Exact reconstruction of a class of nonnegative
measures using model sets
Basarab Matei
Institut Galilée
LIPN – UMR CNRS 7030
Université Paris 13
99 Av. Jean-Baptiste Clément
93430 Villetaneuse, France
Email: matei@lipn.univ-paris13.fr
Abstract—In this paper we are concerned with the
reconstruction of a class of measures on the square from
the sampling of its Fourier coefficients on some sparse
set of points. We show that the exact reconstruction of
a weighted Dirac sum measure is still possible when one
knows a finite number of non-adaptive linear measurements of the spectrum. Surprisingly, these measurements
are defined on a model set, i.e quasicrystal.
I. I NTRODUCTION
A. Background
In image processing, the problem of the exact reconstruction of a positive measure appears in several
applications as image compression, superresolution
problem and image denoising. The pioneering basis
pursuit algorithm is used for the exact reconstruction
of sparse finite dimensional vectors. The basis pursuit
was introduced to the statistics community by Chen,
Donoho and Saunders [6] and by earlier works of
Donoho and Stark [7]. P. Doukhan, E. Gassiat and
P. Gamboa considered in [8] and in [5] the exact
reconstruction of a nonnegative measure in relation
with superresolution problem. More recently, Y. de
Castro and P. Gamboa in [4] considered the exact
reconstruction of a signed measures in one dimensional
case. In their paper, the main result show that the exact
reconstruction of a signed measure from few values of
a finite number of non-adaptive linear measurements, is
possible by using the method of Beurling [1] Beurling
Minimal Extrapolation.
In contrast, the results presented here are intrinsically multidimensional and use the arithmetical properc
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IEEE
ties of simple quasicrystals defined by a cut and project
scheme [15], [14]. The results of the present papers
have deep connections with the remarkable theory
called compressed sensing. The main result of the
paper can be viewed as a simplest version of the deterministic compressed sensing. Indeed, our measures are
defined in a deterministic way by using quasicrystals.
In the compressed sensing theory, the target function is
obtained through a variational minimization procedure
[2], [3]. Following the same principle, in this paper
we developed a related program that recovers all the
nonnegative measures with small supports by using an
irregular sampling in the Fourier domain defined by a
simple quasicrystals.
The result of this work is related with the recently
discovered sampling properties of simple quasicrystals
[13]. The methods are different from those used in [13].
The theory of quasicrystals introduced and develop
by Y. Meyer in [15], [14], was the object of several
researches of J. Lagarias and R. Moody which relate
the arithmetical properties of the quasicrystals Λ to
the analytical
properties of the corresponding measure
P
σΛ = λ∈Λ δλ , see e.g. [10], [16],[17] and [18]. In the
same spirit, one should mention the results of Lev and
Olevskii on the Poisson formula and the quasicrystals
in [19].
The result of the present paper gives a convenient
recipe to the exact reconstruction problem of the
nonnegative measures by using linear non-adaptive
measurements defined by simple quasicrystals.
B. Problem and Solution
Let us explain more precisely what is done in
this paper. The action takes place on Q = [0, 1]2
identified to T2 = (R/Z)2 . In image processing we
often consider that the image is defined on Q and then
extended by periodicity. For each integer N > 0, let
F = {x1 , x2 , ..., xN } be a set of points in the square Q.
We consider MN the class of P
measures on Q defined
N
as follows : ν ∈ MN if ν = j=1 ωj δxj , where the
weights ωj ≥ 0, xj ∈ F for all 1 ≤ j ≤ N and δ is the
Dirac impulse. Note that the definition of a measure ν
in the class MN depends on the parameter N , on the
weights wj , 1 ≤ j ≤ N, and on the set of points F
which are considered arbitrary.
Since the unit square Q has been identified to T2 =
(R/Z)2 , the Fourier transform of an arbitrary measure
µ on Q is the sequence of its Fourier coefficients
defined by
Z
µ̂(k) =
exp(−2πik · x)dµ(x), k ∈ Z2 . (I.1)
Q
We are concerned here with the reconstruction of
ν ∈ MN from the observation of the data a(λ) =
{ν̂(λ), λ ∈ Λ}, i.e. from its Fourier coefficients on
some sampling set Λ ⊂ Z2 . More precisely, in what
follows we prove that for every α ∈ (0, 1/2) there
exists a sparse set Λα such that (a) density Λα = 2α
and (b) the “irregular sampling” ν̂(λ) = a(λ), λ ∈ Λα
on Λα allows to recovery exactly any positive measure
ν ∈ MN .
The construction of the sparse set Λα follows the
theory of “model sets” ([15] and [14]) and is given in
next section. Let us precise the strategy of the proof.
The first step is the unicity. By using the peculiar
structure of the sampling set Λα we show the following
result : if ν ∈ MN and µ is an arbitrary measure
on T2 such that ν̂ = µ̂ on the model set Λα , then
ν = µ over T2 . In the second step we show that
ν ∈ MN has minimum mass among all the measures
µ over T2 . In other words, if we a priori know that the
data a(λ), λ ∈ Λα , are the Fourier coefficients of some
nonnegative measure ν ∈ MN , then ν is the unique
nonnegative solution of the following problem :
arg min{kµk; µ ≥ 0, µ̂(λ) = ν̂(λ}, λ ∈ Λα },
(I.2)
where kµk is the total mass of the measure µ. Note that
we do not impose µ ∈ MN in (I.2). The unknowns of
our problem are the set of points F and the weights
wj , 1 ≤ j ≤ N. Since kµk = µ̂(0) the problem (I.2) is
the simplest version of the minimization problem used
in compressed sensing.
C. Outline
This paper is organized in the following way. Section
II gives a convenient recipe for constructing the sparse
set Λ. Section III formulates the exact reconstruction
of non-negative measures. It should be pointed out
that the proofs in this section are different from those
obtained in [13]. In Section IV we provide several
counter-examples, that prove that our results are sharp.
The construction of the counter-examples follow along
the lines of [13].
II. C ONSTRUCTION OF THE SAMPLING SET
In this section we construct the sampling set Λα
and give some properties of these sets. The recipe
for constructing follows from the “model sets” theory
developed by Y. Meyer (see [14]). Define Π : R 7→
T = R/Z by Π(t) = t (mod 1). Then Z2 can be
embedded in T by the mapping γ ∗ : Z2 7→ T defined
by
√
√
(II.1)
γ ∗ (m, n) = Π(m 2 + n 3).
This mapping γ ∗ is injective with a dense range.
With an obvious abuse of notation we still denote
by Π the canonical mapping from R2 to T2 . Then
2
the dual mapping
√ √γ : Z 7→ T is defined by
γ(k) = Π(k 2, k 3) and its range will be denoted
by Γ. Then Γ is dense in T2 .
Let I ⊂ T an arbitrary interval (or arc) of the
circle. This arc is not necessarily centered in 0 and its
complement in T is also an interval. Define the subset
ΛI ⊂ Z2 by letting
ΛI = {(p, q) ∈ Z2 ; γ ∗ (p, q) ∈ I}.
(II.2)
If I = [a, b] where 0 < a < b < 1,√then Λ
√I =
{(p, q) ∈ Z2 ; ∃r ∈ Z such that a ≤ p 2 + q 3 −
r ≤ b}. If I = [−α, α] we write Λα instead of ΛI .
Note that the set Λα is a ”model set” following the
terminology introduced by Y. Meyer, in [14] and [15].
In [10] more details on the properties of these sets are
obtained. We know from the theory of “model sets”
that the density of ΛI is uniform and equals |I|. This
notion is defined now, following [11].
P∞
Let√ τ be
k=−∞ pk δyk , where yk =
√ the measure
(k 2, k 3)modZ2 . Then
Definition II.1. The upper density D+ (Λ) of Λ
is defined as lim supR→∞ supx∈R2 #{B(x,R)∩Λ}
πR2
and the lower density D− (Λ) is defined
by
lim inf R→∞ inf x∈R2 #{B(x,R)∩Λ}
.
If
πR2
+
D (Λ) = D− (Λ) then the density is called uniform.
τ ∗ρ=0
The density of Λα ⊂ Z2 is uniform and equals 2α.
It means that for every ε > 0 there exists a R(ε)
such that for R ≥ R(ε) and uniformly in x0 ∈ Z2
(2α − ε)πR2 ≤ card{k ∈ Z2 ; Λα ∩ B(x0 , R)} ≤
(2α + ε)πR2 . Here B(x0 , R) is the disc
√ at
√ centered
x0 and radius R. Note that the choice of 2 and 3 is
irrelevant and other irrational numbers ξ1 and ξ2 could
be used as long as ξ1 , ξ2 and 1 are linearly independent
over Q.
III. M AIN R ESULTS
Let α be a fixed constant in (0, 1/2), and Λα ⊂ Z2
be the model set defined by using I = [−α, α]. In
other words
√
√
Λα = {(p, q) ∈ Z2 ; ∃r ∈ Z |p 2 + q 3 − r| ≤ α}.
(III.1)
Let ν be a measure in MN . Then the measure ν is
a sum of atomic masses ωj ≥ 0 on the points F =
{x1 , x2 , ..., xN }. The parameter N > 0, the weights
wj , 1 ≤ j ≤ N and F are arbitrary. We begin our
study by proving the following result :
Theorem III.1. Let ν be the measure defined above
and µ ≥ 0 be a positive measure on the torus T2 such
that µ̂(λ) = ν̂(λ), λ ∈ Λα . Then µ = ν.
Proof. We define
ρ = µ − ν.
(III.2)
By definition ρ̂(λ) = 0, λ ∈ Λα . We will show that
ρ = 0 over T2 . To begin with we prove the following
lemma:
Lemma III.1. Let θ be the triangle function on
T = R/Z supported on [−α, α] defined by θ(0) =
1, θ(α) = θ(−α) = 0, θ being affine on [α, 0] and
[−α, 0]. Then
θ(x) =
∞
X
k=−∞
pk exp (2πikx), where pk ≥ 0.
Proof. By the definition of the measure τ , for all
(p, q) ∈ Z2 , we have
∞
X
τ̂ (p, q) =
√
√
pk exp (2πi(p 2 + q 3)k).
k=−∞
Therefore
√
√
τ̂ (p, q) = θ(p 2 + q 3).
√
√
Note that p 2 + q 3 ∈ J = T \ I for all (p, q) ∈
/
Λα . Since θ vanishes on J it follows that τ̂ (p, q) = 0
whenever (p, q) ∈
/ Λα . Now ρ̂(λ) = 0 whenever λ ∈
Λα . Then τ̂ · ρ̂ = 0, which implies τ ∗ ρ = 0. This ends
the proof of Lemma III.1.
Lemma III.1 show that τ ∗ (µ − ν) = 0 over T2 .
More precisely
(τ ∗ ρ)(x) =
∞
X
pk (µ − ν)(x − yk ) = 0. (III.3)
k=−∞
Let us consider the measure µ̃ = τ ∗ µ. By definition,
the measure µ̃ satisfies the following identity
µ̃(x) =
∞
X
k=−∞
pk (µ)(x − yk ) =
∞ X
N
X
pk ωj δyk +xj .
k=−∞ j=1
(III.4)
Note that the right hand side of (III.4) is an
atomic
supported by Γ + F , where Γ =
√ measure
√
{(k 2, k 3); k ∈ Z} and F = {x1 , x2 , ..., xN }.
The set of points Γ + F may be written as Γ1 ∪ .... ∪
Γm where Γj = Γ + yj , and Γj are disjoints sets of
points for all 1 ≤ j ≤ m, by using a relabeling of F
if necessary.
It follows that the measure µ is absolutely continuous with respect to the measure µ̃. Indeed, we have
µ ≥ 0 , µ̃ ≥ 0 and µ̃ ≥ p0 µ. This follows directly
from the definition of µ̃, the term p0 µ(x) is one of the
terms in the definition of µ̃(x).
Consequently, µ is also an atomic measure supported
on the set of points Γ + F . Finally our measure ρ =
µ − ν is an atomic measure supported on Γ + F . We
decompose now the measure ρ as follows
ρ = ρ1 +....+ρm , where supp(ρj ) ⊂ Γ+yj , 1 ≤ j ≤ m.
It follows that
τ ∗ ρ = τ ∗ ρ1 + .... + τ ∗ ρm
and the support of each part τ ∗ ρj satisfies supp(τ ∗
ρj ) ⊂ Γ + yj , for all 1 ≤ j ≤ m.
Since Γj are disjoints sets of points, by using the
identity τ ∗ ρ = 0 we obtain that τ ∗ ρj = 0 for all
1 ≤ j ≤ m. Let us consider µj (x − yj ) = ρj (x). It
follows that τ ∗ µj = 0, for all 1 ≤ j ≤ m. Then,
µj is a measure supported on Γ and µj is the sum of
atomic masses supported on Γ. Note that by definition
the measure ρj is the restriction of the whole measure
ρ = µ − ν to Γj . From this remark, one conclude that
for all 1 ≤ j ≤ m, all the weights in the definition of
the measure µj are positive, excepting a finite number
of them.
Now we need the following lemma:
Lemma III.2. Let σ be an atomic measure supported
on Γ. We assume that, excepting a finite number, all
the weights in the definition of σ are nonnegatives and
also we assume that τ ∗ σ = 0. Then σ = 0.
The interested reader can find the proof in [12] The
following theorem shows that the measure ν is the
unique solution of the problem (I.2).
Theorem III.2. With ν as above, all measure µ 6= ν
(non necessary positive) satisfying µ̂(λ) = ν̂(λ), λ ∈
Λα verify kµk > kνk.
Proof. In the case ν ≥ 0 and µ a signed measure,
there exist an infinity measures satisfying (I.2). We first
establish kµk ≥ kνk for all these measures. To this
end, we decompose µ as µ = u − v with u ≥ 0 and
v ≥ 0 and the supports of u and v are disjoint Borel
sets. By definition, we have that
Z
Z
µ̂(0) = u − v
while kµk =
Z
kµk =
R
u+
R
Z
u+
v. Therefore
Z
v≥
Z
u−
where we have used 0 ∈ Λα .
v = µ̂(0) = ν̂(0) = kνk,
Assume now that kµk = kνk, we will show that
this implies µ = ν. Since µ̂(λ) = ν̂(λ), λ ∈ Λα and
0 ∈ Λα , we get
Z
Z
Z
Z
kνk = ν̂(0) = µ̂(0) = u− v = u+ v = kµk.
(III.5)
From (III.5) we infer that v = 0. Consequently µ = u
is a positive measure. We are now in the hypothesis
of Theorem III.1, with µ̂(λ) = ν̂(λ), λ ∈ Λα and
0 ∈ Λα , µ ≥ 0, ν ≥ 0 we get µ = ν. Arguing by
contradiction, since µ̂(λ) = ν̂(λ), λ ∈ Λα we can not
have kµk = kνk, whitout µ = ν. Then kµk > kνk.
IV. C OUNTER EXAMPLES
In this section, we ennounce two theorems without
proof, which show that the positivity of the weights
in the definition of measures play a key role. These
theorem follow the lines of similar results obtained in
[13].
Theorem IV.1. The Theorem III.2 is false if ν =
P
N
k=1 αk δxk and αk are not necessarily positives.
Theorem IV.2. Let ν ≥ 0 be an arbitrary positive
measure not necessary of the form of measures in MN
and µ be an arbitrary measure. Then the Theorem III.1
is false.
V. C ONCLUSIONS
The roots of this paper are in Meyer’s deep theory
of “model sets” introduced and developed in several
works, see for example [15], [14]. The problem considered in this paper corresponds to the simplest situation
in the compressed sensing, where we try to reconstruct
a function f from some partial measurements on
fˆ, throught a minimization problem. In the case of
measures the minimization problem of kµk not appear
since µ̂(0) = ν̂(0). In the minimization problem (I.2)
one looks for a minimizer among all signed measures
on Q.
ACKNOWLEDGMENT
The author gratefully acknowledges Yves Meyer for
its constant support and helpful discussions on the
subject. The author acknowledges also John Benedetto
and Joaquim Ortega-Cerdà for theirs encouragement.
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