text: 2 Q. ZHOU, O. MICHAILOVICH, AND Y. RATHI It is known, however, that the modelling capacity of DTI is limited due to its reliance on assuming the ensemble averaged diﬀusion propagator (EAP) to be a unimodal Gaussian. In fact, the above assumption undermines the ability of DTI to provide accurate estimation of the apparent diﬀusivity of white matter at the locations of crossing, diverging, and kissing neural ﬁbre tracts [7, 8, 9]. Alternatively, a parameter-free approach to estimate EAP is provided by Diﬀusion Spectral Imaging (DSI) [10], which allows reconstruction of complex diﬀusivity proﬁles under rather general conditions. Unfortunately, practical implementa- tion of DSI entails acquisition of diﬀusion measurements over a dense Cartesian grid in the q-space, which renders the acquisition requirements of DSI beyond the limits of practically admissible. This problem, however, can be alleviated by restricting the diﬀusion mea- surements to a relatively small number of concentric shells in the q-space. This sampling strategy – known as Multi-Shell Diﬀusion Imaging (MSDI) [11, 12, 13] – has successfully served as a basis for many advanced dMRI methodologies [11, 14, 15, 16, 17, 18, 19, 20]. Although the availability of the EAP is generally preferred, in some cases it is suﬃcient to know the result of its marginalization over the range variable. The resulting proba- bility density is known as the orientation distribution function (ODF), and it quantiﬁes the probability with which water molecules undergo displacement along various spatial directions [21, 22]. On the practical side, a useful approximation of the ODF can be ob- tained by means of q-ball imaging (QBI) [22], which can in turn be based on a single-shell data acquisition scheme, known as High Angular Resolution Diﬀusion Imaging (HARDI) [23, 24, 25, 26, 8, 27, 28]. The nature of single-shell acquisition inherent in HARDI imposes constraints on the directional resolution of estimated ODFs, with higher values of the diﬀusion scintillation parameter b (the so-called b-value) leading to better resolvability between various diﬀusion modes within a given voxel. This fact is exempliﬁed in Fig. 1, which shows a simulated ODF1 (top subplot) corresponding to two ﬁbre tracts crossing each other at an angle of 60o. At the same time, the bottom row of subplots depict the ODFs which have been ∈ { } recovered from the associated HARDI data generated with b 1000, 3000, 5000 s/mm2. One can see that the best directional resolution is attained at the maximum value of b= 5000 s/mm2, as expected. It is also worthwhile noting that the above eﬀect is intrinsic in both the Funk-Radon transform (FRT) [22] and solid angle [27, 29] formulations of QBI. As tempting as it might seem at the ﬁrst glance, working with relatively high b-values is usually avoided in practice for two main reasons. First, an increase in the b-value is typically achieved through using longer mixing times, which unavoidably leads to substantially low values of signal-to-noise ratio (SNR) [30]. Second, using higher b-values makes the diﬀusion data less sensitive to the eﬀects of fast diﬀusion, which is often associated with the concept of “free water” [31], and hence represents a diagnostically important diﬀusion regime. As a result, diﬀusion data are commonly acquired with relatively low values of b, typically around 1000 s/mm2. Needless to add, in such cases, poor directional resolution of ODFs may become an issue, especially for such applications as ﬁbre tractography [32]. 1The ODF was generated using a standard Gaussian mixture model [23] with equal volume fractions, · − FA=0.8 and MD = 7 10 4 mm2/s. ##LTLine##

text: Q.

text: ZHOU,

text: O.

text: MICHAILOVICH,

text: AND

text: Y.

text: RATHI

text: It

text: is

text: known,

text: however,

text: that

text: the

text: modelling

text: capacity

text: of

text: DTI

text: is

text: limited

text: due

text: to

text: its

text: reliance

text: on

text: assuming

text: the

text: ensemble

text: averaged

text: diﬀusion

text: propagator

text: (EAP)

text: to

text: be

text: a

text: unimodal

text: Gaussian.

text: In

text: fact,

text: the

text: above

text: assumption

text: undermines

text: the

text: ability

text: of

text: DTI

text: to

text: provide

text: accurate

text: estimation

text: of

text: the

text: apparent

text: diﬀusivity

text: of

text: white

text: matter

text: at

text: the

text: locations

text: of

text: crossing,

text: diverging,

text: and

text: kissing

text: neural

text: ﬁbre

text: tracts

text: Alternatively,

text: a

text: parameter-free

text: approach

text: to

text: estimate

text: EAP

text: is

text: provided

text: by

text: Diﬀusion

text: Spectral

text: Imaging

text: (DSI)

text: which

text: allows

text: reconstruction

text: of

text: complex

text: diﬀusivity

text: proﬁles

text: under

text: rather

text: general

text: conditions.

text: Unfortunately,

text: practical

text: implementa-

text: tion

text: of

text: DSI

text: entails

text: acquisition

text: of

text: diﬀusion

text: measurements

text: over

text: a

text: dense

text: Cartesian

text: grid

text: in

text: the

text: q-space,

text: which

text: renders

text: the

text: acquisition

text: requirements

text: of

text: DSI

text: beyond

text: the

text: limits

text: of

text: practically

text: admissible.

text: This

text: problem,

text: however,

text: can

text: be

text: alleviated

text: by

text: restricting

text: the

text: diﬀusion

text: mea-

text: surements

text: to

text: a

text: relatively

text: small

text: number

text: of

text: concentric

text: shells

text: in

text: the

text: q-space.

text: This

text: sampling

text: strategy

text: known

text: as

text: Multi-Shell

text: Diﬀusion

text: Imaging

text: (MSDI)

text: has

text: successfully

text: served

text: as

text: a

text: basis

text: for

text: many

text: advanced

text: dMRI

text: methodologies

text: Although

text: the

text: availability

text: of

text: the

text: EAP

text: is

text: generally

text: preferred,

text: in

text: some

text: cases

text: it

text: is

text: suﬃcient

text: to

text: know

text: the

text: result

text: of

text: its

text: marginalization

text: over

text: the

text: range

text: variable.

text: The

text: resulting

text: proba-

text: bility

text: density

text: is

text: known

text: as

text: the

text: orientation

text: distribution

text: function

text: (ODF),

text: and

text: it

text: quantiﬁes

text: the

text: probability

text: with

text: which

text: water

text: molecules

text: undergo

text: displacement

text: along

text: various

text: spatial

text: directions

text: On

text: the

text: practical

text: side,

text: a

text: useful

text: approximation

text: of

text: the

text: ODF

text: can

text: be

text: ob-

text: tained

text: by

text: means

text: of

text: q-ball

text: imaging

text: (QBI)

text: which

text: can

text: in

text: turn

text: be

text: based

text: on

text: a

text: single-shell

text: data

text: acquisition

text: scheme,

text: known

text: as

text: High

text: Angular

text: Resolution

text: Diﬀusion

text: Imaging

text: (HARDI)

text: The

text: nature

text: of

text: single-shell

text: acquisition

text: inherent

text: in

text: HARDI

text: imposes

text: constraints

text: on

text: the

text: directional

text: resolution

text: of

text: estimated

text: ODFs,

text: with

text: higher

text: values

text: of

text: the

text: diﬀusion

text: scintillation

text: parameter

text: b

text: (the

text: so-called

text: b-value)

text: leading

text: to

text: better

text: resolvability

text: between

text: various

text: diﬀusion

text: modes

text: within

text: a

text: given

text: voxel.

text: This

text: fact

text: is

text: exempliﬁed

text: in

text: Fig.

text: which

text: shows

text: a

text: simulated

text: (top

text: subplot)

text: corresponding

text: to

text: two

text: ﬁbre

text: tracts

text: crossing

text: each

text: other

text: at

text: an

text: angle

text: of

text: At

text: the

text: same

text: time,

text: the

text: bottom

text: row

text: of

text: subplots

text: depict

text: the

text: ODFs

text: which

text: have

text: been

text: recovered

text: from

text: the

text: associated

text: HARDI

text: data

text: generated

text: with

text: b

text: One

text: can

text: see

text: that

text: the

text: best

text: directional

text: resolution

text: is

text: attained

text: at

text: the

text: maximum

text: value

text: of

text: b=

text: as

text: expected.

text: It

text: is

text: also

text: worthwhile

text: noting

text: that

text: the

text: above

text: eﬀect

text: is

text: intrinsic

text: in

text: both

text: the

text: Funk-Radon

text: transform

text: (FRT)

text: and

text: solid

text: angle

text: formulations

text: of

text: QBI.

text: As

text: tempting

text: as

text: it

text: might

text: seem

text: at

text: the

text: ﬁrst

text: glance,

text: working

text: with

text: relatively

text: high

text: b-values

text: is

text: usually

text: avoided

text: in

text: practice

text: for

text: two

text: main

text: reasons.

text: First,

text: an

text: increase

text: in

text: the

text: b-value

text: is

text: typically

text: achieved

text: through

text: using

text: longer

text: mixing

text: times,

text: which

text: unavoidably

text: leads

text: to

text: substantially

text: low

text: values

text: of

text: signal-to-noise

text: ratio

text: (SNR)

text: Second,

text: using

text: higher

text: b-values

text: makes

text: the

text: diﬀusion

text: data

text: less

text: sensitive

text: to

text: the

text: eﬀects

text: of

text: fast

text: diﬀusion,

text: which

text: is

text: often

text: associated

text: with

text: the

text: concept

text: of

text: “free

text: water”

text: and

text: hence

text: represents

text: a

text: diagnostically

text: important

text: diﬀusion

text: regime.

text: As

text: a

text: result,

text: diﬀusion

text: data

text: are

text: commonly

text: acquired

text: with

text: relatively

text: low

text: values

text: of

text: b,

text: typically

text: around

text: Needless

text: to

text: add,

text: in

text: such

text: cases,

text: poor

text: directional

text: resolution

text: of

text: ODFs

text: may

text: become

text: an

text: issue,

text: especially

text: for

text: such

text: applications

text: as

text: ﬁbre

text: tractography

text: ODF

text: was

text: generated

text: using

text: a

text: standard

text: Gaussian

text: mixture

text: model

text: with

text: equal

text: volume

text: fractions,

text: and

text: MD

text: ##LTLine##


text: 12 Q. ZHOU, O. MICHAILOVICH, AND Y. RATHI Note that in (16) the minimization is carried out with respect to three variables, namely f, u, and v. A standard approach to solving such equality-constrained problems is based of the use of augmented Lagrangian methods [61]. Particularly, for the case at hand, this approach amounts to the following iterations (starting from some intitial values of Lagrange multipliers p0 and p0, e.g., p0 = p0 = 0)(cid:110). u v u v 1(cid:107) − (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:111) (f(t+1), u(t+1), v(t+1)) = arg min Φf s 2 + λ u + µ v 2 + ν v + f,u,v 2 2 1 a TV δ (cid:107) − (cid:107) δ (cid:107) − (cid:107) (17) +ϕ≥(u) + u f u + p(t) 2 + v f v + p(t) 2 2 u 2 2 v 2 − p(t+1) = p(t) + f(t+1) u(t+1) u u − p(t+1) = p(t) + f(t+1) v(t+1) v v where t stands for an iteration index, while δ > 0 and δ > 0 are some positive u v constants3. Finally, in ADMM, the concurrent minimization with respect to f, u, and v is replaced by sequential minimization with respect to f, u, and v independently. In this case, starting from some f(0) = u(0) = v(0), the update proceeds according to: (cid:110) (cid:111) (18) 1(cid:107) − (cid:107) δ (cid:107) − (cid:107) δ (cid:107) − (cid:107) Step1 : f(t+1) = arg min Φf s 2 + u f u(t) + p(t) 2 + v f v(t) + p(t) 2 f (cid:110)2 2 2 u 2 2 (cid:111) v 2 (19) δ (cid:107) − (cid:107) (cid:107) (cid:107) Step2 : u(t+1) = arg muin(cid:110) 2u f(t+1) u + p(ut) 22 + λ u 1 + ϕ≥(u) (cid:111) (20) δ (cid:107) − (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) Step3 : v(t+1) = arg min v f(t+1) v + p(t) 2 + µ v 2 + ν v , v 2 v 2 a TV followed by the “dual” step of updating the Lagrange multipliers p and p , as speciﬁed u v in (17). Although the ADMM approach has eﬀectively supplanted a single minimization problem (15) by three minimization problems, the latter admit much simpler solutions as detailed below. The optimization problem in (18) is a classical least-square (LS) problem which admits a closed-form solution as given by 3Note that the algorithm is guaranteed to converge for any positive δ and δ . In the present work we u v use δ = δ = 0.5. u v ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Q.

text: ZHOU,

text: O.

text: MICHAILOVICH,

text: AND

text: Y.

text: RATHI

text: Note

text: that

text: in

text: the

text: minimization

text: is

text: carried

text: out

text: with

text: respect

text: to

text: three

text: variables,

text: namely

text: f,

text: u,

text: and

text: v.

text: A

text: standard

text: approach

text: to

text: solving

text: such

text: equality-constrained

text: problems

text: is

text: based

text: of

text: the

text: use

text: of

text: augmented

text: Lagrangian

text: methods

text: Particularly,

text: for

text: the

text: case

text: at

text: hand,

text: this

text: approach

text: amounts

text: to

text: the

text: following

text: iterations

text: (starting

text: from

text: some

text: intitial

text: values

text: of

text: Lagrange

text: multipliers

text: and

text: e.g.,

label: equation

text: u

text: v

text: u

text: v

text: arg

text: min

text: Φf

text: s

text: λ

text: u

text: µ

text: v

text: ν

text: v

text: f,u,v

text: a

text: TV

text: δ

text: δ

text: +ϕ≥(u)

text: u

text: f

text: u

text: p(t)

text: v

text: f

text: v

text: p(t)

text: u

text: v

text: p(t)

text: u

text: u

text: p(t)

text: v

text: v

text: where

text: t

text: stands

text: for

text: an

text: iteration

text: index,

text: while

text: δ

text: and

text: δ

text: are

text: some

text: positive

text: u

text: v

text: Finally,

text: in

text: ADMM,

text: the

text: concurrent

text: minimization

text: with

text: respect

text: to

text: f,

text: u,

text: and

text: v

text: is

text: replaced

text: by

text: sequential

text: minimization

text: with

text: respect

text: to

text: f,

text: u,

text: and

text: v

text: independently.

text: In

text: this

text: case,

text: starting

text: from

text: some

text: the

text: update

text: proceeds

text: according

text: to:

text: δ

text: δ

text: arg

text: min

text: Φf

text: s

text: u

text: f

text: u(t)

text: p(t)

text: v

text: f

text: v(t)

text: p(t)

text: f

text: u

text: v

text: δ

text: arg

text: u

text: p(ut)

text: λ

text: u

text: ϕ≥(u)

text: δ

text: arg

text: min

text: v

text: v

text: p(t)

text: µ

text: v

text: ν

text: v

text: v

text: v

text: a

text: TV

text: followed

text: by

text: the

text: “dual”

text: step

text: of

text: updating

text: the

text: Lagrange

text: multipliers

text: p

text: and

text: p

text: as

text: speciﬁed

text: u

text: v

text: in

text: Although

text: the

text: ADMM

text: approach

text: has

text: eﬀectively

text: supplanted

text: a

text: single

text: minimization

text: problem

text: by

text: three

text: minimization

text: problems,

text: the

text: latter

text: admit

text: much

text: simpler

text: solutions

text: as

text: detailed

text: below.

text: The

text: optimization

text: problem

text: in

text: is

text: a

text: classical

text: least-square

text: (LS)

text: problem

text: which

text: admits

text: a

text: closed-form

text: solution

text: as

text: given

text: by

text: that

text: the

text: algorithm

text: is

text: guaranteed

text: to

text: converge

text: for

text: any

text: positive

text: δ

text: and

text: δ

text: In

text: the

text: present

text: work

text: we

text: u

text: v

text: use

text: δ

text: δ

text: u

text: v

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: SPATIALLY REGULARIZED RECONSTRUCTION OF FIBRE ORIENTATION DISTRIBUTIONS ... 15 A B C D E F Figure 2. (Subplots A-C, left-to-right) The anisotropic, isotropic, and the combined ﬁeld of ODFs associated with in silico HARDI signals generated ◦ with b = 3000 s/mm2, α = 60 , and p = 0.25; (Subplots D-F) Same as ◦ iso above, only with α = 45 and p = 0.75. iso and p = 0.75, correspondingly. (Note that, in the above subplots, the absolute size of the iso glyphs has been optimized for visualization, and thus it does not represent the actual values of the isotropic ODFs.) Finally, Subplot C of Fig. 2 depicts combined ODFs obtained as a result of the summation of the ODFs shown in Subplots A and B, while the result of the summation of the ODFs in Subplots D and E is displayed in Subplot F of the same ﬁgure. The data simulation was repeated for two diﬀerent values of b, viz. b = 1000 s/mm2 and b = 3000 s/mm2, with the corresponding SNRs of Rician noise4 being 20 and 7. It should be noted that the above choice of b-values and their related SNRs is by no means arbitrary, but intended to imitate a real-life situation, in which increasing the value of b comes at the price of a substantially reduced SNR. (The eﬀect of Rician noise is exempliﬁed in Fig. 3 where a noise-free “axial slice” of the simulated HARDI signal (left) is shown along with its noise contaminated versions for SNR = 20 (middle) and SNR=7 (right).) In such a case, the adverse eﬀect of measurement noises could eﬀectively counterpoise the gain in signal bandwidth which comes with exploiting higher b-values. In the context of SD, it is 4In the case at hand, SNR was deﬁned to be a ratio of the mean amplitude of a noise-free HARDI signal to the standard deviation of the Gaussian noise contaminating the complex MR readout. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: SPATIALLY

text: REGULARIZED

text: RECONSTRUCTION

text: OF

text: FIBRE

text: ORIENTATION

text: DISTRIBUTIONS

text: A

font: default

text: B

text: C

text: D

text: E

text: F

text: Figure

text: (Subplots

label: caption

text: A-C,

text: left-to-right)

text: The

text: anisotropic,

text: isotropic,

text: and

text: the

text: combined

text: ﬁeld

text: of

text: ODFs

text: associated

text: with

text: in

text: silico

text: HARDI

text: signals

text: generated

text: with

text: b

text: α

text: and

text: p

text: (Subplots

text: D-F)

text: Same

text: as

text: iso

text: above,

text: only

text: with

text: α

text: and

text: p

text: iso

text: and

text: p

text: correspondingly.

text: (Note

text: that,

text: in

text: the

text: above

text: subplots,

text: the

text: absolute

text: size

text: of

text: the

text: iso

text: glyphs

text: has

text: been

text: optimized

text: for

text: visualization,

text: and

text: thus

text: it

text: does

text: not

text: represent

text: the

text: actual

text: values

text: of

text: the

text: isotropic

text: ODFs.)

text: Finally,

text: Subplot

text: C

text: of

text: Fig.

text: depicts

text: combined

text: ODFs

text: obtained

text: as

text: a

text: result

text: of

text: the

text: summation

text: of

text: the

text: ODFs

text: shown

text: in

text: Subplots

text: A

text: and

text: B,

text: while

text: the

text: result

text: of

text: the

text: summation

text: of

text: the

text: ODFs

text: in

text: Subplots

text: D

text: and

text: E

text: is

text: displayed

text: in

text: Subplot

text: F

text: of

text: the

text: same

text: ﬁgure.

text: The

text: data

text: simulation

text: was

text: repeated

text: for

text: two

text: diﬀerent

text: values

text: of

text: b,

text: viz.

text: b

text: and

text: b

text: with

text: the

text: corresponding

text: SNRs

text: of

text: Rician

text: being

text: and

text: It

text: should

text: be

text: noted

text: that

text: the

text: above

text: choice

text: of

text: b-values

text: and

text: their

text: related

text: SNRs

text: is

text: by

text: no

text: means

text: arbitrary,

text: but

text: intended

text: to

text: imitate

text: a

text: real-life

text: situation,

text: in

text: which

text: increasing

text: the

text: value

text: of

text: b

text: comes

text: at

text: the

text: price

text: of

text: a

text: substantially

text: reduced

text: SNR.

text: (The

text: eﬀect

text: of

text: Rician

text: noise

text: is

text: exempliﬁed

text: in

text: Fig.

text: where

text: a

text: noise-free

text: “axial

text: slice”

text: of

text: the

text: simulated

text: HARDI

text: signal

text: (left)

text: is

text: shown

text: along

text: with

text: its

text: noise

text: contaminated

text: versions

text: for

text: SNR

text: (middle)

text: and

text: (right).)

text: In

text: such

text: a

text: case,

text: the

text: adverse

text: eﬀect

text: of

text: measurement

text: noises

text: could

text: eﬀectively

text: counterpoise

text: the

text: gain

text: in

text: signal

text: bandwidth

text: which

text: comes

text: with

text: exploiting

text: higher

text: b-values.

text: In

text: the

text: context

text: of

text: SD,

text: it

text: is

text: the

text: case

text: at

text: hand,

text: SNR

text: was

text: deﬁned

text: to

text: be

text: a

text: ratio

text: of

text: the

text: mean

text: amplitude

text: of

text: a

text: noise-free

text: HARDI

text: signal

text: to

text: the

text: standard

text: deviation

text: of

text: the

text: Gaussian

text: noise

text: contaminating

text: the

text: complex

text: MR

text: readout.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: 16 Q. ZHOU, O. MICHAILOVICH, AND Y. RATHI Figure 3. Example of a noise-free “axial slice” of the simulated HARDI data (left) along with its noise contaminated versions for SNR = 20 (middle) and SNR =7 (right). therefore important to understand the trade-oﬀ between the values of b and SNR under various regularization schemes. Additionally, real-life HARDI data were acquired by means of a 3T Siemens TrioTim MRI scanner. The diﬀusion-encoded scans were collected from a 45 y.o. healthy volunteer ∈ { } at 2 mm isotropic resolution with: b 1000, 3000 s/mm2, TR=6300 ms, and TE = 85 × ms. The transversal dimensions of the scans were equal to 128 128 pixels, with a total of { } 50 axial slices used. The diﬀusion-encoding directions u K (with K = 64) were deﬁned k k=1 using the method of generalized spirals [63], which provides a closed-form solution to the S problem of quasi-uniform sampling of 2. In this work, we compared the performance of the proposed method against that of a number of alternative approaches. The ﬁrst of these approaches was the positively constrained LS algorithm of [36], which will be referred below to as CSD. Note that this method disregards the presence of isotropic diﬀusion, and its solutions can be produced by solving (15) upon replacing Φ by H and setting λ = µ = ν = 0. In the case when λ > 0, CSD transforms into the method of [42], which was the second reference method L (referred below to as Min- ) used in our comparative study. Note that, just like CSD, L 1 Min- disregards both the presence of isotropic diﬀusion and the spatial-domain regularity 1 of estimated fODFs. While still ignoring the eﬀect of isotropic diﬀusion, the method of [52] takes advantage of an original way to constrain the spatial behaviour of reconstructed fODFs using a ﬁbre continuity model. This method (which we refer to below as CSD- FC) can be also described by the minimization problem in (15) upon replacing Φ by H and setting λ = ν = 0. Further, setting µ = 0 in (15) would result in the SD algorithm recently proposed in [64]. This method explicitly accounts for the presence of isotropic diﬀusion, while constraining the spatial regularity of its related IDM. However, the method ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Q.

text: ZHOU,

text: O.

text: MICHAILOVICH,

text: AND

text: Y.

text: RATHI

text: Figure

text: Example

text: of

text: a

text: noise-free

text: “axial

text: slice”

text: of

text: the

text: simulated

text: HARDI

text: data

text: (left)

text: along

text: with

text: its

text: noise

text: contaminated

text: versions

text: for

text: SNR

text: (middle)

text: and

text: SNR

text: (right).

text: therefore

text: important

text: to

text: understand

text: the

text: trade-oﬀ

text: between

text: the

text: values

text: of

text: b

text: and

text: SNR

text: under

text: various

text: regularization

text: schemes.

text: Additionally,

text: real-life

text: HARDI

text: data

text: were

text: acquired

text: by

text: means

text: of

text: a

text: Siemens

text: TrioTim

text: MRI

text: scanner.

text: The

text: diﬀusion-encoded

text: scans

text: were

text: collected

text: from

text: a

text: y.o.

text: healthy

text: volunteer

text: at

text: mm

text: isotropic

text: resolution

text: with:

text: b

text: ms,

text: and

text: TE

text: ms.

text: The

text: transversal

text: dimensions

text: of

text: the

text: scans

text: were

text: equal

text: to

text: pixels,

text: with

text: a

text: total

text: of

text: axial

text: slices

text: used.

text: The

text: diﬀusion-encoding

text: directions

text: u

text: K

text: (with

text: K

text: were

text: deﬁned

text: k

text: using

text: the

text: method

text: of

text: generalized

text: spirals

text: which

text: provides

text: a

text: closed-form

text: solution

text: to

text: the

text: S

text: problem

text: of

text: quasi-uniform

text: sampling

text: of

text: In

text: this

text: work,

text: we

text: compared

text: the

text: performance

text: of

text: the

text: proposed

text: method

text: against

text: that

text: of

text: a

text: number

text: of

text: alternative

text: approaches.

text: The

text: ﬁrst

text: of

text: these

text: approaches

text: was

text: the

text: positively

text: constrained

text: LS

text: algorithm

text: of

text: which

text: will

text: be

text: referred

text: below

text: to

text: as

text: CSD.

text: Note

text: that

text: this

text: method

text: disregards

text: the

text: presence

text: of

text: isotropic

text: diﬀusion,

text: and

text: its

text: solutions

text: can

text: be

text: produced

text: by

text: solving

text: upon

text: replacing

text: Φ

text: by

text: H

text: and

text: setting

text: λ

text: µ

text: ν

text: In

text: the

text: case

text: when

text: λ

text: CSD

text: transforms

text: into

text: the

text: method

text: of

text: which

text: was

text: the

text: second

text: reference

text: method

text: L

text: (referred

text: below

text: to

text: as

text: Min-

text: used

text: in

text: our

text: comparative

text: study.

text: Note

text: that,

text: just

text: like

text: CSD,

text: L

text: Min-

text: disregards

text: both

text: the

text: presence

text: of

text: isotropic

text: diﬀusion

text: and

text: the

text: spatial-domain

text: regularity

text: of

text: estimated

text: fODFs.

text: While

text: still

text: ignoring

text: the

text: eﬀect

text: of

text: isotropic

text: diﬀusion,

text: the

text: method

text: of

text: takes

text: advantage

text: of

text: an

text: original

text: way

text: to

text: constrain

text: the

text: spatial

text: behaviour

text: of

text: reconstructed

text: fODFs

text: using

text: a

text: ﬁbre

text: continuity

text: model.

text: This

text: method

text: (which

text: we

text: refer

text: to

text: below

text: as

text: CSD-

text: FC)

text: can

text: be

text: also

text: described

text: by

text: the

text: minimization

text: problem

text: in

text: upon

text: replacing

text: Φ

text: by

text: H

text: and

text: setting

text: λ

text: ν

text: Further,

text: setting

text: µ

text: in

text: would

text: result

text: in

text: the

text: SD

text: algorithm

text: recently

text: proposed

text: in

text: This

text: method

text: explicitly

text: accounts

text: for

text: the

text: presence

text: of

text: isotropic

text: diﬀusion,

text: while

text: constraining

text: the

text: spatial

text: regularity

text: of

text: its

text: related

text: IDM.

text: However,

text: the

text: method

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 20 Q. ZHOU, O. MICHAILOVICH, AND Y. RATHI Figure 5. (Left columns of subplots) TP produced by the tested methods for diﬀerent α, p , and b = 1000 s/mm2; (Right columns of subplots) TP iso produced by the tested methods for diﬀerent α, p , and b = 3000 s/mm2. iso and p = 0, dRL is incapable of resolving the crossing ﬁbres of the numerical phantom for ≤ iso◦ α 60 , with the resolvability problem becoming progressively worse with an increase in p . In this case, the values of AAE eﬀectively “mirror” those of α up to the point when iso dRL starts detecting the correct number of ﬁbres, after which AAE becomes a decreasing function of the ﬁbre crossing angle. It also deserves noting that for b = 3000 s/mm2, dRL demonstrates considerably improved performance in terms of AAE, even surpassing CSD L ≥ ◦ ≥ and Min- for α 50 and p 0.5. However, in all the alternative scenarios, the per- 1 iso formance of dRL remains inferior to that of other methods under comparison. A possible explanation to this fact could be that the Lucy-Richardson algorithm exploited by dRL aims at recovering a maximum likelihood estimate under the assumption on measurement noise to be Poissonian. However, such a noise model can hardly be a good approximation to Rician distribution, which is inherent in MRI. Fig. 5 shows the values of TP which have been obtained using the proposed and reference methods for b = 1000 s/mm2 (left column of subplots) and b = 3000 s/mm2 (right column of subplots). One can see that, in the absence of isotropic diﬀusion (i.e., for p = 0), iso the proposed SCSD method performs comparably to CSD-FC for both values of b (with L Min-TV- being the next “best performer”). Yet, the moment p starts growing above 1 iso ##LTFigure## ##LTFigure## ##LTFigure##

text: Q.

text: ZHOU,

text: O.

text: MICHAILOVICH,

text: AND

text: Y.

text: RATHI

text: Figure

text: (Left

text: columns

text: of

text: subplots)

text: TP

text: produced

text: by

text: the

text: tested

text: methods

text: for

text: diﬀerent

text: α,

text: p

text: and

text: b

text: (Right

text: columns

text: of

text: subplots)

text: TP

text: iso

text: produced

text: by

text: the

text: tested

text: methods

text: for

text: diﬀerent

text: α,

text: p

text: and

text: b

text: iso

text: and

text: p

text: dRL

text: is

text: incapable

text: of

text: resolving

text: the

text: crossing

text: ﬁbres

text: of

text: the

text: numerical

text: phantom

text: for

text: iso◦

text: α

text: with

text: the

text: resolvability

text: problem

text: becoming

text: progressively

text: worse

text: with

text: an

text: increase

text: in

text: p

text: In

text: this

text: case,

text: the

text: values

text: of

text: AAE

text: eﬀectively

text: “mirror”

text: those

text: of

text: α

text: up

text: to

text: the

text: point

text: when

text: iso

text: dRL

text: starts

text: detecting

text: the

text: correct

text: number

text: of

text: ﬁbres,

text: after

text: which

text: AAE

text: becomes

text: a

text: decreasing

text: function

text: of

text: the

text: ﬁbre

text: crossing

text: angle.

text: It

text: also

text: deserves

text: noting

text: that

text: for

text: b

text: dRL

text: demonstrates

text: considerably

text: improved

text: performance

text: in

text: terms

text: of

text: AAE,

text: even

text: surpassing

text: CSD

text: L

text: and

text: Min-

text: for

text: α

text: and

text: p

text: However,

text: in

text: all

text: the

text: alternative

text: scenarios,

text: the

text: per-

text: iso

text: formance

text: of

text: dRL

text: remains

text: inferior

text: to

text: that

text: of

text: other

text: methods

text: under

text: comparison.

text: A

text: possible

text: explanation

text: to

text: this

text: fact

text: could

text: be

text: that

text: the

text: Lucy-Richardson

text: algorithm

text: exploited

text: by

text: dRL

text: aims

text: at

text: recovering

text: a

text: maximum

text: likelihood

text: estimate

text: under

text: the

text: assumption

text: on

text: measurement

text: noise

text: to

text: be

text: Poissonian.

text: However,

text: such

text: a

text: noise

text: model

text: can

text: hardly

text: be

text: a

text: good

text: approximation

text: to

text: Rician

text: distribution,

text: which

text: is

text: inherent

text: in

text: MRI.

text: Fig.

text: shows

text: the

text: values

text: of

text: TP

text: which

text: have

text: been

text: obtained

text: using

text: the

text: proposed

text: and

text: reference

text: methods

text: for

text: b

text: (left

text: column

text: of

text: subplots)

text: and

text: b

text: (right

text: column

text: of

text: subplots).

text: One

text: can

text: see

text: that,

text: in

text: the

text: absence

text: of

text: isotropic

text: diﬀusion

text: (i.e.,

text: for

text: p

text: iso

text: the

text: proposed

text: SCSD

text: method

text: performs

text: comparably

text: to

text: CSD-FC

text: for

text: both

text: values

text: of

text: b

text: (with

text: L

text: Min-TV-

text: being

text: the

text: next

text: “best

text: performer”).

text: Yet,

text: the

text: moment

text: p

text: starts

text: growing

text: above

text: iso

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: SPATIALLY REGULARIZED RECONSTRUCTION OF FIBRE ORIENTATION DISTRIBUTIONS ... 3 ORIGINAL ODF ESTIMATED ODF ESTIMATED ODF ESTIMATED ODF b = 1000 s/mm2 b = 3000 s/mm2 b = 5000 s/mm2 Figure 1. (Top) Original ODF; (Bottom) Estimated ODFs obtained from HARDI data generated with (left to right) b=1000, 3000, and 5000 s/mm2. The problem of limited directional resolution of QBI can be addressed using the frame- work of spherical deconvolution (SD) [33]. In this formulation, SD is used to recover a ﬁbre ODF (fODF) which, as opposed to ODF, quantiﬁes the likelihood of a neural ﬁbre to have a (local) tangent vector of a certain orientation. The idea of using SD as a method for improving the directional resolution of QBI was ﬁrst introduced in [34]. In this initial ap- proach, fODFs were recovered through a direct matrix inversion (aka naive deconvolution) preceded by low-pass ﬁltering. A more advanced SD technique based on a least-square (LS) formulation with positivity constraints was proposed by the same authors in [35, 36]. An interesting extension to the later method was also discussed in [37], where the uncer- tainty in fODF estimation was reduced through a bootstrap procedure. In [38], the fOFD is parameterized based on a two-compartment model, followed by its reconstruction using non-regularized (naive) inverse ﬁltering. A similar line of arguments is used in [39], albeit this time with Tikhonov regularization implemented implicitly through damped SVD-based inversion. Moreover, as opposed to many other works in the ﬁeld, this study addressed the important problem of automatic determination of an optimal value of the regularization parameter based on generalized cross validation. Advanced statistical considerations were also employed in [40] to derive a maximum entropy deconvolution algorithm. The same idea of entropy maximization was later adopted by [41]. In the latter work, however, the reconstruction was performed within the framework of blind source separation, which oﬀers the substantial advantage of independence on the knowledge of single ﬁbre response. Un- fortunately, the Infomax approach used by the authors in [41] does not explicitly take into account the distribution and level of measurement noises, which makes it overly dependent on measurement conditions. Finally, we would also like to mention the SD approach of [42], which is close in philosophy to the methods discussed in the present paper. In particular, ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: SPATIALLY

text: REGULARIZED

text: RECONSTRUCTION

text: OF

text: FIBRE

text: ORIENTATION

text: DISTRIBUTIONS

text: ORIGINAL

text: ODF

text: ESTIMATED

text: ODF

text: ESTIMATED

text: ODF

text: ESTIMATED

text: ODF

text: b

text: b

text: b

text: Figure

text: (Top)

text: Original

text: ODF;

text: (Bottom)

text: Estimated

text: ODFs

text: obtained

text: from

text: HARDI

text: data

text: generated

text: with

text: (left

text: to

text: right)

text: and

text: The

text: problem

text: of

text: limited

text: directional

text: resolution

text: of

text: QBI

text: can

text: be

text: addressed

text: using

text: the

text: frame-

text: work

text: of

text: spherical

text: deconvolution

text: (SD)

text: In

text: this

text: formulation,

text: SD

text: is

text: used

text: to

text: recover

text: a

text: ﬁbre

text: ODF

text: (fODF)

text: which,

text: as

text: opposed

text: to

text: ODF,

text: quantiﬁes

text: the

text: likelihood

text: of

text: a

text: neural

text: ﬁbre

text: to

text: have

text: a

text: (local)

text: tangent

text: vector

text: of

text: a

text: certain

text: orientation.

text: The

text: idea

text: of

text: using

text: SD

text: as

text: a

text: method

text: for

text: improving

text: the

text: directional

text: resolution

text: of

text: QBI

text: was

text: ﬁrst

text: introduced

text: in

text: In

text: this

text: initial

text: ap-

text: proach,

text: fODFs

text: were

text: recovered

text: through

text: a

text: direct

text: matrix

text: inversion

text: (aka

text: naive

text: deconvolution)

text: preceded

text: by

text: low-pass

text: ﬁltering.

text: A

text: more

text: advanced

text: SD

text: technique

text: based

text: on

text: a

text: least-square

text: (LS)

text: formulation

text: with

text: positivity

text: constraints

text: was

text: proposed

text: by

text: the

text: same

text: authors

text: in

text: An

text: interesting

text: extension

text: to

text: the

text: later

text: method

text: was

text: also

text: discussed

text: in

text: where

text: the

text: uncer-

text: tainty

text: in

text: fODF

text: estimation

text: was

text: reduced

text: through

text: a

text: bootstrap

text: procedure.

text: In

text: the

text: fOFD

text: is

text: parameterized

text: based

text: on

text: a

text: two-compartment

text: model,

text: followed

text: by

text: its

text: reconstruction

text: using

text: non-regularized

text: (naive)

text: inverse

text: ﬁltering.

text: A

text: similar

text: line

text: of

text: arguments

text: is

text: used

text: in

text: albeit

text: this

text: time

text: with

text: Tikhonov

text: regularization

text: implemented

text: implicitly

text: through

text: damped

text: SVD-based

text: inversion.

text: Moreover,

text: as

text: opposed

text: to

text: many

text: other

text: works

text: in

text: the

text: ﬁeld,

text: this

text: study

text: addressed

text: the

text: important

text: problem

text: of

text: automatic

text: determination

text: of

text: an

text: optimal

text: value

text: of

text: the

text: regularization

text: parameter

text: based

text: on

text: generalized

text: cross

text: validation.

text: Advanced

text: statistical

text: considerations

text: were

text: also

text: employed

text: in

text: to

text: derive

text: a

text: maximum

text: entropy

text: deconvolution

text: algorithm.

text: The

text: same

text: idea

text: of

text: entropy

text: maximization

text: was

text: later

text: adopted

text: by

text: In

text: the

text: latter

text: work,

text: however,

text: the

text: reconstruction

text: was

text: performed

text: within

text: the

text: framework

text: of

text: blind

text: source

text: separation,

text: which

text: oﬀers

text: the

text: substantial

text: advantage

text: of

text: independence

text: on

text: the

text: knowledge

text: of

text: single

text: ﬁbre

text: response.

text: Un-

text: fortunately,

text: the

text: Infomax

text: approach

text: used

text: by

text: the

text: authors

text: in

text: does

text: not

text: explicitly

text: take

text: into

text: account

text: the

text: distribution

text: and

text: level

text: of

text: measurement

text: noises,

text: which

text: makes

text: it

text: overly

text: dependent

text: on

text: measurement

text: conditions.

text: Finally,

text: we

text: would

text: also

text: like

text: to

text: mention

text: the

text: SD

text: approach

text: of

text: which

text: is

text: close

text: in

text: philosophy

text: to

text: the

text: methods

text: discussed

text: in

text: the

text: present

text: paper.

text: In

text: particular,

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: SPATIALLY REGULARIZED RECONSTRUCTION OF FIBRE ORIENTATION DISTRIBUTIONS ... 21 Figure 6. (Left columns of subplots) FP produced by the tested methods for diﬀerent α, p , and b = 1000 s/mm2; (Right columns of subplots) FP iso produced by the tested methods for diﬀerent α, p , and b = 3000 s/mm2. iso L zero, not only CSD and Min- , but also CSD-FC start loosing their ability to accurately 1 estimate the true number of simulated ﬁbres. Note that this is an expected result, con- sidering the fact that neither of the above methods is endowed with facilities to explicitly account for the presence of isotropic diﬃsion, in which case over-estimation errors become inevitable (see below). At the same time, for b = 3000 s/mm2, the proposed SCSD algo- ∈ { } ≥ ◦ rithm provides an ideal detection rate of one for p 0, 0.25, 0.5 and α 30 , as well as ≥ ◦ iso for p = 0.75 and α 35 . It deserves noting that, owing to its properly accounting for iso L the eﬀect of isotropic diﬀusion, the performance of Min-TV- is only marginally inferior ≤ 1 to that of SCSD for b = 3000 s/mm2 and p 0.5. One can also see that a nearly ideal iso TP rate is reached by the dRL algorithm as well. However, it does not happen until after ≥ ◦ α reaches relatively large values (e.g., for α 50 with p = 0.5 and b = 3000 s/mm2). iso One of the principle applications of HARDI is in multi-ﬁbre tractography, in which case SD can be used to improve the resolvability of multiple ﬁbre tracts within each given voxel in a region of interest (ROI). In this situation, overestimating the number of ﬁbres is likely to produce spurious ﬁbre tracts, thereby rendering the resulting reconstructions unreliable. Unfortunately, unless properly regularized, some SD routines tend to amplify the eﬀect of noise, which in turn results in numerous false (local) maxima in reconstructed fODFs – ##LTFigure## ##LTFigure## ##LTFigure##

text: SPATIALLY

text: REGULARIZED

text: RECONSTRUCTION

text: OF

text: FIBRE

text: ORIENTATION

text: DISTRIBUTIONS

text: Figure

text: (Left

text: columns

text: of

text: subplots)

text: FP

text: produced

text: by

text: the

text: tested

text: methods

text: for

text: diﬀerent

text: α,

text: p

text: and

text: b

text: (Right

text: columns

text: of

text: subplots)

text: FP

text: iso

text: produced

text: by

text: the

text: tested

text: methods

text: for

text: diﬀerent

text: α,

text: p

text: and

text: b

text: iso

text: L

text: zero,

text: not

text: only

text: CSD

text: and

text: Min-

text: but

text: also

text: CSD-FC

text: start

text: loosing

text: their

text: ability

text: to

text: accurately

text: estimate

text: the

text: true

text: number

text: of

text: simulated

text: ﬁbres.

text: Note

text: that

text: this

text: is

text: an

text: expected

text: result,

text: con-

text: sidering

text: the

text: fact

text: that

text: neither

text: of

text: the

text: above

text: methods

text: is

text: endowed

text: with

text: facilities

text: to

text: explicitly

text: account

text: for

text: the

text: presence

text: of

text: isotropic

text: diﬃsion,

text: in

text: which

text: case

text: over-estimation

text: errors

text: become

text: inevitable

text: (see

text: below).

text: At

text: the

text: same

text: time,

text: for

text: b

text: the

text: proposed

text: SCSD

text: algo-

text: rithm

text: provides

text: an

text: ideal

text: detection

text: rate

text: of

text: one

text: for

text: p

text: and

text: α

text: as

text: well

text: as

text: iso

text: for

text: p

text: and

text: α

text: It

text: deserves

text: noting

text: that,

text: owing

text: to

text: its

text: properly

text: accounting

text: for

text: iso

text: L

text: the

text: eﬀect

text: of

text: isotropic

text: diﬀusion,

text: the

text: performance

text: of

text: Min-TV-

text: is

text: only

text: marginally

text: inferior

text: to

text: that

text: of

text: SCSD

text: for

text: b

text: and

text: p

text: One

text: can

text: also

text: see

text: that

text: a

text: nearly

text: ideal

text: iso

text: TP

text: rate

text: is

text: reached

text: by

text: the

text: dRL

text: algorithm

text: as

text: well.

text: However,

text: it

text: does

text: not

text: happen

text: until

text: after

text: α

text: reaches

text: relatively

text: large

text: values

text: (e.g.,

text: for

text: α

text: with

text: p

text: and

text: b

text: iso

text: One

text: of

text: the

text: principle

text: applications

text: of

text: HARDI

text: is

text: in

text: multi-ﬁbre

text: tractography,

text: in

text: which

text: case

text: SD

text: can

text: be

text: used

text: to

text: improve

text: the

text: resolvability

text: of

text: multiple

text: ﬁbre

text: tracts

text: within

text: each

text: given

text: voxel

text: in

text: a

text: region

text: of

text: interest

text: (ROI).

text: In

text: this

text: situation,

text: overestimating

text: the

text: number

text: of

text: ﬁbres

text: is

text: likely

text: to

text: produce

text: spurious

text: ﬁbre

text: tracts,

text: thereby

text: rendering

text: the

text: resulting

text: reconstructions

text: unreliable.

text: Unfortunately,

text: unless

text: properly

text: regularized,

text: some

text: SD

text: routines

text: tend

text: to

text: amplify

text: the

text: eﬀect

text: of

text: noise,

text: which

text: in

text: turn

text: results

text: in

text: numerous

text: false

text: (local)

text: maxima

text: in

text: reconstructed

text: fODFs

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 24 Q. ZHOU, O. MICHAILOVICH, AND Y. RATHI A1 -image (axial) A2 FA-image (axial) A3 ODFs: A4 ODFs: B1 -image (coronal) B2 FA-image (coronal) B3 ODFs: B4 ODFs: Figure 9. (Subplots A1-A2) b -image and FA images of an axial slice of 0 an in vivo scan used for presentation of experimental results; (Subplots A3- A4) ODFs corresponding to the localized region indicated by the yellow squares in Subplots A1-A2 for b = 1000 s/mm2 and b = 3000 s/mm2, respectively; (Subplots B1-B2) b -image and FA images of a coronal slice 0 from the same scan; (Subplots B3-B4) ODFs corresponding to the localized region indicated by the yellow squares in Subplots B1-B2 for b = 1000 s/mm2 and b = 3000 s/mm2, respectively. As the next step of our experimental study, real-life estimation has been performed using in vivo diﬀusion data (see Section ). For the sake of the clarity of visualization, we restrict demonstration of the obtained reconstructions to 2-D views (aka “slices”), two examples of which are depicted in Fig 9. In particular, Subplots A1-A2 of the ﬁgure display an axial slice of the acquired b -volume and its corresponding fractional anisotropy (FA) image, 0 respectively, while Subplots B1-B2 show a coronal slice from the same 3-D volume along with its associated FA image, in the same order. Additionally, Subplots A3-A4 of Fig. 9 show the ODFs corresponding to the localized regions indicated by the yellow squares in Subplots A1-A2 for the case of b = 1000 s/mm2 and b = 3000 s/mm2, respectively. (To facilitate the analysis, the ODFs are shown over the background of their associated FA values.) Analogous results pertaining to the coronal view are shown in Subplots B3-B4 of the ﬁgure. It should be noted that, in both cases, the ODFs have been computed by means of the FRACT algorithm of [65], which seems to provide a reasonable balance between ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Q.

text: ZHOU,

text: O.

text: MICHAILOVICH,

text: AND

text: Y.

text: RATHI

text: -image

text: (axial)

text: FA-image

text: (axial)

text: ODFs:

text: ODFs:

text: -image

text: (coronal)

text: FA-image

text: (coronal)

text: ODFs:

text: ODFs:

text: Figure

text: (Subplots

text: b

text: -image

text: and

text: FA

text: images

text: of

text: an

text: axial

text: slice

text: of

text: an

text: in

text: vivo

text: scan

text: used

text: for

text: presentation

text: of

text: experimental

text: results;

text: (Subplots

text: ODFs

text: corresponding

text: to

text: the

text: localized

text: region

text: indicated

text: by

text: the

text: yellow

text: squares

text: in

text: Subplots

text: for

text: b

text: and

text: b

text: respectively;

text: (Subplots

text: b

text: -image

text: and

text: FA

text: images

text: of

text: a

text: coronal

text: slice

text: from

text: the

text: same

text: scan;

text: (Subplots

text: ODFs

text: corresponding

text: to

text: the

text: localized

text: region

text: indicated

text: by

text: the

text: yellow

text: squares

text: in

text: Subplots

text: for

text: b

text: and

text: b

text: respectively.

text: As

text: the

text: next

text: step

text: of

text: our

text: experimental

text: study,

text: real-life

text: estimation

text: has

text: been

text: performed

text: using

text: in

text: vivo

text: diﬀusion

text: data

text: (see

text: Section

text: For

text: the

text: sake

text: of

text: the

text: clarity

text: of

text: visualization,

text: we

text: restrict

text: demonstration

text: of

text: the

text: obtained

text: reconstructions

text: to

text: views

text: (aka

text: “slices”),

text: two

text: examples

text: of

text: which

text: are

text: depicted

text: in

text: Fig

text: In

text: particular,

text: Subplots

text: of

text: the

text: ﬁgure

text: display

text: an

text: axial

text: slice

text: of

text: the

text: acquired

text: b

text: -volume

text: and

text: its

text: corresponding

text: fractional

text: anisotropy

text: (FA)

text: image,

text: respectively,

text: while

text: Subplots

text: show

text: a

text: coronal

text: slice

text: from

text: the

text: same

text: volume

text: along

text: with

text: its

text: associated

text: FA

text: image,

text: in

text: the

text: same

text: order.

text: Additionally,

text: Subplots

text: of

text: Fig.

text: show

text: the

text: ODFs

text: corresponding

text: to

text: the

text: localized

text: regions

text: indicated

text: by

text: the

text: yellow

text: squares

text: in

text: Subplots

text: for

text: the

text: case

text: of

text: b

text: and

text: b

text: respectively.

text: (To

text: facilitate

text: the

text: analysis,

text: the

text: ODFs

text: are

text: shown

text: over

text: the

text: background

text: of

text: their

text: associated

text: FA

text: values.)

text: Analogous

text: results

text: pertaining

text: to

text: the

text: coronal

text: view

text: are

text: shown

text: in

text: Subplots

text: of

text: the

text: ﬁgure.

text: It

text: should

text: be

text: noted

text: that,

text: in

text: both

text: cases,

text: the

text: ODFs

text: have

text: been

text: computed

text: by

text: means

text: of

text: the

text: FRACT

text: algorithm

text: of

text: which

text: seems

text: to

text: provide

text: a

text: reasonable

text: balance

text: between

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 30 Q. ZHOU, O. MICHAILOVICH, AND Y. RATHI The estimated IDMs could serve as an additional input to ﬁbre tractography or an independent observation for further statistical inference. In both scenarios, we believe the spatial regularization is essential to make the reconstructions of fODFs and IDMs be in a better agreement with the anatomical structure of the brain. Furthermore, as long as ﬁbre tractography is concerned, our experimental results indicate the possibility to achieve a better angular resolution using higher values of b (e.g., b = 3000 s/mm2). It is important to note that this outcome would not have been possible without regularizing the IDMs (as explained in Section and experimentally veriﬁed in Section ) due to the adverse eﬀect of measurement noises. The proposed SCSD algorithm has been formulated as a convex minimization problem, which admits a unique and stable minimizer. Moreover, an important contribution of this work is to introduce a computationally eﬃcient scheme for numerical implementation of SCSD. In particular, using ADMM, we have been able to ﬁnd the optimal solution via a sequence of simpler optimization problems, which are both computationally eﬃcient and amenable to parallel computations. The performance of the SCSD method has been quantitatively compared against that of a number of alternative approaches, which have been specially selected to demonstrate the importance of various assumptions and constraints. Although SCSD has been shown to outperform the reference methods in terms of all the comparison metrics used, it still has a few limitations, whose mitigation could potentially produce more accurate reconstructions. Thus, from a Bayesian perspective, the cost functional in (15) suggests the measurement noise to be of an additive Gaussian nature, while in practice it is more likely to be Rician. It should be noted, however, that the Gaussian model can serve as a good approxima- (cid:29) tion to the Rician model for relative high values of SNR (i.e., SNR 5) [67]. Besides, using the Gaussian model is essential for rendering the optimization problem convex and computationally tractable. Yet, we note that, if using the Rician statistics is preferred, it should be straightforward to modify the SCSD algorithm as detailed in [67] for a similar setting. Finally, it should also be mentioned that the present version of SCSD assumes the diﬀusion outside white matter to be purely isotropic, while evidence exists that the diﬀusion within grey matter could exhibit a certain degree of anisotropy [68]. To account for this phenomenon, one could, for instance, replace the constant column of Φ in (7) by spherical harmonics up to second degree inclusive. Exploring such a modiﬁcation and its eﬀect on the quality of SD reconstruction deﬁnes another direction of our future research. Appendix To ﬁnd a computationally eﬃcient way to solve (26), it is convenient to slightly simplify the notations ﬁrst. To this end, we note that the problem has a general form of (cid:26) (cid:27) 1(cid:107) − (cid:107) (cid:107) (cid:107) (32) min w q 2 + τ T w 2 , w 2 2 v 2 ##LTLine##

text: Q.

text: ZHOU,

text: O.

text: MICHAILOVICH,

text: AND

text: Y.

text: RATHI

text: The

text: estimated

text: IDMs

text: could

text: serve

text: as

text: an

text: additional

text: input

text: to

text: ﬁbre

text: tractography

text: or

text: an

text: independent

text: observation

text: for

text: further

text: statistical

text: inference.

text: In

text: both

text: scenarios,

text: we

text: believe

text: the

text: spatial

text: regularization

text: is

text: essential

text: to

text: make

text: the

text: reconstructions

text: of

text: fODFs

text: and

text: IDMs

text: be

text: in

text: a

text: better

text: agreement

text: with

text: the

text: anatomical

text: structure

text: of

text: the

text: brain.

text: Furthermore,

text: as

text: long

text: as

text: ﬁbre

text: tractography

text: is

text: concerned,

text: our

text: experimental

text: results

text: indicate

text: the

text: possibility

text: to

text: achieve

text: a

text: better

text: angular

text: resolution

text: using

text: higher

text: values

text: of

text: b

text: (e.g.,

text: b

text: It

text: is

text: important

text: to

text: note

text: that

text: this

text: outcome

text: would

text: not

text: have

text: been

text: possible

text: without

text: regularizing

text: the

text: IDMs

text: (as

text: explained

text: in

text: Section

text: and

text: experimentally

text: veriﬁed

text: in

text: Section

text: due

text: to

text: the

text: adverse

text: eﬀect

text: of

text: measurement

text: noises.

text: The

text: proposed

text: SCSD

text: algorithm

text: has

text: been

text: formulated

text: as

text: a

text: convex

text: minimization

text: problem,

text: which

text: admits

text: a

text: unique

text: and

text: stable

text: minimizer.

text: Moreover,

text: an

text: important

text: contribution

text: of

text: this

text: work

text: is

text: to

text: introduce

text: a

text: computationally

text: eﬃcient

text: scheme

text: for

text: numerical

text: implementation

text: of

text: SCSD.

text: In

text: particular,

text: using

text: ADMM,

text: we

text: have

text: been

text: able

text: to

text: ﬁnd

text: the

text: optimal

text: solution

text: via

text: a

text: sequence

text: of

text: simpler

text: optimization

text: problems,

text: which

text: are

text: both

text: computationally

text: eﬃcient

text: and

text: amenable

text: to

text: parallel

text: computations.

text: The

text: performance

text: of

text: the

text: SCSD

text: method

text: has

text: been

text: quantitatively

text: compared

text: against

text: that

text: of

text: a

text: number

text: of

text: alternative

text: approaches,

text: which

text: have

text: been

text: specially

text: selected

text: to

text: demonstrate

text: the

text: importance

text: of

text: various

text: assumptions

text: and

text: constraints.

text: Although

text: SCSD

text: has

text: been

text: shown

text: to

text: outperform

text: the

text: reference

text: methods

text: in

text: terms

text: of

text: all

text: the

text: comparison

text: metrics

text: used,

text: it

text: still

text: has

text: a

text: few

text: limitations,

text: whose

text: mitigation

text: could

text: potentially

text: produce

text: more

text: accurate

text: reconstructions.

text: Thus,

text: from

text: a

text: Bayesian

text: perspective,

text: the

text: cost

text: functional

text: in

text: suggests

text: the

text: measurement

text: noise

text: to

text: be

text: of

text: an

text: additive

text: Gaussian

text: nature,

text: while

text: in

text: practice

text: it

text: is

text: more

text: likely

text: to

text: be

text: Rician.

text: It

text: should

text: be

text: noted,

text: however,

text: that

text: the

text: Gaussian

text: model

text: can

text: serve

text: as

text: a

text: good

text: approxima-

text: tion

text: to

text: the

text: Rician

text: model

text: for

text: relative

text: high

text: values

text: of

text: SNR

text: (i.e.,

text: SNR

text: Besides,

text: using

text: the

text: Gaussian

text: model

text: is

text: essential

text: for

text: rendering

text: the

text: optimization

text: problem

text: convex

text: and

text: computationally

text: tractable.

text: Yet,

text: we

text: note

text: that,

text: if

text: using

text: the

text: Rician

text: statistics

text: is

text: preferred,

text: it

text: should

text: be

text: straightforward

text: to

text: modify

text: the

text: SCSD

text: algorithm

text: as

text: detailed

text: in

text: for

text: a

text: similar

text: setting.

text: Finally,

text: it

text: should

text: also

text: be

text: mentioned

text: that

text: the

text: present

text: version

text: of

text: SCSD

text: assumes

text: the

text: diﬀusion

text: outside

text: white

text: matter

text: to

text: be

text: purely

text: isotropic,

text: while

text: evidence

text: exists

text: that

text: the

text: diﬀusion

text: within

text: grey

text: matter

text: could

text: exhibit

text: a

text: certain

text: degree

text: of

text: anisotropy

text: To

text: account

text: for

text: this

text: phenomenon,

text: one

text: could,

text: for

text: instance,

text: replace

text: the

text: constant

text: column

text: of

text: Φ

text: in

text: by

text: spherical

text: harmonics

text: up

text: to

text: second

text: degree

text: inclusive.

text: Exploring

text: such

text: a

text: modiﬁcation

text: and

text: its

text: eﬀect

text: on

text: the

text: quality

text: of

text: SD

text: reconstruction

text: deﬁnes

text: another

text: direction

text: of

text: our

text: future

text: research.

text: Appendix

text: To

text: ﬁnd

text: a

text: computationally

text: eﬃcient

text: way

text: to

text: solve

text: it

text: is

text: convenient

text: to

text: slightly

text: simplify

text: the

text: notations

text: ﬁrst.

text: To

text: this

text: end,

text: we

text: note

text: that

text: the

text: problem

text: has

text: a

text: general

text: form

text: of

text: min

text: w

text: q

text: τ

text: T

text: w

text: w

text: v

text: ##LTLine##


text: SPATIALLY REGULARIZED RECONSTRUCTION OF FIBRE ORIENTATION DISTRIBUTIONS ... 31 with w interpreted as an I-dimensional (row) vector to be optimized over, q is a data R → R vector, τ > 0 is a ﬁxed regularization constant, and T : I I is the operator of ∈ S v directional diﬀerencing in the direction of v 2. The problem (32) is a standard LS problem, whose optimal solution w(cid:63) satisﬁes a system of normal equations of the form ∗ (33) (I + 2τ T T ) w(cid:63) = q, v v ∗ with T standing for the adjoint of T . v v It goes without saying that a straightforward inversion of the matrix on the left-hand side of (33) cannot be accepted as a practical option, even for moderately sized diﬀusion- encoded images. Thus, a more eﬃcient way to solve (33) needs to be found. To this end, we ﬁrst note that, if the practical computation of partial diﬀerences was based on a standard backward-diﬀerencing scheme, then multiplication by T in the spatial domain would be v equivalent to linear ﬁltering with frequency response H (ω) given by (cid:88) v (cid:16) (cid:17) (34) H (ω) = 2 3 v sin ωd e− ωd2−π , ω ∈ R3, v d 2 d=1 with v = (v , v , v ) and ω = (ω , ω , ω ). In this case, by letting wˆ(cid:63)(ω) and qˆ(ω) 1 2 3 1 2 3 denote the discrete-space Fourier transform (DSFT) of w and q, respectively, the optimal solution in (33) could be deﬁned in th(cid:0)e DSFT domain(cid:1) as | | − (35) wˆ(cid:63)(ω) = 1 + 2τ H (ω) 2 1 qˆ(ω), v which suggests that w(cid:63)(ω) is, in fact, a linearly ﬁltered version of q. Although it is deﬁnitely possible to use (35) in practical computations, such an approach would not be recommended for two main reasons. First, FFT-based implementation of linear ﬁltering would entail the use of periodic boundary conditions, which might not be natural for the case at hand. Second, the logarithmic complexity of FFT might still be considered to be prohibitively high (especially taking into account the fact that the ﬁlter (cid:0) (cid:1) needs to be applied J times for J diﬀerent values of v). A practical alternative to an FFT-based computation of wˆ(cid:63)(ω) could be ﬁrst to trans- | | − form the frequency response 1 + 2τ H (ω) 2 1 back into the spatial domain, followed by v truncating the impulse response thus obtained. In our experiments, the impulse responses × × have been truncated to a size of 7 7 7 voxels, with an associated error being less than 1% for a typical choice of the parameter τ (or, equivalently, µ). The resulting ﬁnite im- pulse response ﬁlters have distinctive (directional) low-pass characteristics, and they can be easily applied with a linear complexity under arbitrary boundary conditions. ##LTLine## ##LTLine##

text: SPATIALLY

text: REGULARIZED

text: RECONSTRUCTION

text: OF

text: FIBRE

text: ORIENTATION

text: DISTRIBUTIONS

text: with

text: w

text: interpreted

text: as

text: an

text: I-dimensional

text: (row)

text: vector

text: to

text: be

text: optimized

text: over,

text: q

text: is

text: a

text: data

text: R

text: R

text: vector,

text: τ

text: is

text: a

text: ﬁxed

text: regularization

text: constant,

text: and

text: T

text: I

text: I

text: is

text: the

text: operator

text: of

text: S

text: v

text: directional

text: diﬀerencing

text: in

text: the

text: direction

text: of

text: v

text: The

text: problem

text: is

text: a

text: standard

text: LS

text: problem,

text: whose

text: optimal

text: solution

text: satisﬁes

text: a

text: system

text: of

text: normal

text: equations

text: of

text: the

text: form

text: (I

text: 2τ

text: T

text: T

text: q,

text: v

text: v

text: with

text: T

text: standing

text: for

text: the

text: adjoint

text: of

text: T

text: v

text: v

text: It

text: goes

text: without

text: saying

text: that

text: a

text: straightforward

text: inversion

text: of

text: the

text: matrix

text: on

text: the

text: left-hand

text: side

text: of

text: cannot

text: be

text: accepted

text: as

text: a

text: practical

text: option,

text: even

text: for

text: moderately

text: sized

text: diﬀusion-

text: encoded

text: images.

text: Thus,

text: a

text: more

text: eﬃcient

text: way

text: to

text: solve

text: needs

text: to

text: be

text: found.

text: To

text: this

text: end,

text: we

text: ﬁrst

text: note

text: that,

text: if

text: the

text: practical

text: computation

text: of

text: partial

text: diﬀerences

text: was

text: based

text: on

text: a

text: standard

text: backward-diﬀerencing

text: scheme,

text: then

text: multiplication

text: by

text: T

text: in

text: the

text: spatial

text: domain

text: would

text: be

text: v

text: equivalent

text: to

text: linear

text: ﬁltering

text: with

text: frequency

text: response

text: H

text: (ω)

text: given

text: by

text: v

text: H

text: (ω)

text: v

text: sin

text: ωd

text: e−

text: ωd2−π

text: ω

text: v

text: d

text: with

text: v

text: (v

text: v

text: v

text: and

text: ω

text: (ω

text: ω

text: ω

text: In

text: this

text: case,

text: by

text: letting

text: and

text: qˆ(ω)

text: denote

text: the

text: discrete-space

text: Fourier

text: transform

text: (DSFT)

text: of

text: w

text: and

text: q,

text: respectively,

text: the

text: optimal

text: solution

text: in

text: could

text: be

text: deﬁned

text: in

text: DSFT

text: as

text: 2τ

text: H

text: (ω)

text: qˆ(ω),

text: v

text: which

text: suggests

text: that

text: is,

text: in

text: fact,

text: a

text: linearly

text: ﬁltered

text: version

text: of

text: q.

text: Although

text: it

text: is

text: deﬁnitely

text: possible

text: to

text: use

text: in

text: practical

text: computations,

text: such

text: an

text: approach

text: would

text: not

text: be

text: recommended

text: for

text: two

text: main

text: reasons.

text: First,

text: FFT-based

text: implementation

text: of

text: linear

text: ﬁltering

text: would

text: entail

text: the

text: use

text: of

text: periodic

text: boundary

text: conditions,

text: which

text: might

text: not

text: be

text: natural

text: for

text: the

text: case

text: at

text: hand.

text: Second,

text: the

text: logarithmic

text: complexity

text: of

text: FFT

text: might

text: still

text: be

text: considered

text: to

text: be

text: prohibitively

text: high

text: (especially

text: taking

text: into

text: account

text: the

text: fact

text: that

text: the

text: ﬁlter

text: needs

text: to

text: be

text: applied

text: J

text: times

text: for

text: J

text: diﬀerent

text: values

text: of

text: v).

text: A

text: practical

text: alternative

text: to

text: an

text: FFT-based

text: computation

text: of

text: could

text: be

text: ﬁrst

text: to

text: trans-

text: form

text: the

text: frequency

text: response

text: 2τ

text: H

text: (ω)

text: back

text: into

text: the

text: spatial

text: domain,

text: followed

text: by

text: v

text: truncating

text: the

text: impulse

text: response

text: thus

text: obtained.

text: In

text: our

text: experiments,

text: the

text: impulse

text: responses

text: have

text: been

text: truncated

text: to

text: a

text: size

text: of

text: voxels,

text: with

text: an

text: associated

text: error

text: being

text: less

text: than

text: for

text: a

text: typical

text: choice

text: of

text: the

text: parameter

text: τ

text: (or,

text: equivalently,

text: µ).

text: The

text: resulting

text: ﬁnite

text: im-

text: pulse

text: response

text: ﬁlters

text: have

text: distinctive

text: (directional)

text: low-pass

text: characteristics,

text: and

text: they

text: can

text: be

text: easily

text: applied

text: with

text: a

text: linear

text: complexity

text: under

text: arbitrary

text: boundary

text: conditions.

text: ##LTLine##

text: ##LTLine##


text: SPATIALLY REGULARIZED RECONSTRUCTION OF FIBRE ORIENTATION DISTRIBUTIONS ... 35 [63] E. B. Saﬀ and A. B. J. Kuijlaars. Distributing many points on a sphere. The Mathematical Intelligencer, 19(1):5–11, December 1997. [64] Q. Zhou, O. Michailovich, Y. Rathi, and A. Voineskos. Resolving complex ﬁbre architecture by means of sparse spherical deconvolution in the presence of isotropic diﬀusion. In Proceedings of SPIE Conference on Medical Imaging, San-Diego, CA, 2014. [65] J. P. Haldar and R. M. Leahy. Linear transforms for Fourier data on the sphere: Application to high angular resolution diﬀusion MRI of the brain. NeuroImage, 71:233–247, 2013. [66] I. Daubechies, B. Han, Amos Ron, and Z. Shen. Framelets: MRA-based constructions of wavelet frames. Applied and Computational Harmonic Analysis, 14(1):1–46, Jan 2003. [67] S. Dolui, I. C. Salgado Patarroyo, O. Michailovich, and Y. Rathi. Reconstruction of HARDI using compressed sensing and its application to contrast HARDI. In Proceed. of MMBIA, Breckenridge, CL, 2012. [68] Y. Rathi, O. Pasternak, P. Savadjiev, O. Michailovich, S. Bouix, M. Kubicki, C.-F. Westin, N. Makris, and M. E. Shenton. Gray matter alterations in early aging: A diﬀusion magnetic resonance imaging study. Human Brain Mapping (to appear), 2014.

text: SPATIALLY

text: REGULARIZED

text: RECONSTRUCTION

text: OF

text: FIBRE

text: ORIENTATION

text: DISTRIBUTIONS

label: reference

text: E.

text: B.

text: Saﬀ

text: and

text: A.

text: B.

text: J.

text: Kuijlaars.

text: Distributing

text: many

text: points

text: on

text: a

text: sphere.

text: The

text: Mathematical

text: Intelligencer,

text: December

text: Q.

text: Zhou,

text: O.

text: Michailovich,

text: Y.

text: Rathi,

text: and

text: A.

text: Voineskos.

text: Resolving

text: complex

text: ﬁbre

text: architecture

text: by

text: means

text: of

text: sparse

text: spherical

text: deconvolution

text: in

text: the

text: presence

text: of

text: isotropic

text: diﬀusion.

text: In

text: Proceedings

text: of

text: SPIE

text: Conference

text: on

text: Medical

text: Imaging,

text: San-Diego,

text: CA,

text: J.

text: P.

text: Haldar

text: and

text: R.

text: M.

text: Leahy.

text: Linear

text: transforms

text: for

text: Fourier

text: data

text: on

text: the

text: sphere:

text: Application

text: to

text: high

text: angular

text: resolution

text: diﬀusion

text: MRI

text: of

text: the

text: brain.

text: NeuroImage,

text: I.

text: Daubechies,

text: B.

text: Han,

text: Amos

text: Ron,

text: and

text: Z.

text: Shen.

text: Framelets:

text: MRA-based

text: constructions

text: of

text: wavelet

text: frames.

text: Applied

text: and

text: Computational

text: Harmonic

text: Analysis,

text: Jan

text: S.

text: Dolui,

text: I.

text: C.

text: Salgado

text: Patarroyo,

text: O.

text: Michailovich,

text: and

text: Y.

text: Rathi.

text: Reconstruction

text: of

text: HARDI

text: using

text: compressed

text: sensing

text: and

text: its

text: application

text: to

text: contrast

text: HARDI.

text: In

text: Proceed.

text: of

text: MMBIA,

text: Breckenridge,

text: CL,

text: Y.

text: Rathi,

text: O.

text: Pasternak,

text: P.

text: Savadjiev,

text: O.

text: Michailovich,

text: S.

text: Bouix,

text: M.

text: Kubicki,

text: C.-F.

text: Westin,

text: N.

text: Makris,

text: and

text: M.

text: E.

text: Shenton.

text: Gray

text: matter

text: alterations

text: in

text: early

text: aging:

text: A

text: diﬀusion

text: magnetic

text: resonance

text: imaging

text: study.

text: Human

text: Brain

text: Mapping

text: (to

text: appear),


text: Weak Polarized Electron Scattering 17 frequently because the accelerator needs to be retuned after the reversal. The spectrometers in PVES experiments are designed so that they are as insen- sitive as possible to the beam parameters. By making the apparatus symmetrical, the signal is very insensitive to diﬀerences in beam position and angle. Careful design of apertures can reduce the sensitivity to beam size. In addition, careful tuning of the accelerator reduces the size of the helicity-correlated beam diﬀer- ences at the target. Systematic diﬀerences in the beam parameters are controlled by a set of position- sensitive monitors that together determine the beam position and angle on target. A position monitor in a point of high dispersion determines energy diﬀerences in the beam. The sensitivity of the spectrometer to these beam parameters can be measured by dithering the beam with coils and an RF cavity so corrections can be made for any residual beam diﬀerences (31). Feedback is another tool, especially useful for eliminating the helicity-dependence of the beam intensity. Although the asymmetries measured in PVES experiments are small, in many cases the fractional error δA /A is on the order of 5% and the SoLID exper- LR LR iment proposes to measure δA /A to 0.5%, which is a higher precision than LR LR most cross section experiments. Systematic errors in scale factors, such as the beam polarization P and Q2, become important. e There are two methods for measuring P , each with variations. One is Compton e scattering from a polarized laser beam. Either the scattered electron or photon can be detected; indeed both can be detected to provide independent measure- ments. Theoretical errors for Compton scattering are small, and the beam po- larization can be measured simultaneously with A . Precisions as good as 1% LR

text: Weak

text: Polarized

text: Electron

text: Scattering

text: frequently

text: because

text: the

text: accelerator

text: needs

text: to

text: be

text: retuned

text: after

text: the

text: reversal.

text: The

text: spectrometers

text: in

text: PVES

text: experiments

text: are

text: designed

text: so

text: that

text: they

text: are

text: as

text: insen-

text: sitive

text: as

text: possible

text: to

text: the

text: beam

text: parameters.

text: By

text: making

text: the

text: apparatus

text: symmetrical,

text: the

text: signal

text: is

text: very

text: insensitive

text: to

text: diﬀerences

text: in

text: beam

text: position

text: and

text: angle.

text: Careful

text: design

text: of

text: apertures

text: can

text: reduce

text: the

text: sensitivity

text: to

text: beam

text: size.

text: In

text: addition,

text: careful

text: tuning

text: of

text: the

text: accelerator

text: reduces

text: the

text: size

text: of

text: the

text: helicity-correlated

text: beam

text: diﬀer-

text: ences

text: at

text: the

text: target.

text: Systematic

text: diﬀerences

text: in

text: the

text: beam

text: parameters

text: are

text: controlled

text: by

text: a

text: set

text: of

text: position-

text: sensitive

text: monitors

text: that

text: together

text: determine

text: the

text: beam

text: position

text: and

text: angle

text: on

text: target.

text: A

text: position

text: monitor

text: in

text: a

text: point

text: of

text: high

text: dispersion

text: determines

text: energy

text: diﬀerences

text: in

text: the

text: beam.

text: The

text: sensitivity

text: of

text: the

text: spectrometer

text: to

text: these

text: beam

text: parameters

text: can

text: be

text: measured

text: by

text: dithering

text: the

text: beam

text: with

text: coils

text: and

text: an

text: RF

text: cavity

text: so

text: corrections

text: can

text: be

text: made

text: for

text: any

text: residual

text: beam

text: diﬀerences

text: Feedback

text: is

text: another

text: tool,

text: especially

text: useful

text: for

text: eliminating

text: the

text: helicity-dependence

text: of

text: the

text: beam

text: intensity.

text: Although

text: the

text: asymmetries

text: measured

text: in

text: PVES

text: experiments

text: are

text: small,

text: in

text: many

text: cases

text: the

text: fractional

text: error

text: δA

text: /A

text: is

text: on

text: the

text: order

text: of

text: and

text: the

text: SoLID

text: exper-

text: LR

text: LR

text: iment

text: proposes

text: to

text: measure

text: δA

text: /A

text: to

text: which

text: is

text: a

text: higher

text: precision

text: than

text: LR

text: LR

text: most

text: cross

text: section

text: experiments.

text: Systematic

text: errors

text: in

text: scale

text: factors,

text: such

text: as

text: the

text: beam

text: polarization

text: P

text: and

text: become

text: important.

text: e

text: There

text: are

text: two

text: methods

text: for

text: measuring

text: P

text: each

text: with

text: variations.

text: One

text: is

text: Compton

text: e

text: scattering

text: from

text: a

text: polarized

text: laser

text: beam.

text: Either

text: the

text: scattered

text: electron

text: or

text: photon

text: can

text: be

text: detected;

text: indeed

text: both

text: can

text: be

text: detected

text: to

text: provide

text: independent

text: measure-

text: ments.

text: Theoretical

text: errors

text: for

text: Compton

text: scattering

text: are

text: small,

text: and

text: the

text: beam

text: po-

text: larization

text: can

text: be

text: measured

text: simultaneously

text: with

text: A

text: Precisions

text: as

text: good

text: as

text: LR


text: 36 Erler, Horowitz, Mantry, & Souder (cid:82) (cid:82) to the proton, Z = d3rρ (r), and neutron, N = d3rρ (r), numbers of the p n nucleus. This neglects possible meson exchange currents that are expected to be small because mesons likely transport weak charge only over distances small compared to the nuclear weak radius (106). However, see also the spin-orbit contributions to weak charge densities discussed in (107). The weak single nucleon form factors of the proton, GZ, and neutron, GZ, p n Fourier transformed into coordinate space, are 4GZ = Q Gp + Q Gn − Gs , (50) p p E n E E 4GZ = Q Gp + Q Gn − Gs . (51) n n E p E E p Here the (EM) Electric form factor of the proton is G and the Electric form E factor of the neutron is Gn . Finally, strange quark contributions to the Electric E form factor, Gs , are constrained by several previous measurements and will be E neglected in the following. The weak form factor F (Q2) is deﬁned, W (cid:90) 1 F (Q2) = d3rj (Qr)ρ (r) , (52) W 0 W Q W where j is a spherical Bessel function. The total nuclear weak charge is given by 0 (cid:90) Q = d3rρ (r) = NQ + ZQ . (53) W W n p In the Born approximation, the PV cross-section asymmetry for longitudinally polarized electrons elastically scattered from an unpolarized nucleus, A , is LR 1 Q2 Q F (Q2) W W A ≈ , (54) LR 8πα v2 Z F (Q2) ch where F (Q2) is the Fourier transform of the known charge density and is ch normalized so that F (Q2 = 0) = 1. ch ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Erler,

text: Horowitz,

text: Mantry,

text: Souder

text: to

text: the

text: proton,

text: Z

text: (r),

text: and

text: neutron,

text: N

text: (r),

text: numbers

text: of

text: the

text: p

text: n

text: nucleus.

text: This

text: neglects

text: possible

text: meson

text: exchange

text: currents

text: that

text: are

text: expected

text: to

text: be

text: small

text: because

text: mesons

text: likely

text: transport

text: weak

text: charge

text: only

text: over

text: distances

text: small

text: compared

text: to

text: the

text: nuclear

text: weak

text: radius

text: However,

text: see

text: also

text: the

text: spin-orbit

text: contributions

text: to

text: weak

text: charge

text: densities

text: discussed

text: in

text: The

text: weak

text: single

text: nucleon

text: form

text: factors

text: of

text: the

text: proton,

text: GZ,

text: and

text: neutron,

text: GZ,

text: p

text: n

text: Fourier

text: transformed

text: into

text: coordinate

text: space,

text: are

text: Q

text: Gp

text: Q

text: Gn

text: Gs

text: p

text: p

text: E

text: n

text: E

text: E

text: Q

text: Gp

text: Q

text: Gn

text: Gs

text: n

text: n

text: E

text: p

text: E

text: E

text: p

text: Here

text: the

text: (EM)

text: Electric

text: form

text: factor

text: of

text: the

text: proton

text: is

text: G

text: and

text: the

text: Electric

text: form

text: E

text: factor

text: of

text: the

text: neutron

text: is

text: Gn

text: Finally,

text: strange

text: quark

text: contributions

text: to

text: the

text: Electric

text: E

text: form

text: factor,

text: Gs

text: are

text: constrained

text: by

text: several

text: previous

text: measurements

text: and

text: will

text: be

text: E

text: neglected

text: in

text: the

text: following.

text: The

text: weak

text: form

text: factor

text: F

text: is

text: deﬁned,

text: W

text: F

text: (Qr)ρ

text: (r)

text: W

text: W

text: Q

text: W

text: where

text: j

text: is

text: a

text: spherical

text: Bessel

text: function.

text: The

text: total

text: nuclear

text: weak

text: charge

text: is

text: given

text: by

text: Q

text: (r)

text: NQ

text: ZQ

text: W

text: W

text: n

text: p

text: In

text: the

text: Born

text: approximation,

text: the

text: PV

text: cross-section

text: asymmetry

text: for

text: longitudinally

text: polarized

text: electrons

text: elastically

text: scattered

text: from

text: an

text: unpolarized

text: nucleus,

text: A

text: is

text: LR

text: Q

text: F

text: W

text: W

text: A

text: LR

text: 8πα

text: Z

text: F

text: ch

text: where

text: F

text: is

text: the

text: Fourier

text: transform

text: of

text: the

text: known

text: charge

text: density

text: and

text: is

text: ch

text: normalized

text: so

text: that

text: F

text: ch

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Weak Polarized Electron Scattering 7 metry (11) in elastic polarized ep scattering, (cid:126)e−p → e−p, dσ − dσ s g ep (cid:2) (cid:3) Aep ≡ L R = − AV y + O(y2) Fep (Q2, y) , (6) LR dσ + dσ v2 4πα QED L R where s = Q2/y is the center-of-mass energy, α the EM ﬁne structure constant, and Fep (Q2, y) is a QED correction factor. Here and in the following, y is the QED fractional energy transfer from the electrons to the hadrons which is perturba- tively small for Qweak kinematics, y ≈ 0.0082, and will be smaller yet for the experiment at the MESA facility in Mainz. The analysis of the Qweak commis- sioning data (about 4% of the total) translates into the constraint, 2 g eu + g ed = −0.032 ± 0.006 , (7) AV AV to be compared to SM prediction of −0.0355. The ﬁnal result is expected to be four times as precise, and even greater precision will be possible in Mainz. If additionally elastic scattering oﬀ isoscalar nuclei like 12C was used to extract g ep + g en , one could disentangle g eu and g ed from PVES alone. AV AV AV AV The analogous asymmetry in Møller scattering (12), s g ee 2y(1 − y) Aee = V A Fee (Q2, y) , (8) LR v2 4πα (1 − y + y2)2 QED has been obtained by the SLAC–E158 Collaboration (13) from which one can extract (9), gee = 0.0190 ± 0.0027 , (9) AV while the SM predicts the value 0.0225. The MOLLER Collaboration (14) at Jeﬀerson Lab aims to reduce the uncertainty in gee by a factor of ﬁve by taking AV advantage of the energy upgraded CEBAF. Deep inelastic PVES (PVDIS) experiments (2,15) are sensitive to the interfer- ence of the quark-level amplitudes corresponding to Lef with photon exchange. NC ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Weak

text: Polarized

text: Electron

text: Scattering

text: metry

text: in

text: elastic

text: polarized

text: ep

text: scattering,

text: e−p,

text: dσ

text: dσ

text: s

text: g

text: ep

text: Aep

text: L

text: R

text: AV

text: y

text: Fep

text: y)

text: LR

text: dσ

text: dσ

text: 4πα

text: QED

text: L

text: R

text: where

text: s

text: is

text: the

text: center-of-mass

text: energy,

text: α

text: the

text: EM

text: ﬁne

text: structure

text: constant,

text: and

text: Fep

text: y)

text: is

text: a

text: QED

text: correction

text: factor.

text: Here

text: and

text: in

text: the

text: following,

text: y

text: is

text: the

text: QED

text: fractional

text: energy

text: transfer

text: from

text: the

text: electrons

text: to

text: the

text: hadrons

text: which

text: is

text: perturba-

text: tively

text: small

text: for

text: Qweak

text: kinematics,

text: y

text: and

text: will

text: be

text: smaller

text: yet

text: for

text: the

text: experiment

text: at

text: the

text: MESA

text: facility

text: in

text: Mainz.

text: The

text: analysis

text: of

text: the

text: Qweak

text: commis-

text: sioning

text: data

text: (about

text: of

text: the

text: total)

text: translates

text: into

text: the

text: constraint,

text: g

text: eu

text: g

text: ed

text: AV

text: AV

text: to

text: be

text: compared

text: to

text: SM

text: prediction

text: of

text: The

text: ﬁnal

text: result

text: is

text: expected

text: to

text: be

text: four

text: times

text: as

text: precise,

text: and

text: even

text: greater

text: precision

text: will

text: be

text: possible

text: in

text: Mainz.

text: If

text: additionally

text: elastic

text: scattering

text: oﬀ

text: isoscalar

text: nuclei

text: like

text: was

text: used

text: to

text: extract

text: g

text: ep

text: g

text: en

text: one

text: could

text: disentangle

text: g

text: eu

text: and

text: g

text: ed

text: from

text: PVES

text: alone.

text: AV

text: AV

text: AV

text: AV

text: The

text: analogous

text: asymmetry

text: in

text: Møller

text: scattering

text: s

text: g

text: ee

text: y)

text: Aee

text: V

text: A

text: Fee

text: y)

text: LR

text: 4πα

text: y

text: QED

text: has

text: been

text: obtained

text: by

text: the

text: Collaboration

text: from

text: which

text: one

text: can

text: extract

text: gee

text: AV

text: while

text: the

text: SM

text: predicts

text: the

text: value

text: The

text: MOLLER

text: Collaboration

text: at

text: Jeﬀerson

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text: uncertainty

text: in

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text: a

text: factor

text: of

text: ﬁve

text: by

text: taking

text: AV

text: advantage

text: of

text: the

text: energy

text: upgraded

text: CEBAF.

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text: inelastic

text: PVES

text: (PVDIS)

text: experiments

text: are

text: sensitive

text: to

text: the

text: interfer-

text: ence

text: of

text: the

text: quark-level

text: amplitudes

text: corresponding

text: to

text: Lef

text: with

text: photon

text: exchange.

text: NC

text: ##LTLine##

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text: 10 Erler, Horowitz, Mantry, & Souder Table 1: Apparates for selected PV experiments. For magnets, Q (D) refers to quadrupole (dipole). Experiment Magnets Detector Count Angles Physics SLAC–E122 Dipoles Pb Glass No 12◦ geq , geq AV V A Mainz None Air C No 146◦ geq AV MIT–Bates Q Lucite C No 20◦ geq AV SAMPLE None Air C No 146◦ Strange FF HAPPEX QQDDQ Pb-Lucite No 15◦ Strange FF GØ Toroid Scintillator Yes 20◦-50◦ Strange FF Mainz–A4 None PbF Yes 30◦ Strange FF 2 SLAC–E158 QQQQ Cu-Quartz No 2◦ gee AV HAPPEX–He QQDDQ Cu-Quartz No 5◦ Strange FF PREX QQDDQ Quartz No 5◦ R n PREX-II QQDDQ Quartz No 5◦ R n CREX QQDDQ Quartz No 4◦ R n JLab–Hall A QQDDQ Pb Glass Yes 19◦ geq V A Qweak Toroid Pb-Quartz No 5◦ geq AV MOLLER Toroid Quartz No 1◦ gee AV SoLID Solenoid Package Yes 22◦-35◦ geq , CSV, HT V A Mainz–P2 Solenoid Quartz No 20◦ geq AV Mainz–C Solenoid Quartz No 40◦ geq AV For PREX, the spectrometers needed suﬃcient energy resolution to reject in- elastically scattered electrons that had lost more than a few MeV. The HRS detector package was not used, but rather elastic events were focussed onto a ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Erler,

text: Horowitz,

text: Mantry,

text: Souder

text: Table

text: Apparates

text: for

text: selected

text: PV

text: experiments.

text: For

text: magnets,

text: Q

text: (D)

text: refers

text: to

text: quadrupole

text: (dipole).

text: Experiment

label: table

text: Magnets

text: Detector

text: Count

text: Angles

text: Physics

text: Dipoles

text: Pb

text: Glass

text: No

text: geq

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text: AV

text: V

text: A

text: Mainz

text: Air

text: C

text: No

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text: AV

text: MIT–Bates

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text: HAPPEX

text: QQDDQ

text: Pb-Lucite

text: No

text: Strange

text: FF

text: GØ

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text: Strange

text: FF

text: PbF

text: Yes

text: Strange

text: FF

text: QQQQ

text: Cu-Quartz

text: No

text: 2◦

text: gee

text: AV

text: HAPPEX–He

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text: 5◦

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text: R

text: n

text: PREX-II

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text: Quartz

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text: R

text: n

text: CREX

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text: JLab–Hall

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text: Pb

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text: Solenoid

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text: No

text: geq

text: AV

text: For

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text: spectrometers

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text: in-

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