Weighted Radon transforms for which the Chang approximate inversion formula is precise

0.07 MB·English

by  Roman Novikov

Weighted Radon transforms for which
the Chang approximate inversion formula is precise
R.G. Novikov
CNRS (UMR 7641), Centre de Math´ematques Appliqu´ees, Ecole Polytechnique,
91128 Palaiseau, France and
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IIEPT RAS - MITPAN, Profsoyuznaya str., 84/32, Moscow 117997, Russia
1
0 e-mail: [email protected]
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n Abstract. We describe all weighted Radon transforms on the plane for which the
a
J Chang approximate inversion formula is precise. Some subsequent results, including the

7 Cormack type inversion for these transforms, are also given.
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1.Introduction

]
A We consider the weighted ray transformation P defined by the formula
W
F
. ⊥ ⊥
h P f(s,θ) = W(sθ +tθ,θ)f(sθ +tθ)dt,
W
t Z
a R (1)
m
[ s ∈ R, θ = (θ1,θ2) ∈ S1, θ⊥ = (−θ2,θ1),

2
v where W = W(x,θ) is the weight, f = f(x) is a test function, x ∈ R, θ ∈ S1. Up to change
5 of variables, P is known also as weighted Radon transform on the plane.
W
2
7 Werecallthatindefinition(1)theproductR×S1 isinterpretedasthesetofalloriented
1 straight lines in R2. If γ = (s,θ) ∈ R × S1, then γ = {x ∈ R2 : x = sθ⊥ + tθ, t ∈ R}
.
1 (modulo orientation) and θ gives the orientation of γ.
0
We assume that
1
1
: W is complex−valued,
v
i W ∈ C(R2 ×S1)∩L∞(R2 ∩S1),
X
(2)
ar w (x) d=ef 1 W(x,θ)dθ 6= 0, x ∈ R2,
0
2π Z
S1
where dθ is the standard element of arc length on S1.
If W ≡ 1, then P is known as the classical ray (or Radon) transform on the plane.
W
If
W(x,θ) = exp(−Da(x,θ)),
+∞
(3)
Da(x,θ) = a(x+tθ)dt,
Z
0
where a is a complex-valued sufficiently regular function on R2 with sufficient decay at
infinity, then P is known as the attenuated ray (or Radon) transform.
W
The classical Radon transform arises, in particular, in the X-ray transmission tomog-
raphy. The attenuated Radon transform (at least, with a ≥ 0) arises, in particular, in the
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single photon emission computed tomography (SPECT). Some other weights W also arise
in applications. For more information in this connection see, for example, [Na], [K].
Preciseandsimultaneouslyexpliciteinversionformulasfortheclassicalandattenuated
Radon transforms were given for the first time in [R] and [No], respectively. For some other
weights W precise and simultaneously explicite inversion formulas were given in [BS], [G].
On the other hand, the following Chang approximate inversion formula for P , where
W
W is given by (3) with a ≥ 0, is used for a long time, see [Ch], [M], [K]:
1 d
′ ⊥ ′
f (x) = h (xθ ,θ)dθ, h (s,θ) = h(s,θ),
appr
4πw (x) Z ds
0
S1
(4)
1 P f(t,θ)
h(s,θ) = p.v. W dt, s ∈ R, θ ∈ S1, x ∈ R2,
π Z s−t
R
where w is defined in (2). It is known that (4) is efficient as the first approximation in
0
SPECT reconstructions and, in particular, is sufficiently stable to the strong Poisson noise
of SPECT data. The results of the present note consist of the following:
(1) In Theorem, under assumptions (2), we describe all weights W for which the Chang
approximate inversion formula (4) is precise, that is f ≡ f on R2;
appr
(2) For P with W of Theorem we give also the Cormack type inversion (see Remark A)
W
and inversion from limited angle data (see Remark B).
These results are presented in detail in the next section. In addition, we give also an
explanation of efficiency of the Chang formula (4) as the first approximation in SPECT
reconstructions (on the level of integral geometry).
2. Results
Let
C (R2) denote the space of continuous
0
(5)
compactly supported functions on R2.
Let
L∞,σ(R2) = {f : Mσf ∈ L∞(R2)},
(6)
Mσf(x) = (1+|x|)σf(x), x ∈ R2, σ ≥ 0.
Theorem. Let assumptions (2) hold and let f (x) be given by (4). Then
appr
f = f (in the sense of distributios) on R2 for all f ∈ C (R2) (7)
appr 0
if and only if
W(x,θ)−w (x) ≡ w (x)−W(x,−θ), x ∈ R2, θ ∈ S1. (8)
0 0
(This result remains valid with C (R2) replaced by L∞,σ(R2) for σ > 1.)
0
Theorem 1 is based on the following facts:
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• Formula (4) coincides with the classical Radon inversion formula if W ≡ 1 and, as a
corollary, is precise if W ≡ w .
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• Formula (4) is equivalent to the symmetrized formula
1
f (x) = g′(xθ⊥,θ)dθ, x ∈ R2,
appr
4πw (x) Z
0
S1
(9)
1 P f(t,θ)+P f(−t,−θ)
g(s,θ) = p.v. W W dt, (s,θ) ∈ R×S1.
2π Z s−t
R
• The following formula holds:
1
(P f(s,θ)+P f(−s,−θ)) = P f(s,θ),(s,θ)∈ R×S1,
W W W
2 sym
(10)
1
W (x,θ) = (W(x,θ)+W(x,−θ)), x ∈ R2, θ ∈ S1.
sym
2
• If
q ∈ C(R×S1), suppq is compact, q(s,θ) = q(−s,−θ),
1 q(t,θ)
g(s,θ) = p.v. dt, (s,θ) ∈ R×S1,
π Z s−t
R (11)
g′(xθ⊥,θ)dθ ≡ 0 as a distribution of x ∈ R2,
Z
S1
then q ≡ 0 on R×S1.
The statement that, under the assumptions of Theorem, property (8) implies (7) can
be also deduced from considerations developed in [K].
Using that W ≡ w under condition (8), we obtain also the following
sym 0
Remarks. Let conditions (2), (8) be fulfiled. Let f ∈ C (R2). Then:
0
(A) P f on Ω(D) uniquely determines f (or more precisely w f) on R2\D via (10) and
W 0
the Cornack inversion from P f on Ω(D), where D is a compact in R2, Ω(D)
W
sym
denotes the set of all straight lines in R2 which do not intersect D;
(B) PWf on R×(S+ ∪S−) uniquely determines f on R2 via (10) and standard inversion
from the limited angle data P f on R×S , where S is an arbitrary nonempty
W + +
sym
open connected subset of S1, S− = −S+.
For the case when W is given by (3) under the additional conditions that a ≥ 0
and suppa ⊂ D, where D is some known bounded domain which is not too big, and for
f ∈ C(R2), f ≥ 0, suppf ⊂ D, the transform P f is relatively well approximated by
W
P f, where W (x,θ) = w (x)+(1/2)(W(x,θ)−W(x,−θ)). In addition, this W
W appr 0 appr
appr
already satisfies (8). This explains the efficiency of (4) as the first aproximation in SPECT
reconstructions (on the level of integral geometry).
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References
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