashing pumpkins': How noise distorts diffusion weighted MR data[END_REF]. This is particularly problematic when characterizing the non-Gaussianity of the diffusion signal profile, associated to restricted diffusion, since a high b-value is generally required [START_REF] Basser | Relationships between diffusion tensor and q-space MRI[END_REF]. Indeed a nonlinear dependence between the log-transformed DW intensity and the diffusion-weighting (b-value), as the sole result of noise, has been previously reported [START_REF] Jones | Squashing peanuts and smashing pumpkins': How noise distorts diffusion weighted MR data[END_REF]. This can lead to misinterpretations of the underlying diffusion process which, for instance, can be erroneously thought as restricted, thus ascribing to the tissue more structural complexity than exists. However, one of the major sources of error in magnetic resonance imaging (MRI), and even more accentuated in DW-MRI due to the inherent low SNR, is the noise.

In MRI the signal, acquired for each coil in quadrature, is complex with an additive thermal noise that can be considered to be derived from a bivariate normal distribution N(0, σ ) [START_REF] Henkelman | Measurement of signal intensities in the presence of noise in MR images[END_REF]. However, when the magnitude of the complex signal is computed, the noise becomes Rician distributed [START_REF] Gudbjartsson | The Rician distribution of noisy MRI data[END_REF] and a not-negligible noise floor, the minimum signal measurable, appears. The bias introduced by the Rician distribution leads to the distortion of estimated quantitative diffusion parameters. For instance, the noise floor causes the DW signal to be overestimated, leading to underestimation of the Apparent Diffusion Coefficient (ADC) [START_REF] Dietrich | Noise correction for the exact determination of apparent diffusion coefficients at low SNR[END_REF]. Other noise-related issues have also been reported, such as orientationally dependent deviation from Gaussianity of the ADC profile, underestimation of diffusion anisotropy indices and correlation between mean diffusivity and diffusion anisotropy [START_REF] Jones | Squashing peanuts and smashing pumpkins': How noise distorts diffusion weighted MR data[END_REF]. Hence, denoising is essential in DW-MRI.

To properly denoise magnitude signals, the noise distribution should be taken into account. However, the distribution of the noise affecting magnitude DW images (DWIs) changes depending on the number of coils used for the acquisition and on the employed reconstruction method. For instance, when magnitude images are obtained via the sum-of-squares reconstruction from multiple coils, the noise follows a non-central χ distribution [START_REF] Constantinides | Signal-to-noise measurements in magnitude images from NMR phased arrays[END_REF]. Nevertheless, the complex diffusion signal is still affected by noise with a Gaussian distribution, which can be exploited via any Gaussian-based denoising technique or fitting procedure.

In this paper we analyze the theoretical performance gain given by considering the complex signal instead of just using the magnitude. The complex signal is synthetically generated, according to [START_REF] Özarslan | Anisotropy induced by macroscopic boundaries: surfacenormal mapping using diffusion-weighted imaging[END_REF], by considering a voxel located close to a boundary, where the underlying diffusion process is restricted. In fact, by exploiting the asymmetry, due to the presence of the boundary, of the displacement density probability of the water molecules, also known as Ensemble Average Propagator (EAP) [START_REF] Callaghan | High-resolution q-space imaging in porous structures[END_REF][START_REF] Stejskal | Use of Spin Echoes in a Pulsed Magnetic-Field Gradient to Study Anisotropic Restricted Diffusion and Flow[END_REF], it is possible to obtain a complex valued signal. The complex signal is generated, along one gradient direction, for a voxel located in the proximity of a single infinite plate. The noisy complex and magnitude signals are then obtained and the reconstruction of the magnitude diffusion signal is performed from each of them. The reconstructed signal, as the linear combination of basis functions, leads to an effective characterization of the diffusion properties and is a useful tool for measuring noise-related performances. In this respect we employed the Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE) [START_REF] Özarslan | Simple harmonic oscillator based estimation and reconstruction for one-dimensional q-space MR[END_REF]. Within this framework, a Maximum Likelihood Estimator (MLE), for the reconstruction based on the Rician magnitude signal, is also provided for performance comparison.

Methods

In this section we present the theoretical framework. First, we present the diffusion signal equation for a voxel in a position close to a single infinite plate. Then we recapitulate the SHORE formulation for the signal fitting. Finally we describe the integration of the MLE in the SHORE.

Signal in the proximity of a single infinite plate

When considering a voxel located in the proximity of a restricting boundary such as an infinite plate, with voxel's dimensions significantly smaller than the separation distance between the plate and any other boundary, the magnetization in the voxel is influenced only by the boundary in the vicinity [START_REF] Özarslan | Anisotropy induced by macroscopic boundaries: surfacenormal mapping using diffusion-weighted imaging[END_REF]. A graphical representation of this scenario is shown in Fig. 1a, where an infinite plate is represented with its normal aligned with the z-axis, and a voxel with height z 2z 1 is located at a distance z 1 from the plate itself. Using the notation in [START_REF] Özarslan | Anisotropy induced by macroscopic boundaries: surfacenormal mapping using diffusion-weighted imaging[END_REF] it is convenient to define the dimensionless position variable

ζ = z u ( 1
)
where u is the characteristic diffusion length given by

u = 4D 0 ∆ (2)
with D 0 being the free diffusion coefficient and ∆ the diffusion time. Fig. 1a also shows the gradient vector which, in this case, is assumed to be aligned with the positive z-axis. The gradient vector magnitude is then given by q = γδ G/2π where γ is the gyromagnetic ratio, δ is the diffusion pulse duration and G is the diffusion gradient strength. The considered experiment is the pulsed gradient spin echo (PGSE) sequence with pulse duration δ small compared to ∆ . Finally, after defining the dimensionless wave-number κ = πqu, the complex signal is given by [START_REF] Özarslan | Anisotropy induced by macroscopic boundaries: surfacenormal mapping using diffusion-weighted imaging[END_REF] Fig. 1 (a) An infinite plate lies along the x-axis, with normal along the z-axis. A voxel is located at a distance z 1 from the infinite plate and has a height given by z 2z 1 . The gradient vector q makes a right angle with the x-axis toward the positive z-axis; adapted from [START_REF] Özarslan | Anisotropy induced by macroscopic boundaries: surfacenormal mapping using diffusion-weighted imaging[END_REF]. (b) Magnitude, real and imaginary parts of the signal generated according to Eq. 3 for a voxel with z 1 = 0 and dimensionless height of 0.25; δ = 3 ms, ∆ = 100 ms, D 0 = 2.299 × 10 -9 m 2 /s and G max = 350 mT /m.

E([ζ 1 , ζ 2 ], κ) = e -κ 2 F(ζ 2 ) -F(ζ 1 ) 2(ζ 2 -ζ 1 ) (3)
where ζ 1 and ζ 2 are the dimensionless coordinates of the voxel corresponding to z 1 and z 2 via Eq. 1, and

F(ζ ) is F(ζ ) =ζ + 1 √ π e -(ζ -iκ) 2 + e i4κζ i4κ [1 -erf(ζ + iκ)] -ζ -i κ + 1 4κ erf(-ζ + iκ). (4)

Signal reconstruction

The signal fitting is performed with SHORE [START_REF] Özarslan | Simple harmonic oscillator based estimation and reconstruction for one-dimensional q-space MR[END_REF], a promising signal reconstruction method suitable for q-space magnetic resonance. Within this framework the signal is represented as the linear combination of orthogonal basis functions, result of the multiplication between an exponential and an Hermite polynomial

φ n (u, q) = i -n u √ π 2 n-2 n! e -2π 2 q 2 u 2 H n (2πqu) (5)
where n is the order of the basis, H n (x) is the nth-order Hermite polynomial and u is the characteristic data dependent length or scaling factor to be determined, for instance by fitting the signal to an exponential decay according to E(q) = exp(-2π 2 q 2 u 2 ). Our formulation of the basis functions in Eq. 5 differs from the one given in [START_REF] Özarslan | Simple harmonic oscillator based estimation and reconstruction for one-dimensional q-space MR[END_REF] with the introduction of the normalizing factor 2 u √ π, which renders the bases orthonormal. The bases are well suited for representing the signal in the complex domain: the even order basis functions are real valued and evenly symmetric whereas the odd order basis functions are imaginary and show odd symmetry, which is precisely the case of the real and the imaginary parts of the diffusion signal. The normalized diffusion signal in the SHORE representation can thus be expressed by

E(q) = N-1 ∑ n=0 a n φ n (u, q) ( 6
)
where N is the maximum allowed order in the reconstruction and a n are the coefficients corresponding to the respective bases. The choice of N directly affects the signal reconstruction: in the case of noisy data, a high order will potentially cause the reconstruction to follow the noise, whereas a low order will inherently enforce a smoothing effect. For a given order N the signal reconstruction in the complex domain is performed by considering the even and odd coefficients for the real and imaginary parts respectively. In the case of the magnitude signal reconstruction, only the even coefficients are taken into account. In any case, a fitting procedure with a Linear Least Squares approach is used to estimate the coefficients.

Maximum likelihood estimation

To better take into account the Rician distribution of the noise affecting the diffusion magnitude signal, the estimation of the coefficients can be performed with the Maximum Likelihood Estimator (MLE). In the case of Rician noise, the MLE has been introduced in [START_REF] Sijbers | Maximum-likelihood estimation of Rician distribution parameters[END_REF]. Normally several noisy realizations of each signal sample are required to properly estimate via the MLE. However in the case of one noisy realization per sample, the signal samples can be interpreted as noisy realizations of a function, thus the likelihood is given by

L = n ∏ i=1 M i σ 2 e - M 2 i +[Ac] 2 i 2σ 2 I 0 [Ac] i M i σ 2 (7)
where n is the number of samples of the magnitude signal, M i is the i-th sample, σ 2 is the noise variance, A is the real SHORE design matrix, c is the even coefficients vector and I 0 is the modified zeroth order Bessel function of the first kind. The MLE is then defined as the estimator maximizing L or equivalently log

L ĉML = argmax c (log L) . ( 8
)
The performance of the estimator is expected to increase with the increasing number of samples of the signal. It should then be noticed that the MLE requires σ 2 to be given. Thus a prior estimation of the signal noise variance is necessary.

Experimental results

This section describes the parameters adopted for the generation of the complex signal (see Sect. 2.1) and how from this the noisy complex and magnitude signals are obtained. Then, the fitting procedures for the magnitude signal reconstruction based on the SHORE, in the complex domain and for the magnitude (with both LLS and MLE), are discussed. Finally we present the way the performances of the magnitude signal reconstructions are compared. We generated the complex diffusion signal, according to Eq. 3, for a voxel adjacent to an infinite plate and with dimensionless height of 0.25 (Fig. 1b). A total of n equally spaced samples of the signal, with n ∈ {1, 2, . . . , 100}, is generated along the direction normal to the infinite plate (Fig. 1a). The sequence parameters are fixed to G max = 350 mT /m, δ = 3 ms, ∆ = 100 ms and the physical quantities to γ = 2.675 × 10 8 rad/sT and D 0 = 2.299 × 10 -9 m 2 /s. Uncorrelated Gaussian noise with equal variance is added on the real and imaginary parts of the signal, obtaining the noisy complex signal. From this the magnitude is computed, obtaining the noisy magnitude signal. The standard deviation of the noise is calculated for several different SNR values by σ = SNR -1 relatively to the non-weighted signal sample (G = 0).

The noisy complex and magnitude signals are then fitted via SHORE according to Eq. 6. The even (real) and odd (imaginary) SHORE bases are used for the complex signal fitting whereas only the even bases are used in the case of the magnitude signal fitting. Moreover the magnitude signal is fitted also with the MLE, obtaining the vector of the SHORE coefficients according to Eq. 8. To observe the influence of the maximum allowed order in the SHORE reconstruction N (see Sect. 2.2 and Eq. 6), two orders are tested: one relatively low order N = 6 and one relatively high order N = 10. However the following discussion is referred to N = 6 and a comparison with N = 10 will be given later in Sect. [START_REF] Dietrich | Noise correction for the exact determination of apparent diffusion coefficients at low SNR[END_REF]. In order to render the fitting of the complex and magnitude signals independent from the estimation of the scaling factor u, in every case the u estimated on the noisy magnitude is also used in the complex fitting.

After the complex based fitting a real and an imaginary vectors of coefficients are obtained, leading to a real and an imaginary reconstruction respectively. These reconstructions are then used to compute the magnitude reconstruction from the complex fitting (M C ), which is then compared to the one reconstructed from the noisy magnitude signal via both LLS (M M ) and MLE (M M ML ) based fittings.

We performed the comparison by calculating the Root Sum of Squares (RSS) value of the residuals between the magnitude reconstructions (M C , M M ,M M ML ) and the ground truth M GT = |E(q)|. The whole procedure is performed with 1000 different noise realizations for each couple of SNR and number of samples n (sampling resolution). Hence a triple of averaged RSS values, RSS C ± SD RSS C , RSS M ± SD RSS M and RSS M ML ± SD RSS M ML (RSS ML , SD RSS ML from now on and in the figures) is obtained for each pair (SNR, n). For each value of n, the RSS values and their standard deviations can be represented as functions of the SNR, obtaining the curves {RSS} and {SD RSS } as shown in Fig. 2a and 2b respectively (n = 31). We also compared the reconstruction performances as function of the signal sampling resolution, i.e. as function of the number of samples n. Fig. 3 shows the integral of the difference between the RSS (light blue) and SD RSS (magenta) curves of different techniques, as function of the number of samples n. Precisely, it shows the integral values of {RSS M } -{RSS C } and {SD RSS M } -SD RSS C (Fig. 3a), {RSS ML } -{RSS C } and {SD RSS ML } -SD RSS C (Fig. 3b), and finally {RSS M } -{RSS ML } and {SD RSS M } -{SD RSS ML } (Fig. 3c). Thus a positive value globally indicates a better reconstruction (less overall reconstruction error) for M C compared to M M (Fig. 3a), M C compared to M M ML (Fig. 3b) and M M ML compared to M M (Fig. 3c). It should be noticed that Fig. 3c substantially describes the difference between Fig. 3a and3b.

Figure 4 shows the inter-technique comparisons of the RSS values (a,b,c) and their standard deviations (d,e,f ), as function of both the SNR and the sampling resolution (i.e. n). In detail it shows the differences RSS M -RSS C (Fig. 4a) and SD RSS M -SD RSS C (Fig. 4d), RSS ML -RSS C (Fig. 4b) and SD RSS ML -SD RSS C (Fig. 4e), and finally RSS M -RSS ML (Fig. 4c) and SD RSS M -SD RSS ML (Fig. 4f ). Thus a positive value (red) indicates a better reconstruction M C compared to M M (Fig. 4a,d), M C compared to M M ML (Fig. 4b,e) and M M ML compared to M M (Fig. 4c,f ). It should be noticed that the range of values in the color bars are different. For instance the amplitude of the range of the performance gain for the complex reconstruction, compared to the magnitude based one, is higher when the comparison is made with respect to the LLS fitting (Fig. 4a) than when the MLE is considered (Fig. 4b). The opposite holds for the performance loss. On the contrary we observe that the range of the performance gain/loss in the standard deviation is higher when the comparison refers to the MLE (Fig. 4e) rather than the LLS (Fig. 4d). Finally Fig. 5 shows the map of the best reconstruction technique for each pair of SNR and n, that is the technique giving less reconstruction error (RSS value). In order to show the influence of the maximum reconstruction order of SHORE N on the results, maps are generated for N = 6 (Fig. 5a) and N = 10 (Fig. 5b).

Fig. 3 In light blue the differences between the areas under the RSS curves of two different compared techniques (i.e. the curves in Fig. 2a); in magenta the differences between the areas under the SD RSS curves (i.e. the curves in Fig. 2b We performed the reconstructions also for a signal generated with a set of parameters achievable in experimental conditions. In detail the voxel size is set to 50 µm, G max = 60 mT /m, δ = 15 ms and ∆ = 50 ms. Results are shown in Fig. 6 for N = 6. More precisely it shows the comparison between the RSS value of the complex based reconstruction and that of the magnitude based reconstruction for a LLS approach Fig. 6a and for a MLE apporach Fig. 6b. The range of SNR values [START_REF] Callaghan | High-resolution q-space imaging in porous structures[END_REF]20] in the figures is set to that showing major differences (values different from zero) in the results and corresponds to that generally adopted [START_REF] Jones | Squashing peanuts and smashing pumpkins': How noise distorts diffusion weighted MR data[END_REF]. The number of samples n range has been focused to [START_REF] Callaghan | High-resolution q-space imaging in porous structures[END_REF]15], in fact at higher values the trend is similar to that shown in Fig. 5. Figure 6c finally shows the best reconstruction technique for SNR values up to 50 and is representative also for higher SNR up to 100.

Discussion of the results

The noisy complex and magnitude signals have been fitted with the SHORE obtaining the complex M C and the magnitude based M M reconstructions respectively. The noisy magnitude has also been reconstructed by using the MLE, M M ML . In each of the three cases M C , M M ML and M M the reconstruction error has been calculated and compared as function of the SNR and the number of samples n.

The magnitude reconstruction performed by fitting the complex domain signal, M C , globally shows a better performance (less reconstruction error) compared to the one based on the LLS fitting on the magnitude, M M , for n > 5 (Fig. 3a). The performance gain for the complex reconstruction is particularly evident for SNR ≤ 5 (Fig. 4a), as expected. Indeed at higher SNR values the Rician distribution is well approximated by a Gaussian [START_REF] Pajevic | Parametric and non-parametric statistical analysis of DT-MRI data[END_REF][START_REF] Salvador | Formal characterization and extension of the linearized diffusion tensor model[END_REF], thus the performances of magnitude and complex based reconstructions should be equivalent. However an opposite trend is registered in case of n = 4, 5.

Moreover a performance dependence with the sampling resolution is observed. In first place, by increasing the sampling resolution (n) the complex reconstruction M C shows less error compared to the magnitude based one M M also at SNR > 5; with this regard the range of SNR values in which the performance gain is observed also increases (Fig. 4a). Secondly, the amount of error reduction in the case of M C with respect to M M follows the same trend.

Similar observations apply when the complex based reconstruction M C is compared to that obtained via MLE based fitting of the magnitude M M ML , but in this case for n > 11 (Fig. 3b). Indeed the MLE for the magnitude based fitting, slightly improves the overall performance with respect to LLS at any sampling resolution for low SNR values (Fig. 3c). However it should be noticed that the MLE requires an estimation of the noise variance, and in the present case the exact value was supplied. The reconstruction improvement given by the MLE over the LLS also happens for SNR < 5 (Fig. 4c), as clearly represented in the example of Fig. 2a. In addition the amplitude of the error reduction obtained with MLE over LLS increases with the sampling resolution (n), whereas the range of SNR within which a performance gain is observed reduces (Fig. 4c).

Moreover, a performance loss for the MLE on the magnitude when compared to the LLS is observed within a region with n ≤ 6 and SNR ∈ [START_REF] Henkelman | Measurement of signal intensities in the presence of noise in MR images[END_REF][START_REF] Stejskal | Use of Spin Echoes in a Pulsed Magnetic-Field Gradient to Study Anisotropic Restricted Diffusion and Flow[END_REF] at different grades (Fig. 4c). However the limits of this region seem to be dependent on the maximum reconstruction order N allowed in the SHORE, as shown in Fig. 5a for N = 6 and 5b for N = 10. In fact when N = 10 the range of the SNR values within which the performance gain of the MLE over the LLS is observed reduces. On the other hand the range of the number of samples n within which M M ML shows less error than M C , at very low SNR, slightly increases. Otherwise, the region in which the complex based reconstruction is the most performing do not seems to be affected by the choice of N.

Finally the complex based fitting leads to a reconstruction with an error standard deviation SD RSS C lower than SD RSS M for n > 6 (Fig. 3a and4d). Similar observations apply when SD RSS C is compared to SD RSS ML for n > 8 (Fig. 3b), although at low SNR the MLE shows a lower standard deviation (Fig. 4e and4f ).

Very similar considerations apply when the reconstructions are performed and evaluated in the case of signal generated with different voxel and pulse sequence parameters as shown in Fig. 6. However in this case the region in which the magnitude based reconstruction shows better performances (Fig. 6c) is almost entirely defined by the performance of the MLE (Fig. 6b). Indeed almost everywhere in this region the performance of the LLS is comparable or inferior to that of the complex based reconstruction (Fig. 6a). Finally at higher SNR values the complex based reconstruction is generally the one showing the best performance (Fig. 6c).

Hence, as shown in Fig. 3b, 5a and 6c, at low SNR values it might be better to perform a reconstruction on the noisy magnitude signal when n ≤ 10 with order N = 6 and n ≤ 15 with order N = 10 (Fig. 5b). Within this region the MLE generally shows better performances for SNR ≤ 5. However in the rest of the cases a reconstruction performed in the complex domain generally leads to less reconstruction error.

Conclusions

We have presented a comparative analysis between the diffusion signal reconstruction on the magnitude and in the complex domain. For the magnitude signal reconstruction we relied on the SHORE bases for both the noisy complex and magnitude fittings. In the case of the noisy (Rice distributed) magnitude fitting we introduced the maximum likelihood based (MLE) coefficients estimation for the SHORE, as an alternative to the standard Linear Least Squares based approach (LLS). We compared the performance of the reconstructions as a function of the SNR value and the number of samples in the signal n (sampling resolution).

Our results show that the choice of which reconstruction technique to adopt mainly depends on the SNR and on the number of samples of the signal, as shown in Fig. 5a,b and Fig. 6c. Depending on these parameters, the best magnitude signal reconstruction can be achieved for any of the compared techniques. However the complex based reconstruction reveals to be the most performing at any SNR with the increasing total number of samples of the diffusion signal. To further improve its performance, regularization constraints could also be implemented. This would be beneficial for the estimation of the diffusion parameters.

Despite the difficulty that may be encountered in clinical conditions in achieving an acquisition setup such as the one here adopted, the presented results might be directly useful and potentially validated in experimental setups.

Fig. 2

 2 Fig. 2 Root sum of squares values RSS C , RSS M and RSS ML (a), and standard deviations SD RSS C , SD RSS M and SD RSS ML (b), as functions of the SNR. Each point represents the averaged value over 1000 noise realizations with the corresponding SNR. G max = 350 mT /m, with n = 31.

  Fig.3In light blue the differences between the areas under the RSS curves of two different compared techniques (i.e. the curves in Fig.2a); in magenta the differences between the areas under the SD RSS curves (i.e. the curves in Fig.2b). Values are shown as a function of the number of samples n. A positive value indicates a globally better reconstruction (less overall reconstruction error) for M C over M M (a), M C over M M ML (b) and M M ML over M M (c). G max = 350 mT /m.

Fig. 4

 4 Fig. 4 Differences between RSS values (a,b,c) and correspondent SD RSS (d,e,f ) for different compared techniques, as function of the SNR and of the number of samples n. A positive value (red) indicates a better reconstruction for M C compared to M M (a), M C compared to M M ML (b) and M M ML compared to M M (c), or less error variance (d,e,f ). Each pixel represents the average value over 1000 noise realizations for the corresponding SNR and number of samples n. G max = 350 mT /m.

Fig. 5

 5 Fig. 5 The best reconstruction technique as a function of the SNR and the number of samples n, for a maximum reconstruction order N = 6 (a) and N = 10 (b), see Sect. 2.2 and Eq. 6 for reference. Red, green and blue colors indicate that the signal reconstruction showing less error is the one performed by fitting the noisy complex signal, M C , the noisy magnitude signal with the Maximum Likelihood Estimator (MLE), M M ML , and with Linear Least Squares (LLS), M M , respectively. A purple color indicates that the best reconstruction is based on the noisy magnitude signal fitting, without any preference regarding the estimation technique.

Fig. 6

 6 Fig. 6 Differences between RSS values for different compared techniques (a,b) and best reconstruction technique as function of n and SNR (c). Results reported for a voxel with side 50 µm and with G max = 60 mT /m, δ = 15 ms, ∆ = 50 ms. A positive value (red) indicates a better reconstruction for M C compared to M M (a) and for M C compared to M M ML (b). (c) The best reconstruction technique is given for each pair (n,SNR). Red, green and blue colors indicate that the signal reconstruction showing less error is the one performed by fitting the noisy complex signal, M C , the noisy magnitude signal with the Maximum Likelihood Estimator (MLE), M M ML , and with Linear Least Squares (LLS), M M , respectively. A purple color indicates that the best reconstruction is based on the noisy magnitude signal fitting, without any preference regarding the estimation technique.

The final publication will be available at http://link.springer.com/book/10.1007/978-3-319-11182-7, c Springer-Verlag

2015.

Acknowledgments

The authors would like to express their thanks to Olea Medical and to the Provence-Alpes-Côte d'Azur (P.A.C.A.) Regional Council for providing grant and support for this work. The authors also thank Demian Wassermann for the careful reading.