Non-abelian Radon transform and its applications

25 Pages·2017·0.13 MB·English

by  Roman Novikov*

Non-abelian Radon transform and its applications
∗
Roman Novikov
∗ CNRS,CentredeMath´ematiques Appliqu´ees, EcolePolytechnique
March 23, 2017
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Consider the equation
θ∂ ψ+A(x,θ)ψ = 0, x Rd, θ Sd−1, (1)
x
∈ ∈
where A is a sufficiently regular function on Rd Sd−1 with
×
sufficient decay as x .
| | → ∞
We assume that A and ψ take values in that is in n n
n,n
M ×
complex matrices.
Consider the ”scattering” matrix S for equation (1):
S(x,θ) = lim ψ+(x +sθ,θ), (x,θ) TSd−1, (2)
s→+∞ ∈
where
TSd−1 = (x,θ) Rd Sd−1 : xθ = 0 (3)
{ ∈ × }
and ψ+(x,θ) is the solution of (1) such that
lim ψ+(x +sθ,θ)= Id, x Rd, θ Sd−1. (4)
s→−∞ ∈ ∈
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We interpret TSd−1 as the set of all rays in Rd. As a ray γ we
understand a straight line with fixed orientation.
If γ = (x,θ) TSd−1, then γ = y Rd : y = x +tθ, t R
∈ { ∈ ∈ }
(up to orientation) and θ gives the orientation of γ.
We say that S is the non-abelian Radon transform along oriented
straight lines (or the non-abelian X-ray transform) of A.
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We consider the following inverse problem:
Problem 1. Given S, find A.
Note that S does not determine A uniquely, in general. One of the
reasons is that S is a function on TSd−1, whereas A is a function
on Rd Sd−1 and
×
dimRd Sd−1 = 2d 1 > dimTSd−1 = 2d 2.
× − −
In particular, for Problem 1 there are gauge type non-uniqueness,
non-uniqueness related with solitons,
and Boman type non-uniqueness.
Equation (1), the ”scattering” matrix S and Problem 1 arise, for
example, in the following domains:
I. Tomographies:
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A. The classical X-ray transmission tomography:
n = 1, A(x,θ)= a(x), x Rd, θ Sd−1, (5a)
∈ ∈
S(γ) = exp[ Pa(γ)], Pa(γ) = a(x+sθ)ds, γ = (x,θ) TSd−1,
− Z ∈
R
(5b)
where a is the X-ray attenuation coefficient of the medium, P is
the classical Radon transformation along straight lines (classical
ray transformation), S(γ) describes the X-ray photograph along γ.
In this case, for d 2,
≥
S uniquely determines a , (6)
TS1(Y) Y
(cid:12) (cid:12)
(cid:12) (cid:12)
where Y is an arbitrary two-dimensional plane in Rd, TS1(Y) is
the set of all oriented straight lines in Y. In addition, this
determination can be implemented via the Radon inversion formula
for P in dimension d = 2.
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B. Single-photon emission computed tomography (SPECT):
In SPECT one considers a body containing radioactive isotopes
emitting photons. The emission data p in SPECT consist in the
radiation measured outside the body by a family of detectors
during some fixed time. The basic problem of SPECT consists in
finding the distribution f of these isotopes in the body from the
emission data p and some a priori information concerning the
body. Usually this a priori information consists in the photon
attenuation coefficient a in the points of body, where this
coefficient is found in advance by the methods of the classical
X-ray transmission tomography.
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Problem 1 arises as a problem of SPECT in the framework of the
following reduction [R.Novikov 2002 a]: n = 2,
A = a(x), A = f(x), A = 0, A = 0, x Rd, (7a)
11 12 21 22
∈
S =exp[ P a], S = P f, S = 0, S = 1, (7b)
11 0 12 a 21 22
− −
P f(γ) = exp[ Da(x +sθ,θ)]f(x +sθ)ds, γ = (x,θ) TSd−1,
a
Z − ∈
R
(8)
+∞
Da(x,θ)= a(x +sθ)ds, x Rd, θ Sd−1,
Z ∈ ∈
0
where f 0 is the density of radioactive isotopes, a 0 is the
≥ ≥
photon attenuation coefficient of the medium, P is the attenuated
a
Radon transformation (along oriented straight lines), P f describes
a
the expected emission data.
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In this case (as well as for the case of the classical X-ray
transmission tomography), for d 2,
≥
S uniquely determines a and f , (9)
TS1(Y) Y Y
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
where Y is an arbitrary two-dimensional plane in Rd, TS1(Y) is
the set of all oriented straight lines in Y. In addition, this
determination can be implemented via the following inversion
formula [R.Novikov 2002b]:
f = P−1g, where g = P f,
a a
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1
P−1g(x) = θ⊥∂ exp[ Da(x, θ)]g˜ (θ⊥x) dθ, (10a)
a 4π Z x − − θ
(cid:0) (cid:1)
S
g˜ (s) = exp(A (s))cos(B (s))H(exp(A )cos(B )g )(s)+
θ θ θ θ θ θ
exp(A (s))sin(B (s))H(exp(A )sin(B )g )(s), (10b)
θ θ θ θ θ
⊥ ⊥
A (s) = (1/2)P a(sθ ,θ), B (s) = HA (s), g (s) = g(sθ ,θ),
θ 0 θ θ θ
(10c)
1 u(t)
Hu(s) = p.v. dt,
π Z s t
R −
x R2, θ⊥ = ( θ ,θ ) for θ = (θ ,θ ) S1, s R.
2 1 1 2
∈ − ∈ ∈
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C. Tomographies related with weighted Radon transforms:
We consider the weighted Radon transformations P defined by
W
the formula
P f(x,θ)= W(x +sθ,θ)f(x +sθ)ds, (x,θ) TSd−1, (11)
W Z ∈
R
where W = W(x,θ) is the weight, f = f(x) is a test function. We
assume that
W C(Rd Sd−1),
∈ ×
W = W¯, 0 < c W c , (12)
0 1
≤ ≤
lim W(x +sθ,θ)= w±(x,θ), (x,θ) TSd−1.
s→±∞ ∈
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