text: 7 (500, 150) GeV benchmarks considered in Figure 2. Again, for µ2 = µ2 (Top) H → ZZ searches provide min 0 In Figure 4 we show the limits from H searches for the only meaningful constraint, which gets signiﬁcantly 0 m = 300 GeV, for Type I/II (Left/Right). Here, the weakened in the hierarchical scenario m −m (cid:29) m . prHes0ence of the decay H → hh requires us to take into For µ2 = µ2 the LHC searches for HH0 → hAh0 in b¯bZb¯b 0 max 0 account the µ2 dependence in the limit extraction, and and b¯bγγ are the most constraining, being particularly we show the limits for µ2 = µ2 (Top) and µ2 = µ2 sensitive around t ∼ 1, and the limits only get mildly min max β (Bottom). In the former case the strongest limits come weakened in the hierarchical 2HDM scenario. In addi- from H → ZZ searches, with H → hh playing no rel- tion, for m = 500 GeV there is no appreciable dif- 0 0 H0 evant role because of its suppressed branching fraction. ference between Types I and II for low and moderate t , β Moreover, in this case the presence of a sizable m −m with b¯b-associated production of H in H → ττ searches H0 A0 0 0 splitting does lead to a signiﬁcant reduction of the limits constraining the t (cid:29) 1 region in Type II. β on the 2HDM parameter space from these searches. In Contrary to Figure 2, Figures 4-5 do not show the contrast, for µ2 = µ2 the H → hh searches provide would-be limits on the (c − , t ) plane from searches of max 0 β α β the dominant constraint for low and moderate t , and H in regions which are not viable theoretically: These β 0 these limits do not change signiﬁcantly in a hierarchical limits depend crucially on the value of µ2 (in contrast 2HDM scenario, as the branching fraction of H → hh with the situation for A searches discussed in Section ), 0 0 is still the dominant one in this case. A similar situa- and the theoretical bounds correspond precisely to the tion occurs for m = 500 GeV, as shown in Figure 5. absence of an allowed µ2 range. H0 Type I Type II 30 30 M = 150 GeV H0 MA0 = 150 GeV LHC exclusion 10 10 Unphysical 5 5 t β 3 3 H0 → ZZ H0 → γγ, µ2min bb¯ (H0 → ττ) 1 1 H0 → ττ H0 → γγ, µ2max ATLAS 0.4 0.4 CMS -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 c c β−α β−α F15I0G.G3e.VCaunrdren2tH9D5M%TCy.Lpe. eIx/cIlIu(dLeedftr/eRgiiognht()i,ncgormeeinng) bfryomATsLeaArSch(essoloifdglign,ebs¯b)→andH0CM→SW(dWas,hZedZli(nbelas)c,krelisnpeesc)t,ivgegly→foAr 0m→H0τ=τ (red lines), b¯b → A0 → ττ (light-blue lines) and gg → A0 → γγ respectively for µ2 = µ2min/µ2 = µ2max (yellow/pink lines). The grey region is theoretically excluded. Figures 4-5 highlight that, for ∆2 < 0 (and m2H± = conditions for cβ−α = 0 read m2 ), the theoretical bounds from stability, unitarity anHd0perturbativity are signiﬁcantly more important than m2 Ω2 for the previously discussed ∆2 > 0 case, and in particu- λ1 = v2h − tβ(1 + t2β) v2 > 0 lar constrain the alignment limit cβ−α = 0. 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text: 18 THOMAS DESCHATRE − − Proposition 20(cid:104). Le(cid:16)t t > 0(cid:17). We(cid:16)denote b(cid:17)y(cid:105)X (ρ) Y N (ρ) th(cid:16)e process X Y N when(cid:17) the correlation of the model is equal to ρ and the upper barrier to η. Let η > z > 0. We have − − ∀ ∈ √z z √2η ∃ ∈ − P (cid:16) − (cid:17) ≥ x 0, Φ + Φ , ρ [ 1, 1] : X (ρ) Y Nt (ρ) z = x 2 t 2 t t t (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) P − ≥ Contrary to Proposition 14, it is not possible to control X Y Nt z between it bounds ; t t we can only achieve Φ −√z + Φ z−√2η and not 2Φ −√z . It would be true if z could be equal 2 t 2 t 2 t (cid:16) (cid:17) to η. The problem comes from a discontinuity when ρ = 1 and z = η. As z gets closer to η, the bound grows but we can not reach z = η. However, the values that we can achieve in this model are still better than in a Gaussian copula case where the supremum is equal to Φ −√z . In this 2 t model, the distribution of the diﬀerence between the two Brownian motions is less degenerated than the one of Proposition 14 in the sense that there exists two distinct sources of randomness at all time. Results of Proposition 18 are illustrated in Figure 5a. (a) t = 1 (b) t = 20 − Figure 5. Survival function of X Y n in the multi-barrier model at time t with parameters ν = 0, η = 0.5 and ρ = 0.9 for diﬀerent values of n The case n = 0 corresponds to the Gaussian case. We can see that in [ν, η], the survival function is increasing with n. In Figure 5a, the curves for n = 5, n = 10 and n = 50 are the same. At time t = 1, the probability to cross more than 5 barrier is very weak then the Brownian reﬂection reﬂects less than 5 times with a high probability. The convergence in n at small time is fast. In Figure 5b, we can observe the diﬀerence between the cases n = 5, n = 10 and n = 50 at time t = 20. The survival function continues to grow. − The results are conﬁrmed with Figure 6. The higher the number of reﬂections is, more X Y n − is concentrated in the region [ν, η]. However, in the positive part of the plan, X Y n take lower ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: THOMAS

text: DESCHATRE

text: Proposition

text: t

text: (ρ)

text: Y

text: N

text: (ρ)

text: process

text: X

text: Y

text: N

text: the

text: correlation

text: of

text: the

text: model

text: is

text: equal

text: to

text: ρ

text: and

text: the

text: upper

text: barrier

text: to

text: η.

text: Let

text: η

text: z

text: We

text: have

text: √z

text: z

text: √2η

text: P

text: x

text: Φ

text: Φ

text: ρ

text: X

text: (ρ)

text: Y

text: Nt

text: (ρ)

text: z

text: x

text: t

text: t

text: t

text: t

text: P

text: Contrary

text: to

text: Proposition

text: it

text: is

text: not

text: possible

text: to

text: control

text: X

text: Y

text: Nt

text: z

text: between

text: it

text: bounds

text: t

text: t

text: we

text: can

text: only

text: achieve

text: Φ

text: −√z

text: Φ

text: z−√2η

text: and

text: not

text: 2Φ

text: −√z

text: It

text: would

text: be

text: true

text: if

text: z

text: could

text: be

text: equal

text: t

text: t

text: t

text: to

text: η.

text: The

text: problem

text: comes

text: from

text: a

text: discontinuity

text: when

text: ρ

text: and

text: z

text: η.

text: As

text: z

text: gets

text: closer

text: to

text: η,

text: the

text: bound

text: grows

text: but

text: we

text: can

text: not

text: reach

text: z

text: η.

text: However,

text: the

text: values

text: that

text: we

text: can

text: achieve

text: in

text: this

text: model

text: are

text: still

text: better

text: than

text: in

text: a

text: Gaussian

text: copula

text: case

text: where

text: the

text: supremum

text: is

text: equal

text: to

text: Φ

text: −√z

text: In

text: this

text: t

text: model,

text: the

text: distribution

text: of

text: the

text: diﬀerence

text: between

text: the

text: two

text: Brownian

text: motions

text: is

text: less

text: degenerated

text: than

text: the

text: one

text: of

text: Proposition

text: in

text: the

text: sense

text: that

text: there

text: exists

text: two

text: distinct

text: sources

text: of

text: randomness

text: at

text: all

text: time.

text: Results

text: of

text: Proposition

text: are

text: illustrated

text: in

text: Figure

text: (a)

text: t

text: (b)

text: t

text: Figure

text: Survival

text: function

text: of

text: X

text: Y

text: n

text: in

text: the

text: multi-barrier

text: model

text: at

text: time

text: t

text: with

text: parameters

text: ν

text: η

text: and

text: ρ

text: for

text: diﬀerent

text: values

text: of

text: n

text: The

text: case

text: n

text: corresponds

text: to

text: the

text: Gaussian

text: case.

text: We

text: can

text: see

text: that

text: in

text: [ν,

text: η],

text: the

text: survival

text: function

text: is

text: increasing

text: with

text: n.

text: In

text: Figure

text: the

text: curves

text: for

text: n

text: n

text: and

text: n

text: are

text: the

text: same.

text: At

text: time

text: t

text: the

text: probability

text: to

text: cross

text: more

text: than

text: barrier

text: is

text: very

text: weak

text: then

text: the

text: Brownian

text: reﬂection

text: reﬂects

text: less

text: than

text: times

text: with

text: a

text: high

text: probability.

text: The

text: convergence

text: in

text: n

text: at

text: small

text: time

text: is

text: fast.

text: In

text: Figure

text: we

text: can

text: observe

text: the

text: diﬀerence

text: between

text: the

text: cases

text: n

text: n

text: and

text: n

text: at

text: time

text: t

text: The

text: survival

text: function

text: continues

text: to

text: grow.

text: The

text: results

text: are

text: conﬁrmed

text: with

text: Figure

text: The

text: higher

text: the

text: number

text: of

text: reﬂections

text: is,

text: more

text: X

text: Y

text: n

text: is

text: concentrated

text: in

text: the

text: region

text: [ν,

text: η].

text: However,

text: in

text: the

text: positive

text: part

text: of

text: the

text: plan,

text: X

text: Y

text: n

text: take

text: lower

text: ##LTFigure##

text: ##LTFigure##

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text: 32 THOMAS DESCHATRE Let fξ be the density of ξ. Let B be a Brownian motion independent from ξ. We enlarge the ﬁltration of B to take into account ξ. We consider the reﬂection of the Brownian motion B˜ξ. We have: (cid:16) (cid:17) (cid:104) (cid:16) (cid:17)(cid:105) √ √ √ √ (17) P B ≤ tΦ−1 (u) , B˜ξ ≤ tΦ−1 (v) = E P B ≤ tΦ−1 (u) , B˜ξ ≤ tΦ−1 (v) | ξ . t t t t (cid:16) (cid:17) Since B is independent from ξ, we have according to Proposition 2: √ √ P Bt ≤ tΦ−1 (u) , B˜tξ ≤ tΦ−1 (v) | ξ =v1Φ−(cid:16)1(u)−Φ−1(v)≥√2ξ + W (u, v)(cid:17)1Φ−1(u)−Φ−1(v)<√2ξ t t − − − √2ξ + Φ Φ 1 (M (u, 1 v)) t 1Φ−1(u)−Φ−1(v)<√2ξt Thus, the right hand si(cid:104)de of Equation (17) i(cid:105)s the su(cid:16)m of the three followin(cid:17)g terms: (i) √ − − − E Φ 1 (u) Φ 1 (v) (cid:104) v1Φ−1(u)−Φ−1(v)≥√2ξt (cid:105) = vFξ t (cid:16) 2 , (cid:17) (ii) √ − − − E ξ Φ 1 (u) Φ 1 (v) W (u, v)(cid:104)1Φ(cid:16)−1(u)−Φ−1(v)<√2ξ = W (u,(cid:17)v) F t 2(cid:105) , t (iii) E − − − √2ξ Φ Φ 1 (M (u, 1 v)) t 1Φ−1(u)−Φ−1(v)<√2ξt , that we denot(cid:90)e by I. (cid:16) (cid:17) We have: ∞ I = √+ (cid:16)Φ Φ−1 (M (u, 1 − v))(cid:17)− √2h fξ (h) dh t(cid:90)Φ−−1(u)−2Φ−1(ξv) √ Φ−(cid:16)1 (u) − Φ−1 (v) t (cid:17) = M (1 u, v) F t 2 ∞ − √2 √+ Φ(cid:48) Φ−1 (M (u, 1 − v)) − √2h Fξ (h) dh. t tΦ−1(u)−Φ−1(v) t (cid:90) 2 (cid:16) − (cid:17) Adding the three terms of Equation (17), since M (1 u, v) + W (u, v) = v, we obtain: ∞ Ctξ (u, v) = v − (cid:90)√2t √+tΦ−1(u)−Φ−1(v) Φ(cid:48) Φ(cid:16)−√1 (M(cid:0) (u, 1 − v)) − √2ht Fξ ((cid:1)h(cid:17)) dh 2 − − = v − Φ 1(M(1 u,v)) Φ(cid:48) (h) Fξ t Φ−1 (M (u, 1 − v)) − h dh −∞ 2 with Φ(cid:48) (x) = e√−2x2 , which achieves the proof. 2π ##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: THOMAS

text: DESCHATRE

text: Let

text: fξ

text: be

text: the

text: density

text: of

text: ξ.

text: Let

text: B

text: be

text: a

text: Brownian

text: motion

text: independent

text: from

text: ξ.

text: We

text: enlarge

text: the

text: ﬁltration

text: of

text: B

text: to

text: take

text: into

text: account

text: ξ.

text: We

text: consider

text: the

text: reﬂection

text: of

text: the

text: Brownian

text: motion

text: B˜ξ.

text: We

text: have:

label: equation

text: P

text: B

text: tΦ−1

text: (u)

text: B˜ξ

text: tΦ−1

text: (v)

text: E

text: P

text: B

text: tΦ−1

text: (u)

text: B˜ξ

text: tΦ−1

text: (v)

text: ξ

text: t

text: t

text: t

text: t

text: Since

text: B

text: is

text: independent

text: from

text: ξ,

text: we

text: have

text: according

text: to

text: Proposition

text: P

text: Bt

text: tΦ−1

text: (u)

text: B˜tξ

text: tΦ−1

text: (v)

text: ξ

text: W

text: (u,

text: t

text: t

text: √2ξ

text: Φ

text: Φ

text: (M

text: (u,

text: v))

text: t

text: Thus,

text: the

text: right

text: hand

text: of

text: Equation

text: the

text: of

text: the

text: three

text: terms:

text: (i)

text: E

text: Φ

text: (u)

text: Φ

text: (v)

text: vFξ

text: t

text: (ii)

text: E

text: ξ

text: Φ

text: (u)

text: Φ

text: (v)

text: W

text: (u,

text: W

text: F

text: t

text: t

text: (iii)

text: E

text: √2ξ

text: Φ

text: Φ

text: (M

text: (u,

text: v))

text: t

text: that

text: we

text: by

text: I.

text: We

text: have:

text: I

text: Φ−1

text: (M

text: (u,

text: fξ

text: (h)

text: dh

text: (u)

text: Φ−1

text: (v)

text: t

text: M

text: u,

text: v)

text: F

text: t

text: √2

text: Φ−1

text: (M

text: (u,

text: v))

text: Fξ

text: (h)

text: dh.

text: t

text: t

text: Adding

text: the

text: three

text: terms

text: of

text: Equation

text: since

text: M

text: u,

text: v)

text: W

text: (u,

text: v)

text: v,

text: we

text: obtain:

text: Ctξ

text: (u,

text: v)

text: v

text: (u,

text: v))

text: Fξ

text: dh

text: v

text: Φ

text: u,v))

text: (h)

text: Fξ

text: t

text: Φ−1

text: (M

text: (u,

text: v))

text: h

text: dh

text: with

text: (x)

text: which

text: achieves

text: the

text: proof.

text: 2π

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

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text: CONTROL THE DIFFERENCE BETWEEN TWO BROWNIAN MOTIONS WITH DYNAMIC COPULAE 33 [8, Proposition 2.1] states that (cid:90) (cid:16) (cid:16) (cid:0) (cid:1) (cid:17)(cid:17) (18) P (X +(cid:16)Y ≤ x)(cid:16)= 1(cid:0)∂ C(cid:1) u, FY(cid:17)(cid:17)x − FX −1 (u) du, x ∈ R. u 0 − − ∈ The existence of ∂ C u, FY x FX 1 (u) for u [0, 1] is assured by [8, Lemma 2.1]. u We also have − − − ∈ R (19) F Y (y) = 1 FY ( y) , y and, − − − ∈ (20) CX, Y (u, v) = u C (u, 1 v) , (u, v) [0, 1] , − − with CX, Y the copula of (X, Y ). − Equation (18) is also valid for(cid:90)(X(cid:104), Y ). Us(cid:16)ing Equ(cid:16)a(cid:0)tion(cid:1)(19) and E(cid:17)q(cid:17)u(cid:105)ation (20), we have (21) P (X − Y ≤ x) = 1 1 − ∂ C u, FY FX −1 (u) − x du, x ∈ R u 0 and (cid:90) (cid:16) (cid:16)(cid:0) (cid:1) (cid:17)(cid:17) (22) P (X − Y > x) = 1 ∂ C u, FY FX −1 (u) − x du, x ∈ R. u 0 ∈ C Let us suppose that CY,X and that X and Y have the same continous marginal distribution s function F. Let CY,X be the copula of (Y, X). We have CY,X (u, v) = CX,Y (v, u). However, CX,Y (v, u) = CX,Y (u, v) then CY,X (u, v) = CX,Y (u, v) and (cid:90) (cid:104) (cid:16) (cid:16) (cid:17)(cid:17)(cid:105) P − ≥ P − ≤ − (X Y x) = (Y X x) = (cid:90) 1 (cid:104)1 − ∂ CY,X (cid:16)u, F (cid:16)(F)−1 (u) + x(cid:17)(cid:17)(cid:105)du u 0 1 − − = 1 ∂ CX,Y u, F (F) 1 (u) + x du u P0 − ≤ − = (X Y x) (cid:16) (cid:17) using Equation (21). (cid:0) (cid:1) ∈ − √ − (i) Let ρ [ 1, 1]. We have S˜η (CG,ρ) = Φ η− . This function is decreasing in ρ and then −2(1 ρ) − the extremum are achieved for ρ = 1 and ρ = 1 and are equal to 0 and Φ η . 2 (ii) This is a direct application of the results of [15] where superior and inferior bounds on P (X + Y < η) are found and where X and Y are two random variables with known marginals. As P − ≥ − P − 1) sup (X Y η) = 1 inf (X Y < η) , ∈C C ∈C C −C C 2) Y and Y have the same law, ##LTLine## ##LTLine## ##LTLine##

text: CONTROL

text: THE

text: DIFFERENCE

text: BETWEEN

text: TWO

text: BROWNIAN

text: MOTIONS

text: WITH

text: DYNAMIC

text: COPULAE

text: Proposition

text: states

text: that

text: P

text: (X

text: u,

text: FX

text: −1

text: (u)

text: du,

text: x

text: R.

text: u

text: The

text: existence

text: of

text: C

text: u,

text: FY

text: x

text: FX

text: (u)

text: for

text: u

text: is

text: assured

text: by

text: Lemma

text: u

text: We

text: also

text: have

text: R

text: F

text: Y

text: (y)

text: FY

text: y)

text: y

text: and,

text: CX,

text: Y

text: (u,

text: v)

text: u

text: C

text: (u,

text: v)

text: (u,

text: v)

text: with

text: CX,

text: Y

text: the

text: copula

text: of

text: (X,

text: Y

text: Equation

text: is

text: also

text: valid

text: Y

text: and

text: we

text: have

text: P

text: (X

text: Y

text: x)

text: C

text: u,

text: FY

text: FX

text: −1

text: (u)

text: x

text: du,

text: x

text: R

text: u

text: and

text: P

text: (X

text: Y

text: x)

text: C

text: u,

text: FY

text: FX

text: −1

text: (u)

text: x

text: du,

text: x

text: R.

text: u

text: C

text: Let

text: us

text: suppose

text: that

text: CY,X

text: and

text: that

text: X

text: and

text: Y

text: have

text: the

text: same

text: continous

text: marginal

text: distribution

text: s

text: function

text: F.

text: Let

text: CY,X

text: be

text: the

text: copula

text: of

text: (Y,

text: X).

text: We

text: have

text: CY,X

text: (u,

text: v)

text: CX,Y

text: (v,

text: u).

text: However,

text: CX,Y

text: (v,

text: u)

text: CX,Y

text: (u,

text: v)

text: then

text: CY,X

text: (u,

text: v)

text: CX,Y

text: (u,

text: v)

text: and

text: P

text: P

text: (X

text: Y

text: x)

text: (Y

text: X

text: x)

text: CY,X

text: F

text: (u)

text: u

text: CX,Y

text: u,

text: F

text: (F)

text: (u)

text: x

text: du

text: u

text: (X

text: Y

text: x)

text: using

text: Equation

text: (i)

text: Let

text: ρ

text: We

text: have

text: S˜η

text: (CG,ρ)

text: Φ

text: η−

text: This

text: function

text: is

text: decreasing

text: in

text: ρ

text: and

text: then

text: ρ)

text: the

text: extremum

text: are

text: achieved

text: for

text: ρ

text: and

text: ρ

text: and

text: are

text: equal

text: to

text: and

text: Φ

text: η

text: (ii)

text: This

text: is

text: a

text: direct

text: application

text: of

text: the

text: results

text: of

text: where

text: superior

text: and

text: inferior

text: bounds

text: on

text: P

text: (X

text: Y

text: η)

text: are

text: found

text: and

text: where

text: X

text: and

text: Y

text: are

text: two

text: random

text: variables

text: with

text: known

text: marginals.

text: As

text: P

text: P

text: sup

text: (X

text: Y

text: η)

text: inf

text: (X

text: Y

text: η)

text: ∈C

text: C

text: ∈C

text: C

text: −C

text: C

text: Y

text: and

text: Y

text: have

text: the

text: same

text: law,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:0) (cid:1) CONTROL THE DIFFERENCE BETWEEN TWO BROWNIAN MOTIONS WITH DYNAMIC COPULAE 35 (cid:55)→ − Then ρ S˜ Cp is continuous on [ 1, 1], which achieves the proof. η ρ C (i) As the copulae of d are of the form (C ) ≥ , the demonstration of this part of the proposition G G,ρt t 0 is similar to the one of the static framework. (ii) Let (B1, B2) be two Brownian mo{tion≥ with copula} Cref,η2{. ≥B2 is the−n the reﬂ}ection of B1 according to the st(cid:16)opping (cid:17)time τ = inf t 0 : B1 = η (cid:16)= inf t 0(cid:17): B1 (cid:16)B2 =(cid:17)η . For t < τ, − ≥ − t 2 t t B1 B2 < η and for t τ, B1 B2 = η. Thus, we have: t t t t − Sη,t Cref,η2 = PCref,η2 (t ≥ τ) = PCref,η2 ssu≤pt Bs1 ≥ η2 = 2Φ 2√ηt according to Lemma 23. (cid:0) (cid:1) (cid:0) (cid:1) (cid:16) (cid:17) ∈ C C If C , the copula C is in and then according to Proposition 13 B t − P − ≥ ≤ P − ≥ √η sup B1 B2 η sup B1 B2 η = 2Φ , C∈CB C t t (cid:2) C(cid:0)∈C (cid:1)C(cid:3) t t 2 t (cid:104) (cid:16) (cid:17)(cid:105) which concludes this part of the proof. − C (iii) We want to prove that for all x in 0, 2Φ η , there exists C in such that S˜ (C) = x. Let ∈ (cid:104) −(cid:16) (cid:17)(cid:105) 2 η (cid:16) (cid:17) x 0, 2Φ √η . 2 T If x ∈ (cid:104)0,−Φ(cid:16) 2−√η(cid:17)t , w(cid:16)e con(cid:17)s(cid:105)ider the Gaussian dynamic copula with (ρs)s≥0 = 1 − 21t Φ−η1(x) 2 which is in [ 1, 1] and we have S (C ) = x. η,t G,ρ ∈ − − If x Φ √η , 2Φ √η , we(cid:16)cons(cid:17)ider the copula(cid:16)Cexp,h,λ de(cid:17)ﬁned by Equation (3). After 2 T 2 t some alge(cid:0)bra, we ﬁnd th(cid:1)at (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) √ PCexp,h,λ Bt1 − Bt2 ≥ x (cid:16)=  Φ2Φ(cid:17)2−√xt2−√x+t (cid:18)Φe−λx22x−√+(cid:19)4tλhh −−Φ(cid:18)Φ 2−x√2−x√t4t−h −λ2λt√2(cid:19)teλe82tλ8−2tλ−4xλ+4xλ+hλh iiff xx ≥< 22hh Then, √ − − (cid:0) Sη,t C(cid:1)exp,η2,λ = 2Φ 2√ηt − Φ 2√ηt − λ2 t eλ82t+λ4η As we hav(cid:0)e: (cid:1) (cid:16) (cid:17) (cid:55)→ ∞ 1) λ (cid:0)Sη,t Cex(cid:1)p,η2,λ is co(cid:16)ntinu(cid:17)ous on [0, ), −→ − 2) Sη,t Cexp,η2,λ λ→0 Φ 2√ηt , −→ − 3) Sη,t Cexp,η2,λ λ→∞ 2Φ 2√ηt , we can conclude. (cid:112) (i) This part of the proof can be done by induction. ≤ ≤ − − − (ii) For τ0 = 0 t(cid:16) τ1, Xt Y(cid:17)tn(cid:0)= (1 + ρ) B(cid:1) tX (cid:112) 1 ρ(cid:0)2BtY . The(cid:1)equality holds for k = 0. Let us suppose that the property true at rank k < n + 1, that is − − (cid:112) − − − − ≤ ≤ X Y n = 1 + ( 1)k ρ BX BX 1 ρ2 BY BY + α , τ t τ . t t t τk t τk k k k+1 ≤ ≤ − If τ t τ , Y n = ρB˜k + 1 ρ2BY then k k+1 t t t ##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: CONTROL

text: THE

text: DIFFERENCE

text: BETWEEN

text: TWO

text: BROWNIAN

text: MOTIONS

text: WITH

text: DYNAMIC

text: COPULAE

text: Then

text: ρ

text: S˜

text: Cp

text: is

text: continuous

text: on

text: which

text: achieves

text: the

text: proof.

text: η

text: ρ

text: C

text: (i)

text: As

text: the

text: copulae

text: of

text: d

text: are

text: of

text: the

text: form

text: (C

text: the

text: demonstration

text: of

text: this

text: part

text: of

text: the

text: proposition

text: G

text: G,ρt

text: t

text: is

text: similar

text: to

text: the

text: one

text: of

text: the

text: static

text: framework.

text: (ii)

text: Let

text: be

text: two

text: Brownian

text: mo{tion≥

text: with

text: copula}

text: is

text: the−n

text: the

text: reﬂ}ection

text: of

text: according

text: to

text: the

text: τ

text: inf

text: t

text: η

text: inf

text: t

text: For

text: t

text: τ,

text: t

text: t

text: t

text: η

text: and

text: for

text: t

text: τ,

text: η.

text: Thus,

text: we

text: have:

text: t

text: t

text: t

text: t

text: Sη,t

text: (t

text: τ)

text: ssu≤pt

text: η2

text: 2Φ

text: 2√ηt

text: according

text: to

text: Lemma

text: C

text: C

text: If

text: C

text: the

text: copula

text: C

text: is

text: in

text: and

text: then

text: according

text: to

text: Proposition

text: B

text: t

text: P

text: P

text: √η

text: sup

text: η

text: sup

text: η

text: 2Φ

text: C∈CB

text: C

text: t

text: t

text: t

text: t

text: t

text: which

text: concludes

text: this

text: part

text: of

text: the

text: proof.

text: C

text: (iii)

text: We

text: want

text: to

text: prove

text: that

text: for

text: all

text: x

text: in

text: 2Φ

text: η

text: there

text: exists

text: C

text: in

text: such

text: that

text: S˜

text: (C)

text: x.

text: Let

text: η

text: x

text: 2Φ

text: √η

text: T

text: If

text: x

text: the

text: Gaussian

text: dynamic

text: copula

text: with

text: which

text: is

text: in

text: and

text: we

text: have

text: S

text: (C

text: x.

text: η,t

text: G,ρ

text: If

text: x

text: Φ

text: √η

text: 2Φ

text: √η

text: the

text: by

text: Equation

text: After

text: T

text: t

text: some

text: we

text: ﬁnd

text: PCexp,h,λ

text: x

text: iiff

text: xx

text: Then,

text: Sη,t

text: 2Φ

text: 2√ηt

text: Φ

text: 2√ηt

text: λ2

text: t

text: As

text: we

text: λ

text: is

text: on

text: Sη,t

text: λ→0

text: Φ

text: 2√ηt

text: Sη,t

text: λ→∞

text: 2Φ

text: 2√ηt

text: we

text: can

text: conclude.

text: (i)

text: This

text: part

text: of

text: the

text: proof

text: can

text: be

text: done

text: by

text: induction.

text: (ii)

text: For

text: τ0

text: Xt

text: ρ)

text: tX

text: holds

text: for

text: k

text: Let

text: us

text: suppose

text: that

text: the

text: property

text: true

text: at

text: rank

text: k

text: n

text: that

text: is

text: X

text: Y

text: n

text: ρ

text: BX

text: BX

text: ρ2

text: BY

text: BY

text: α

text: τ

text: t

text: τ

text: t

text: t

text: t

text: τk

text: t

text: τk

text: k

text: k

text: If

text: τ

text: t

text: τ

text: Y

text: n

text: ρB˜k

text: then

text: k

text: t

text: t

text: t

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text: Fig. 6. Success rate and variance averaged by shelf level (L), orientation (C) and object (R) similarities were noticed between the performance of the The experiments carried out also showed the variation in 2 and 3-ﬁngered hand conﬁgurations, and between both 4- the grasping performance according to the objects pose. Fig. ﬁnger conﬁgurations. While the former pair achieved around 8 details the success rate according to the object orientation, 80% of successful grasp, the latter remained at around only averaged along the three shelf levels. Again the 2-ﬁngered 50%. hand achieves the highest success rate across all orientations. Concerning the middle level shelf, the 2-ﬁngered and In an overall picture, all hands performed best when the both of the hands with 4 ﬁngers showed roughly the same object has an orientation closer to 0 and 180 degrees (object’s performance with respect to the previous level. A more largest axis was perpendicular to the palm). interesting results is observed in the 3-ﬁngered hand, for which the success rate drops nearly 20% with relation to 1 2 finger the previous shelf. Finally, the top level surface stood as 0.9 3 finger 4 finger the most challenging level to grasp objects. At this level all 0.8 4 finger II the hand conﬁgurations struggle to grab objects; while the 0.7 two 4-ﬁngered hands kept achieving low scores, the 2 and 3- 0.6 ﬁngered hands experienced a 20% drop respect to the success 0.5 0.4 rate obtained in the previous level. 0.3 0.2 1 2 finger 0.1 0.9 3 finger 0.8 44 ffiinnggeerr II 0 0 90 180 270 θ [deg] 0.7 0.6 Fig. 8. Average success per hand conﬁguration according to the object orientation 0.5 0.4 The worst performance across all hands was obtained 0.3 0.2 when the objects orientation was closer to 270 degrees. One 0.1 might expect, as was the case in 0 and 180 degrees, a similar 0 performance between 270 and 90 degrees; while 3 out 4 Low Mid Top hands showed this expected behaviour, for the 2-ﬁngered Fig. 7. Average success of all hand conﬁgurations according to shelf level hand that was not the case. The difference in that perfor- where the grasp is attempted. mance (about 18%) is associated to the irregularities in the This plot shows that the worst outcome in all hand objects set shapes, strongly inﬂuenced by the performance at conﬁgurations is obtained at the top level surface. The reason the top-level surface. Another important thing to notice is that of this low success rate was observed to be that many objects the 3-ﬁngered conﬁguration remains as the most consistent were out of reach for all conﬁgurations. Due to the angle in along all orientations with a success rate around 60%, its which the ﬁngers were located at the start of the grasp, they variance was previously shown in Fig. 6. fail to even touch the object. An example of this situation An wider view of the results is shown in Fig. 9, which for the 2-ﬁngered hand is shown in Fig. 12, which depicts shows success rate for combinations of shelf level and object a failed case when trying to grab a memory stick at the orientation. From these graphs it can bee seen that the 2- top-level shelf. Due to the constrained wrist posture small ﬁngered hand’s highest success rate was achieved at the object generally were very few times grasped. While these low-level shelf with objects oriented around 90 degrees; conditions appear overly harsh, recall we are on purpose not while the worst performance resulted in a tie between the considering any planning but rather interested to evaluate two 4-ﬁngered hands on the mid-level shelf with objects at precisely performance in the absence of higher intelligence, 270 degrees. Whereas the low performance does not results or should it be relevant, when planning has great uncertainty. very surprising, the highest performance does. In fact, the four cases with the highest performance are observed when B. Object pose grasping objects at 90 degrees. This is related to the fact ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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: Fig.

text: Success

label: caption

text: rate

text: and

text: variance

text: averaged

text: by

text: shelf

text: level

text: (L),

text: orientation

text: (C)

text: and

text: object

text: (R)

text: similarities

text: were

text: noticed

text: between

text: the

text: performance

text: of

text: the

text: The

text: experiments

text: carried

text: out

text: also

text: showed

text: the

text: variation

text: in

text: and

text: hand

text: conﬁgurations,

text: and

text: between

text: both

text: the

text: grasping

text: performance

text: according

text: to

text: the

text: objects

text: pose.

text: Fig.

text: ﬁnger

text: conﬁgurations.

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text: finger

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text: Fig.

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text: 2 I. INTRODUCTION A clinical dual-energy computed tomography (DECT) scan comprises two images, corresponding to a map of photon attenuation coeﬃcients of diﬀerent spectral weighting. While primarily intended for beneﬁts in diagnostic radiological applications [1], DECT has become of particular interest in radiotherapy in recent years [2]. The combined information in the two images can be used to extract an electron-density and an eﬀective-atomic-number 3D map of the patient [3]. Such quantities can be employed in a physics-based approach to convert CT numbers to the required input quantity of treatment planning systems: electron density for conventional photon therapy; stopping-power ratio (SPR) for proton or heavier ion therapy. In particular for the latter, DECT methods oﬀer the potential to reduce treatment uncertainties [4–7]. For physics-based SPR prediction, an accurate and robust determination of the electron density is crucial, as it is responsible for about 95% of the variability in SPR. In contrast, determination of an eﬀective atomic number is not necessary for SPR prediction [8, 9]. Various algorithms for the determination of electron density and/or eﬀective atomic number with a clinical DECT scanner have been proposed [5, 10–14]. Most of these algorithms use the CT numbers as provided by the scanner in the two images for the high and low tube voltage as input. This decouples – to a certain extent – the source of uncertainty in the CT numbers originating from scanner technology and image reconstruction and the inherent accuracy of the algorithm to derive electron density. Driven by the demand for high accuracy, the complexity of published algorithms tends to increase, e.g. employing a sophisticated parameterization of the photon attenuation cross section [14]. This is in contrast to the requirements of clinical implementation, which favors a simple and robust approach in terms of practical calibration, required input parameters or computational demands. A correlation between complexity and practical clinical accuracy is not evident. We therefore investigated the seemingly simplest of available algorithms, which is a one-parametric linear superpo- sition of the two DECT images, referred to in this manuscript as ‘alpha blending’. Such an approach was originally empirically postulated by Saito [11], therein referred to as ‘dual-energy subtraction’. Meanwhile, the algorithm is com- mercially available in a clinical software product (syngo.via rho/Z, Siemens Healthineers, Forchheim, Germany). It was also successfully experimentally validated by Hu¨nemohr et al.[5], where a mean absolute deviation of measured from reference electron densities of 0.4% was found for tissue substitutes. Studies of other algorithms reported similar results [13, 14], without any clear advantage for one or the other method. While such studies comparing reference values and measurements of tissue substitutes are important as a ﬁrst performance test for a new algorithm, the determined accuracy cannot be easily generalized and translated one-to- one to clinical conditions. On the one hand, the reference electron density of tissue substitutes is itself subject to an uncertainty of often unknown, yet likely not negligible magnitude. On the other hand, results are inﬂuenced by CT-related uncertainties which depend on the speciﬁc experimental setup and scan protocol (e.g. type and number of used inserts, type/geometry/diameter of phantom, beam hardening correction). The results of such tissue-substitute measurements are thus not directly transferable to a patient scan. For example, the experimental setup – unlike the patient setup – is often optimized in some way (e.g. simple geometries, inserts always on central axis). Consequently, we suggest to separate the sources of uncertainty in the evaluation of algorithms for image-based electron-density determination. Assuming ideal CT numbers as input, we quantiﬁed the purely methodological uncer- tainty of the alpha-blending approach for arbitrary mixtures of human tissue. Secondly, we evaluated the uncertainties of a proposed calibration method. II. METHODS A. Electron-density assessment with dual-energy CT The core of the studied approach is a simple parameterization for the electron density relative to water, n(cid:98), linear superposition of the spectrally weighted attenuation coeﬃcient relative to water µ(cid:98) (µ(cid:98) ) for the higher (lower) tube h l voltage with a single parameter α: n(cid:98) = αµ(cid:98) + (1 − α)µ(cid:98) . (1) h l We chose this particular representation of equation 1, which is in image processing referred to as alpha blending, as it best illustrates the mathematical structure of an aﬃne combination and allows for an elegant form of the derived equations in section . Furthermore, the correspondence to the calculation of pseudo-monoenergetic CT images is obvious in this form [15]. Please note, that in the case of electron density as studied here, equation 1 actually corresponds to a weighted subtraction of the two attenuation coeﬃcients, as α > 1. Also, equation 1 can be (cid:48) transformed into equivalent formulations by transforming the parameter α, e.g. α = α − 1 as used by Saito [11].

text: I.

label: section

text: INTRODUCTION

text: A

text: clinical

text: dual-energy

text: computed

text: tomography

text: (DECT)

text: scan

text: comprises

text: two

text: images,

text: corresponding

text: to

text: a

text: map

text: of

text: photon

text: attenuation

text: coeﬃcients

text: of

text: diﬀerent

text: spectral

text: weighting.

text: While

text: primarily

text: intended

text: for

text: beneﬁts

text: in

text: diagnostic

text: radiological

text: applications

text: DECT

text: has

text: become

text: of

text: particular

text: interest

text: in

text: radiotherapy

text: in

text: recent

text: years

text: The

text: combined

text: information

text: in

text: the

text: two

text: images

text: can

text: be

text: used

text: to

text: extract

text: an

text: electron-density

text: and

text: an

text: eﬀective-atomic-number

text: map

text: of

text: the

text: patient

text: Such

text: quantities

text: can

text: be

text: employed

text: in

text: a

text: physics-based

text: approach

text: to

text: convert

text: CT

text: numbers

text: to

text: the

text: required

text: input

text: quantity

text: of

text: treatment

text: planning

text: systems:

text: electron

text: density

text: for

text: conventional

text: photon

text: therapy;

text: stopping-power

text: ratio

text: (SPR)

text: for

text: proton

text: or

text: heavier

text: ion

text: therapy.

text: In

text: particular

text: for

text: the

text: latter,

text: DECT

text: methods

text: oﬀer

text: the

text: potential

text: to

text: reduce

text: treatment

text: uncertainties

text: For

text: physics-based

text: SPR

text: prediction,

text: an

text: accurate

text: and

text: robust

text: determination

text: of

text: the

text: electron

text: density

text: is

text: crucial,

text: as

text: it

text: is

text: responsible

text: for

text: about

text: of

text: the

text: variability

text: in

text: SPR.

text: In

text: contrast,

text: determination

text: of

text: an

text: eﬀective

text: atomic

text: number

text: is

text: not

text: necessary

text: for

text: SPR

text: prediction

text: Various

text: algorithms

text: for

text: the

text: determination

text: of

text: electron

text: density

text: and/or

text: eﬀective

text: atomic

text: number

text: with

text: a

text: clinical

text: DECT

text: scanner

text: have

text: been

text: proposed

text: Most

text: of

text: these

text: algorithms

text: use

text: the

text: CT

text: numbers

text: as

text: provided

text: by

text: the

text: scanner

text: in

text: the

text: two

text: images

text: for

text: the

text: high

text: and

text: low

text: tube

text: voltage

text: as

text: input.

text: This

text: decouples

text: to

text: a

text: certain

text: extent

text: the

text: source

text: of

text: uncertainty

text: in

text: the

text: CT

text: numbers

text: originating

text: from

text: scanner

text: technology

text: and

text: image

text: reconstruction

text: and

text: the

text: inherent

text: accuracy

text: of

text: the

text: algorithm

text: to

text: derive

text: electron

text: density.

text: Driven

text: by

text: the

text: demand

text: for

text: high

text: accuracy,

text: the

text: complexity

text: of

text: published

text: algorithms

text: tends

text: to

text: increase,

text: e.g.

text: employing

text: a

text: sophisticated

text: parameterization

text: of

text: the

text: photon

text: attenuation

text: cross

text: section

text: This

text: is

text: in

text: contrast

text: to

text: the

text: requirements

text: of

text: clinical

text: implementation,

text: which

text: favors

text: a

text: simple

text: and

text: robust

text: approach

text: in

text: terms

text: of

text: practical

text: calibration,

text: required

text: input

text: parameters

text: or

text: computational

text: demands.

text: A

text: correlation

text: between

text: complexity

text: and

text: practical

text: clinical

text: accuracy

text: is

text: not

text: evident.

text: We

text: therefore

text: investigated

text: the

text: seemingly

text: simplest

text: of

text: available

text: algorithms,

text: which

text: is

text: a

text: one-parametric

text: linear

text: superpo-

text: sition

text: of

text: the

text: two

text: DECT

text: images,

text: referred

text: to

text: in

text: this

text: manuscript

text: as

text: ‘alpha

text: blending’.

text: Such

text: an

text: approach

text: was

text: originally

text: empirically

text: postulated

text: by

text: Saito

text: therein

text: referred

text: to

text: as

text: ‘dual-energy

text: subtraction’.

text: Meanwhile,

text: the

text: algorithm

text: is

text: com-

text: mercially

text: available

text: in

text: a

text: clinical

text: software

text: product

text: (syngo.via

text: rho/Z,

text: Siemens

text: Healthineers,

text: Forchheim,

text: Germany).

text: It

text: was

text: also

text: successfully

text: experimentally

text: validated

text: by

text: Hu¨nemohr

text: et

text: where

text: a

text: mean

text: absolute

text: deviation

text: of

text: measured

text: from

text: reference

text: electron

text: densities

text: of

text: was

text: found

text: for

text: tissue

text: substitutes.

text: Studies

text: of

text: other

text: algorithms

text: reported

text: similar

text: results

text: without

text: any

text: clear

text: advantage

text: for

text: one

text: or

text: the

text: other

text: method.

text: While

text: such

text: studies

text: comparing

text: reference

text: values

text: and

text: measurements

text: of

text: tissue

text: substitutes

text: are

text: important

text: as

text: a

text: ﬁrst

text: performance

text: test

text: for

text: a

text: new

text: algorithm,

text: the

text: determined

text: accuracy

text: cannot

text: be

text: easily

text: generalized

text: and

text: translated

text: one-to-

text: one

text: to

text: clinical

text: conditions.

text: On

text: the

text: one

text: hand,

text: the

text: reference

text: electron

text: density

text: of

text: tissue

text: substitutes

text: is

text: itself

text: subject

text: to

text: an

text: uncertainty

text: of

text: often

text: unknown,

text: yet

text: likely

text: not

text: negligible

text: magnitude.

text: On

text: the

text: other

text: hand,

text: results

text: are

text: inﬂuenced

text: by

text: CT-related

text: uncertainties

text: which

text: depend

text: on

text: the

text: speciﬁc

text: experimental

text: setup

text: and

text: scan

text: protocol

text: (e.g.

text: type

text: and

text: number

text: of

text: used

text: inserts,

text: type/geometry/diameter

text: of

text: phantom,

text: beam

text: hardening

text: correction).

text: The

text: results

text: of

text: such

text: tissue-substitute

text: measurements

text: are

text: thus

text: not

text: directly

text: transferable

text: to

text: a

text: patient

text: scan.

text: For

text: example,

text: the

text: experimental

text: setup

text: unlike

text: the

text: patient

text: setup

text: is

text: often

text: optimized

text: in

text: some

text: way

text: (e.g.

text: simple

text: geometries,

text: inserts

text: always

text: on

text: central

text: axis).

text: Consequently,

text: we

text: suggest

text: to

text: separate

text: the

text: sources

text: of

text: uncertainty

text: in

text: the

text: evaluation

text: of

text: algorithms

text: for

text: image-based

text: electron-density

text: determination.

text: Assuming

text: ideal

text: CT

text: numbers

text: as

text: input,

text: we

text: quantiﬁed

text: the

text: purely

text: methodological

text: uncer-

text: tainty

text: of

text: the

text: alpha-blending

text: approach

text: for

text: arbitrary

text: mixtures

text: of

text: human

text: tissue.

text: Secondly,

text: we

text: evaluated

text: the

text: uncertainties

text: of

text: a

text: proposed

text: calibration

text: method.

text: II.

text: METHODS

text: A.

text: Electron-density

text: assessment

text: with

text: dual-energy

text: CT

text: The

text: core

text: of

text: the

text: studied

text: approach

text: is

text: a

text: simple

text: parameterization

text: for

text: the

text: electron

text: density

text: relative

text: to

text: water,

text: linear

text: superposition

text: of

text: the

text: spectrally

text: weighted

text: attenuation

text: coeﬃcient

text: relative

text: to

text: water

text: for

text: the

text: higher

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text: 11 [11] Masatoshi Saito. Potential of dual-energy subtraction for converting CT numbers to electron density based on a single linear relationship. Medical Physics, 39(4):2021, 2012. [12] Guillaume Landry, Joao Seco, Mathieu Gaudreault, and Frank Verhaegen. Deriving eﬀective atomic numbers from DECT based on a parameterization of the ratio of high and low linear attenuation coeﬃcients. Physics in Medicine and Biology, 58(19):6851–6866, oct 2013. [13] Alexandra E. Bourque, Jean-Fran¸cois Carrier, and Hugo Bouchard. A stoichiometric calibration method for dual energy computed tomography. Physics in medicine and biology, 59(8):2059–88, 2014. [14] Joanne K van Abbema, Marc-Jan van Goethem, Marcel J W Greuter, Arjen van der Schaaf, Sytze Brandenburg, and Emiel R van der Graaf. Relative electron density determination using a physics based parameterization of photon interac- tions in medical DECT. Physics in Medicine and Biology, 60(9):3825, 2015. [15] Stefan Kuchenbecker, Sebastian Faby, Stefan Sawall, Michael Lell, and Marc Kachelrieß. Dual energy CT: How well can pseudo-monochromatic imaging reduce metal artifacts? Medical physics, 42(2):1023, 2015. [16] H Q Woodard and D R White. The composition of body tissues. The British Journal of Radiology, 59(708):1209–1218, dec 1986. [17] D. R. White, H. Q. Woodard, and S. M. Hammond. Average soft-tissue and bone models for use in radiation dosimetry. The British Journal of Radiology, 60(717):907–913, sep 1987. [18] M J Berger, J H Hubbell, S M Seltzer, J Chang, J S Coursey, R Sukumar, D S Zucker, and K Olsen. Xcom: Photon cross sections database. NIST Standard Reference Database, 8:87–3597, 1998. [19] JCGM. Evaluation of measurement data – guide to the expression of uncertainty in measurement (GUM). Report 100:2008, BIPM, S´evres: Joint Committee for Guides in Metrology / WG 1, 2008. [20] Masayoshi Tsukihara, Yoshiyuki Noto, Takahide Hayakawa, and Masatoshi Saito. Conversion of the energy-subtracted CT number to electron density based on a single linear relationship: an experimental veriﬁcation using a clinical dual-source CT scanner. Physics in Medicine and Biology, 58(9):N135–N144, may 2013. [21] Guillaume Landry, Brigitte Reniers, Patrick Vincent Granton, Bart Van Rooijen, Luc Beaulieu, Joachim E. Wildberger, and Frank Verhaegen. Extracting atomic numbers and electron densities from a dual source dual energy CT scanner: Experiments and a simulation model. Radiotherapy and Oncology, 100(3):375–379, 2011. [22] Daphne F. Jackson and D. J. Hawkes. X-ray attenuation coeﬃcients of elements and mixtures. Physics Reports, 70(3):169– 233, 1981. [23] Robert E Alvarez and Albert Macovski. Energy-selective reconstructions in X-ray computerized tomography. Physics in medicine and biology, 21(5):733–744, 1976.

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text: Medical

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text: Guillaume

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