Reconstruction from Fourier measurements using
compactly supported shearlets
Jackie Ma
Technische Universität Berlin
Department of Mathematics
Straße des 17. Juni 136, 10623 Berlin
Email: ma@math.tu-berlin.de
Abstract—We study the reconstruction problem of a compactly
supported function from its Fourier coefficients using a compactly
supported shearlet system. We assume that only finitely many
Fourier samples are accessible and the reconstruction can only
be constructed using finitely many shearlets. Our main result
shows that stable recovery of the signal is possible provided the
number of measurements is up to a constant equal to the number
of shearlet elements that are used for the approximation.
Numerical experiments with MR data are presented which
emphasize the advantages of shearlets compared to other systems
such as wavelets.
I. I NTRODUCTION
A general task in sampling theory is the reconstruction of
an object from finitely many measurements. Such a problem
also appears in signal processing and medical imaging the
latter being a main motivation of this paper with view to
applications. Further, we model the reconstruction problem in
a Hilbert space H with inner product h·, ·i. The measurements
are assumed to be acquired with respect to a fixed sampling
system {s1 , s2 , . . .} ⊂ H and the reconstruction is computed
with respect to another system {r1 , r2 , . . .} ⊂ H which is
assumed to form a frame. Recall that a set of vectors (ϕi )i∈I ⊂
H is called a frame for H if there exist 0 < A ≤ B < ∞
such that
X
Akf k2 ≤
|hf, ϕi|2 ≤ Bkf k2 , ∀ f ∈ H.
i∈I
The concrete reconstruction problem is then formulated
as follows: Given finitely many linear measurements
hf, s1 i, . . . , hf, sM i, M ∈ N of a function f ∈ H, find
a reconstruction fN ∈ span{r1 , . . . , rN }, N ∈ N that is
numerically stable and whose error converges to zero as N
tends to infinity.
Reconstruction problems of this type have already been
studied, e.g. by Unser and Aldroubi in [1] and later also
by Eldar in [2], [3] for the case N = M which is known
as consistent reconstruction. We study this reconstruction
problem by using generalized sampling (GS) which can be
seen as an extension of consistent reconstruction.
II. G ENERALIZED SAMPLING
Generalized sampling was introduced by Adcock et al. in
[4], [5], [6] and studies the reconstruction problem explained
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
above given that the sampling system and reconstruction
system form frames. The key difference between generalized
sampling and consistent reconstruction is to allow the number
of measurements M to vary independently of the number of
reconstruction elements N . This flexibility enables generalized
sampling to overcome some of the barriers of consistent
reconstruction [4].
A. GS reconstruction method
Given a sampling system {s1 , s2 , . . .} ⊂ H we define
the sampling space S = span{sk : k ∈ N} ⊂ H
and the finite dimensional version of this sampling space is
denoted by SM = span{s1 , . . . , sM }, M ∈ N. Analogously
for a reconstruction system {r1 , r2 , . . .} ⊂ H we define
the reconstruction space R = span{rk : k ∈ N} ⊂ H
and, likewise, its finite dimensional version is defined as
RN = span{r1 , . . . , rN }, N ∈ N.
We will assume the sampling system to form an orthonormal
basis and the reconstruction system to be a frame, since this
will exactly be our setup, see Section IV. Further it is natural
to assume a certain subspace condition, indeed, we require
R ∩ S ⊥ = {0}
and R + S is closed.
(1)
This guarantees a well posedness of the finite dimensional
reconstruction problem, cf. Theorem II.1.
Theorem II.1 ([6]). For any N, M ∈ N let SM and RN be
as above and PSM be the following finite rank operator
PSM : H → SM ,
f 7→
M
X
hf, sk isk .
k=1
Let N ∈ N be fixed. If (1) holds, then there exists an M ∈ N
such that the system of equations
hPSM fN,M , rj i = hPSM f, rj i,
j = 1, . . . , N
(2)
has a unique solution fN,M ∈ RN . Moreover, the smallest
M ∈ N such that the system is uniquely solvable is the least
number M0 ∈ N so that
cN,M := inf kPSM f k > 0.
f ∈RN ,
kf k=1
(3)
Furthermore,
kf − PRN f k ≤ kf − fN,M k ≤
A. Assumptions on the generator
1
cN,M
kf − PRN f k,
(4)
where PRN : H → RN denotes the orthogonal projection
onto RN .
Let φ and ψ be compactly supported functions in L2 (R2 )
e 1 , x2 ) := ψ(x2 , x1 ). We assume there exist
and define ψ(x
some constants C1 , C2 > 0 such that
1
,
(1 + |ξ1 |)r (1 + |ξ2 |)r
min{1, |ξ1 |α }
b 1 , ξ2 )| ≤ C2 ·
|ψ(ξ
.
(1 + |ξ1 |)r (1 + |ξ2 |)r
b 1 , ξ2 )| ≤ C1 ·
|φ(ξ
Definition II.2 ([6]). The solution fN,M in Theorem II.1 is
called generalized sampling reconstruction.
Definition II.3 ([1], [7]). The quantity cN,M in (3) is called
the infimum cosine angle between the subspaces RN and SM .
B. Stable sampling rate
In light of Theorem II.1 it is crucial to have control over
cN,M . This motivates the definition of the stable sampling rate.
Definition II.4. For any fixed N ∈ N and θ > 1 the stable
sampling rate Θ(N, θ) is defined as
1
Θ(N, θ) = min M ∈ N : cN,M >
.
θ
The stable sampling rate determines the number of measurements M that are needed in order to find an approximation
fN ∈ RN or equivalently, find N coefficients, such that the
angle cN,M is controlled by the threshold θ.
III. S HEARLETS
Shearlet systems were first introduced by K. Guo, G.
Kutyniok, D. Labate, W.-Q Lim and G. Weiss in [8], [9]
and the following notations have became standard in shearlet
theory. The parabolic scaling matrices with respect to scale
j ∈ N ∪ {0} are denoted by
j
j/2
2
0
2
0
e
A2j :=
,
A2j :=
,
0 2j/2
0
2j
and the shearing matrices with parameter k ∈ Z are
1 k
Sk =
.
0 1
These operations, together with the standard integer shift of
functions in L2 (R2 ) are then used to define the cone adapted
discrete shearlet system.
Definition III.1 ([10]). Let φ, ψ, ψe ∈ L2 (R2 ) be the generating functions and c = (c1 , c2 ) ∈ (R+ )2 . Then the (cone
adapted discrete) shearlet system is defined as
e c)
SH(φ, ψ, ψ,
= {φ(· − c1 m) : m ∈ Z2 }
n
o
∪ 23j/4 ψ ((Sk A2j ) · −cm) : j ≥ 0, |k| ≤ 2j/2 , m ∈ Z2
n
o
e2j ) · −e
∪ 23j/4 ψe (SkT A
cm : j ≥ 0, |k| ≤ 2j/2 , m ∈ Z2
where the multiplication of c and e
c with the translation
parameter m should be understood pointwise.
Under some regularity condition, see (5) and (6) in Subsection III-A, the shearlet system defined above can form a
frame and an explicit construction of such systems and their
generators φ, ψ is given in [10].
(5)
(6)
where the regularity parameters α, r > 0 are large enough so
that the shearlet system gives rise to a frame for L2 (R2 ), [10].
B. Assumptions on finite dual shearlets
In [11] it was shown that the frame operator for continuous
shearlets acts as a Fourier multiplier and the existence of a
dual shearlet that has the same asymptotic Fourier decay as
the original shearlet was given. In fact, similar computations
as those in Lemma 4.2 in [11] can be performed for discrete
shearlets as defined in Definition III.1 for appropriate sampling
constants c. In the following we make this result more precise
for our setup.
Let (ψλ )λ∈N be a compactly supported shearlet frame for
H = L2 (R2 ) and for N ∈ N let HN := span{ψλ : λ ≤ N }.
Note that (ψλ )λ forms a frame for HN . Now we assume that
there exist finite dual shearlets (ψλd,N )λ≤N that asymptotically have the same behavior as the original shearlet atoms
(ψλ )λ≤N , i.e. we assume
cλ (ξ)| as N → ∞
|(ψλd,N )∧ (ξ)| |ψ
(7)
for ξ ∈ R2 . Note that by definition of a dual frame, the finite
dual shearlet system (ψλd,N )λ≤N can be used to represent any
f ∈ HN by
X
X
f=
hf, ψλd,N iψλ =
hf, ψλ iψλd,N ,
λ≤N
λ≤N
see [12] for the definition of dual frames.
Assumption (7) is, for instance, fulfilled for digital shearlets
[13]. In fact, for these shearlets, dual shearlets are given by
d
b
d
φ(ξ)
ψ
ψej,k (ξ)
j,k (ξ) [
[
d,N
d,N
[
d,N (ξ) =
, ψj,k
(ξ) =
, ψej,k
(ξ) =
,
φ
Γ(ξ, J)
Γ(ξ, J)
Γ(ξ, J)
where
X X d
2
2
d
e
|ψj,k (ξ)| + |ψj,k (ξ)|
b 2+
Γ(ξ, J) = |φ(ξ)|
j≤J k≤2j/2
is bounded from above and below, see [14] and the references
therein. NJ indicates that only shearlets up to a certain scale
J are considered, cf. Subsection IV-A. Further, it was shown
in [10] that
X X
X X d
2
b +
d
A ≤ φ(ξ)
|ψ
|ψej,k (ξ)|2 ≤ B
j,k (ξ)| +
j≥0 k≤2j/2
j≥0 k≤2j/2
where A and B are the lower and upper frame bounds of
the full shearlets system (ψλ )λ∈N , respectively. Therefore the
assumption made in (7) can be justified.
We wish to mention that recently a new type of shearlet
systems has been introduced in [15] where a corresponding
dual shearlet can be given explicitly. These dual shearlets
corresponding to the new dualizable shearlets also show the
same asymptotic decay as the dualizable shearlet generators
similar to (7). Thus, emphasizing the rationality of such
assumptions.
IV. S TABLE SHEARLET RECONSTRUCTIONS FROM
F OURIER MEASUREMENTS
Finally, for M = (M1 , M2 ) ∈ N × N let
o
n
(ε)
(ε)
SM = span s` : ` = (`1 , `2 ) ∈ Z2 , |`i | ≤ Mi , i = 1, 2
be the finite dimensional sampling space. Note that M =
(M1 , M2 ) determines the size of the grid and thus, the total
number of possible measurements are asymptotically of order
M1 · M2 . In the event that the stable sampling rate grows
linearly we would have M1 · M2 = O(22J ) as for the wavelet
case shown in [16], [17].
A. Shearlet reconstruction and Fourier sampling space
B. Almost linear stable sampling rate
Without loss of generality we can assume that the generating
scaling function φ and the shearlets ψ, ψe are compactly
supported in [0, a]2 where a is some positive integer. Then we
consider all scaling functions whose support intersect [0, a]2
and denote this index set by Ω, i.e.
Ω = m ∈ Z2 : supp φm ∩ [0, a]2 6= ∅ .
Theorem IV.1 shows that the angle between the shearlet
reconstruction space and the Fourier sampling space can be
controlled with an almost linear sampling ratio.
Similarly, we consider all shearlets whose support intersect
[0, a]2 . For this, let J − 1 ∈ N ∪ {0} be a fixed scale. Then
by ΛJ we denote the following paramater set
n
o
ΛJ = (j, k, m) : 0 ≤ j ≤ J − 1, |k| ≤ 2j/2 , m ∈ Ωj,k ,
where Ωj,k = {m ∈ Z2 : supp ψj,k,m ∩ [0, a]2 6= ∅}. Note
that Ωj,k is of finite cardinality. Analogously we set
n
o
e J = (e
ee e
Λ
j, e
k, m)
e : 0≤e
j ≤ J − 1, |e
k| ≤ 2j/2 , m
e ∈Ω
j,k
2
e e e = {m
with Ω
e ∈ Z2 : supp ψeej,ek,m
j,k
e ∩[0, a] 6= ∅} and define
RNJ := span {φm : m ∈ Ω} ∪ {ψj,k,m : (j, k, m) ∈ ΛJ }
n
o
e
eJ
e
∪ ψeej,ek,m
:
(
j,
k,
m)
e
∈
Λ
.
(8)
e
e be a compactly supTheorem IV.1 ([18]). Let SH(φ, ψ, ψ)
e We further
ported shearlet frame with generators φ, ψ, and ψ.
assume the generators to be as in Subsection III-A and with
(finite) duals as in Subsection III-B. Let N ≤ NJ = O(22J ).
Then for all θ > 1 there exists Sθ > 0 such that
1
ε fk ≥
, as N → ∞
cN,M = inf kPSM
f ∈RN
θ
kf k=1
where M = (M1 , M2 ) ∈ N × N obeys
Mi = dSθ · 2J(1+δ) /εe, i = 1, 2
2
with δ ≥ 2r−1
and r > 0 is the regularity parameter from (5)
and (6). Further, the constant Sθ does not depend on N but
on θ, ω, α, and r.
Remark IV.2. The dependence of the constant Sθ on all
other relevant variables can be explicitly computed. Moreover,
without the assumptions made in Subsection III-A, the result
holds true with Sθ depending on the lower frame bound AN
of the finite shearlet frame (ψλ )λ≤N .
Up to a fixed scale J − 1 we, asymptotically, have NJ =
V. N UMERICAL EXPERIMENTS
O(22J ) many generating functions in RNJ as J → ∞. Note
In this section we provide some reconstructions from MR
that by construction at each scale we only have finitely many
data using complex exponentials (Fourier inversion), wavelets
elements, therefore, an ordering can be performed quite natuand compactly supported shearlets. Although our main result
rally namely we order the system along scales and within the
guarantees stable and convergent reconstructions when the
2
scales, the translations in Z are ordered in a lexicographical
sampling rate is almost linear it is not efficient (and also not
manner.
By R we denote the reconstruction space that contains all necessary) to acquire that many samples in practice. Thus, we
will subsample the Fourier data according to some common
shearlets across all scales, i.e.
subsampling patterns used in applications such as spirals and
R = span {φm : m ∈ Ω} ∪ {ψj,k,m : (j, k, m) ∈ ΛJ , J ≥ 0} radial lines, see Figure 2a and Figure 2b.
A. Data
n
o
e
eJ , J ≥ 0
e
∪ ψeej,ek,m
:
(
j,
k,
m)
e
∈
Λ
.
The underlying Fourier data, also called k-space, was
e
acquired by a multi-channel acquisition consisting of four
To define the sampling space we first choose T1 , T2 > 0 so channels. All four k-spaces had a fixed resolution of 128×128
1
that R ⊂ L2 ([−T1 , T2 ]2 ). Let ε ≤ T1 +T
< 1 determine the which results in an 128 × 128 image in time by applying an
2
sampling density. Then we define the sampling vectors on a inverse Fourier transform, see Figure 1a.
(ε)
uniform grid by s` = εe2πiεh`,·i , ` ∈ Z2 over [−T1 , T2 ]2 and
The single channel images are combined by using the sumthe sampling space S by
of-squares method, [19]. The resulting sum-of-squares image
n
o
from all four channels of the original k-space is used as the
(ε)
S (ε) = span s` : ` ∈ Z2 .
reference image and is depicted in Figure 1.
20
20
20
40
40
40
60
60
60
80
80
80
100
100
100
120
120
120
50
100
150
200
250
(a) 4 channel reconstructions
(b) Combined image by sumof-squares
Fig. 1: Reference images computed with complete data
B. Methods
We mimic an acceleration factor of almost 5 by using a
0 − 1 mask to subsample the k-space, cf. Fig. 2a and Fig. 2b.
We then compute each channel image separately and combine
them by sum-of-squares.
The following problems and methods are used to obtain the
single channel images
• Shearlets: the reconstructions are obtained by solving the
following standard analysis formulation
β e
− bk22 + λkSH(u)k1 ,
min kFu
u 2
where Fe is the subsampled Fourier operator, b is the
subsampled input data, SH is the shearlet transform, β
and λ are some parameters larger than zero, and u is
the image. For our numerical tests we used the shearlab
package available at
(a) Phyllotaxis spiral mask:
20.37 %
(b) Radial mask: 20.74%
20
20
40
40
60
60
80
80
100
100
120
120
(c) Reconstruction: shearlets,
spiral
(d) Reconstruction: shearlets, radial
20
20
40
40
60
60
80
80
100
100
120
120
(e) Reconstruction: wavelets,
spiral
(f) Reconstruction: wavelets,
radial
20
20
40
40
60
60
http://www.shearlab.org
•
A three scale shearlet transform was used
with number of directions set to [0 0 1], see
http://www.shearlab.org and also [23], [20] for
more interest in the algorithmic realization of shearlets.
Wavelets: reconstructions are computed using the
SparseMRI package from Lustig et al. [21] which is
downloaded from
http://www.eecs.berkeley.edu/˜mlustig/Software.html
•
Fourier: computed by simple Fourier inversion of the
subsampled data.
C. Results
The shearlet reconstructions sare shown in Figure 2 and
the relative errors are given in Table I. As we can see,
the shearlet reconstruction show reduced artifacts without
loosing fine details. This can be explained in terms of the
sparse approximation rate of cartoon-like-functions [22], more
precisely, shearlets provide an almost optimal rate of order
N −2 for cartoon-like functions, while wavelets only reach
N −1 and Fourier N −1/2 , respectively, see [23].
Relative error kf − fN k/kf k
Shearlets Wavelets
Phyllotaxis Spiral
0.0653
0.1145
Radial
0.0914
0.1531
Fourier
0.2570
0.2733
TABLE I: Relative error for Figure 2
80
80
100
100
120
120
(g) Reconstruction: Fourier
inversion, spiral
(h) Reconstruction: Fourier
inversion, radial
Fig. 2: Reconstructions using a spiral and radial mask.
ACKNOWLEDGMENT
The author would like to thank Gitta Kutyniok for helpful
discussions. Moreover, the author thanks Dr. Sina Straub from
German Cancer Research Center (DKFZ) for providing the
MRI data and Dr. Matthias Dieringer from Siemens AG,
Healthcare Sector for providing a code to read the raw data
into MATLAB. Further, the author acknowledges support
from the Berlin Mathematical School as well as the DFG
Collaborative Research Center TRR 109 ”Discretization in
Geometry and Dynamics”.
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