Compressed sensing with local structure: theory,
applications and benefits
Ben Adcock
Anders C. Hansen
Bogdan Roman
Department of Mathematics
Simon Fraser University
Burnaby, BC Canada
Email: ben adcock@sfu.ca
DAMTP
University of Cambridge
Cambridge, United Kingdom
Email: ach70@cam.ac.uk
DAMTP
University of Cambridge
Cambridge, United Kingdom
Email: abr28@cam.ac.uk
Abstract—We present a generalized framework for compressed
sensing, introduced originally in [1], [2], that is more relevant to
practical applications where incoherence is lacking and sparsity
can be shown to be too crude a signal model. This framework, based on new local notions of sparsity, coherence and
subsampling, provides the theoretical underpinnings as to why
compressed sensing works in many such applications. Moreover,
it explains a number of key effects that are witnessed in practice,
yet are not explained by the standard theory. We also demonstrate
how this framework can lead to marked gains in a number of
other applications by leveraging the structured sparsity of typical
images through structured sampling strategies.
I. I NTRODUCTION
Compressed sensing (CS) asserts that a vector x ∈ CN that
is s-sparse in an orthonormal basis Φ ∈ CN ×N can be recovered from roughly s log N well-chosen linear measurements.
To enable this recovery, such measurements should be incoherent with the sparsifying transformation Φ. Unfortunately,
in many applications of CS the measurements are imposed by
a particular physical device and may in fact be highly coherent
with the sparsifying transformation Φ. This is the case notably
in medical imaging – Magnetic Resonance Imaging (MRI)
and X-ray Computerized Tomography (CT), for example –
but also applications such as Thermoacoustic, Photoacoustic
and Electrical Impedance Tomography, Electron Microscopy,
and Radio Interferometry. Yet CS has been applied in some
of these areas, often with a large degree of success.
The purpose of this paper is twofold. First, we present
the theoretical underpinnings that explain why CS works in
such applications, where the standard theory does not apply.
To do this, we use a new framework for CS, introduced
in [1], which relaxes the notion of incoherence to so-called
local incoherence in levels. Surprisingly, as we explain, the
underlying reason why CS works in such applications is not
due solely to the sparsity of the object x in the basis Φ. The
structure of the sparsity – that is, the ordering of the entries of
x – also plays a central role. This dependency is encapsulated
in this framework via a new signal model, namely, sparsity in
levels. A numerical experiment known as the flip test confirms
that this is a more appropriate signal model for analyzing
such applications. Moreover, when combined with so-called
multilevel random subsampling, this framework provides nearoptimal recovery guarantees for such applications.
As was first discussed in detail in [2], this new framework
also explains several phenomena that are observed in such
applications but that run contrary to the standard intuition in
CS. We highlight several of these in this paper. In particular,
the dependence of the best sampling strategy on the sparsity
structure of x and the influence of the problem resolution on
the reconstruction quality. These phenomena provide critical
insight into important issues in such applications, including
the design of optimal sampling strategies.
In the second part of this paper we focus on a different class
of applications where one has freedom to choose the measurements. Examples include compressive imaging (e.g. singlepixel and lensless imaging) and fluorescence microscopy. The
framework we describe in the first part of the paper explains
the key role structured sparsity plays in CS recovery from
locally incoherent measurements. However, popular sparsifying transforms such as wavelets and their generalizations
are known to give rise to a characteristic asymptotic sparsity structure for all natural images. This begs the question:
given the freedom to design measurements, should one seek
incoherence? The answer is no. By designing asymptotically
incoherent measurements (a specific type of local incoherence)
matching the asymptotic sparsity structure of the coefficients
we are able to get noticeable gains in reconstruction quality
over approaches based on incoherent sensing.
It has long been known that structured sparsity persists
in many applications. In CS this has led to a long line of
developments in so-called structured recovery algorithms [3],
[4], [5], [6], wherein incoherent measurements are used and
standard CS recovery algorithms (such as thresholding or
greedy methods) are modified to exploit the additional structure. Our approach is different. It uses the standard recovery
procedure of `1 minimization, but modifies the measurements
to match the underlying structure. As we briefly discuss, this
structured sampling approach yields substantial performance
gains in practice over current structured recovery techniques.
Full details are presented in [2].
II. S TANDARD COMPRESSED SENSING
Consider the following standard CS setup. Let x ∈ CN be
the object to recover and Φ ∈ CN ×N and orthonormal sparsifying transformation. We shall assume that the measurements
arise by subsampling the rows of an isometry Ψ ∈ CN ×N ;
the sampling operator. That is, the (noisy) measurements
are y = PΩ Ψ∗ x + e ∈ Cm , where PΩ ∈ Cm×N is the
projection matrix that selects the rows of Ψ∗ corresponding
to the index set Ω ⊆ {1, . . . , N }, |Ω| = m, and the vector
e satisfies kek ≤ η for some η ≥ 0. Given Φ and Ψ we let
U = (uij ) = Ψ∗ Φ ∈ CN ×N (note that U is an isometry) and
define the coherence µ(U ) = maxi,j=1,...,N |uij |2 ∈ [N −1 , 1].
In order to recover x from the measurements y we use `1
minimization. That is, we solve
min kΦ∗ zk1
z∈CN
subject to ky − PΩ Ψ∗ zk ≤ η,
(1)
where k·k1 is the `1 norm. Let x ∈ CN and 0 < < e−1
be given and suppose that Ω is chosen uniformly at random.
Then a standard result [1], [7] gives that
√
kx − x̂k . σs (Φ∗ x) + sη,
where σs (z) is the error of the best `1 -norm approximation
of the vector z by an s-sparse vector, provided the number of
measurements m satisfies
m & s · µ(U ) · N · log(−1 ) · log(N ),
uniform random
multilevel
Fig. 1. Top row: subsampling maps Ω, where m/N = 12.5%. Each
dot corresponds to a frequency sampled in Fourier space. Bottom row: CS
reconstruction using (1).
(2)
In particular, if U is incoherent, i.e. µ(U ) . 1/N , then the
number of measurements required is on the order of s log(N ).
III. E XAMPLE : F OURIER SAMPLING WITH WAVELETS
In applications such as MRI, X-ray CT and radio interferometry, measurements arise by sampling the Fourier transform
ΨDFT and the sparsifying transformation is usually taken to be
a wavelet transform ΦDWT . Unfortunately, µ(U ) = O (1) as
N → ∞ in this case, due to the high coherence of coarse scale
wavelets with low frequency Fourier measurements. Although
only a sufficient condition, (2) suggests that subsampling
uniformly at random may lead to poor results in this case.
In fact, this is well-known in the CS MRI literature [8], [9],
and is shown explicitly in Fig 1.
Nonetheless, it has long been observed empirically that
excellent CS reconstruction is indeed possible in MRI using
the same total number of measurements, provided the Fourier
transform is sampled in a different way [8], [9]. In particular,
the sampling strategy must be adapted to the coherence pattern
of U . Fig 1 illustrates that distributing more of the samples
near the low-frequency regime, where the coherence is higher,
leads to a vastly superior reconstruction in this case.
Perhaps surprisingly, in addition to being adapted to the
coherence pattern, the subsampling strategy must also be
adapted to the sparsity structure of the coefficients Φ∗ x. To
show this, we perform the following experiment, introduced in
[1], [2] and known as the flip test. First, let Ω be an appropriate
sampling strategy for x (e.g. as in the top-right panel of Fig
1) and let x̂ be the CS reconstruction of x from (1). Given
x, let x0 = ΦP Φ∗ x, where P ∈ CN ×N is a permutation
matrix that flips the entries of a vector, i.e. (P z)n = zN −n+1 ,
n = 1, . . . , N . Now compute the reconstruction x̂0 of x̂
in the same way using the same sampling strategy Ω and
reconstruction (1), and then reverse the permutation to get
another reconstruction of x given by x̌ = ΦP Φ∗ x̂0 . Notice
that the best approximation error σs (Φ∗ x) = σs (P Φ∗ x) is
unaffected by permutations. Hence, if sparsity alone governs
the reconstruction quality of x, then both x̂ and x̌ should give
rise to roughly the same errors.
Two examples of the flip test, arising from different CS
applications, are shown in Fig 2 (further illustrations, including
from applications such as electron microscopy and fluorescence microscopy, can be found in [2]). As is evident, in both
cases the flipped reconstruction x̌ is drastically worse than the
unflipped reconstruction x̂. We conclude that the structure,
in other words, the ordering, of the coefficients Φ∗ x plays a
crucial role in the reconstruction quality, not just sparsity.
IV. A LEVEL - BASED THEORY OF COMPRESSED SENSING
The standard CS result (2) explains neither the good recovery seen in Fig 1 when sampling according to the coherence,
nor the role of the sparsity structure witnessed in Fig 2. We
now present a new framework that explains both. Full details
can be found in [1].
A. New concepts
Standard CS is built on three global properties: sparsity,
incoherence and uniform random subsampling. Conversely,
Figs 1 and 2 demonstrate that local behaviour matters. In
particular, local coherence and local sampling (Fig 1), and
local sparsity (Fig 2). Hence, we first introduce extensions of
sparsity, incoherence and uniform random subsampling that
take into account such local behaviour.
Our construction is based on levels. Let r ∈ N be the
number of levels and define two vectors of parameters N =
(N1 , . . . , Nr ) ∈ Nr , 1 ≤ N1 < . . . < Nr = N and
M = (M1 , . . . , Mr ) ∈ Nr , 1 ≤ M1 < . . . < Mr = N .
1, . . . , r. In the simple case where U is block diagonal, i.e.
U (j,k) = 0 for j 6= k, the minimization (1) decouples into r
separate problems. Applying the standard CS estimate (2) to
each problem, one deduces that mj must be on the order of
sj · µN,M (j, j) · (Nj − Nj−1 ) (up to log factors) to ensure
recovery of x up to the noise η and the best approximation
error σs,M (Φ∗ x).
In practice, U is not block diagonal and this complicates
the recovery estimates substantially. In particular, there may
be interference between different sparsity levels. To handle
this, we need the notion of a relative sparsity:
subsampling map
x̂
x̌
Fig. 2. Examples of the flip test. Top row: MRI with N = 256 × 256 and
m/N = 20%. Bottom row: Radio interferometry with N = 512 × 512 and
m/N = 15%.
We refer to N and M as the sampling and sparsity levels
respectively. For convenience, we write M0 = N0 = 0.
Definition 1 (Local coherence in levels). Let U ∈ CN ×N be
an isometry. For j, k = 1, . . . , r the (j, k)th local of coherence
of U with respect to the levels N and M is
r N
N
M
µN,M (j, k) = µ PNjj−1 U PMkk−1 µ PNjj−1 U .
Here we use the notation Pba = P{a+1,...,b} for 1 ≤ a <
b ≤ N . Note that the global coherence is a special case of
local coherence corresponding to one level.
Definition 2 (Sparsity in levels). Let c ∈ CN . Given M and
s = (s1 , . . . , sr ) ∈ Nr with sj ≤ Mj − Mj−1 , j = 1, . . . , r
we say that z is (s, M)-sparse if
|{j : cj 6= 0} ∩ {Mj−1 + 1, . . . , Mj }| ≤ sj ,
j = 1, . . . , r.
Definition 4 (Relative sparsity). Given an isometry U ∈
CN ×N , levels N and M and local sparsities s the j th relative
sparsity Sj = Sj (N, M, s) is defined by
o
n
N
Sj = max kPNjj−1 U ck2 : c ∈ Σs,M , kck∞ ≤ 1 .
Note that Sj ≤ s1 + . . . + sr for j = 1, . . . , r. Our main
result is now as follows (see [1] for a proof):
Theorem 5. Fix x ∈ CN , isometries Ψ, Φ ∈ CN ×N and
0 < ≤ e−1 . Let Ω = ΩN,m be an (N, m)-multilevel
subsampling scheme, and suppose that there are sparsity levels
M and local sparsities s satisfying


r
Nj − Nj−1 X
·
µN,M (j, k) · sk  · log(−1 ) · log(N ),
1&
mj
j=1
for j = 1, . . . , r and mj & m̂j ·log(−1 )·log(N ), j = 1, . . . , r,
where m̂j is such that
1&
r X
Nj − Nj−1
− 1 · µN,M (j, k) · s̃j ,
m̂j
j=1
k = 1, . . . , r,
for all s̃1 , . . . , s̃r > 0 satisfying
The set of (s, M)-sparse vectors is denoted by Σs,M .
Note that sparsity is a special case of sparsity in levels corresponding to one level. Similarly, we define the best (s, M)term approximation error as σs,M (z) = minc∈Σs,M kz − ck1 .
Observe also that (s, M)-sparsity is a local quantity within the
levels. Unlike s-sparsity, it is not preserved under the flipping
permutation of the flip test.
Definition 3 (Multilevel random subsampling). Given N, let
m = (m1 , . . . , mr ) ∈ Nr with mj ≤ Nj − Nj−1 , ∀j. Suppose
that Ωj ⊆ {Nj−1 + 1, . . . , Nj }, |Ωj | = mj , j = 1, . . . , r, are
chosen uniformly at random. We say that the set Ω = ΩN,m =
Ω1 ∪ · · · ∪ Ωr is an (N, m)-multilevel sampling scheme.
Unlike uniform random subsampling, multilevel random
sampling permits different degrees of sampling locally within
the pre-specified levels. Examples are shown in Figs 1 and 2.
B. Theoretical results
The sampling and sparsity levels N and M divide the matrix
N
M
U into rectangular blocks U (j,k) = PNjj−1 U PMkk−1 , j, k =
s̃1 + . . . + s̃r ≤ s1 + . . . + sr ,
s̃j ≤ Sj .
Suppose that x̂ ∈ CN is a minimizer of
min kΦ∗ zk1
z∈CN
√
subject to ky − PΩ Ψ∗ zk ≤ η/ K,
where K = maxj=1,...,r {(Nj − Nj−1 )/mj }. Then with probability exceeding 1 − s, where s = s1 + . . . + sr , we have
that
√
kx − x̂k . σs,M (Φ∗ x) + sη.
If mj = Nj − Nj−1 for j = 1, . . . , r then this holds with
probability 1.
This result is general, applying to any isometries Φ, Ψ and
multilevel subsampling schemes Ω, with the standard CS estimate (2) becoming a corollary of this theorem. Moreover, the
general estimates are also sharp in the sense that they reduce
down to known information-theoretic limits in a number of
key cases [1].
C. Example: Fourier sampling with Haar wavelets
E. Background
To illustrate this result, consider the example of Fourier
sampling Ψ = ΨDFT with Haar wavelets Φ = ΦHWT in one
dimension. In this case, if the sparsity levels are chosen to be
the wavelet scales – in particular, so that Nj −Nj−1 = 2j – and
if the sampling levels describe the dyadic bands in frequency
space B1 , . . . , Br , where B1 = {0, 1} and
It has long been known that wavelet coefficients possess
additional structure beyond sparsity, and there are a number
of CS algorithms that exploit the so-called connected tree
structure of wavelet coefficients. We discuss these further in
Sec V-C. However, we note in passing that local sparsity is a
more general signal model than tree structure, in that it neither
assumes the sparsifying transform is of wavelet type nor that
there are any inter-level dependencies between coefficients.
There are also a number of different characterizations of
non-uniform incoherence in the CS literature [11], [12], [13],
[14]. What differentiates local coherence in levels is that it
allows one to capture local behaviour in the different sparsity
levels, and in particular, capture the near block diagonal
structure of the Fourier/wavelet matrix. This is necessary
in order to get recovery guarantees such as (3) that are in
agreement with the flip test.
Furthermore, we note that the idea of sampling low-order
coefficients of an image differently dates back to some of the
earliest work in CS. For a in-depth discussion we refer to [2].
Bj+1 = {−2j + 1, . . . , −2j−1 } ∪ {2j−1 + 1, . . . , 2j },
for j = 0, . . . , r − 1, then the following can be shown [10].
First, the local coherences satisfy µN,M (j, j) . 2−j and
µN,M (j, k) . µN,M (j, j)2−|j−k| ,
j, k = 1, . . . , r.
Loosely speaking, we refer to the decreasing nature of the
coherences as j or k increases as asymptotic
Pr incoherence.
Second, the relative sparsities satisfy Sj . k=1 2−|j−k|/2 sk ,
j = 1, . . . , r. Substituting these bounds into Theorem 5 now
reveals the following estimate
!
r
X
−|j−k|/2
mj &
2
sj · log(−1 ) · log(N ).
(3)
k=1
This confirms near-optimal recovery of Haar wavelet coefficients from Fourier samples. In the j th sampling level the
number of measurements is proportional to the corresponding
sparsity sj plus exponentially-decaying terms in sk , k 6= j.
The presence of such terms is due to the aforementioned
interference between sparsity levels.
This estimate also explains the observations witnessed in
Figs 1 and 2. First, note that wavelet coefficients of natural
images are asymptotically sparse: that is, sj /(Nj − Nj−1 )
decreases with increasing j [1]. Hence (3) dictates that a higher
fraction of samples be taken for small j (i.e. in low-frequency
regimes in Fourier space) than for large j, which is exactly
the characteristic of the sampling strategy deployed in Fig 1.
Second, note that if the coefficients Φ∗ x are flipped as in Fig 2
then the local sparsities sj change. Worse recovery is expected
in the flipped case, since not enough measurements are taken
at high frequencies to recover the flipped image x0 .
This result extends to arbitrary wavelets, in which case the
effect of the interference is lessened as the smoothness and
number of vanishing moments increases. See [1].
D. Further theoretical results
In problems such as MRI, X-ray and electron microscopy
measurements actually arise from sampling the continuous
Fourier transform. In this case, an infinite-dimensional CS
framework [15] can be used to avoid issues such measurement
mismatch which can otherwise degrade reconstruction accuracy. Fortunately, this framework and, in particular, Theorem
5 can be extended to this setting. For details we refer to [1].
The reader will have noticed that Theorem 5 is a nonuniform
recovery guarantee. In order to prove uniform guarantees,
a level-based notion of the restricted isometry property was
introduced in [16].
V. C ONSEQUENCES
We now discuss several consequences of this framework. For
a comprehensive list and thorough discussion, we refer to [2].
A. Best sampling strategies
The flip test and the Theorem 5 both demonstrate that the
best sampling strategy for an image x depends on not just
its global sparsity, but also the ordering of its coefficients
c = Φ∗ x. Unlike in the standard setting of incoherent sensing
(where the measurements are chosen independently of the
entries of c, in the locally incoherent setting both the sampling
levels N and subsampling fractions mj must be chosen
depending on the local sparsities sj and sparsity levels M.
As mentioned, the measurements in MRI are constrained
to be Fourier measurements with wavelets (or one of their
generalizations) being the sparsifying transformation, which
is an ideal fit for this framework. This effect means that in
practical MR setups the chosen sampling strategy must be
tailored to application-specific features (e.g. brain imaging).
No universal optimal sampling strategy exists.
B. Reconstruction and resolution
Regardless of the sampling basis and subsampling scheme,
in the asymptotically sparse and asymptotically incoherent
setting the quality of the reconstruction increases as resolution
increases. This effect is shown in Fig 3. At higher resolution,
the reconstruction from the same total number of measurements is substantially better than at lower resolutions. This
has consequences for numerous applications, such as MRI.
Rather than reducing scanning times at low resolutions, the
effectiveness of CS for such applications lies in enhancing
quality in higher resolution scans. Practical verification of this
phenomenon has been shown in [17].
We note that the best subsampling strategy must also depend
on the resolution in addition to the signal structure [2].
Fig. 3. Reconstruction from 5122 = 262144 Fourier measurements. Left:
reconstruction from lowest 5122 frequencies. Right: reconstruction from
multilevel subsampled frequencies using same number of measurements.
C. Structured sampling algorithms for compressive imaging
The connected tree structure of wavelet coefficients has
previously been incorporated into CS reconstructions via modification of a suitable CS recovery algorithm (e.g. matching
pursuit or iterative thresholding) rather than the measurements.
Examples of this are model-based CS [3], Bayesian CS [5] and
TurboAMP [6], all of which are based around incoherent sensing, typically with random Gaussian or Bernoulli matrices. We
refer to these approaches as structured recovery algorithms.
Due to the requirement of incoherent sensing, these algorithms
are restricted to applications where the sensing process can be
tailored as opposed to applications where the sensing process
is fixed (e.g. medical imaging).
Conversely, in this paper we consider structured sampling
algorithms: the structured sparsity (specifically, the asymptotic sparsity) of wavelet coefficients is incorporated into
the sampling procedure via multilevel subsampling of an
asymptotically incoherent sensing operator such as the Fourier
or Hadamard matrix, and then combined with a standard CS
recovery procedure such as `1 minimization. Although this
approach was initially motivated by problems such as MRI,
there is no reason why it cannot be applied to problems such
as compressive imaging.
In Fig 4 we present a comparison of these approaches
– see [2] for further details. As is evident, the structured
sampling approach yields a substantial benefit over structured
recovery algorithms. Moreover, this approach has two additional benefits. First, it is fast and not memory intensive since
it uses Hadamard/Fourier matrices. Second, unlike the other
approaches it can immediately be adapted to other sparsifying
transforms, such as curvelets, shearlets, etc (see Fig 4).
R EFERENCES
[1] B. Adcock, A. C. Hansen, C. Poon, and B. Roman, “Breaking the coherence barrier: A new theory for compressed sensing,” arXiv:1302.0561,
2014.
[2] B. Roman, A. C. Hansen, and B. Adcock, “On asymptotic structure in
compressed sensing,” arXiv:1406.4178, 2014.
[3] R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hedge, “Model-based
compressive sensing,” IEEE Trans. Inform. Theory, vol. 56, no. 4, pp.
1982–2001, 2010.
[4] D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithms for compressed sensing,” Proc. Natl Acad. Sci. USA, vol. 106,
no. 45, pp. 18 914–18 919, 2009.
Fig. 4. Comparison of structured recovery and structure sampling algorithms
for 12.5% subsampling at 256 × 256 resolution. Top row: `1 minimization
with random Bernoulli (Err = 16.0%), model-based CS with random Bernoulli
(Err = 17.0%), TurboAMP with random Bernoulli (Err = 13.1%). Bottom row:
Bayesian CS with random Bernoulli (Err = 12.6%), `1 minimization with
multilevel Hadamard and db4 wavelets (Err = 9.5%), `1 minimization with
multilevel Hadamard and DT-CWT redundant frame [18] (Err = 8.6 %).
[5] L. He, H. Chen, and L. Carin, “Tree-structured compressive sensing with
variational Bayesian analysis,” IEEE Signal Process. Letters, vol. 17,
no. 3, pp. 233–236, 2010.
[6] S. Som and P. Schniter, “Compressive imaging using approximate
message passing and a markov-tree prior,” IEEE Trans. Signal Process.,
vol. 60, no. 7, pp. 3439–3448, 2012.
[7] E. J. Candès and Y. Plan, “A probabilistic and RIPless theory of
compressed sensing,” IEEE Trans. Inform. Theory, vol. 57, no. 11, pp.
7235–7254, 2011.
[8] M. Lustig, D. L. Donoho, J. M. Santos, and J. M.
Pauly, “Compressed Sensing MRI,” IEEE Signal Process. Mag.,
vol. 25, no. 2, pp. 72–82, March 2008. [Online]. Available:
http://dx.doi.org/10.1109/MSP.2007.914728
[9] M. Lustig, D. L. Donoho, and J. M. Pauly, “Sparse MRI: the application
of compressed sensing for rapid MRI imaging,” Magn. Reson. Imaging,
vol. 58, no. 6, pp. 1182–1195, 2007.
[10] B. Adcock, A. C. Hansen, and B. Roman, “A note on compressed sensing of structured sparse wavelet coefficients from subsampled Fourier
measurements,” arXiv:1403.6541, 2014.
[11] D. L. Donoho and M. Elad, “Optimally sparse representation in general
(non-orthogonal) dictionaries vi `1 minimizatiob,” Proc. Natl Acad. Sci.
USA, vol. 100, pp. 2197–2002, 2003.
[12] S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive
Sensing. Birkhauser, 2013.
[13] J. A. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Trans. Inform. Theory, vol. 50, no. 10, pp. 2231–2242,
2004.
[14] ——, “Just relax: convex programming methods for identifying sparse
signals in noise,” IEEE Trans. Inform. Theory, vol. 52, no. 3, pp. 1030–
1051, 2006.
[15] B. Adcock and A. C. Hansen, “Generalized sampling and infinitedimensional compressed sensing,” Technical report NA2011/02, DAMTP,
University of Cambridge, 2011.
[16] A. Bastounis and A. C. Hansen, “On the absence of the RIP in
real-world applications of compressed sensing and the RIP in levels,”
arXiv:1411.4449, 2014.
[17] Q. Wang, M. Zenge, H. E. Cetingul, E. Mueller, and M. S. Nadar, “Novel
sampling strategies for sparse mr image reconstruction,” Proc. Int. Soc.
Mag. Res. in Med., no. 22, 2014.
[18] N. Kingsbury, “Complex wavelets for shift invariant analysis and filtering of signals,” J Appl Comp Harmonic An, 2001.