Jiayin Gao

Takayuki Arai

Revisiting laryngeal features with a dynamic approach

This paper questions the phonological analyses of laryngeal features at the segmental and suprasegmental levels. We address the limits of a static and synchronic analysis of phonological features. We instead propose a dynamic approach taking into account positional variation and diachronic evolution. Empirical data come from Tokyo Japanese. Acoustic investigations were conducted on a total of 2664 utterances produced by 9 female speakers aged 20 to 27.

[voice]: when phonetic evidence contradicts phonological arguments

In languages with a two-way laryngeal contrast based on VOT as a main cue for plosives, [voice] is employed as a distinctive feature in a classic approach (hereafter Approach A) (Keating, 1984). This embraces at least two types of languages: (1) "aspiration", realized mainly as a long-lag vs. short-lag or negative VOT contrast, such as English or German; (2) "true voicing", realized mainly as a shortlag vs. negative VOT contrast, such as French and Italian. In this approach, the different realizations are viewed as language-specific phonetic implementations of the same phonological [voice] contrast.

Another approach (hereafter Approach B) consists in using different and privative phonological features, [spread glottis] vs. ∅ for (1), and ∅ vs. [voice] for (2) (Iverson & Salmons, 1995). ∅ represents the short-lag plosive series in both types, which is underspecified (or unmarked).

Phonetic and phonological criteria have been put forward to determine whether a language belongs to type (1) or (2). One of the phonetic criteria is passive vs. active voicing (Beckman et al., 2013). If one plosive series is consistently and robustly voiced in all contexts, as in French, the active feature is [voice]. If, for example, one plosive series is usually voiceless in utterance-initial position, but (variably) voiced in utterance-medial position, this voicing is considered as "passive voicing". This is the case in English and German, hence [spread glottis] is proposed to be an active feature.

Japanese has long been analyzed as a type (2) language. The most widely accepted phonological argument is the active use of [voice] in different phonological rules involved in compounding, or Rendaku (Itô and Mester, 1986). Phonetically, it has been assumed by many that Japanese has a shortlag vs. negative VOT contrast (e.g., Iverson & Salmons, 1995).

) cross-generational study on word-initial voiced stops in Japanese, suggesting a trend towards devoicing these stops. Meanwhile, in UI, VOTs overlap between the two series, but remain significantly different. Therefore, in modern Tokyo Japanese, unlike a "true voicing" language, word-initial voiceless plosives are moderately aspirated, and voiced plosives are not robustly voiced in utterance-initial position. Their voicing in word-medial position may be considered as passive voicing. This phonetic evidence contradicts the phonological arguments in favor of a [voice] analysis.

F0 perturbation

Onset-induced F0 perturbation is attested in many languages. It is viewed in Approach A as part of a [voice] feature, including both type (1) and (2) languages, in that F0 at the vowel onset is lowered by /b,d,g/ compared to /p,t,k/ (Kingston & Diehl, 1994). Approach B considers that [spread glottis] raises F0 and [voice] lowers F0 (Iverson & Salmons, 1995), leading to the same relative F0 difference.

, right two panels) plosives have much less effect on the following F0. The position-dependent F0 perturbation pattern in Tokyo Japanese suggests that F0 perturbation is neither a purely automatic effect of the phonetic voicing, (otherwise, it would disappear with a "devoiced" stop), nor phonologically involved in [voice] or [spread glottis] feature, (otherwise, we would expect similar F0 perturbation effect across all positions).

Fig. 1: Normalized F0 curves using z-score transformation after /p,t,k/, prevoiced /b,d,g/, and devoiced /b,d,g/, in UI and WM, in two pitch-accent contexts (L and H), computed by SS-ANOVA.

Proposal: a dynamic approach 3.1. Positional variation

Modern Tokyo Japanese exhibits a differential use of cues to distinguish its two plosive series, depending on the prosodic positions: F0 (with overlapping short-vs. medium-lag VOTs) in word-, especially utterance-initial position, and short-lag vs. negative VOTs in word-medial position. We question the use of [voice] as a phonological feature, which does not account for the utterance/wordinitial realization. We may of course propose a phonological rule which aspirates /p,t,k/ and devoices /b,d,g/ while lowering the following F0 in word-initial position. But, what prevents an analysis in the other way around: voicing of an L syllable in word-medial position? As a comparison, the three plosive series in Seoul Korean are all voiceless in word-initial position. One of them have lower F0 on the following vowel in word-initial position, and are voiced in word-medial position. Contrary to Japanese, in Seoul Korean, [voice] has been long avoided as a phonological feature, while wordmedial voicing is considered as a phonological rule or a phonetic effect.

In Tokyo Japanese, F0 and VOT cues do not have a classic trading relation or perceptual equivalence because their use is first conditioned by position. But at the current stage, the differential use of these two cues mapped in different prosodic positions points to a same phonological category.

Diachronic evolution

While this proposal sounds similar to the [voice/L] feature proposed by Bradshaw (1999), our proposal is more dynamic in terms of diachronic evolution. We believe that multiple cues of one phonological categories are not bound to be tied together forever. The diachronic evolution from [voice] to L has taken place in Vietnamese, Thai and many Chinese languages. In these languages, when the evolution has been completed, L is completely dissociated from [voice].

From a diachronic point of view, in utterance-initial position, the devoicing of voiced plosives, the aspiration of voiceless plosives, and the remarkable F0 difference might suggest that Tokyo Japanese is undergoing one of the following sound changes: (a) Germanic shift: p > p h > f; (b) tonogenesis (if "pitch-accent" is not considered as tone), as proposed for Seoul Korean (e.g., Silva, 2006). During the transitional stage of a sound change, phonological categories are, we believe, characterized by a bundle of cues (Mazaudon, 2012), difficult to sort out using a purely static and synchronic approach.

  Beckman, J., Jessen, M., & Rigen, C. (2013). Empirical evidence for laryngeal features: aspirating vs. true voice languages. J.Linguistics, 49(2), 259-284. || Bradshaw, M. M. (1999). A crosslinguistic study of consonant-tone interaction. Ph.D thesis. The Ohio State Univ. ||Itô, J., & Mester, A. (1986). The phonology of voicing in Japanese: theoretical consequences for morphological accessibility. Linguistic Inquiry, 17(1), 49-73. || Iverson, G., & Salmons, J. (1995). Aspiration and laryngeal representations in Germanic. Phonology, 12(3), 369-396. || Keating, P. A. (1984). Phonetic and phonological representation of stop consonant voicing. Language, 60(2), 286. || Kingston, J., & Diehl, R. L. (1994). Phonetic Knowledge. Language, 70(3), 419. || Mazaudon, M. (2012). Paths to tone in the Tamang branch of Tibeto-Burman (Nepal). In Vogelaer & Seiler (Eds.), The Dialect Laboratory, (pp. 139-175). John Benjamins. || Riney, T. J., Takagi, N., Ota, K., & Uchida, Y. (2007). The intermediate degree of VOT in Japanese initial voiceless stops. J. Phonetics, 35(3), 439-443. || Shimizu, K. (1996). A cross-language study of voicing contrasts of stop consonants in six Asian languages. Tokyo: Seibido. || Silva, D. J. (2006). Acoustic evidence for the emergence of tonal contrast in contemporary Korean. Phonology, 23, 287-308. || Takada, M., Kong, E. J., Yoneyama, K., & Beckman, M. E. (2015). Loss of prevoicing in modern Japanese /g , d , b/. Proc. 18th ICPhS, paper N° 873, 1-5.


Patrice Berthod

Yassin Hamini

Lionel Aranda

Influence of tantalum on the rates of high temperature oxidation and chromia volatilization for cast (Fe and/or Ni)-30Cr-0.4C alloys

LCSM Laboratoire de Chimie du Solide Minéral [UMR 7555]

The equiaxed superalloys obtained by classical foundry process are often reinforced by interdendritic carbides appeared during solidification. It is notably the case of cobalt-base alloys, but this principle of strengthening can also be used for nickel and iron-base alloys. Thus, in addition to chromium, the (well known) role of which is to bring a good resistance against high temperature oxidation to the conventionally cast superalloys, tantalum, which is a very oxidable element too, can also play a role in the high temperature oxidation of (Fe and/or Ni)base alloys. In this work the high temperature oxidation behaviours of two iron-base alloys and two iron-nickel base alloys, containing 0.4 wt.%C and two amounts of tantalum 3 wt.%Ta and 6 wt.%Ta, are studied, with comparison with results obtained previously for ternary M30Cr0.4C alloys and two nickel-base alloys with 0.4%C and 3% to 6%Ta . .s -1 ) and Kv (x 10 Bold characters: excellent fit between the experimental mass gain curv e and the mathematical one Normal characters: good fit between the experimental mass gain curve and the mathematical one

Italic characters: not good fit between the experimental mass gain curve and the mathematical one

When present in Fe-base and (Fe,Ni)-base carbon-containing alloys, tantalum leads to a TaC carbides network which seems to influence the high temperature oxidation behaviour. Their mass gains are higher than for Ta-free alloys, especially for the highest temperatures. Moreover the oxidation of Ta leads to difficulties for determining the parabolic and chromia volatilization constants by using the {m × dm/dt = Kp -Kv × m}-treatment. This is particularly true in the case of iron-base alloys in which the diffusion of Ta towards the oxidation front is the easiest because of its availability in grain boundaries. Thus, an efficient reinforcement by tantalum carbides is probably easier to obtain in Fe-base and (Fe,Ni)-base alloys than in Nibase alloys, but unfortunately with a significant decrease of the resistance against high temperature oxidation of alloys. Tantalum diffuses from the carbides towards the oxidation front to be oxidized in internal oxides. This promotes the development of a carbide-free zone, the depth of which is higher for a higher temperature of oxidation, but also higher for a lower Ta-content in the alloy. Oxidation involves almost the same quantity of tantalum, then promotes, in the Fe-base alloy with 3%Ta, the disappearance of tantalum carbides over a double depth compared to the Fe-base alloy with 6%Ta.

The ratio of carbide-free depths between the (Fe,Ni)-base alloys with 3%Ta and 6%Ta, or between the Ni-base alloys with 3%Ta and 6%Ta, are not the same. A significant part of Ta is also contained in the matrix of these alloys while Ta is almost only present in the form of carbides in Fe-base alloys. The location of tantalum in the alloy seems to have consequences on the importance of internal oxidation, more precisely on the oxidation of tantalum. When this element is in grain boundaries as tantalum carbides (Fe-base alloys: higher equivalent thickness of tantalum oxide and deeper carbide-free zones) its diffusion towards the oxidation front seems easier than when a part of tantalum is in solid solution in the matrix ((Fe,Ni)-base and Ni-base alloys). Oxidation of tantalum leads to higher mass gains at 1,200 and 1,100°C, and to difficulties for the (Kp+Kv) determinations and validations. The fast oxidation of the Fe-base alloys, compared to the (Fe,Ni)-base and Ni-base alloys, can be related both to the distribution of Ta in the alloy (more present in interdendritic carbides and less in matrix) and to the crystallographic structure of the matrix at the test temperatures, i.e. BCC (ferritic) for the Fe-base alloys and FCC (austenitic) for the others. The nature of the major element itself probably plays a role since oxidation is also slightly faster for a (Fe,Ni)-base alloy than for its corresponding Ni-base alloy. The decrease of the parabolic constant of a Fe-base alloy when the Ta content is changed from 3 to 6%, is also different to what it is encountered for the austenitic alloys. There is probably an effect of the BCC-FCC difference and/or of the base element difference here too. The obtained Kv values are in accordance with data from literature.

7th

  International Symposium on High-Temperature Corrosion and Protection of Materials (HTCPM 2008) -May 18-23, 2008 -Les Embiez (France) Values of the average thicknesses of external chromia and of the equivalent (Cr,Ta) oxide assessed by image analysis (in each case average of 3 measures +/-1 std deviation) 0%Ta 3%Ta 6%Ta

  divided by 2 ) (italic characters: the oxidation time for Ni30Cr0.4C3or6Ta was of 100h [3] and not 50 hours) (n. o. r. : no external oxide remained on the sample after cooling) of the FeNi30Cr0.4CyTa (y = 3 and 6) alloys at 1,200 (a), 1,100 (b), 1,000°C (c

  dm/dt) (g2/cm4/s) {m × (dm/dt) = Kp -Kv × m}-type treatment of the curve for FeNi30Cr0.4C6Ta at 1,100°C

1200 21 ± 3 n. o. r. 5.9 ± 0.7 (4.2*) n. o. r. 4.8 ± 2.3 (3.4*) 1100 11 ± 1 13 ± 4 (9.2*) 2.9 ± 0.7 (2.1*) 8.6 ± 1.0 (6.1*) 6.0 ± 0.6 (4.2*) Ni30Cr0.4C 1000 4.6 ± 0.7 13 ± 4 (9.2*) 1.9 ± 0.2 (1.3*) 6.0 ± 1.0 (4.2*) 2.1 ± 0.2 (1.5*)

	Av. thick.	T
	of Cr oxide Equ. thick. of Ta oxide	(°C)	Cr2O3 (µm)	Cr2O3 (µm)	(Cr,Ta) ox. (µ m)	Cr2O3 (µm)	(Cr,Ta) ox. (µ m)
		1200 18 ± 3	32 ± 6	7.4 ± 0.4	16 ± 4	8.0 ± 2.0
	Fe30Cr0.4C	1100 18 ± 3	22 ± 2	5.3 ± 1.0	19 ± 3	7.3 ± 0.8
		1000 6.6 ± 0.7	11 ± 1	0.7 ± 0.2	8.3 ± 1	2.2 ± 0.3
		1200	/	30 ± 2	4.8 ± 0.3	31 ± 2	9.2 ± 1.1
	FeNi30Cr0.4C	1100	/	13 ± 2	1.8 ± 0.3	12 ± 2	4.4 ± 0.9
		1000	/	10 ± 1	0.7 ± 0.3	6.2 ± 2	1.1 ± 0.4
	Average depth of		0%Ta				3%Ta	6%Ta
	the carbide-free zone	Temperature	dcfz (µm)			dcfz (µm)	dcfz (µm)
			1200°C	333 ± 3 3			272 ± 11	91 ± 9
	Fe30Cr0.4C	1100°C	144 ± 1 7			98 ± 2	65 ± 4
			1000°C	41 ± 2				36 ± 6	20 ± 2
			1200°C	/				103 ± 10	107 ± 6
	FeNi30Cr0.4C	1100°C	/				49 ± 7	40 ± 4
			1000°C	/				19 ± 5	11 ± 1
			1200°C	143 ± 2			211 ± 10 (149*)	190 ± 8 (134*)
	Ni30Cr0.4C	1100°C	66 ± 5				79 ± 8 (56*)	76 ± 4 (54*)
			1000°C	26 ± 3				30 ± 2 (21*)	21 ± 4 (15*)
	(*: values deduced from the former multiplicated by	50	h	/	h 100	, i.e.

(*: values deduced from the former multiplicated by h 100 / h 50 , i.e. divided by 2 ) (italic characters: the oxidation time for Ni30Cr0.4C3or6Ta was of 100h [3] and not 50 hours) (n. o. r. : no external oxide remained on the sample after cooling) Values of the average depths of the carbide-free zones (average of 3 measures +/-1 std deviation)


I would like to express my gratitude to the reviewers and committee members who accepted to evaluate this work. I am also truly thankful to all the collaborators who contributed to this research, in particular to the

Introduction

This manuscript presents a review of our contributions to research in diffusion magnetic resonance imaging (MRI) methods over the past decade. It is organized in four chapters presenting original research, and a concluding chapter in which we discuss the perspectives.

In Chapter 1, we present mathematical methods we developed to represent and process the signal in q-space as well as B-tensor encoding. We show that proposing an adapted, continuous signal expansion is important since it may provide a compact representation useful for denoising, inverse problem solving and efficient sampling. In continuation, observing that commonly used metrics are derived from rotation-invariant features, we also proposed a mathematical framework to systematically characterize a family of such invariants.

In Chapter 2, we consider biophysical models, which are nonlinear models relating tissue properties to the diffusion signal. The estimation of model parameters is in general ill-posed; however the accuracy of estimated parameters can be improved with a proper acquisition design. The use of generalized diffusion-encoding gradient waveforms have shown promising results in this context. The optimization of these waveforms is however tedious due to the possibly infinite dimension of the space of admissible gradient waveforms. We developed two family of methods, based on discrete sampling and sparse representation in the one hand, and on an optimization framework that increases the sensitivity to specific microstructure parameters compared to using the pulsed gradients as in the original Stejskal-Tanner sequence in the second hand.

In Chapter 3, we were interested in the reconstruction of brain structural connectivity using tractography. We believe validation is key to a larger endorsement by the community and its translation towards clinical applications. We have organized several competitions gathering international teams to exchange and compare their solutions to some of the many challenges raised by tractography and ultimately the reconstruction of a quantitative, microstructureinformed connectome. As a by-product of these competitions, we have released several open-source softwares and datasets to provide the community with easily accessible tools to validate their methods, using simulated data with a ground truth.

In Chapter 4 we present two clinical applications to which we contributed. The first application is related to diffusion MRI tractography in patients with brain tumour. We proposed estimation 1

Adapted representation and sampling

The diffusion-weighted signal attenuation depends on the encoding gradient and is related to the diffusion properties of water molecules and the microscopic organization of biological tissues within which they diffuse. To exploit information in the signal, different families of mathematical models have been proposed, some of which aim at estimating general properties such as the ensemble average propagator or the diffusion tensor distribution. For these methods, we have shown that proposing an adapted, continuous signal expansion in q-space is important since it may provide a compact representation useful for denoising, inverse problem solving and efficient sampling (Caruyer and Deriche, 2012b;[START_REF] Merlet | Impact of radial and angular sampling on multiple shells acquisition in diffusion MRI[END_REF][START_REF] Caruyer | Design of multishell sampling schemes with uniform coverage in diffusion MRI[END_REF][START_REF] Rutger | MAPL: Tissue Microstructure Estimation Using Laplacian-Regularized MAP-MRI and its Application to HCP Data[END_REF][START_REF] Truffet | Optimal selection of diffusion-weighting gradient waveforms using compressed sensing and dictionary learning[END_REF]. More recently, we have extended these qspace representations to the context of multidimensional diffusion encoding, to study the properties of the signal and give recommendations for sampling the space of axisymmetric B-tensors [START_REF] Bates | A 4D Basis and Sampling Scheme for the Tensor Encoded Multi-Dimensional Diffusion MRI Signal[END_REF]. Last, observing that commonly used metrics are derived from rotation-invariant features, we proposed a mathematical framework to systematically characterize a family of such features [START_REF] Caruyer | On facilitating the use of HARDI in population studies by creating rotationinvariant markers[END_REF], which could be used as building blocks for elaborating new biomarkers.

Introduction

Digital signal processing emerged in the second half of the twentieth century and has found applications in audio, image or video processing to name a few. Each of these applications requires specific adaptations due to the mathematical domain on which the signal is defined. The main objectives of signal processing range from harmonic analysis, filtering to sampling and low dimension (sparse) representations. Similarly, in diffusion MRI different mathematical domains need to be considered, depending on the acquisition strategy and applications. High angular resolution diffusion imaging (HARDI) makes use of signal and fiber orientation distribution function representations on the sphere; the natural dual representation is spherical harmonics. In continuation, we have worked on natural representations for the signal in q-space and B-tensor encoding. We present in this chapter applications to signal regularization and optimal sampling. Last, we show how to take into account the geometry of the space by defining rotation-invariant features in HARDI.

Adapted representations for q-space diffusion MRI

The diffusion properties of water molecules within a voxel V can be summarized by the so-called ensemble average propagator (EAP)

P(r; τ) = V p(r + r 0 ; r 0 , τ)dr 0 ,
which is the density of the average spin displacement over a time period τ. In a spin-echo MRI sequence, a pair of pulsed diffusion-encoding gradients can be added to measure diffusion (see Fig. 1.1). Under the short pulse assumption [START_REF] Paul T Callaghan | Principles of nuclear magnetic resonance microscopy[END_REF], the signal attenuation E is related to the propagator via a Fourier transform

E(q) = P(r; τ) exp(-2iπq • r)dr, (1.1)
where the diffusion time τ and the wavevector q are defined by τ = ∆δ/3 and q = γδg.

To exploit the link between the signal and the propagator, several strategies had been proposed -either based on the discrete Fourier transform and associated sampling of E(q) on a regular grid [START_REF] Van J Wedeen | Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging[END_REF], or by using intermediate continuous representations [START_REF] Assemlal | Efficient and robust computation of pdf features from diffusion mr signal[END_REF][START_REF] Cheng | Model-free and analytical eap reconstruction via spherical polar fourier diffusion mri[END_REF][START_REF] Descoteaux | Multiple q-shell diffusion propagator imaging[END_REF][START_REF] Özarslan | Mean apparent propagator (map) mri: a novel diffusion imaging method for mapping tissue microstructure[END_REF][START_REF] Pasha Hosseinbor | Bessel fourier orientation reconstruction (BFOR): An analytical diffusion propagator reconstruction for hybrid diffusion imaging and computation of q-space indices[END_REF]Caruyer and Deriche, 2012a). We present in this section several contributions to the latter strategy and their impact on optimal acquisition design.

Laplacian-regularized MAP-MRI

Mean apparent propagator (MAP) MRI is a method based on the projection of the signal E(q) onto an orthogonal basis of 3 dimensional, homogeneous Gauss-Hermite polynomials (Özarslan et al., 2013)

E a (q) = N max ∑ N=0 ∑ (n 1 ,n 2 ,n 3 )∈E N a n 1 n 2 n 3 ψ n 1 (q x ; u x )ψ n 2 (q y ; u y )ψ n 3 (q z ; u z ),
where a is the vector of coefficients, u j are scale factors and the set

E N = {(n 1 , n 2 , n 3 ) s.t. n 1 + n 2 + n 3 = N}.
The Gauss-Hermite polynomial of degree n is defined as

ψ n (q; u) = 1 √ 2 n+1 πn!u exp - q 2 2u 2 H n ( q u
).

(1.

2)

The basis is adapted (orientation and scale) to the diffusion properties in each voxel, by first fitting a diffusion tensor model. The propagator, together with quantitative parameters such as the return to the origin probability (RTOP), return to the axis probability (RTAP) and return to the plane probability (RTPP), can be directly estimated from the coefficients a.

Observing that this approach is subject to noise and usually requires a large number of samples for an accurate estimation, we proposed to add a regularization constraint based on the norm of the Laplacian operator [START_REF] Rutger | MAPL: Tissue Microstructure Estimation Using Laplacian-Regularized MAP-MRI and its Application to HCP Data[END_REF], a strategy that had been used successfully for other types of signal representations in diffusion MRI [START_REF] Descoteaux | Regularized, fast, and robust analytical q-ball imaging[END_REF]Caruyer and Deriche, 2012a). The regularized least squares estimation problem is

ã = arg min a K ∑ k=1 (E(q k ) -E a (q k )) 2 + λ |∆E a (q)| 2 dq.
We show that the Laplacian has an analytical and compact expression in the coefficients in the basis, which incorporates seamlessly in the least squares estimation as a quadratic regularization term. In particular, this can be combined with the reconstruction under positivity constraint proposed in the original method (Özarslan et al., 2013). The Laplacian-regularized solution is more robust to noise than the signal reconstructed taking into account the positivity constraint solely, as illustrated on Fig. 1.2. We also demonstrate that, beyond the EAP and derived scalar parameters (RTxP), reconstruction with MAPL can be used as a pre-processing step for denoising, interpolating and extrapolating the signal prior to biophysical model fitting. With this framework, we show that indices of axon diameter from the Axcaliber model [START_REF] Assaf | AxCaliber: a method for measuring axon diameter distribution from diffusion MRI[END_REF] and the orientation dispersion index from the NODDI model [START_REF] Zhang | Noddi: practical in vivo neurite orientation dispersion and density imaging of the human brain[END_REF] can both be estimated more accurately than fitting these models to the original data. MAPL is implemented in Dipy1 .

Multi-dimensional diffusion MRI

In the previous section, we saw that the ensemble average propagator (EAP) can be reconstructed with q-space diffusion MRI. Note that since it is ensemble average, this description lacks to capture the full heterogeneity of diffusion properties in a voxel. To circumvent this, several approaches were proposed to model the micro-environments in a voxel with a diffusion tensor distribution, f (D), that corresponds to a continuum of diffusive compartments -each of which is modeled with a diffusion tensor [START_REF] Jian | A novel tensor distribution model for the diffusion-weighted mr signal[END_REF][START_REF] Zhang | Noddi: practical in vivo neurite orientation dispersion and density imaging of the human brain[END_REF][START_REF] Scherrer | Characterizing brain tissue by assessment of the distribution of anisotropic microstructural environments in diffusion-compartment imaging (diamond)[END_REF]. The average diffusion tensor, which is the first order moment of f (D), is equivalent to the diffusion tensor defined classically. Higher order moments of the distribution give access to new and useful indices to characterize this heterogeneity.

Recently, it was shown that conventional diffusion encoding, using pulsed magnetic field gradients with constant orientations, are unable to reconstruct the full covariance of this distribution [START_REF] Westin | Q-space trajectory imaging for multidimensional diffusion mri of the human brain[END_REF][START_REF] Topgaard | Multidimensional diffusion mri[END_REF][START_REF] Reymbaut | Magic diamond: Multi-fascicle diffusion compartment imaging with tensor distribution modeling and tensor-valued diffusion encoding[END_REF]. Q-space trajectory imaging, employing gradient waveforms g(t) with time-varying orientation, was proposed to access these parameters, from which we can define new indices such as the microscopic fractional anisotropy (µFA) or the orientation dispersion (OD). Within this framework, the signal attenuation is related to the diffusion tensor distribution through a Laplace transform

E(B) = S + n f (D) exp(-B : D)dD,
where the integral is defined on the space of positive semi-definite matrices S + n . The matrix B above, generally referred to as the Btensor, summarizes the acquisition parameters. It is defined from the gradient waveform g(t) by B = TE 0 q(t)q(t) T dt, where

q(t) = γ t 0 g(t ′ )dt ′ .
In the notations above, TE is the echo time and γ is the proton gyromagnetic ratio. From a numerical perspective, contrarily to the Fourier transform, the inverse Laplace transform is an ill-posed problem [START_REF] Topgaard | Multidimensional diffusion mri[END_REF]. Similarly to what we contributed to develop in q-space, we proposed an orthogonal basis for representing the signal E(B) [START_REF] Bates | A 4D Basis and Sampling Scheme for the Tensor Encoded Multi-Dimensional Diffusion MRI Signal[END_REF]. Observing that most methods proposed for white matter imaging employ encoding B-matrices with cylindrical geometry [START_REF] Westin | Q-space trajectory imaging for multidimensional diffusion mri of the human brain[END_REF][START_REF] Topgaard | Multidimensional diffusion mri[END_REF], we restricted the representation to the 4 dimensional space of symmetric positive matrices with cylindrical geometry, meaning that two of the three eigenvalues are equal. The basis we propose can be seen as an extension of the spherical polar Fourier basis [START_REF] Assemlal | Efficient and robust computation of pdf features from diffusion mr signal[END_REF]:

E(B) = P ∑ p=0 N ∑ n=0 L ∑ ℓ=0 ℓ ∑ m=-ℓ c pnℓm X p (b || )X n (b ⊥ )Y m ℓ (θ, ϕ) (1.3)
where b || and b ⊥ are the axial and radial eigenvalues of B, respectively, and θ, ϕ are the polar and azimuthal angle defining the axis of symmetry of B. The functions X n are defined by

X n (b) = n! ζ 3 (n + 2)! exp(- b 2ζ )L 2 n ( b ζ )
where L 2 n are the n-th generalized Laguerre polynomials of parameter 2, ζ is a scale factor, Y m ℓ (θ, ϕ) are the real spherical harmonics (SH) of maximum degree L [START_REF] Descoteaux | Regularized, fast, and robust analytical q-ball imaging[END_REF] and N is the maximum order. Using quadrature formulas, we also derive a tight sampling scheme for the truncated expansion (the minimal number of samples required is equal to the dimension of the basis) and an analytical reconstruction scheme. This provides an analytical, numerically stable estimation framework for the coefficients c pnlm in Eq. 1.3. .3). This is an opportunity to further reduce the number of acquisitions, by adapting the number of directions to the angular resolution. In total, we show that a sampling scheme with a minimal number of samples K = 280 is sufficient for an accurate reconstruction of the signal in this basis, for B-tensor eigenvalues up to 2000 s mm -2 .

Rotation-invariant measures from diffusion data

Equally important to the proper mathematical representation of the signal is the ability to extract meaningful information from this representation and construct imaging biomarkers. A first step towards this is the definition of rotation-invariant features; in diffusion tensor imaging, several such measures were developed, such as the fractional anisotropy (FA), the mean diffusivity (MD) and beyond [START_REF] Westin | Processing and visualization for diffusion tensor mri[END_REF][START_REF] Daniel | Orthogonal tensor invariants and the analysis of diffusion tensor magnetic resonance images[END_REF]. When it comes to characterize complex white matter with high angular resolution diffusion imaging (HARDI), the angular stucture is either represented using higher order tensors [START_REF] Barmpoutis | Approximating symmetric positive semidefinite tensors of even order[END_REF][START_REF] Özarslan | Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging[END_REF] or spherical harmonics [START_REF] Lawrence R Frank | Characterization of anisotropy in high angular resolution diffusion-weighted mri[END_REF][START_REF] Descoteaux | Regularized, fast, and robust analytical q-ball imaging[END_REF]. Some groups had proposed to extract rotation invariant measures from HARDI [START_REF] Gur | Generalized hardi invariants by method of tensor contraction[END_REF][START_REF] Schwab | Rotation invariant features for hardi[END_REF]Ghosh et al., 2012a,b), but there had been no systematic search for an independent set of invariants.

We introduced a method to derive all rotation-invariant features expressed as an order-t polynomial in the coefficients of the rank-L spherical harmonics representation [START_REF] Caruyer | On facilitating the use of HARDI in population studies by creating rotationinvariant markers[END_REF]. We show that finding such polynomials is equivalent to solving a large system of equations; solutions were found numerically with an efficient implementation taking advantage of the sparse structure of the linear system of equations. From the set of solutions, using a pruning algorithm, we isolated a family of 12 (respectively 25) algebraically independant invariants for the spherical harmonics representation up to rank L = 4 (respectively L = 6). These invariant features show unique contrast in white matter (see Fig. 1.4). On a test-retest dataset, we showed that these new measures are reproducible across repetitions of the same subject, and exhibit subject-specific features. This set of invariants offers a new rotation-invariant representation of the HARDI signal, from which biomarkers could be constructed.

Conclusion

In this chapter, we have shown that understanding the structure of the signal is important, irrespective of the application. This has a positive impact on acquisition design, pre-processing and interpretation of the diffusion signal. The methods we derived are based on very general concepts (band-limitedness of the diffusion signal, regularity, rotation invariance). This prevents the rapid obsolescence of data acquired or processed using these concepts. Microstructure-driven acquisition design Biophysical modeling methods in diffusion MRI propose to relate the diffusion signal to microscopic tissue properties within the voxel. This family of methods therefore promise to provide very specific biomarkers and have been increasingly employed in clinical research for this reason. The estimation of model parameters is however in general ill-posed and the accuracy of estimated parameters can be improved with a proper acquisition design. The use of optimized diffusionencoding gradient parameters and in particular generalized gradient waveforms have shown promising results in this context. This optimization is however tedious due to the possibly infinite dimension of the space of admissible gradient waveforms. In a first attempt to better characterize this space of gradient waveforms, we have proposed a dictionary learning and optimal sampling scheme, showing that we could successfully sub-sample the signal acquired for a set of gradient waveforms and yet recover the full signal as in the original dictionary [START_REF] Truffet | Optimal selection of diffusion-weighting gradient waveforms using compressed sensing and dictionary learning[END_REF]. For the specific problem of biophysical modeling, we have proposed in continuation an optimization framework [START_REF] Truffet | An evolutionary framework for microstructuresensitive generalized diffusion gradient waveforms[END_REF], showing that we can increase the sensitivity to specific microstructure parameters compared to using the pulsed gradients as in the original Stejskal-Tanner sequence. Being based on Monte-Carlo simulations, this framework is general and can be adapted to a large family of models and parameters of interest.

Introduction

When the diffusion is restricted, the expression of the spin-echo attenuation is relatively well known for simple confining geometries such as parallel planes, spheres or cylinders. While analytical formulas are known for pulsed gradients [START_REF] Ch Neuman | Spin echo of spins diffusing in a bounded medium[END_REF][START_REF] Paul T Callaghan | Pulsed-gradient spin-echo nmr for planar, cylindrical, and spherical pores under conditions of wall relaxation[END_REF], computing the response to generalized gradient encoding waveforms is done either via the matrix formalism (Callaghan, 1997) or the multiple correlation function approach [START_REF] Denis S Grebenkov | Nmr survey of reflected brownian motion[END_REF]. For the hindered, extra-cellular space though, except in specific, periodic configurations [START_REF] Moutal | Diffusion nmr in periodic media: efficient computation and spectral properties[END_REF], the only method to predict the diffusion signal is usually via Monte-Carlo simulations [START_REF] Matt | Convergence and parameter choice for monte-carlo simulations of diffusion mri[END_REF][START_REF] Rafael-Patino | Robust monte-carlo simulations in diffusionmri: Effect of the substrate complexity and parameter choice on the reproducibility of results[END_REF]. Besides, for arbitrary gradients, the acquisition domain is complex since the space of admissible gradient waveforms is virtually infinite dimensional. This makes the notion of regularity of the signal harder to capture than for parametric acquisition domains such as q-space or B-tensor encoding.

In this section, we first present a work that constructs a representation of the signal corresponding to non parametric gradient waveforms; this makes use of sparse transform and dictionary learning. Then, using Monte-Carlo simulations, we propose a framework to optimize gradient waveforms towards a higher Fisher information for specific microstructure parameters of interest. In this work, we proposed to start from a discrete set of K pseudorandom gradient waveforms (see Fig. 2.1) and evaluate the possibility to sub-sample the corresponding signal, while being able to predict the unseen data [START_REF] Truffet | An evolutionary framework for microstructuresensitive generalized diffusion gradient waveforms[END_REF]. Using a set of example microstructure configurations modeling white matter fibers, that consist on parallel cylinders with random packing and gammadistributed radii, we built a testing and training dataset of Monte-Carlo simulated signals using Camino (Hall and Alexander, 2009).

Sparse dictionary learning & optimal waveforms selection

The training dataset was used to learn a sparse representation [START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF], D, by optimizing arg min

x n ,D N ∑ n=1 1 2 ||y n -Dx n || 2 2 + λ||x n || 1 , (2.1)
where y n is the K-dimensional signal vector corresponding to the microstructure configuration n, D is the K × M learnt dictionary and

x n is the M-dimensional vector of coefficients, the sparsity of which is promoted with the ℓ 1 regularization penalty term in Eq. 2.1. The regularization weight was fixed to λ = 0.15 and the number of atoms (the size of the dictionary) was set to M = 200. As a proof-of-concept, we propose to consider gradient waveforms g k defined with a constant direction, u k , modulated by a pseudorandom piecewise constant function, taking values in {-g max , 0, g max } (see Fig. 2.1). We generated 65 such temporal functions; combined with 40 directions uniformly spread on a sphere [START_REF] Jones | Optimal strategies for measuring diffusion in anisotropic systems by magnetic resonance imaging[END_REF][START_REF] Caruyer | Design of multishell sampling schemes with uniform coverage in diffusion MRI[END_REF] this makes a total of K = 2600 gradient waveforms. We then find the best subset of measurements, Ω ⊂ {1, . . . , K} minimizing the correlation between lines of the dictionary:

arg min Ω f (Ω) = ∑ k,l∈Ω M ∑ m=1 Dkm Dlm 2 ,
where D is the dictionary D with its lines centered and reduced. With this sub-sampling strategy, we show on our testing data that the coefficients x estimated with a fraction of the original measurements, |Ω| ≪ K, and the corresponding, extracted dictionary D Ω , can be used to reconstruct the full signal with y = Dx with high accuracy. The root mean squared error (RMSE) is about 0.005 for the signal reconstructed using a number of |Ω| = 30 measurements. These results suggest that, despite the apparent complexity of the sampling domain and the diffusion-weighted attenuation, a sparse dictionary learning based representation can represent the regularity of the signal.

Optimization of microstructure-sensitive waveforms

Besides representing the signal, an important problem is that of optimizing gradient waveforms for the estimation of biophysical model parameters. In this work, we separated the orientation from the diffusion encoding by proposing a rotation-invariant method based on a genetic algorithm to maximize the sensitivity of the signal to specific microstructure features of interest [START_REF] Truffet | An evolutionary framework for microstructuresensitive generalized diffusion gradient waveforms[END_REF]. We constructed microstructural substrates ready for Monte-Carlo simulation (Rafael-Patino et al., 2020) so that partial derivatives with respect to microstructure parameters, such as the mean axon radius or the intra-axonal volume fraction (IAVF), can be approximated using finite differences. From these derivatives, we can in turn compute the Fisher information associated to a specific acquisition protocol. Previous studies had targeted the same objective [START_REF] Daniel | A general framework for experiment design in diffusion mri and its application in measuring direct tissuemicrostructure features[END_REF][START_REF] Drobnjak | Optimizing gradient waveforms for microstructure sensitivity in diffusionweighted MR[END_REF][START_REF] Drobnjak | Optimising time-varying gradient orientation for microstructure sensitivity in diffusionweighted mr[END_REF][START_REF] Drobnjak | PGSE, OGSE, and sensitivity to axon diameter in diffusion MRI: Insight from a simulation study[END_REF]; a major difference is that these prior studies were based on analytical models of the diffusion signal [START_REF] Assaf | Composite hindered and restricted model of diffusion (charmed) mr imaging of the human brain[END_REF][START_REF] Assaf | AxCaliber: a method for measuring axon diameter distribution from diffusion MRI[END_REF], where the diffusion in the extra-axonal compartment is approximated by an anisotropic Gaussian diffusion. In contrast, Monte-Carlo simulations offer greater flexibility on the definition of the substrate and a more realistic signal model [START_REF] Rensonnet | Towards microstructure fingerprinting: estimation of tissue properties from a dictionary of monte carlo diffusion mri simulations[END_REF][START_REF] Rafael-Patino | Robust monte-carlo simulations in diffusionmri: Effect of the substrate complexity and parameter choice on the reproducibility of results[END_REF].

(a) (b) (c) (d)
Figure 2.3: The cross-over is an important operation of the genetic algorithm; here we illustrate the result of the crossover between two gradient waveforms, g 1 (t) and g 2 (t). NB: only the part of the waveform for t ∈ [0, TE/2] is represented, the second part is simply defined by symmetry. Adapted from [START_REF] Truffet | An evolutionary framework for microstructuresensitive generalized diffusion gradient waveforms[END_REF].

The objective was to optimize a set of diffusion encoding gradient waveforms, g(t) = g(t)u, defined in a constant direction, u. The set of directions were chosen to spread the sphere uniformly [START_REF] Jones | Optimal strategies for measuring diffusion in anisotropic systems by magnetic resonance imaging[END_REF], and the optimization was performed uniquely on the gradient waveforms g(t). Since the optimization domain is high dimensional, we anticipated a high probability of local minima; we therefore performed a stochastic optimization based on a genetic algorithm. Waveforms can be generated pseudo-randomly using a Markov chain (see for instance Fig. 2.3). For the genetic evolution of the algorithm, cross-overs (generation of a new waveform as a "mixture" of two parent waveforms) are defined as illustrated on Fig. 2.3.

Interestingly, the b-value associated to the waveforms provides a first approximation on the optimality of the waveform (see Fig. 2.4); but there also remains a large variability of Fisher information for a given b-value. We decided to limit the search in the b-value range [1500, 3100] s mm -2 . The genetic algorithm consisted in "generations" of 100 waveforms; the first generation contains only random waveforms, while starting from generation 2, most waveforms are obtained via cross-over of pairs of waveforms from the previous generation. After only 30 generations, we observe that the distribution of Fisher information stabilizes. The best candidate waveforms provide significantly higher Fisher information (around 1900) than the best PGSE acquisitions (around 1300). These results show that we can optimize waveforms and obtain, as a result, a better sensitivity to the IAVF than what was possible with PGSE. Since the waveforms are pseudo-randomly generated, interpreting the resulting optimized waveforms is somewhat complicated though; a local search optimization (gradient descent) would be interesting to further the optimization in this respect.

Conclusion

While representing the signal as a function of diffusion-encoding gradient waveforms or trajectories is complex, we have proposed a method using dictionary learning offering a compact representation of the signal. With this dictionary-based representation, we show that the signal acquired for a pseudo-random set of waveforms can be optimally sub-sampled, in such a way that unseen data can be predicted with good precision. In continuation, we have proposed a framework for the generation of diffusion encoding gradient waveforms optimized for the estimation of microstructure parameters of interest. The method is illustrated for the estimation of the IAVF in a model of white matter fibers; but since the framework makes use of Monte-Carlo simulations it can be generalized to a large family of substrates and microstructure parameters.

Validation of diffusion MRI-based tractography

We believe validation of tractography is key to a larger endorsement by the community and its translation towards clinical applications. We have organized several competitions gathering international teams to exchange and compare their solutions to some of the many challenges raised by tractography and ultimately the reconstruction of a quantitative, microstructure-informed connectome. As a by-product of these competitions, we have released several open-source softwares and datasets to provide the community with easily accessible tools to validate their methods, using simulated data with a ground truth.

Introduction

The reconstruction of major white matter fiber bundles using tractography was first proposed 30+ years ago [START_REF] Mori | Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging[END_REF]; since then, the technique had a major impact on our knowledge of brain anatomy [START_REF] Mori | MRI atlas of human white matter[END_REF] and how it is affected by pathology. This method relies on the indirect relationship between the local, voxelwise diffusion-weighted measurements and the axonal bundles pathways connecting distant grey matter regions. The reconstruction of these pathways is the result of a complex processing pipeline, the major steps of which are summarized in Fig. 3 [START_REF] Delettre | Comparison be-tween diffusion mri tractography and histological tract-tracing of cortico-cortical structural connectivity in the ferret brain[END_REF][START_REF] Girard | On the cortical connectivity in the macaque brain: A comparison of diffusion tractography and histological tracing data[END_REF] or postmortem dissection [START_REF] Zemmoura | Fibrascan: a novel method for 3d white matter tract reconstruction in mr space from cadaveric dissection[END_REF][START_REF] Hau | Revisiting the human uncinate fasciculus, its subcomponents and asymmetries with stem-based tractography and microdissection validation[END_REF], see [START_REF] Drobnjak | Physical and digital phantoms for validating tractography and assessing artifacts[END_REF] for a detailed review.

Over the last decade, we have contributed to this effort by organizing several competitions gathering international teams; we also released open-source software and datasets to provide the community with flexible validation methods and tools for the reconstruction of structural connectivity. This chapter proposes a tour of these contributions, putting them in perspective.

Diffusion MRI reconstruction challenges

Following the success of previous editions (MICCAI Fibercup challenge (2009) [START_REF] Fillard | Quantitative evaluation of 10 tractography algorithms on a realistic diffusion mr phantom[END_REF], MICCAI DTI tractography challenge (2011 and 2012) [START_REF] Pujol | The DTI challenge: toward standardized evaluation of diffusion tensor imaging tractography for neurosurgery[END_REF], IEEE ISBI Workshop on HARDI reconstruction (2012) (Daducci et al., 2014a)), we co-organized several diffusion reconstruction challenges 1,2,3 in a collaborative ef- http://hardi.epfl.ch/static/ events/2021_challenge fort to evaluate tractography algorithms. A common objective of these challenges was to propose a collection of simulated datasets with a known ground truth. Concomitant to the shift of interest of the community and the progress made in simulation, the focus of these three editions evolved towards the evaluation of quantitative, microstructure-informed tractography. We summarize in this section the organization and conclusions of these three challenges.

The HARDI reconstruction challenge (ISBI 2013)

The objective of the 2013 HARDI reconstruction challenge was to evaluate the effects of the estimation accuracy of intra-voxel fiber configurations on the quality of subsequent connectivity analyses. We created a digital phantom of spherical shape comprising 27 fiber bundles, connecting 53 regions together (see Fig. 3.2). Participants requested diffusion-weighted data customized to their own acquisition scheme within a predefined budget, the diffusion weighted images were simulated using a multicompartment diffusion model (see Section 3.3 for a detailed description) and corrupted with Rician distributed noise. In total, the challenge received 17 submissions from 8 different groups; the task was to detect and estimate intra-voxel fiber orientations. We evaluated the impact on the quality of subsequent connectivity analyses using the Tractometer 4 [START_REF] Côté | Tractometer: towards validation of tractography pipelines[END_REF]. In a 4 The Tractometer -a tractography evaluation tool http://tractometer.org/ nutshell the strategy is to perfom tracking with standard algorithms in the literature, and compute statistics (average, best/worst case) on the computed tractograms, such as the number of valid/invalid bundles. We also computed two scores measuring the local accuracy of the fiber orientation estimates: the correct estimation of the number of estimated fiber compartments and their angular precision.

One of the main findings [START_REF] Houde | How should tractography go forward? A Tractometer evaluation of local reconstruction and tracking[END_REF] of this edition is that we cannot predict the performance of tractography looking solely at the average score in local reconstruction. This suggests that more than an average performance in local fiber direction, deciphering local fiber orientation in key areas is critical for tractography. When comparing the different strategies of reconstruction, we also noticed that denoising had a positive impact on tractography. Another important conclusion of this challenge is that even using ground truth fiber orientation distributions (FOD) as an input, all tractography pipelines find a ratio of nearly 3 invalid bundles reconstructed per valid bundles.

The 2013 challenge was based on artificial fiber geometries. This has provided a valuable feedback on the weaknesses of current tractography method, however the applicability of these findings to in vivo white matter tractography needed to be confirmed.

The ISMRM 2015 tractography challenge

To get an evaluation of tractography in a more realistic scenario, we proposed in the 2015 edition to simulate images starting from white matter fiber trajectories obtained with tractography in a real subject [START_REF] Maier-Hein | The challenge of mapping the human connectome based on diffusion tractography[END_REF]. After a full brain tractography of one of the HCP subject [START_REF] Stamatios N Sotiropoulos | Advances in diffusion mri acquisition and processing in the human connectome project[END_REF], 25 fiber bundles were manually segmented by an anatomical expert (see Fig. This challenge gathered 20 team who submitted 96 tractography results in total. In line with the findings of the previous edition, the main outcome of the challenge was that most tractography methods were able to detect the 25 ground truth bundles, at the cost of a very large number of false positives. In trying to explain these false positives, we identified several regions where more than 3 fiber bundles were gathering -something which has later been referred to as the bottleneck effect [START_REF] Schilling | Prevalence of white matter pathways coming into a single white matter voxel orientation: The bottleneck issue in tractography[END_REF].

This challenge has exhibited several weaknesses in tractography as a detection problem. In the meantime, in a connectomics approach, there has been a growing interest in quantifying the connection strength between two regions, taking advantage of the microstructure information embedded in the MRI signal.

The 2021 DiSCo (diffusion-simulated connectivity) challenge

In order to evaluate quantitative connectivity reconstruction methods, we designed a collection of 3 phantoms (for training, validation and testing respectively) comprising in the order of 12 000 numerical tubular hollow fibers, with diameters ranging from 1.4 to 4.2 µm. From this model of microscopic fibers with macroscopic connections (each phantom fits in a 1 mm diameter sphere) connections (see Fig.

Phantomas: simulated diffusion MR phantom for the evaluation of structural connectivity pipelines

For the needs of the ISBI 2013 challenge, we needed a flexible solution to create user-defined fiber geometries with some degree of control and simulate corresponding diffusion-weighted images. In contrast, existing solutions, such as the numerical fiber generator (NFG) [START_REF] Thomas G Close | A software tool to generate simulated white matter structures for the assessment of fibre-tracking algorithms[END_REF], had been designed to provide a fully automated solution to the creation of phantom geometries. After the challenge, we decided to make the code open-source: Phantomas [START_REF] Caruyer | Phantomas: a flexible software library to simulate diffusion MR phantoms[END_REF] was first released in January 2014.

In Phantomas, a fiber bundle is defined as a tubular-shape object wrapped around its centerline; the trajectory of the centerline is fully specified by a set of control points using piecewise polynomials. A web application5 was also implemented to help define fiber bundle http://emmanuelcaruyer.com/ phantomas-web/ trajectories with a graphical interface and mouse interaction. We also defined isotropic regions, to simulate contamination with cerebrospinal fluid (CSF) or free water, such as edematous regions that may surrounding brain tumor in patients [START_REF] Parker | Freewater estimatoR using iNtErpolated ini-Tialization (FERNET): Characterizing peritumoral edema using clinically feasible diffusion MRI data[END_REF]. From a collection of fiber trajectories and isotropic regions, T1and T2-weighted images can be simulated, along with diffusionweighted images. In every voxel, the diffusion is modeled by a distribution of Gaussian compartments. It is also possible to compute the ground truth fiber orientation distribution (FOD); a summary of these output is presented on Fig. 3.5.

Conclusion

We contributed to the field of validation of diffusion MRI tractography with the development of simulation methods to synthesize diffusion-weighted images together with ground truth structural connectivity. The software and datasets derived from this work are opensource and have already been extensively used by the community. With DiSCo, the dataset that is created is unique since it combines macro-scale connectivity with the micro-scale complexity of interwoven axons. We believe this has contributed to expand our knowledge of current limits in tractography. In the future, we can expect this simulation field will also benefit from recent progress made possible by electron microscopy and synchrotron X-ray imaging.

Applications in clinical research

In the previous chapters, we have presented methods towards a better understanding of the measurement of diffusion in MRI and its validation in neuroimaging. In this chapter we present two clinical applications to which we contributed. The first application is related to diffusion MRI tractography in patients with brain tumour. We proposed estimation method of the diffusion tensor to mitigate the effect of edema surrounding certain brain tumours. The second application concerns diffusion MRI of the spinal cord in patients with multiple sclerosis. We compared pre-processing methods for the correction of image distortions and evaluated the test-retest repeatability of quantitative diffusion MRI of the spine, an important feature for the longitudinal analysis of tissue changes induced by multiple sclerosis (MS).

Free water elimination in peritumoral edema and improed tractography using clinical diffusion MRI

Characterization of healthy versus pathological tissue in the peritumoral area is confounded by the presence of edema, making free water estimation the key concern in modeling tissue microstructure [START_REF] Pierpaoli | Removing csf contamination in brain dtmris by using a two-compartment tensor model[END_REF][START_REF] Pasternak | Free water elimination and mapping from diffusion mri[END_REF]. In these areas, we rely on a two-compartment model, for which the predicted signal is

E(b, u; D, f ) = f exp(-bu T Du) + (1 -f ) exp(-bd),
where b is the diffusion weighting factor, u is the encoding gradient direction, D is the diffusion tensor modeling the tissue compartment and f is the tissue volume fraction. One of the difficulties in fitting this model from clinical data is that estimating a two compartment model is an ill-posed problem if the acquisition consisted only in a single shell with a unique b-value [START_REF] Pasternak | Free water elimination and mapping from diffusion mri[END_REF][START_REF] Scherrer | Why multiple b-values are required for multi-tensor models. evaluation with a constrained logeuclidean model[END_REF]. This underscores the need for a robust free water elimination (FWE) method that estimates free water in pathological tissue but can be used with clinically prevalent single-shell diffusion tensor imaging data.

The solution to free water elimination (FWE) requires optimization, which relies on an initialization step. We proposed a novel initialization approach for FWE [START_REF] Parker | Freewater estimatoR using iNtErpolated ini-Tialization (FERNET): Characterizing peritumoral edema using clinically feasible diffusion MRI data[END_REF], called Freewater estimatoR using iNtErpolated iniTialization (FERNET), which improves the estimation of free water in edematous and infiltrated peritumoral regions, using single-shell diffusion MRI data. The first strategy for the initialization of f , as found in the original paper [START_REF] Pasternak | Free water elimination and mapping from diffusion mri[END_REF], was solely based on the T 2 -weighted information in the non diffusion-weighted (b = 0) image. We completed this with a second strategy, that uses the orientation average information in the diffusion-weighted signal and the prior information that the mean diffusivity in the white matter tissue compartment is expected around 0.6 × 10 -3 mm2 s -1 . The initialization which is proposed in FERNET is an interpolate between these two strategies. [START_REF] Pasternak | Free water elimination and mapping from diffusion mri[END_REF] and FERNET in a patient with a metastatic brain tumor. The corrected FA map obtained with FERNET shows better agreement between the peritumoral region and the contralateral, healthy white matter compared to the original initialization. Adapted from [START_REF] Parker | Freewater estimatoR using iNtErpolated ini-Tialization (FERNET): Characterizing peritumoral edema using clinically feasible diffusion MRI data[END_REF].

The method has been extensively investigated on data simulated with Phantomas [START_REF] Caruyer | Phantomas: a flexible software library to simulate diffusion MR phantoms[END_REF] and healthy dataset. Additionally, it has been applied to clinically acquired data from brain tumor patients (see Fig. 4.1) to characterize the peritumoral region and improve tractography in it.

Diffusion MRI within the cervical spinal cord in patients with multiple sclerosis

Multiple sclerosis (MS) is a neuro-inflammatory disease and a major source of disability in young adults. The disease is associated with a range of clinical symptoms and progressive physical impairment. While brain imaging is now a standard diagnosis tool in clinical routine, there has been a recent interest in evaluating the extent of tissue damage in the spinal cord associated with the progression of symptoms in MS. Diffusion MRI using echo-planar imaging is challenging in the spinal cord; local changes in magnetic susceptibility due to the vicinity of air and bones cause strong field inhomogeneities, source of imaging artefacts.

Comparison of distortion correction methods

Post-processing methods were developed to reduce the impact of distortion, using a pair of images acquired with reverse phase encoding directions. However, these methods were mainly validated on brain images, more scarcely on spinal cord images. In this work [START_REF] Snoussi | Geometric evaluation of distortion correction methods in diffusion MRI of the spinal cord[END_REF][START_REF] Snoussi | Evaluation of distortion correction methods in diffusion mri of the spinal cord[END_REF], we compared the results of 3 distortion correction methods: HySCO [START_REF] Ruthotto | Diffeomorphic susceptibility artifact correction of diffusion-weighted magnetic resonance images[END_REF] as implemented in ACID/SPM Toolbox 1 , Topup [START_REF] Jesper Lr Andersson | How to correct susceptibility distortions in spin-echo echo-planar images: application to diffusion tensor imaging[END_REF] as imple- We evaluated the performance of distortion correction algorithm with different, complementary metrics. Diffusion-weighted images (DWI) were acquired on a group of N = 95 subjects (29 controls and 66 MS patients); the protocol consists in thirty non-collinear DWI at b = 900 s mm -2 ; six non-DWI measurements and one non-DWI with an opposite phase encoding direction were also acquired. The resolution is 2 × 2 × 2 mm 3 and the acquisition is sagittal.

The first score measures the local alignment of the apparent centerline of the spine with the principal direction of diffusion. Anatomically, it is expected that the microscopic fiber direction is aligned with the macroscopic geometry of the spine. By expressing the principal axis of the diffusion in the local Frenet frame of the centerline, we can compute a concentration parameter [START_REF] Kanti V Mardia | Directional statistics[END_REF], which is a normalized measure in the range [0, 1]. The higher the concentration, the better the centerline of the spine is aligned with the diffusion direction. We also computed a cross-correlation score with the co-registered T 2 -weighted image. Last, 3 experts were asked to rank the corrected images for each subject (blind assessment), presented along with the uncorrected image. HySCO and Voss perform best in terms of subjective evaluation, followed with BM and last Topup. All methods improve the realignment of the centerline of the spine with the diffusion direction; improvement is most significant with BM and Voss. Last, cross-correlation with the T 2 -weighted image is improved (p < 0.05) with HySCO and BM. Based on these results, we would recommend HySCO for images acquired with a similar protocol.

Reproducibility of diffusion MRI scalar measures in spinal cord

One of the objectives of the EMISEP project 4 was to monitor the changes in spinal cord damage associated with the progression of motor deficit in MS. Reproducible measurements are the crux of longitudinal analyses; in this study [START_REF] Snoussi | Reproducibility and evolution of diffusion MRI measurements within the cervical spinal cord in multiple sclerosis[END_REF] we investigated on a test-retest dataset the reproducibility of scalar measurements on regions of interests defined as intervertebral levels. Being based on a clinical dataset, we focused on the diffusion tensor and ball-and-stick models. For the former, we estimated the fractional anisotropy (FA), axial, radial and mean diffusivities (AD, RD, MD); for the latter, we estimated the intrinsic diffusivity (ID), defined as the only positive eigenvalue of the stick, and the free water weight (FWW).

Intervertebral levels were segmented on the T 1 -weighted image using the spinal cord toolbox 5 [START_REF] De Leener | SCT: Spinal cord toolbox, an open-source software for processing spinal cord mri data[END_REF] and the PAM50 5 The spinal cord toolbox https:// spinalcordtoolbox.com/ template [START_REF] De Leener | Pam50: Unbiased multimodal template of the brainstem and spinal cord aligned with the icbm152 space[END_REF]; the labels were then co-registered to the diffusion images. The first observation is that there is a larger inter-subject variability in all metrics over vertebral levels at both ends of the imaging window when centered on the cervical spinal cord (C1-C2 and C6-C7); in continuation we decided to focus on the central part of the image (C3-C5). On the set of control subjects (N = 8), we computed the Bland-Altman plots and the 95% confidence interval, which enables to detect significant evolutions in patients between the baseline image (M0) and the follow-up scan (M12). Using these confidence intervals, we can therefore follow the longitudinal evolution of the same metrics for each patient, and identify abnormal trajectories associated with the pathology. Comparing metrics based on DTI and Ball-and-Stick suggests that both models provide complementary information. This suggests that even for clinical data, multi-compartment models provide novel information about the evolution of tissue microstructure, and should be included in the processing workflow.

Conclusion

We have presented in this chapter two clinical applications of the methods we developed for diffusion MRI. The first method, which is related to multi-compartment modeling, has a direct application for brain tumors since it provides better free water elimination using clinical diffusion images. The maps generated and the subsequent tractography provide a valuable improvement in the context of presurgical planning. The second set of methods aim at improving diffusion MRI of the spinal cord, in terms of pre-processing and statistical analysis. In the context of MS, images of the cervical spinal cord offer complementary biomarkers, which may predict physical impairment better than when using brain lesions solely. Perspectives

We have presented in this manuscript a summary of our research activity in the past decade. We have developed applied mathematical methods for the acquisition design, signal regularization, microstructure modeling and quantitative connectivity with diffusion MRI. We also translated some of these methods to clinical applications, mainly for brain tumors and spinal cord imaging in MS. Many challenges remain in the field of microstructure imaging and structural connectivity with MRI. The rapid uptake of machine learning in this area has provided successful practical response to some of these problems; this has also brought legitimate questions on the potential biases or inaccuracies that may be introduced by an inadequate model or training. We propose to contribute to this emerging field, building upon our expertise in simulation and mathematical representations in diffusion MRI. We will also develop rotation invariant reconstruction methods in microstructure, taking into account the full information in the spherical diffusion signal. Last, in order to fully benefit from the many degrees of freedom in gradient waveforms, we will propose a mathematical framework for the representation of this acquisition domain and facilitate acquisition design.

Introduction

As in many scientific domains, there has been a rapid rise of machine learning for estimating microstructure parameters, detect abnormalities related to pathology, or drive tractography. This can be partly explained by the improvement of simulation tools, which have become more realistic and more efficient. Yet, there is a pressing need to develop machine learning methods which are adapted to the constraints of diffusion MRI. One of these constraint is rotation invariance: we ambition to develop algorithms that leverage machine learning to identify a minimal set of rotation invariant features to estimate microstructure parameters of interest. In continuation, to take advantage of our expertise in simulation of large substrates that mimic brain connectivity [START_REF] Caruyer | Phantomas: a flexible software library to simulate diffusion MR phantoms[END_REF][START_REF] Rafael-Patino | The diffusion-simulated connectivity (DiSCo) dataset[END_REF], we ambition to create a training dataset of such configurations and train a quantitative connectivity predictor using machine learning. Last, understanding the appropriate structure of the sampling domain is difficult for complex acquisitions that make use of B-tensor and gradient trajectories encoding. We will develop repre-sentations of these sampling domains, taking into account their geometrical properties, defined by the invariance properties of diffusion measurement with MRI.

Microstructure parameters and rotation invariants

The diffusion tensor distribution (DTD) describes the intra-voxel diffusion as a continuum of isolated micro-environments, each of which characterized by a Gaussian diffusion. Although this method is not novel [START_REF] Jian | A novel tensor distribution model for the diffusion-weighted mr signal[END_REF], the recent development of multi-dimensional diffusion encoding gave fresh impetus to the DTD [START_REF] Topgaard | Multidimensional diffusion mri[END_REF][START_REF] Westin | Q-space trajectory imaging for multidimensional diffusion mri of the human brain[END_REF] since the former provides unique access to specific properties such as the microscopic fractional anisotropy (µFA) or the orientation dispersion (OD). An estimation method commonly used, called the spherical mean technique or powder averaging [START_REF] Kaden | Multi-compartment microscopic diffusion imaging[END_REF][START_REF] Topgaard | Multidimensional diffusion mri[END_REF], consists in first computing the rotationinvariant spherical mean of the signal and use it to estimate these statistics of the DTD. The same method has also been applied to the estimation of microstructure parameters such as the intra-axonal volume fraction [START_REF] Li | Linking spherical mean diffusion weighted signal with intraaxonal volume fraction[END_REF] or the [START_REF] Afzali | Direction-averaged diffusion-weighted mri signal using different axisymmetric b-tensor encoding schemes[END_REF][START_REF] Andersson | Does powder averaging remove dispersion bias in diffusion mri diameter estimates within real 3d axonal architectures?[END_REF].

Provided that the original sampling scheme contains a sufficient number of directions, adapted to the angular band limit of the signal and that these directions form a spherical design (Caruyer and Deriche, 2012b), there is a guarantee that the spherical mean (or powder average) of the signal is invariant to rotations, which is a desirable property. However, this is arguably just one of the many invariants of a spherical signal [START_REF] Caruyer | On facilitating the use of HARDI in population studies by creating rotationinvariant markers[END_REF]; for this reason the spherical mean may not be a sufficient statistic for the estimation of the parameters of interest. As a result, this estimation technique may be suboptimal, in that it does not take advantage of all the relevant information contained in the acquired signal. We propose to systematically search for the existence of sufficient statistics for parameters of interest in the DTD and microstructure models, and consequently propose minimal acquisition requirements for the estimation of these parameters.

This will benefit to a better acquisition and modelling in white matter and gray matter. Among the applications, we plan to investigate imaging biomarkers of gray matter neurodegeneration in Alzheimer's disease.

Towards a more quantitative tractography

Tractography filtering methods such as SIFT (Smith et al., 2013), LiFE (Caiafa and[START_REF] Cesar | Multidimensional encoding of brain connectomes[END_REF] or COMMIT (Daducci et al., 2014b) have significantly improved connectivity estimation by reducing the number of false-positive connexions typically found by tractography algorithms. Yet a number of confounding factors remain for tractography to be truly quantitative and we believe that machine learning is an interesting lead towards improving structural connectivity.

The recent development of efficient simulation methods provides a unique tool for the generation of arbitrary complex, microscopically realistic configurations of fiber bundles and their associated simulated diffusion-weighted images [START_REF] Rafael-Patino | The diffusion-simulated connectivity (DiSCo) dataset[END_REF]. We ambition to create a supervised training/testing dataset and use it to train a network in order to predict structural connectivity. Due to the computational cost of simulation, one of the challenges will be to robustly train a model with a limited number of examples. We will also try to propose a hybrid approach, in which only part of the tractography pipeline will be trained. We will need to find a standardized way to represent the inputs of the tractography filtering method, so that this can be abstracted in a large dataset of training examples.

Acquisition design and quantitative tractography

In voxel-wise analysis, we have shown in the previous chapters that acquisition design needs to be adapted to the reconstruction problem at hand, in particular for microstructure modelling. Recently, the use of B-tensor encoding was also shown to be useful in the estimation of the fiber orientation distribution [START_REF] Rensonnet | A signal peak separation index for axisymmetric b-tensor encoding[END_REF][START_REF] Jeurissen | Multi-tissue spherical deconvolution of tensor-valued diffusion mri[END_REF]. Similarly, downstream the processing pipeline, the quantification of structural connectivity is impacted by acquisition parameters; however to our knowledge few groups have systematically investigated sampling strategies optimized for connectivity estimation. One of the advantages of dictionary-based representation methods such as COMMIT (Daducci et al., 2014b) is the direct relationship between the diffusion signal and the fiber bundles in the tractogram. We will evaluate the impact of different acquisition protocols on the accuracy of the reconstructed connectome. Extending our preliminary work in [START_REF] Truffet | Optimal selection of diffusion-weighting gradient waveforms using compressed sensing and dictionary learning[END_REF], our search for an optimized solution will be driven by properties of the dictionary necessary for a stable reconstruction in the theory of sparse reconstruction. For this optimization to be as exhaustive as possible, we will also work on efficient representation of gradient waveforms. The diffusion signal as a function of diffusion gradient encoding exhibit symmetries which need to be considered (shift invariance, antipodal symmetry). We will incorporate these in the definition of a metric in the space of gradient trajectories. Complementary to the discrete approach taken in [START_REF] Truffet | Optimal selection of diffusion-weighting gradient waveforms using compressed sensing and dictionary learning[END_REF], we propose to investigate signal processing on graphs (see illustration on Fig. 5.1) using the novel metric we will define. We anticipate to lay the basis of an optimization of diffusion encoding towards a better estimation of the structural connectivity.

Conclusion

We conclude this manuscript with a research project, which aims at improving the clinical benefits of measuring water diffusion with MRI. Our project will focus on the development of new methods for quantitative tractography, both in terms of acquisition design and reconstruction. We will use data-driven approaches, combined with mathematical methods embedding the natural structure of the signal and the acquisition domain.

Figure 1 . 1 :

 11 Figure1.1: Simplified chronogram of a spin-echo magnetic resonance sequence, with a pair of pulsed magnetic gradients. δ is the pulse duration, while ∆ denotes the pulse separation.

Figure 1 . 2 :

 12 Figure 1.2: Example of a synthetic signal corrupted with Rician noise (SNR = 25) corresponding to a two-tensor model; the ground truth signal is represented (a) and the acquisition scheme, comprising 60 diffusion-weighted measurements over three shells (b = 1000, 2000, 3000 s mm -2 ) is overlaid. We compare the reconstruction and interpolation with MAP-MRI without (b) and with (c) Laplace regularization.

Figure 1

 1 Figure 1.3: Angular band-limit of the signal (worst case across all microstructural substrates considered) represented versus the eigenvalues of the B-tensor. The angular resolution can opportunely be tuned to the shape of the B-tensor. Adapted from Bates et al. (2019).

  1

Figure 1 . 4 :

 14 Figure 1.4: Rotation invariant polynomials of different degrees, t, computed from the SH coefficients at different ranks, L. Here, the SH coefficients represent the apparent diffusion coefficient (ADC) profile; note that the first invariant (L = 0, t = 1) is similar to the mean diffusivity. Adapted from Caruyer and Verma (2015).

Figure 2

 2 Figure2.1: Example of the pseudorandom gradient waveforms used for the training and testing of sparse representation with dictionary learning. As a proof-of-concept, we propose to look at the signal response for piecewise constant gradient waveforms, taking values in the set {-g max , 0, g max }.

Figure 2 . 2 :

 22 Figure2.2: Example of a substrate that allows simulation of both intra-axonal and extra-axonal compartments. The procedure to study the local dependence (partial derivative) with respect to the IAVF is illustrated: the base configuration is transformed by scaling the voxel and the position of the cylinders, but without scaling the cylinder's radii. Adapted from[START_REF] Truffet | An evolutionary framework for microstructuresensitive generalized diffusion gradient waveforms[END_REF]

Figure 2 . 4 :

 24 Figure 2.4: Fisher information as a function of the b-value; the bars correspond to the spread in Fisher information over 100 randomly generated waveforms for each b-value. Adapted from Truffet et al. (2020).

Figure 3

 3 Figure 3.1: A typical diffusion MRIbased tractography pipeline; only the major steps are represented here. There is a large variability in tractography outcomes.

Figure 3 . 2 :

 32 Figure 3.2: The 27 fiber bundles geometry of the ISBI 2013 HARDI reconstruction challenge and the "cortical" connected regions (outlined in white).

  3.3), by defining inclusion and exclusion regions of interest (ROI). From these, T1and diffusion-weighted images were simulated with Fiberfox[START_REF] Peter F Neher | Fiberfox: facilitating the creation of realistic white matter software phantoms[END_REF].

Figure 3 . 3 :

 33 Figure 3.3: The ground truth fibers of the ISMRM 2015 tractography challenge, comprising projection, association and commisural fibers. A focus is shown on (from left to right) the left frontopontine tract (sagittal view), the left uncinate fasciculus (axial view) and the anterior commissure (coronal view).

  3.4, diffusion-weighted images of dimension 40 × 40 × 40 were synthesized using Monte-Carlo simulation (Rafael-Patino et al., 2021).

Figure 3 . 4 :

 34 Figure 3.4: The ground truth "cortical" regions, connectivity matrix and fibers of the DiSCo challenge testing data.

  Figure 3.5: Digital phantom in diffusion MRI: fiber geometries (a), T1weighted image (b), ground truth fiber orientation distribution (c), an example of fiber tracking result (d). Adapted from Caruyer et al. (2014).

Figure 4 . 1 :

 41 Figure 4.1: Comparison of the free water weight (FWW) and the FA of the tissue tensor obtained with the original initialization in (Pasternak et al.,

  and block-matching () as implemented in Anima 3 . The result of ap-

Figure 4 . 2 :

 42 Figure 4.2: Result of different distortion correction methods applied to the same dataset. We overlay the spinal cord mask computed from the T 2 -weighted image rigidly registered to the diffusion image.

Figure 5 . 1 :

 51 Figure 5.1: Example of the harmonic analysis on a discretely sampled sphere (N = 60 samples) represented as a graph; the 6 first eigenvectors of the graph Laplacian are illustrated. Adapted from an image courtesy of Constance Bocquillon.

DiPy, an open-source software library for medical image processing https:// dipy.org/

A web interface to help define fiber bundle trajectories

ACID -Artefact correction in diffusion MRIhttp://www.diffusiontools. org/ mented in FSL

, Voss[START_REF] Henning | Fiber tracking in the cervical spine and inferior brain regions with reversed gradient diffusion tensor imaging[END_REF] (in-house implementation)2 FMRIB Software Library (FSL), https://fsl.fmrib.ox.ac.uk/fsl/ fslwiki

Acknowledgements