Fast NMR Spectroscopy
DOI: 10.1002/anie.201100370
Accelerated NMR Spectroscopy by Using Compressed Sensing**
Krzysztof Kazimierczuk and Vladislav Yu. Orekhov*
There is increased interest in high-resolution, fast multi-
dimensional NMR spectroscopy for studying molecular
structure, interactions, and dynamics. The distinct feature of
the contemporary NMR spectroscopy, namely the possibility
to observe hundreds of atoms in complex macromolecules
simultaneously, finds its foundation in the invention of
multidimensional experiments in the mid 1970s.[1] However,
the ultimate resolution obtained in these experiments comes
at the high price of the long data collection times needed to
systematically sample the large multidimensional data sets.
The number of measured data points increases polynomialy
with desired spectral resolution and exponentially with a
number of dimensions.[2] The problem of lengthy sampling
often compromises or even prohibits many applications of the
multidimensional spectroscopy in chemistry and molecular
biology. Fortunately, the field of fast NMR spectroscopy
offers a number of solutions.[3–9] A common approach is to
replace the time-consuming systematic sampling of the signal
on the fine Nyquist grid by the random non-uniform sampling
(NUS).[4] For many years, however, NUS was associated with
the inherent loss of the spectrum quality, such as the presence
of spectral artefacts and false peaks.
Recently, Cands et al.[10] formulated a new NUS theo-
rem, which states that for most of the practical cases, a
significantly smaller number of data points in comparison to
the size the full Nyquist grid is sufficient for obtaining the
exact reconstruction of the spectrum. The theorem evoked
the rapidly growing group of signal processing methods,
referred to as the compressed sensing (CS) or compressive
sampling. A number of CS applications has been recently
demonstrated in various fields of science and technology,
including the striking results obtained for fast magnetic
resonance imaging (MRI).[11] Herein, we demonstrate CS as
an effective tool for obtaining high-quality spectra from the
NUS data and present the first experimental examples of
compressed sensing in NMR spectroscopy (CS-NMR).
According to the classical Nyquist–Shannon sampling
theorem, sampling at the constant rate, which is equal or
larger than the spectral bandwidth, is the necessary condition
for the exact reconstruction of the spectrum. This theorem is
based on an implicit pessimistic assumption that every point
in the spectrum carries important information. However, in
the most of the practical cases, including the NMR spectros-
copy, the peaks occupy only a small fraction of the spectrum,
while the rest is the baseline. In other words, we say that the
spectrum is sparse.
NMR signal sampled in the time domain and represented
as a vector s is associated with a frequency-domain spectrum S
in the following way:
FS ¼ s ð1Þ
where F stands for the inverse of discrete Fourier transform.
Equation (1) can be defined for one or multiple dimensions. If
the signal is sampled regularly on the Nyquist grid, the
number of unknown points in the spectrum S is equal to the
number of elements in the time-domain signal s, and
Equation (1) has a unique solution. In the NUS strategy,
measurements are performed only for a small fraction of the
randomly selected points from the grid. This saves much
experimental time, but the system of Equations (1) becomes
underdetermined and thus has the infinite number of
solutions. To choose the right solution and obtain the good
spectrum, additional assumptions need to be introduced. The
processing methods that have been developed over last three
decades for dealing with the NUS signal differ primarily in the
kind of assumptions and are more or less successful depend-
ing on its relevance for the particular type of spectrum and the
level of sparseness. Thus, the simplest approach is to choose
the solution that features the minimal power (or l2 norm). As
this is equivalent to assuming that signal is equal to zero in all
the grid points that were omitted in the experiment, the
spectrum can be obtained by the direct discrete Fourier
transform.[6] Another traditional approach looks for the
solution having maximum entropy.[7] The SIFT method[8]
offers the third alternative, where the number of unknowns
in the linear system of Equation (1) is significantly reduced by
setting some points of spectrum to be equal to zero. Finally, an
alternative to the Fourier basis set can be used that contains
fewer basis functions. Thus, MDD describes the spectrum by
the small number of tensor products of one-dimensional
vectors.[9] Unfortunately, none of above methods was math-
ematically proven to give the exact, that is, artifact-free
reconstruction of the spectrum.
In the CS approach, it is assumed that the best among the
solutions fulfilling Equation (1) is the sparsest one, that is,
containing the highest number of zeros. This corresponds to
the minimal l0 norm of the spectrum. Unfortunately, finding
such solution is the NP-hard task, and typically unachievable
in a reasonable computational time. Here the power of the CS
theory comes into play, which states that the sparsest solution
[*] K. Kazimierczuk, V. Yu. Orekhov
Swedish NMR Centre, University of Gothenburg
Box 465, 40530 Gteborg (Sweden)
Fax: (+46)31-786-3886
E-mail: vladislav.orekhov@nmr.gu.se
K. Kazimierczuk
Faculty of Chemistry, University of Warsaw
Pasteura 1, 02093 Warsaw (Poland)
[**] The Swedish Research Council (grant 2008-4299), Foundation for
Polish Science (KOLUMB stipend) for supporting K.K., Bruker
BioSpin, and EC Bio-NMR project no. 261863 are acknowledged.
Supporting information for this article is available on the WWW
under http://dx.doi.org/10.1002/anie.201100370.
Communications
5556  2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2011, 50, 5556 –5559

can be almost always found by the minimization of penalty
function involving the lp norm of the spectrum
[12] with p= 1
[Equation (2)]:
FS sk k2l2þl Sk k
p
lp ð2Þ
The penalty function is convex and thus has only one
(global) minimum. Moreover, the reconstruction of the
spectrum from NUS signal is exact (that is, the same as
from the full dataset) with overwhelming probability, and no
other reconstruction method can in general perform better
than CS.[13] We should note, however, that strictly speaking,
neither the NMR signal nor the accompanying Gaussian noise
are sparse in the frequency domain. Nevertheless, the theory
predicts that CS can be also applied to approximately sparse
and/or noisy signals (for a comprehensive review, see
Ref. [14]). Notably, the l1 norm penalty function was recently
successfully employed in the reconstruction of NMR spectra
by the forward maximum entropy method.[5] The penalty
functions [Eq. (2)] with the norm lp, where p less than 1 are
non-convex, that is, may have more than one local minimum.
However, in real applications the solution is at least as good as
the one obtained for the l1 norm. Besides, spectral recon-
structions using a norm with 0< p< 1 may display better
convergence[15] and require fewer measurements in compar-
ison with the l1 norm.
Finally, the matrix F in Equations (1) and (2) can be easily
defined to span only a fraction of the multidimensional
spectrum, and it is possible to apply the CS approach to
reconstruct regions of high-dimensional spectra when signal
positions in some of dimensions are known.[16] This would
correspond to assuming, in a similar fashion to the SIFT
method,[8] that there are no peaks beyond the selected
regions.
Herein we present results obtained using two different CS
algorithms: iterative soft thresholding (IST)[17] and iterative
re-weighted least squares (IRLS)[18] (for details of the
algorithms, see the Supporting Information). The IST algo-
rithm has been demonstrated on simulated NUS datasets[19] in
combination with the wavelet transform and for spectrum
reconstruction from a spatially encoded signal.[20] It was also
used for suppressing sinc-“wiggles” in the spectra of truncated
signals.[21] IST is reminiscent of various algorithms for
cleaning aliasing artefacts in the NMR spectra obtained
from sparse data.[22] Contrary to the IST, which is equivalent
to l1 norm minimization, the IRLS algorithm allows also the
minimization of norms less than 1 (later referred to as the lp!0
norm).[23] When IRLS algorithm is used for the l1 norm
penalty function, the convergence and results are similar to
IST, albeit with longer computational times.
We demonstrate results of the IST and IRLS algorithms
for three two-dimensional spectra: 1H–15N HSQC of a
globular 14 kDa protein azurin[24] (1 mm, temperature 25 8C,
900 MHz Varian UNITY Inova with cryoprobe) and 2D
NOESY and 2D DQF-COSY of human ubiquitin (1 mm,
temperature 25 8C, 600 MHz Varian UNITY Inova). It should
be emphasized that despite the modest computational costs,
the low-dimensional spectra of complex molecules, such as
proteins, are among the most difficult for the fast sampling
methods. Thus, for two-dimensional spectra, the reduced
dimensionality is not applicable and the NUS methods are
rarely used because 2D spectra are not significantly sparse
and the low absolute number of points leads to the poor
statistics for the random sampling.[25]
Figure 1a–c shows the comparison of HSQC spectrum
obtained by the Fourier transform of a full dataset (120
complex points) with the CS reconstructions using 18.3% of
the indirect dimension samples, chosen randomly according
to the exponentially biased probability distribution.[26] Recon-
structions were performed using the lp!0 norm (IRLS) and l1
norm (IST) minimizations. Figure 1d–f presents results for
2D NOESY with undersampling at the level of 39% (other
regions of the NOESY spectrum are shown in the Supporting
Information, Figure S3). Figure 2 illustrates linearity of the
peak intensities in the IRLS reconstruction of the NOESY
spectrum. The number of iterations in IRLS was set to 10.
Results for IST show that the algorithm works well for
spectra with good signal-to-noise ratios (S/N) and moderate
dynamic range; for example, the HSQC depicted in Fig-
ure 1a–c. In the more demanding situations, such as the
NOESY-type spectra (Figure 1d–f) with high dynamic range
and many weak peaks, the convergence of the algorithm
becomes slow. Although the IST finally converged to the
artefact-free spectrum, it required about 5000 iterations, and
thus significantly longer calculations than required for the
IRLS. Similarity of the spectra obtained using l1 (IST) and lp!0
(IRLS) norms is in line with the CS theory.[12] The limited
performance of the IST is in line with observations reported
for some of the thresholding-based artefact-cleaning algo-
Figure 1. a–c) 15N HSQC spectrum of azurin: a) Full spectrum
(120 pts), b) IRLS reconstruction (22 pts, 10 iterations), c) IST recon-
struction (22 pts, 900 iterations). d–f) 2D NOESY spectrum of ubiqui-
tin: d) Full spectrum (512 pts), e) IRLS reconstruction (200 pts, 10
iterations), f) IST reconstruction (200 pts, 1500 iterations). Residual
artifacts originating from a strong diagonal peak can be observed in
panel (f). Exponentially decaying distributions of points were used in
both NUS experiments.
5557Angew. Chem. Int. Ed. 2011, 50, 5556 –5559  2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org