Microwave Ghost Imaging via LTE-DL Signals

Microwave Ghost Imaging via LTE-DL Signals

                               Ziqian Zhang∗ , Ruichen Luo† , Xiaopeng Wang‡ and Zihuai Lin§
                                        ∗†‡§ School of Electrical and Information Engineering
                                     The University of Sydney, Sydney, N.S.W. 2006, Australia
                                               ∗ Email: ziqian.zhang@sydney.edu.au
                                                   † Email: rui.luo@sydney.edu.au
                                             ‡ Email: xiaopeng.wang@sydney.edu.au
                                                 § Email: zihuai.lin@sydney.edu.au


    Abstract—In this paper, we propose a long-term-evolution           LTE-DL signal to perform microwave GI has been deduced.
(LTE) downlink (DL) signal-based microwave ghost imaging (GI)          Then a signal selection scheme is proposed to transform the
scheme. Motivated by its waveform structures and base stations         existing LTE communication system into a LTE-DL based
(BSs) distributions, LTE-DL signals conventionally employed for        microwave GI system. Numerical simulation results show that
communication applications are transformed and adopted into            the proposed scheme can effectively achieve the reconstruction
the scenario of microwave GI. Numerical simulation validates
that the proposed LTE-DL based microwave GI scheme can
                                                                       of objects. Since the requirement of the purposely deployment
effectively obtain the reconstruction of objects. Since the proposed   of transmitters and receivers is avoided, the microwave GI
system takes the advantage of existing LTE BSs as its transceivers,    system complexity and operational cost has been significantly
microwave GI system complexity and operational cost have been          reduced. To the best of our knowledge, this is the first time for
significantly reduced.                                                 microwave GI to be implemented by non-purposely designed
                                                                       illuminating signals and sources. It is also the first time
  Keywords—Long-term-evolution (LTE), LTE-downlink (LTE-               for LTE signals which are originally designed and used for
DL) signals, microwave ghost imaging.
                                                                       communications to be applied in the scenario of microwave
                                                                       GI.
                      I.   I NTRODUCTION
                                                                           The remaining of this paper is organized as follows. In
    Microwave ghost imaging (GI) is originated from quantum            Section II, the motivation about applying LTE-DL signals for
and optical areas [1], [2], [3], [4], [5], [6]. Compared with          microwave GI is presented, followed by a typical LTE-DL-
conventional microwave imaging methods, it possesses some              based GI scenario in Section III. In Section IV, the signal
unique features such as nonlocal reconstruction [6], non-              selection method is presented together with the reconstruction
scanning [7], super-resolution [8], [9], etc. Besides, it also         process of the proposed LTE-DL signal based microwave GI
benefits from the penetration ability of microwave spectrum,           system. Then in Section V, numerical simulation results are
which enables microwave GI systems with a immunity towards             shown to verify the effectiveness of the proposed scheme.
weather and illumination conditions [9], as well as obstacles          Finally in Section VI, some conclusion remarks are drawn.
[10]. However, due to incoherent requirements of the illumina-
tion fields, both signal waveforms and transmitter deployment                                II.   M OTIVATION
should be deliberately designed [11] in such a system. Thus,
the system complexity and operational cost for previously                   Time-space incoherent fields are essential to both optical
proposed microwave GI schemes [9], [10], [11] should be                and microwave GI [17]. In order to satisfy this requirement,
further reduced.                                                       signals that are used for illumination should (1) possess a
                                                                       randomly modulated waveform; (2) be orthogonal to each
    Long-term-evolution (LTE) is a representative technology           other when waveforms are from different transmitters [11].
which is widely used in the fourth generation (4G) wireless            Microwave GI systems previously discussed in the literature
communication system [12]. LTE-downlink (LTE-DL) sig-                  are employing signals obtained from stochastic processes, such
nals are orthogonal-frequency-division-multiplexing (OFDM)             as Gaussian random processes [11] and nonlinear chaotic
structured, with an operating frequency varying from 800MHz            processes [18]. However, theses deliberately designed signals
to 3.5GHz and a bandwidth upto 20MHz [13]. Although                    are not easy to be generated [9]. Thus the system complexity
the LTE-DL signal is originally designed for communication             and operational cost of conventional microwave GI schemes
usage, it has been adopted into many other applications re-            still need to be further optimized.
cently. For example, due to deterministic features contained
in the signal, it is proposed to perform range and Doppler                 Besides, waveforms that already exist in the environment
estimation in [14], [15]. As another example, based on the             also possess a similar random feature, especially for those
forward scattering radar technique, it is also used for vehicle        employed in mobile communications such as FDD-LTE and
recognition in [16].                                                   TDD-LTE in the 4G systems. Both FDD and TDD based
                                                                       LTE systems contain two independent links for data transfer,
    Motivated by its pseudo-random feature, we propose a               namely the LTE-up-link (LTE-UL) and the LTE-DL. LTE-UL
LTE-DL signal based microwave GI scheme in this paper.                 refer to the links from battery powered mobile devices to their
Based on a further analysis of both the signal structure and           individually associated BSs while LTE-DL are those links from
distribution of LTE base stations (BSs), the feasibility of using      BSs to mobile devices. In this paper, we only consider using
Radio frames
                                                                                                                                                BS                                                        BS




                          One Radio frame = 10 ms
                                                                                                                                                                   BS
                                                                                                                                                                                       OFDM Symbols Only Contain Scrambled Dat
                                                                                                          Objects                                                                                    BS             Base Stations
              0       1    2    4   5   6         . . . . . . . . . .       18 19
                                                                                                                                    BS


                               One Slot = 0.5 ms                                                    z
                                                                                                                                                                                      BS
                                                                                                                      B




                                                                                                                                                             Subcarrier (Frequency)
                  0             1       2          3           4        5      6                y
                                                                                                            x


                               One OFDM Symbol
                                                                                          Fig. 2. A typical 3D scenario of the proposed LTE-based microwave GI
              Cyclic prefix                 OFDM Symbol Data
                                                                                          scheme.

                                                                                                                      Symbol time duration T              Time period of OFDM symbol data

Fig. 1.   LTE-DL signal structure.                                                                        1 st   CP      OFDM Symbol Data            CP        OFDM Symbol Data
                                                                                                          2 nd
                                                                                                                      CP     OFDM Symbol Data               CP                   OFDM Symbol Data

                                                                                                                 Delay
LTE-DL signals as the illumination sources for microwave GI                                               Last               CP          OFDM Symbol Data                         CP          OFDM Symbol Data               OFDM Symbols (Time)
as BSs always have stable power supply. As illustrated in Fig.                                                                                                                                                     Time
                                                                                                                                  Useable time duration                                    Useable time duration
1, a typical LTE-DL signal radio frame consists of 10 sub-
frames. Each of the sub-frame lasts for 1ms. One sub-frame                                Fig. 3.       LTE-DL signals time selection.
can be further divided into 2 slots, while each slot contains 7
OFDM symbols for short cyclic prefix (CP). Data requested by
mobile users are contained in those OFDM symbols together                                      IV.         T HE LTE-DL S IGNAL - BASED M ICROWAVE GI
with other repeated signal elements for identification, channel
estimation, synchronization, etc.                                                             In order to perform microwave GI by using LTE-DL
                                                                                          signals, the illumination fields should satisfy the time-space
    In order to evenly distribute energy upon the carrier band                            independent requirement [17]. Since the generation of fields
and decrease the error probability, users’ requested data is                              are characterised by both LTE-DL signals and the localizations
processed by the scrambler before being passed to the OFDM                                of corresponding BSs, in this section, we mainly discuss the
modulator. A scrambler is to convert the original input data into                         selection of LTE-DL signals and the distribution of BSs.
a pseudo-random state to avoid long data sequences appear
the same values. Denote scrambled data symbols in the kth
subcarrier by Xk , k ∈ {1, 2, ..., K}, where K is the total                               A. LTE-DL Signal Selection
number of sub-carriers, the resulting data part waveform can                                  In order to generate the incoherent field for microwave GI,
be written as,                                                                            the signal source should be random [11]. However, due to
                                        K−1                                               the LTE-DL signal structure, the coherent components such as
                                                                                          pilots and synchronization sequences containing in the signal
                                        X
                           v(t) =               Xk ej2π(fc +k∆f )t                  (1)
                                                                                          will lead to the incoherence destruction of the illumination
                                        k=0
                                                                                          fields and the quality of reconstruction. Although the scram-
where fc is the carrier frequency and ∆f is the subcarrier                                bling process introduces randomness in the generated LTE-DL
spacing.                                                                                  signals, not all the OFDM symbols contain the scrambled data.
                                                                                          For instance, the CP is a repetition of each OFDM symbol’s
    Apparently that since Xk is processed by the scrambler, the                           end. Obviously this repetition will decrease the randomness of
generated waveform can be recognized as a randomly mod-                                   the resulting time domain LTE-DL signals. Similarly, repetitive
ulated signal. In addition, according to the LTE regulations,                             modulated signals contained in sub-frames, such as the syn-
different scrambling sequences will be assigned to different                              chronization sequence [19], will also reduce the randomness.
users in different BSs. In other words, signals transmitted by
different BSs are orthogonal to each other. Consequently, the                                In order to ensure that the EM fields used for reconstruction
condition of performing microwave GI is satisfied.                                        only consist of the data part of LTE-DL signals, here we
                                                                                          propose a signal selection method. As illustrated in Fig. 3, only
                                                                                          OFDM symbols containing users data are considered while the
                                III.        S YSTEM M ODEL
                                                                                          CP part should be avoided. This usable time duration can be
    A typical three-dimensional (3D) LTE-based microwave                                  expressed as
GI scenario is shown in Fig. 2. The investigation area B is
inhomogeneous and contains several objects to be imaged. The
objects in B are distinguished by their scattering coefficients                                                                      τu = τdata − τmax                                                                              (2)
σ(ro ), ro ∈ B, where ro is a vector containing the locations                             where τdata is the time period of the OFDM symbol data,
for all the objects in the area of B. The investigation area B                            τmax is maximum propagation delay from BSs to pixels on
is under the illumination from LTE BSs whose locations are                                the imaging plane B. Assuming each OFDM symbol with the
expressed as ri , i = 1, 2, ..., I, where I is the total number of                        scrambled data only has a time duration of T , then the window
BSs. The deployment height from BSs to B are identical and                                period twindow suitable for microwave GI reconstruction can
denoted as h. Signals transmitted from the ith BS is denoted                              be written as
by Si . Reflected signals from objects are collected by a single
receiving BS located in the centre of B.                                                                                   τmax + τCP < twindow < T                                                                                 (3)
TABLE I.         PARAMETERS OF SIMULATION
   In other words, measurements should be conducted within
the above window period, in order to obtain a high quality                                       Parameters            Settings
reconstruction of objects.                                                                 Frame Structure Type     FDD-LTE-DL
                                                                                                 Bandwidth             20 MHz
                                                                                              Central Frequency        2.6 GHz
B. Distribution of Base Stations
                                                                                        Number of Resource Blocks        100
     Although the LTE BSs distributions are normally modelled                              Sub-carriers Spacing        15 KHz
by placing the BS on a regular hexagonal lattice or a square                                      CP Type              Normal
lattice in standard regulations [20], distribution of base stations                        Constellation Mapper       256-QAM
is difficult to remain uniform in practice [21]. Instead, BSs                                   Average ISD             50 m
distribution is normally modelled as Poisson point process
(PPP) with an intensity η, which can be written as,
                                    1 2
                           η=(         )                       (4)
                                   2dε
where dε is the average inter-site distance (ISD).
   Consider the EM field on plane B under the illumination
by the LTE BSs,
                                                                                        (a)                                    (b)
                            I
                            X
              E(∆x, t) =          Si (t − τi,∆x )`i,∆x         (5)
                            i=1

where E(∆x, t) is the EM state of the single point ∆x on the
imaging plane B, ∆x ∈ B, τi,∆x is the propagation delay from
the ith signal transmitter to ∆x, and `i,∆x is the propagation
attenuation. Apparently, the background EM field is not only
affected by the illumination signal, but also determined by                             (c)                                    (d)
the propagation delay induced by the BSs distribution. Thus
the spatial incoherence of the generated EM field can be
further enhanced by the randomness introduced by the irregular        Fig. 4. Normalized correlation results with and without proposed signal
                                                                      selection method. (a) Autocorrelation result of the original LTE-DL signal.
propagation delays provided by non-uniform distributed BSs.           (b) Autocorrelation result after the proposed method is applied. (c) Cross-
                                                                      correlation result of the original LTE-DL signal from different transmitters.
                                                                      (d) Cross-correlation result after the proposed method is applied.
C. Image Reconstruction
   Assuming the imaging area B is divided into N pixels with
P rows and Q columns, where N = P × Q. According to the               where k · k22 represents for the square of the Euclidean norm,
Born approximation [22], the overall imaging equation can be          and [σo ] is the unknown optimization variable. For solving this
expressed as                                                          optimization problem, algorithms such as gradient projection
                      [y] = [E][σ][ρ]                    (6)          [24], genetic algorithm [25], singular-value decomposition
                                                                      [26], and iteratively reweighted norm algorithm [27] can be
where                1                                              used here.
                                 1             1
                     E1,1      E2,1     . . . EP,Q
                     2          2             2 
                    E1,1      E2,1     . . . EP,Q                                       V.      S IMULATION R ESULTS
              [E] = 
                     ..        ..      ..     ..             (7)
                                            .
                                                   
                     .          .              .                        In this section, numerical simulation results are presented
                       N
                     E1,1        N
                               E2,1            N
                                        . . . EP,Q                    to validate the effectiveness of our proposed scheme. The
                                                                      investigation area B is set to be 420m×420m and discrete
                                                                      into sub-grids with a size of 10m×10m. The total number of
                [σ] = [σ1,1 , σ2,1 , ..., σP,Q ]T              (8)    BSs is 19 with a distance of h = 10m. The distribution of BSs
                [ρ] = diag[ρ1,1 , ρ2,1 , ..., ρP,Q ]           (9)    is configured according to the PPP model [21]. Other details
                                                                      of the simulation are listed in the Table I.
                [y] = [y1 , y2 , . . . , yN ]T                (10)
         n                                                                Self-correlation of the LTE-DL signal and cross-correlation
where Ep,q   is the background EM field at the corresponding          between LTE-DL signals transmitted by different BSs are eval-
pixel in the nth measurement, σp,q the scattering coefficient,        uated and as shown in Fig. 4. We can see from Fig. 4(a) that
yn is the receiving signal from the nth illumination and ρp,q         the original LTE-DL signal suffers greatly from the repeated
is the propagation attenuation from the pixel to the receiver.        signal elements, resulting in a series of significant side-lobes.
    Assuming ρp,q is known, then the image reconstruction of          However, after the proposed signal selection method is applied
the object can be achieved by solving the following convex            to the LTE-DL signal, those side-lobes have been effectively
optimization problem [23]                                             compressed and the signal self-correlation level has been
                                                                      dramatically increased, as shown in Fig. 4(b). In addition, from
                minimize k[y] − [E][σo ][ρ]k22                (11)    the result shown in Fig. 4(c) and (d) we can see that the cross-
400
                                                                                                                                                                 is applied as a quantitive evaluation metric, the reconstruction




                                                      Normalized Spatial Correlation
                                                                                         1


            300                                                                        0.75
                                                                                                                                                                 performance of the proposed LTE-based microwave GI is
                                                                                                                                                                 shown in Fig. 6(d). We can see that the proposed method is
     Y(m)


                                                                                        0.5
            200                                                                                                                                                  not sensitive to the signal-to-noise ratio (SNR) condition when
            100
                                                                                       0.25                                                                      SNR is high but the MSE performance decreases dramatically
                                                                                          0
                                                                                                                                                                 especially when SNR is lower than 5dB.
                                                                                        400
              0                                                                                   200                                                      400
                                                                                                          0                                          200
                  0       100    200    300    400                                                            -200                          0
                                                                                                                     -400            -200
                                                                                                                              -400
                                 X(m)                                                               Y(m)                                    X(m)                                            VI.     C ONCLUSION
                                (a)                                                                                     (b)
                                                                                                                                                                     In this paper, we proposed a novel LTE-DL signal based
                                                                                                                                                                 microwave GI scheme. To the best of our knowledge, this
Fig. 5. BSs distribution and spatial-correlation of the generated illumination                                                                                   is the first time for LTE-DL signals which are originally
field. (a) PPP-based BSs distribution. (b) Spatial-correlation of the illumination                                                                               designed for communication purposes to be adopted in the
field.
                                                                                                                                                                 framework of microwave GI. It is also the first time for
                      0           0.5           1                                                     0                        0.5                         1     microwave GI to be implemented by using signals and trans-
                                                                                                                                                                 mitters that originally designed for communication purpose.
            400                                                                          400                                                                     Numerical simulation results showed that the proposed LTE-
            300                                                                          300
                                                                                                                                                                 DL signal based microwave GI scheme can effectively obtain
                                                                                                                                                                 the image of objects. Compared to conventional microwave GI,
     Y(m)




                                                      Y(m)




            200                                                                          200                                                                     the proposed method possesses a significantly reduced system
                                                                                                                                                                 complexity and operational cost, since the purposely deployed
            100                                                                          100
                                                                                                                                                                 signals and transceivers are avoided.
                  0                                                                               0
                      0   100    200    300   400                                                     0         100           200         300        400
                                 X(m)                                                                                         X(m)                                                              R EFERENCES
                                (a)                                                                                     (b)                                       [1]   T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging
                  0               0.5           1
                                                                                                                                                                        by means of two-photon quantum entanglement,” Physical Review A,
                                                                                                                                                                        vol. 52, no. 5, p. R3429, 1995.
                                                                                       0.2
            400                                                                                                                                                   [2]   J. Cheng and S. Han, “Incoherent coincidence imaging and its appli-
                                                                                                                                                                        cability in x-ray diffraction,” Physical review letters, vol. 92, no. 9, p.
                                                                              0.15
            300                                                                                                                                                         093903, 2004.
                                                                                                                                                                  [3]   G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of
                                                     MSE
     Y(m)




            200                                                                        0.1
                                                                                                                                                                        chaotic light be considered as correlation of intensity fluctuations?”
                                                                                                                                                                        Physical review letters, vol. 96, no. 6, p. 063602, 2006.
            100                                                               0.05
                                                                                                                                                                  [4]   Y. Shih, “Quantum imaging,” IEEE Journal of Selected Topics in
                                                                                                                                                                        Quantum Electronics, vol. 13, no. 4, pp. 1016–1030, 2007.
              0                                                                          0
                  0       100    200    300   400                                             0            5            10           15         20         25     [5]   J. Liu and Y. Shih, “N th-order coherence of thermal light,” Physical
                                 X(m)                                                                                       SNR (dB)                                    Review A, vol. 79, no. 2, p. 023819, 2009.
                                (c)                                                                                     (d)                                       [6]   J. Liu, J. Zhu, C. Lu, and S. Huang, “High-quality quantum-imaging
                                                                                                                                                                        algorithm and experiment based on compressive sensing,” Optics letters,
                                                                                                                                                                        vol. 35, no. 8, pp. 1206–1208, 2010.
Fig. 6. Reconstruction results with PPP-based BSs distribution and proposed                                                                                       [7]   C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han,
signal selection method. (a) Original scenario. (b) Reconstructed results when                                                                                          “Ghost imaging lidar via sparsity constraints,” Applied Physics Letters,
SNR = 10 dB. (c) Reconstructed results when SNR = 0 dB. (d) Reconstruction                                                                                              vol. 101, no. 14, p. 141123, 2012.
MSE curve.
                                                                                                                                                                  [8]   M. DAngelo, A. Valencia, M. H. Rubin, and Y. Shih, “Resolution of
                                                                                                                                                                        quantum and classical ghost imaging,” Physical Review A, vol. 72, no. 1,
                                                                                                                                                                        p. 013810, 2005.
correlations between LTE-DL signals transmitted by different                                                                                                      [9]   X. Wang and Z. Lin, “Microwave surveillance based on ghost imaging
BSs without and with proposed method, respectively, stays                                                                                                               and distributed antennas,” IEEE Antennas and Wireless Propagation
low and does not change significantly with the applications of                                                                                                          Letters, vol. 15, pp. 1831–1834, 2016.
the proposed signal selection method, indicating an inherently                                                                                                   [10]   ——, “High-resolution through-wall ghost imaging algorithm using
orthogonal relationship for LTE-DL signals.                                                                                                                             chaotic modulated signal,” in Acoustics, Speech and Signal Processing
                                                                                                                                                                        (ICASSP), 2015 IEEE International Conference on. IEEE, 2015, pp.
    The PPP-modelled BSs distribution and the spatial-                                                                                                                  1642–1646.
correlation of the generated illumination fields are illustrated in                                                                                              [11]   D. Li, X. Li, Y. Qin, Y. Cheng, and H. Wang, “Radar coincidence
Fig. 5(a) and (b), respectively. It shows that when the proposed                                                                                                        imaging: an instantaneous imaging technique with stochastic signals,”
signal selection method is applied, the resulting microwave                                                                                                             IEEE Transactions on Geoscience and Remote Sensing, vol. 52, no. 4,
                                                                                                                                                                        pp. 2261–2277, 2014.
fields on the investigation area B is incoherent, which satisfy
                                                                                                                                                                 [12]   S. Sesia, M. Baker, and I. Toufik, LTE-the UMTS long term evolution:
the requirement of performing microwave GI.                                                                                                                             from theory to practice. John Wiley & Sons, 2011.
    In order to validate the effectiveness of the proposed LTE-                                                                                                  [13]   S. Ahmadi, “An overview of 3gpp long-term evolution radio access
based microwave GI, a single square shaped target is located                                                                                                            network,” New Directions in Wireless Communications Research, pp.
                                                                                                                                                                        431–465, 2009.
in the center of the investigation area B, as shown in Fig.
                                                                                                                                                                 [14]   A. Salah, R. R. Abdullah, A. Ismail, F. Hashim, and N. A. Aziz,
6(a). As shown in Fig. 6(b) and (c), the proposed method can                                                                                                            “Experimental study of lte signals as illuminators of opportunity for
obtain the object reconstruction when SNR=10 dB and 0 dB                                                                                                                passive bistatic radar applications,” Electronics Letters, vol. 50, no. 7,
with clear edges. In addition, when mean-square-error (MSE)                                                                                                             pp. 545–547, 2014.
[15]   A. Evers and J. A. Jackson, “Experimental passive sar imaging exploit-
       ing lte, dvb, and dab signals,” in Radar Conference, 2014 IEEE. IEEE,
       2014, pp. 0680–0685.
[16]   R. R. Abdullah, A. A. Salah, N. A. Aziz, and N. A. Rasid, “Vehicle
       recognition analysis in lte based forward scattering radar,” in Radar
       Conference (RadarConf), 2016 IEEE. IEEE, 2016, pp. 1–5.
[17]   J. W. Goodman, Speckle phenomena in optics: theory and applications.
       Roberts and Company Publishers, 2007.
[18]   T. D. Rogers and D. C. Whitley, “Chaos in the cubic mapping,”
       Mathematical Modelling, vol. 4, no. 1, pp. 9–25, 1983.
[19]   A. A. Salah, R. R. Abdullah, A. Ismail, F. Hashim, C. Leow, M. Roslee,
       and N. A. Rashid, “Feasibility study of lte signal as a new illuminators
       of opportunity for passive radar applications,” in RF and Microwave
       Conference (RFM), 2013 IEEE International. IEEE, 2013, pp. 258–
       262.
[20]   M. Series, “Guidelines for evaluation of radio interface technologies for
       imt-advanced,” Report ITU, no. 2135-1, 2009.
[21]   J. G. Andrews, F. Baccelli, and R. K. Ganti, “A tractable approach
       to coverage and rate in cellular networks,” IEEE Transactions on
       Communications, vol. 59, no. 11, pp. 3122–3134, 2011.
[22]   R. H. Brown and R. Q. Twiss, “Correlation between photons in two
       coherent beams of light,” Nature, vol. 177, no. 4497, pp. 27–29, 1956.
[23]   S. Boyd and L. Vandenberghe, Convex optimization.             Cambridge
       university press, 2004.
[24]   M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection
       for sparse reconstruction: Application to compressed sensing and other
       inverse problems,” IEEE Journal of selected topics in signal processing,
       vol. 1, no. 4, pp. 586–597, 2007.
[25]   K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist
       multiobjective genetic algorithm: Nsga-ii,” IEEE transactions on evo-
       lutionary computation, vol. 6, no. 2, pp. 182–197, 2002.
[26]   J. C. Nash, Compact numerical methods for computers: linear algebra
       and function minimisation. CRC press, 1990.
[27]   B. Wohlberg and P. Rodrı́guez, “An iteratively reweighted norm al-
       gorithm for minimization of total variation functionals,” IEEE Signal
       Processing Letters, vol. 14, no. 12, pp. 948–951, 2007.
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