Elementary Inversion of Riesz Potentials and Radon-John Transforms

0.14 MB·English

by  Boris Rubin

ELEMENTARY INVERSION OF RIESZ POTENTIALS
AND RADON-JOHN TRANSFORMS
1
B. RUBIN
1
0
2
Abstract. New simple proofs are given to some elementary ap-

n proximateandexplicitinversionformulaeforRieszpotentials. The
a
results are applied to reconstruction of functions from their inte-
J
grals over Euclidean planes in integral geometry.
6
2

]
A
F 1. Introduction
.
h The Riesz potential of order α of a sufficiently good function f on
t
a Rn is defined by
m
[ 1 f(y)dy 2απn/2Γ(α/2)
(Iαf)(x) = , γ (α) = , (1.1)
1 γn(α) ZRn |x−y|n−α n Γ((n−α)/2)
v
5 Reα > 0, α 6= n,n+2,n+4,... .
0
1 This operator can be regarded (in a certain sense) as a negative power
5
of “minus-Laplacian”, namely,
.
1
0 ∂2 ∂2 ∂2
1 Iα = (−∆)−α/2, ∆ = + +...+ . (1.2)
∂x2 ∂x2 ∂x2
1 1 2 n
:
v If f ∈ Lp(Rn), then the integral (1.1) converges a.e. provided that
i
X 1 ≤ p < n/Reα. This condition is sharp. Explicit and approxi-
r mate inversion formulae for Iα are of great importance. We refer to
a
[7, 12, 21, 23, 26, 27, 30] for the basic properties and applications of
Riesz potentials. In this article we restrict to real α ∈ (0,n). This case
reflects basic features and is sufficient for integral geometric applica-
tions in Section 4.
Our mainworking toolsareasuitably chosen auxiliary functionw(x)
(we call it a reconstructing function) and its scaled version w (x) =
t
t−nw(x/t), t > 0. In the modern literature such a function is called a
2000 Mathematics Subject Classification. Primary 44A12; Secondary 47G10.
Keywordsandphrases. Rieszpotentials,Radontransforms,Wavelettransforms.
The work was supported by the NSF grants PFUND-137 (Louisiana Board of
Regents), DMS-0556157,and the Hebrew University of Jerusalem.
1
2 B. RUBIN
wavelet and a convolution (f ∗w )(x) is called the continuous wavelet
t
transform of f; cf. [2, 3]. We choose
w(x) = (−∆)m[(1+|x|2)m−(n+α)/2], w˜(x) = (−∆w)(x), (1.3)
m ∈ N, and keep this notation throughout the paper.
The main results of the article are presented by Theorem A below
and Theorem B in Section 4. The latter gives an example of applica-
tion of Theorem A to elementary inversion of the k-plane Radon-John
transform in integral geometry and falls into the general scope of the
convolution-backprojection method (cf. [15, 22]). Radon transforms
and their generalizations are the basic tools in tomography and numer-
ous related areas of pure and applied mathematics; see [1, 5, 6, 7, 8, 9,
10, 11, 14, 15, 20, 18] and references therein.
Theorem A. Let 2m > α, f ∈ Lp(Rn), 1 ≤ p < n/α. Then
c f = lim(Iαf ∗t−αw ), c = γ (2m−α), (1.4)
α,m t α,m n
t→0
∞ Iαf ∗w˜
t
d f = dt, d = (2m−α)c , (1.5)
α,m Z t1+α α,m α,m
0
∞ ∞
where (...) = lim (...) and γ (·) has the same meaning as in
0 ε n
ε→0
(1.1). TRhe limit in botRh formulae exists in the Lp-norm and in the a.e.
sense. If, moreover, f ∈ C (Rn), then it exists in the sup-norm.
0
We recall that C (Rn) denotes the space of continuous functions on
0
Rn vanishing at infinity.
Some comments are in order. Statements of this kind are not new.
Regarding (1.4), we observe that approximate inversion of operators of
thepotentialtypewasinitiatedbyZavolzhenskii, NoginandSamko[32,
17, 24, 25, 26]. Conceptually this approach is close to the convolution-
backprojection method in tomography [15]. An elegant formula (1.4)
is due to Samko [26, p. 325], [24, 25]; see also [19]. Below we give a
new simple proof of it.
Regarding (1.5), wavelet-like inversion formulae for Riesz potentials
are also well-known; see, e.g., [21, Section 17]. Our aim is to show that
elementary reconstructing functions (1.3) work perfectly in the wavelet
inversion scheme and the relevant justification is much simpler than
in the general theory; cf. [21]. The case α = 0 in (1.5), when Iα is
substituted by the identity operator, represents a variant of Caldero´n’s
reproducing formula [2, 3].
The validity of formulae (1.4) and (1.5) can be formally explained in
the language of the Fourier transform. We have
lim(Iαf ∗t−αw )∧(ξ) = fˆ(ξ) lim(t|ξ|)−αwˆ(tξ) = c fˆ(ξ) (1.6)
t w
t→0 t→0
ELEMENTARY INVERSION OF RIESZ POTENTIALS 3
provided that the limit c = lim |ξ|−αwˆ(ξ) exists. Moreover, since w˜
w
ξ→0
in (1.3) is a radial function, we easily get
∞ Iαf ∗w˜ ∧ 1 wˆ(y)dy
t ˆ
dt (ξ) = d f(ξ), d = , (1.7)
hZ0 t1+α (cid:21) w w σn−1 ZRn |y|n+α−2
where σ =2πn/2 Γ(n/2) is the surface area of the unit sphere |x|=1.
n−1
This simple argu(cid:14)ment does not work for arbitrary Lp-functions. To
cover this case, we develop another technique and give the final answer
without using the Fourier transform.
2. Preliminaries
We recall some known facts. The Fourier transform of a function
f ∈ L1 = L1(Rn) is defined by
fˆ(ξ) = f(x)eix·ξdx, x·ξ = x ξ +...+x ξ . (2.1)
1 1 n n
ZRn
Let S = S(Rn) be the Schwartz space of all C∞ functions which, to-
gether with derivatives of all orders, vanish at infinity faster than any
inverse power of |x| = (x2 + ... + x2)1/2. We endow S(Rn) with a
1 n
standard topology generated by the sequence of norms
||ϕ|| = max(1+|x|)m |(∂jϕ)(x)|, m = 0,1,2,... .
m
x
X
|j|≤m
The following spaces adapted to Riesz potentials were introduced by
Semyanistyi [28]; further generalizations due to Lizorkin and Samko
can be found in [13, 26].
Let Ψ = Ψ(Rn) be the subspace of S, consisting of functions ψ(ξ)
vanishing at ξ = 0 with all derivatives, and let Φ = Φ(Rn) be the
Fourier image of Ψ. We equip Φ with the induced topology of S. Then
Φ becomes a topological vector space which is isomorphic to Ψ under
the action of the Fourier transform. We denote by Φ′ the space of
distributions over Φ.
The main reason for introducing the spaces Ψ and Φ is that Ψ is
invariant under multiplication by |ξ|α for any α ∈ C and therefore, the
Riesz potential Iα is an automorphism of Φ and Φ′.
Proposition 2.1. [21, p. 21 ] Let f ∈ Lr, 1 ≤ r < ∞, and g ∈ Lp,
1 ≤ p < ∞. If f = g in the Φ′-sense, then f = g almost everywhere.
4 B. RUBIN
3. Proof of Theorem A
The following auxiliary propositions form the heart of the proof.
Lemma 3.1. Let w∈L1(Rn) be such that h = Iαw has an integrable
decreasing radial majorant. If f∈Lp(Rn), 1≤p<n/α, then
lim(Iαf ∗t−αw ) = c (α)f, c (α) = h(x)dx, (3.1)
t w w
t→0 ZRn
where the limit exists in the Lp-norm and in the a.e. sense. If, more-
over, f ∈ C (Rn), then the limit exists in the sup-norm.
0
Proof. We have t−α(Iαw )(x) = t−nh(x/t) = h (x). Then
t t
Iαf ∗t−αw = f ∗t−αIαw = f ∗h ,
t t t
whichyieldsanapproximateidentity[30, p. 62]. Theaboveapplication
of Fubini’s theorem is permissible because (Iα|f|)∗|w| < ∞ a.e. (cid:3)
There are many ways to choose a reconstructing function w. Usually
w or Iαw or both are expressed analytically in a pretty complicated
way. As we shall see below, anadvantage ofthe choice (1.3) isthat both
w and h = Iαw are elementary and a constant c (α) can be different
w
from zero.
Lemma 3.2. Let w(x) = (−∆)m[(1 + |x|2)m−(n+α)/2]. If α > 0, then
w∈L1(Rn). If, moreover, α < min(n,2m) and h = Iαw, then
Γ((n+2m−α)/2)
h(x)=a (1+|x|2)(α−n)/2−m, a =22m−α ,
α,m α,m
Γ((n+α−2m)/2)
so that h ∈ L1(Rn) and
22m−απn/2Γ(m−α/2)
c ≡ h(x)dx = = γ (2m−α);
α,m n
ZRn Γ((n+α−2m)/2)
cf. the normalizing constant in (1.1). If α is not an integer, then
c 6= 0 for any m > α/2. If α ∈ {1,2,...,n−1}, then c 6= 0
α,m α,m
provided, e.g., that m = [(α+2)/2].
Proof. The first statement can be easily checked by differentiation. For
instance, we can write ∆ in polar coordinates to get
d d
w(x)=(Lmψ)(|x|2), L=4r1−n/2 rn/2 , ψ(r)=(1+r)m−(n+α)/2.
dr dr
This gives w(x) = O((1+|x|2)−(α+n)/2) ∈ L1(Rn). To prove the second
statement we invoke the Bessel-McDonald kernel
K (|ξ|) 21−(α+n)/2
(n−α)/2
G (ξ) = λ , λ = , (3.2)
α α |ξ|(n−α)/2 α πn/2Γ(α/2)
ELEMENTARY INVERSION OF RIESZ POTENTIALS 5
where K (·) denotes the modified Bessel function (or the McDonald
ν
function) of order ν with the property
K (z) = K (z); (3.3)
ν −ν
see, e.g., [31]. It is known that
|ξ|(α−n−1)/2e−|ξ| if |ξ| > 1,
|ξ|α−n if |ξ| < 1, α < n,
G (ξ) ≤ c  (3.4)
α α 
1 if |ξ| < 1, α > n,
1+log(1/|ξ|) if |ξ| < 1, α = n,




where c is a continuous function of α; see, e.g., [21, p. 257], [16, p.
α
285]. Furthermore, for any α > 0 theFourier transform of(1+|x|2)−α/2
in the sense of distributions is computed by the formula
((1+|x|2)−α/2,φ(x)) = (2π)−n(G (ξ),φˆ(ξ)), φ ∈ S(Rn), (3.5)
α
[16, Section 8.1]. Let us prove that
(Iαw,φ) = a ((1+|x|2)(α−2m−n)/2,φ(x)) (3.6)
α,m
for any test function φ in the space Φ(Rn). Once this is done, the
required pointwise equality will follow owing to Proposition 2.1. We
have
(Iαw,φ) = (w,Iαφ) = ((1+|x|2)(2m−n−α)/2,(−∆)m(Iαφ)(x))
= (2π)−n(G (ξ),|ξ|2m−αφˆ(ξ))
n+α−2m
K (|ξ|)
= (2π)−nλ m−α/2 ,|ξ|2m−αφˆ(ξ) .
n+α−2m (cid:18) |ξ|m−α/2 (cid:19)
Using (3.3) and (3.5), we continue:
(2π)−nλ
(Iαw,φ) = n+α−2m (G (ξ),φˆ(ξ))
2m+n−α
λ
2m+n−α
= a ((1+|x|2)(α−2m−n)/2,φ(x)).
α,m
Evaluation of c is straightforward. To complete the proof, we note
α,m
that c 6= 0 if (n+α−2m)/2 6= 0,−1,−2,.... The latter is guaran-
α,m
teed under the afore-mentioned choice of m. (cid:3)
Proof of Theorem A. The first statement follows immediately from
Lemmas 3.1 and 3.2. To prove the second statement, we first note
that w˜ = −∆w belongs to L1. Hence, the convolution Iαf ∗ t−αw˜ is
t
˜
well-defined in the Lebesgue sense and can be written as f ∗ h with
t
6 B. RUBIN
h˜ = Iαw˜ = Iα(−∆)w. Furthermore, for any test function φ ∈ Φ(Rn),
(3.6) yields
(h˜,φ) = (Iα(−∆)w,φ) = −(Iαw,∆φ)
= −a (∆[(1+|x|2)(α−n)/2−m],φ).
α,m
Since ∆[(1+|x|2)(α−n)/2−m] = O((1+|x|2)(α−n)/2−m−1) ∈ L1, then, by
Proposition 2.1, we have a pointwise equality
h˜(x) = −a ∆[(1+|x|2)(α−n)/2−m].
α,m
Denote
∞ Iαf ∗w˜
t
J = dt, ε > 0.
ε Z t1+α
ε
Then
∞ f ∗h˜ ∞ h˜ (x)
t t
J = dt = f ∗ψ , ψ (x) = dt.
ε ε ε
Z t Z t
ε ε
Setting h˜(x) = h˜ (|x|2), we get
0
∞ h˜ (|x|2/t2) 1 x|2/ε2
ψ (x) = 0 dt = h˜ (r)rn/2−1dr.
ε Z tn+1 2|x|n Z 0
ε 0
Thisisascaledversionofthefunctionψ(x)=2−1|x|−n |x|2h˜ (r)rn/2−1dr.
0 0
Observing that R
d d
h˜ (r) = −a L[(1+r)(α−n)/2−m], L=4r1−n/2 rn/2 ,
0 α,m
dr dr
we have
2a x|2 d d
ψ(x) = − α,m rn/2 [(1+r)(α−n)/2−m]dr
|x|n Z dr dr
0
2a d
= − α,m rn/2 [(1+r)(α−n)/2−m]
|x|n (cid:26) dr (cid:27)
r=|x|2
= −a (α−n−2m)(1+|x|2)(α−n)/2−m−1.
α,m
This gives lim J = d f(x), where
ε α,m
ε→0
22m−α+1πn/2Γ(m+1−α/2)
d = ψ(x)dx = = (2m−α)γ (2m−α),
α,m n
ZRn Γ((n+α−2m)/2)
and the limit is understood in the required sense.
ELEMENTARY INVERSION OF RIESZ POTENTIALS 7
4. Inversion of the Radon-John Transform
We recall basic definitions. More information can be found in [5, 7,
23]. Let G and G be the affine Grassmann manifold of all non-
n,k n,k
oriented k-dimensional planes τ in Rn and the ordinary Grassmann
manifold of k-dimensional linear subspaces ζ of Rn, respectively. Each
k-plane τ ∈ G is parameterized as τ = (ζ, u), where ζ ∈ G and
n,k n,k
u ∈ ζ⊥ (the orthogonal complement of ζ in Rn). The k-plane Radon-
John transform of a function f on Rn is defined by
(R f)(τ) ≡ (R f)(ζ, u) = f(y +u)dy, (4.1)
k k
Z
ζ
where dy is the induced Lebesque measure on the subspace ζ ∈ G .
n,k
This transform assigns to a function f a collection of integrals of f over
all k-planes in Rn. The corresponding dual transform of a function ϕ
on G is defined as the mean value of ϕ(τ) over all k-planes τ through
n,k
x ∈ Rn:
(R∗ϕ)(x) = ϕ(σζ +x)dσ, x ∈ Rn. (4.2)
k 0
Z
O(n)
Here ζ ∈ G is an arbitrary fixed k−plane through the origin. If
0 n,k
f ∈ L (Rn), then R f is finite a.e. on G if and only if 1 ≤ p < n/k
p k n,k
[29, 23].
AvarietyofinversionproceduresareknownforR f; see[5,7,22,23].
k
Oneof the most important algorithms relies onthe Fuglede formula [4],
R∗R f = d Ikf, d = (2π)kσ /σ , (4.3)
k k k,n k,n n−k−1 n−1
with the Riesz potential Ikf on the right-hand side. Hence, Theorem
A yields the following result. We denote
w(x)=(−∆)m[(1+|x|2)m−(n+k)/2], m=[(k+2)/2], w˜(x)=(−∆w)(x);
4mπ(n+k)/2Γ(n/2)Γ(m−k/2)
λ = , δ = (2m−k)λ .
k k k
Γ((n−k)/2)Γ((n+k)/2−m)
Theorem B. If ϕ = R f, f ∈ Lp(Rn), 1 ≤ p < n/k, then
k
∞ R∗ϕ∗w˜
λ f = lim(R∗ϕ∗t−kw ), δ f = k t dt, (4.4)
k t→0 k t k Z t1+k
0
∞ ∞
where (...) = lim (...). The limit in both formulae exists in the
0 ε→0 ε
Lp-norRm and in the aR.e. sense. If, moreover, f ∈ C (Rn), then it exists
0
in the sup-norm.
8 B. RUBIN
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DepartmentofMathematics, LouisianaStateUniversity,BatonRouge,
Louisiana 70803, USA
E-mail address: [email protected]