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Research article Special Issues

Histogram tomography

  • In many tomographic imaging problems the data consist of integrals along lines or curves. Increasingly we encounter "rich tomography" problems where the quantity imaged is higher dimensional than a scalar per voxel, including vectors tensors and functions. The data can also be higher dimensional and in many cases consists of a one or two dimensional spectrum for each ray. In many such cases the data contain not just integrals along rays but the distribution of values along the ray. If this is discretized into bins we can think of this as a histogram. In this paper we introduce the concept of "histogram tomography". For scalar problems with histogram data this holds the possibility of reconstruction with fewer rays. In vector and tensor problems it holds the promise of reconstruction of images that are in the null space of related integral transforms. For scalar histogram tomography problems we show how bins in the histogram correspond to reconstructing level sets of function, while moments of the distribution are the x-ray transform of powers of the unknown function. In the vector case we suggest a reconstruction procedure for potential components of the field. We demonstrate how the histogram longitudinal ray transform data can be extracted from Bragg edge neutron spectral data and hence, using moments, a non-linear system of partial differential equations derived for the strain tensor. In x-ray diffraction tomography of strain the transverse ray transform can be deduced from the diffraction pattern the full histogram transverse ray transform cannot. We give an explicit example of distributions of strain along a line that produce the same diffraction pattern, and characterize the null space of the relevant transform.

    Citation: William R. B. Lionheart. Histogram tomography[J]. Mathematics in Engineering, 2020, 2(1): 55-74. doi: 10.3934/mine.2020004

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  • In many tomographic imaging problems the data consist of integrals along lines or curves. Increasingly we encounter "rich tomography" problems where the quantity imaged is higher dimensional than a scalar per voxel, including vectors tensors and functions. The data can also be higher dimensional and in many cases consists of a one or two dimensional spectrum for each ray. In many such cases the data contain not just integrals along rays but the distribution of values along the ray. If this is discretized into bins we can think of this as a histogram. In this paper we introduce the concept of "histogram tomography". For scalar problems with histogram data this holds the possibility of reconstruction with fewer rays. In vector and tensor problems it holds the promise of reconstruction of images that are in the null space of related integral transforms. For scalar histogram tomography problems we show how bins in the histogram correspond to reconstructing level sets of function, while moments of the distribution are the x-ray transform of powers of the unknown function. In the vector case we suggest a reconstruction procedure for potential components of the field. We demonstrate how the histogram longitudinal ray transform data can be extracted from Bragg edge neutron spectral data and hence, using moments, a non-linear system of partial differential equations derived for the strain tensor. In x-ray diffraction tomography of strain the transverse ray transform can be deduced from the diffraction pattern the full histogram transverse ray transform cannot. We give an explicit example of distributions of strain along a line that produce the same diffraction pattern, and characterize the null space of the relevant transform.


    In this article, we study the oscillatory behavior of the fourth-order neutral nonlinear differential equation of the form

    {(r(t)Φp1[w(t)])+q(t)Φp2(u(ϑ(t)))=0,r(t)>0, r(t)0, tt0>0, (1.1)

    where w(t):=u(t)+a(t)u(τ(t)) and the first term means the p-Laplace type operator (1<p<). The main results are obtained under the following conditions:

    L1: Φpi[s]=|s|pi2s, i=1,2,

    L2: rC[t0,) and under the condition

    t01r1/(p11)(s)ds=. (1.2)

    L3: a,qC[t0,), q(t)>0, 0a(t)<a0<, τ,ϑC[t0,), τ(t)t, limtτ(t)=limtϑ(t)=

    By a solution of (1.1) we mean a function u C3[tu,), tut0, which has the property r(t)(w(t))p11C1[tu,), and satisfies (1.1) on [tu,). We assume that (1.1) possesses such a solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [tu,), and otherwise it is called to be nonoscillatory. (1.1) is said to be oscillatory if all its solutions are oscillatory.

    We point out that delay differential equations have applications in dynamical systems, optimization, and in the mathematical modeling of engineering problems, such as electrical power systems, control systems, networks, materials, see [1]. The p-Laplace equations have some significant applications in elasticity theory and continuum mechanics.

    During the past few years, there has been constant interest to study the asymptotic properties for oscillation of differential equations with p-Laplacian like operator in the canonical case and the noncanonical case, see [2,3,4,11] and the numerical solution of the neutral delay differential equations, see [5,6,7]. The oscillatory properties of differential equations are fairly well studied by authors in [16,17,18,19,20,21,22,23,24,25,26,27]. We collect some relevant facts and auxiliary results from the existing literature.

    Liu et al. [4] studied the oscillation of even-order half-linear functional differential equations with damping of the form

    {(r(t)Φ(y(n1)(t)))+a(t)Φ(y(n1)(t))+q(t)Φ(y(g(t)))=0,Φ=|s|p2s, tt0>0,

    where n is even. This time, the authors used comparison method with second order equations.

    The authors in [9,10] have established sufficient conditions for the oscillation of the solutions of

    {(r(t)|y(n1)(t)|p2y(n1)(t))+ji=1qi(t)g(y(ϑi(t)))=0,j1, tt0>0,

    where n is even and p>1 is a real number, in the case where ϑi(t)υ (with rC1((0,),R), qiC([0,),R), i=1,2,..,j).

    We point out that Li et al. [3] using the Riccati transformation together with integral averaging technique, focuses on the oscillation of equation

    {(r(t)|w(t)|p2w(t))+ji=1qi(t)|y(δi(t))|p2y(δi(t))=0,1<p<, , tt0>0.

    Park et al. [8] have obtained sufficient conditions for oscillation of solutions of

    {(r(t)|y(n1)(t)|p2y(n1)(t))+q(t)g(y(δ(t)))=0,1<p<, , tt0>0.

    As we already mentioned in the Introduction, our aim here is complement results in [8,9,10]. For this purpose we discussed briefly these results.

    In this paper, we obtain some new oscillation criteria for (1.1). The paper is organized as follows. In the next sections, we will mention some auxiliary lemmas, also, we will use the generalized Riccati transformation technique to give some sufficient conditions for the oscillation of (1.1), and we will give some examples to illustrate the main results.

    For convenience, we denote

    A(t)=q(t)(1a0)p21Mp1p2(ϑ(t)), B(t)=(p11)εϑ2(t)ζϑ(t)r1/(p11)(t), ϕ1(t)=tA(s)ds,R1(t):=(p11)μt22r1/(p11)(t),ξ(t):=q(t)(1a0)p21Mp2p11ε1(ϑ(t)t)3(p21),η(t):=(1a0)p2/p1Mp2/(p12)2t(1r(δ)δq(s)ϑp21(s)sp21ds)1/(p11)dδ,ξ(t)=tξ(s)ds, η(t)=tη(s)ds,

    for some μ(0,1) and every M1,M2 are positive constants.

    Definition 1. A sequence of functions {δn(t)}n=0 and {σn(t)}n=0 as

    δ0(t)=ξ(t), and σ0(t)=η(t),δn(t)=δ0(t)+tR1(t)δp1/(p11)n1(s)ds, n>1σn(t)=σ0(t)+tσp1/(p11)n1(s)ds, n>1. (2.1)

    We see by induction that δn(t)δn+1(t) and σn(t)σn+1(t) for tt0, n>1.

    In order to discuss our main results, we need the following lemmas:

    Lemma 2.1. [12] If the function w satisfies w(i)(ν)>0, i=0,1,...,n, and w(n+1)(ν)<0  eventually. Then, for every ε1(0,1), w(ν)/w(ν)ε1ν/n eventually.

    Lemma 2.2. [13] Let u(t) be a positive and n-times differentiable function on an interval [T,) with its nth derivative u(n)(t) non-positive on [T,) and not identically zero on any interval of the form [T,), TT and u(n1)(t)u(n)(t)0, ttu then there exist constants θ, 0<θ<1  and ε>0 such that

    u(θt)εtn2u(n1)(t),

    for all sufficient large t.

    Lemma 2.3 [14] Let uCn([t0,),(0,)). Assume that u(n)(t) is of fixed sign and not identically zero on [t0,) and that there exists a t1t0 such that u(n1)(t)u(n)(t)0 for all tt1. If limtu(t)0, then for every μ(0,1) there exists tμt1 such that

    u(t)μ(n1)!tn1|u(n1)(t)| for ttμ.

    Lemma 2.4. [15] Assume that (1.2) holds and u is an eventually positive solution of (1.1). Then, (r(t)(w(t))p11)<0 and there are the following two possible cases eventually:

    (G1) w(k)(t)>0, k=1,2,3,(G2) w(k)(t)>0, k=1,3, and w(t)<0.

    Theorem 2.1. Assume that

    liminft1ϕ1(t)tB(s)ϕp1(p11)1(s)ds>p11pp1(p11)1. (2.2)

    Then (1.1) is oscillatory.

    proof. Assume that u be an eventually positive solution of (1.1). Then, there exists a t1t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for tt1. Since r(t)>0, we have

    w(t)>0, w(t)>0, w(t)>0, w(4)(t)<0 and (r(t)(w(t))p11)0, (2.3)

    for tt1. From definition of w, we get

    u(t)w(t)a0u(τ(t))w(t)a0w(τ(t))(1a0)w(t),

    which with (1.1) gives

    (r(t)(w(t))p11)q(t)(1a0)p21wp21(ϑ(t)). (2.4)

    Define

    ϖ(t):=r(t)(w(t))p11wp11(ζϑ(t)). (2.5)

    for some a constant ζ(0,1). By differentiating and using (2.4), we obtain

    ϖ(t)q(t)(1a0)p21wp21(ϑ(t)).wp11(ζϑ(t))(p11)r(t)(w(t))p11w(ζϑ(t))ζϑ(t)wp1(ζϑ(t)).

    From Lemma 2.2, there exist constant ε>0, we have

    ϖ(t)q(t)(1a0)p21wp2p1(ϑ(t))(p11)r(t)(w(t))p11εϑ2(t)w(ϑ(t))ζϑ(t)wp1(ζϑ(t)).

    Which is

    ϖ(t)q(t)(1a0)p21wp2p1(ϑ(t))(p11)εr(t)ϑ2(t)ζϑ(t)(w(t))p1wp1(ζϑ(t)),

    by using (2.5) we have

    ϖ(t)q(t)(1a0)p21wp2p1(ϑ(t))(p11)εϑ2(t)ζϑ(t)r1/(p11)(t)ϖp1/(p11)(t). (2.6)

    Since w(t)>0, there exist a t2t1 and a constant M>0 such that

    w(t)>M.

    Then, (2.6), turns to

    ϖ(t)q(t)(1a0)p21Mp2p1(ϑ(t))(p11)εϑ2(t)ζϑ(t)r1/(p11)(t)ϖp1/(p11)(t),

    that is

    ϖ(t)+A(t)+B(t)ϖp1/(p11)(t)0.

    Integrating the above inequality from t to l, we get

    ϖ(l)ϖ(t)+ltA(s)ds+ltB(s)ϖp1/(p11)(s)ds0.

    Letting l and using ϖ>0 and ϖ<0, we have

    ϖ(t)ϕ1(t)+tB(s)ϖp1/(p11)(s)ds.

    This implies

    ϖ(t)ϕ1(t)1+1ϕ1(t)tB(s)ϕp1/(p11)1(s)(ϖ(s)ϕ1(s))p1/(p11)ds. (2.7)

    Let λ=inftTϖ(t)/ϕ1(t) then obviously λ1. Thus, from (2.2) and (2.7) we see that

    λ1+(p11)(λp1)p1/(p11)

    or

    λp11p1+(p11)p1(λp1)p1/(p11),

    which contradicts the admissible value of λ1 and (p11)>0.

    Therefore, the proof is complete.

    Theorem 2.2. Assume that

    liminft1ξ(t)tR1(s)ξp1/(p11)(s)ds>(p11)pp1/(p11)1 (2.8)

    and

    liminft1η(t)t0η2(s)ds>14. (2.9)

    Then (1.1) is oscillatory.

    proof. Assume to the contrary that (1.1) has a nonoscillatory solution in [t0,). Without loss of generality, we let u be an eventually positive solution of (1.1). Then, there exists a t1t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for tt1. From Lemma 2.4 there is two cases (G1) and (G2).

    For case (G1). Define

    ω(t):=r(t)(w(t))p11wp11(t).

    By differentiating ω and using (2.4), we obtain

    ω(t)q(t)(1a0)p21wp21(ϑ(t))wp11(t)(p11)r(t)(w(t))p11wp1(t)w(t). (2.10)

    From Lemma 2.1, we get

    w(t)w(t)3ε1t.

    Integrating again from t to ϑ(t), we find

    w(ϑ(t))w(t)ε1ϑ3(t)t3. (2.11)

    It follows from Lemma 2.3 that

    w(t)μ12t2w(t), (2.12)

    for all μ1(0,1) and every sufficiently large t. Since w(t)>0, there exist a t2t1 and a constant M>0 such that

    w(t)>M, (2.13)

    for tt2. Thus, by (2.10), (2.11), (2.12) and (2.13), we get

    ω(t)+q(t)(1a0)p21Mp2p11ε1(ϑ(t)t)3(p21)+(p11)μt22r1/(p11)(t)ωp1/(p11)(t)0,

    that is

    ω(t)+ξ(t)+R1(t)ωp1/(p11)(t)0. (2.14)

    Integrating (2.14) from t to l, we get

    ω(l)ω(t)+ltξ(s)ds+ltR1(s)ωp1/(p11)(s)ds0.

    Letting l and using ω>0 and ω<0, we have

    ω(t)ξ(t)+tR1(s)ωp1/(p11)(s)ds. (2.15)

    This implies

    ω(t)ξ(t)1+1ξ(t)tR1(s)ξp1/(p11)(s)(ω(s)ξ(s))p1/(p11)ds. (2.16)

    Let λ=inftTω(t)/ξ(t) then obviously λ1. Thus, from (2.8) and (2.16) we see that

    λ1+(p11)(λp1)p1/(p11)

    or

    λp11p1+(p11)p1(λp1)p1/(p11),

    which contradicts the admissible value of λ1 and (p11)>0.

    For case (G2). Integrating (2.4) from t to m, we obtain

    r(m)(w(m))p11r(t)(w(t))p11mtq(s)(1a0)p21wp21(ϑ(s))ds. (2.17)

    From Lemma 2.1, we get that

    w(t)ε1tw(t) and hence w(ϑ(t))ε1ϑ(t)tw(t). (2.18)

    For (2.17), letting mand using (2.18), we see that

    r(t)(w(t))p11ε1(1a0)p21wp21(t)tq(s)ϑp21(s)sp21ds.

    Integrating this inequality again from t to , we get

    w(t)ε1(1a0)p2/p1wp2/p1(t)t(1r(δ)δq(s)ϑp21(s)sp21ds)1/(p11)dδ, (2.19)

    for all ε1(0,1). Define

    y(t)=w(t)w(t).

    By differentiating y and using (2.13) and (2.19), we find

    y(t)=w(t)w(t)(w(t)w(t))2y2(t)(1a0)p2/p1M(p2/p1)1t(1r(δ)δq(s)ϑp21(s)sp21ds)1/(p11)dδ, (2.20)

    hence

    y(t)+η(t)+y2(t)0. (2.21)

    The proof of the case where (G2) holds is the same as that of case (G1). Therefore, the proof is complete.

    Theorem 2.3. Let δn(t) and σn(t) be defined as in (2.1). If

    limsupt(μ1t36r1/(p11)(t))p11δn(t)>1 (2.22)

    and

    limsuptλtσn(t)>1, (2.23)

    for some n, then (1.1)is oscillatory.

    proof. Assume to the contrary that (1.1) has a nonoscillatory solution in [t0,). Without loss of generality, we let u be an eventually positive solution of (1.1). Then, there exists a t1t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for tt1. From Lemma 2.4 there is two cases.

    In the case (G1), proceeding as in the proof of Theorem 2.2, we get that (2.12) holds. It follows from Lemma 2.3 that

    w(t)μ16t3w(t). (2.24)

    From definition of ω(t) and (2.24), we have

    1ω(t)=1r(t)(w(t)w(t))p111r(t)(μ16t3)p11.

    Thus,

    ω(t)(μ1t36r1/(p11)(t))p111.

    Therefore,

    limsuptω(t)(μ1t36r1/(p11)(t))p111,

    which contradicts (2.22).

    The proof of the case where (G2) holds is the same as that of case (G1). Therefore, the proof is complete.

    Corollary 2.1. Let δn(t) and σn(t) be defined as in (2.1). If

    t0ξ(t)exp(tt0R1(s)δ1/(p11)n(s)ds)dt= (2.25)

    and

    t0η(t)exp(tt0σ1/(p11)n(s)ds)dt=, (2.26)

    for some n, then (1.1) is oscillatory.

    proof. Assume to the contrary that (1.1) has a nonoscillatory solution in [t0,). Without loss of generality, we let u be an eventually positive solution of (1.1). Then, there exists a t1t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for tt1. From Lemma 2.4 there is two cases (G1) and (G2).

    In the case (G1), proceeding as in the proof of Theorem 2, we get that (2.15) holds. It follows from (2.15) that ω(t)δ0(t).  Moreover, by induction we can also see that ω(t)δn(t) for tt0, n>1. Since the sequence {δn(t)}n=0 monotone increasing and bounded above, it converges to δ(t). Thus, by using Lebesgue's monotone convergence theorem, we see that

    δ(t)=limnδn(t)=tR1(t)δp1/(p11)(s)ds+δ0(t)

    and

    δ(t)=R1(t)δp1/(p11)(t)ξ(t). (2.27)

    Since δn(t)δ(t), it follows from (2.27) that

    δ(t)R1(t)δ1/(p11)n(t)δ(t)ξ(t).

    Hence, we get

    δ(t)exp(tTR1(s)δ1/(p11)n(s)ds)(δ(T)tTξ(s)exp(sTR1(δ)δ1/(p11)n(δ)dδ)ds).

    This implies

    tTξ(s)exp(sTR1(δ)δ1/(p11)n(δ)dδ)dsδ(T)<,

    which contradicts (2.25). The proof of the case where (G2) holds is the same as that of case (G1). Therefore, the proof is complete.

    Example 2.1. Consider the differential equation

    (u(t)+12u(t2))(4)+q0t4u(t3)=0,  (2.28)

    where q0>0 is a constant. Let p1=p2=2, r(t)=1, a(t)=1/2, τ(t)=t/2, ϑ(t)=t/3 and q(t)=q0/t4. Hence, it is easy to see that

    A(t)=q(t)(1a0)(p21)Mp2p1(ϑ(t))=q02t4, B(t)=(p11)εϑ2(t)ζϑ(t)r1/(p11)(t)=εt227

    and

    ϕ1(t)=q06t3,

    also, for some ε>0, we find

    liminft1ϕ1(t)tB(s)ϕp1/(p11)1(s)ds>(p11)pp1/(p11)1.liminft6εq0t3972tdss4>14q0>121.5ε.

    Hence, by Theorem 2.1, every solution of Eq (2.28) is oscillatory if q0>121.5ε.

    Example 2.2. Consider a differential equation

    (u(t)+a0u(τ0t))(n)+q0tnu(ϑ0t)=0, (2.29)

    where q0>0 is a constant. Note that p=2, t0=1, r(t)=1, a(t)=a0, τ(t)=τ0t, ϑ(t)=ϑ0t  and q(t)=q0/tn.

    Easily, we see that condition (2.8) holds and condition (2.9) satisfied.

    Hence, by Theorem 2.2, every solution of Eq (2.29) is oscillatory.

    Remark 2.1. Finally, we point out that continuing this line of work, we can have oscillatory results for a fourth order equation of the type:

    {(r(t)|y(t)|p12y(t))+a(t)f(y(t))+ji=1qi(t)|y(σi(t))|p22y(σi(t))=0,tt0, σi(t)t, j1,, 1<p2p1<.

    The paper is devoted to the study of oscillation of fourth-order differential equations with p-Laplacian like operators. New oscillation criteria are established by using a Riccati transformations, and they essentially improves the related contributions to the subject.

    Further, in the future work we get some Hille and Nehari type and Philos type oscillation criteria of (1.1) under the condition υ01r1/(p11)(s)ds<.

    The authors express their debt of gratitude to the editors and the anonymous referee for accurate reading of the manuscript and beneficial comments.

    The author declares that there is no competing interest.



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