658 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 36, NO. 4, OCTOBER 2011
Optimized Simulated Annealing Algorithm forThinning and Weighting Large Planar Arrays
in Both Far-Field and Near-FieldPeng Chen, YaYu Zheng, and Wei Zhu
Abstract—In this paper, an optimized simulated annealing algo-rithm is proposed for thinning and weighting large planar arraysin both far-field and near-field of underwater 3-D sonar imagingsystems. This optimized algorithm is designed for the large planararray with a fixed sidelobe peak and a fixed current taper ratiounder a narrowband excitation. It applies the approximationfor time delay in the near-field beam pattern, and extends thesimulated annealing algorithm to both far-field and near-field bydefining a new “energy” function. One example of large planararray was used to evaluate the accuracy and efficiency of theoptimized method.
Index Terms—Beam steering, Fresnel approximation, planararray signal processing, sonar signal processing, underwater 3-Dsonar imaging system.
I. INTRODUCTION
T HREE DIMENSIONAL (3-D) real-time acousticalimaging systems have been used extensively in many
applications, such as underwater construction research, envi-ronmental studies, survey of shipwrecks, dredging, and offshoreoil detection [1]–[3]. To prevent grating lobes, half-wavelength
spacing between the transducers of the array shouldnot be exceeded. At the same time, the array should have awide spatial extension to obtain a fine lateral resolution. The
-condition with the fine resolution requirement often resultsin a 2-D array with thousands of transducers [4]. Critical issuesin the development of high-resolution 3-D sonar imagingsystems are 1) the cost of hardware associated with the hugenumber of transducers in the planar array, and 2) the compu-tational load of signal processing [5]. Chirp Zeta Transform[6], [7] and a fast beamforming algorithm [8] are presentedto reduce the computing load. Field-programmable gate array(FPGA) platforms [9], [10] are proposed for the sonar signalprocessing.
In the process of thinning and weighting a large planar array,some transducers need to be turned off and some weighting
Manuscript received April 20, 2011; accepted August 08, 2011. Date of pub-lication September 22, 2011; date of current version October 21, 2011. Thiswork was supported by the Natural Science Foundation of Zhejiang Provinceunder Grants Y1110175, Y1110532, and Y1100803.
Associate Editor: W. Carey.The authors are with the College of Information and Engineering, Zhejiang
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JOE.2011.2164957
parameters are changed in an evenly spaced or periodic arrayto minimize the number of active transducers, and to obtain afixed sidelobe peak (SLP) and a fixed current taper ratio (CTR)(the ratio between the maximum and minimum weight coeffi-cient). Simulated annealing (SA) [11], [12] and genetic algo-rithms [13], [14] are optimization methods for thinning arrays.SA models the annealing (slow cooling) process of metals froma liquid to a solid state, while genetic algorithms model evo-lution and genetic recombination in nature. Genetic algorithmsare generally limited to the optimization of moderately sized ar-rays because of the large number of genes in large planar arrays.The essential elements of the SA algorithm were invented byMetropolis et al. [15]; it was developed to simulate the behaviorof the molecules of a pure substance during the slow coolingthat results in the formation of a perfect crystal (minimum en-ergy state) [11]. It is a stochastic methodology to solve multiob-jective optimization problems. SA algorithm is a Markov chainmethod because of the discrete jump induced by new iteration.If the new configuration causes the value of the “energy” func-tion to decrease, it is accepted as the next entry in the currentMarkov chain. Conversely, if the new configuration increasesthe value of the “energy,” it is accepted with a probability de-pending on the system “temperature,” in accordance with theBoltzmann distribution.
The efficiently optimized SA algorithm that is suited for onlyfar-field has been proposed in our previous work [16]. Thispaper proposes a novel optimized SA algorithm for large sparsearray in both far-field and near-field of underwater 3-D sonarimaging systems. The optimized algorithm applies the approxi-mation for time delay in the near-field beam pattern, and extendsthe SA algorithm to both far-field and near-field by defininga new “energy” function. This work is organized as follows.Section II presents the detailed scheme of the optimized SA al-gorithm. Section III illustrates the experimental results of thin-ning and weighting a large planar array. Section IV gives thediscussion about the thinning results and the thinning process.Finally, Section V provides the conclusion.
II. OPTIMIZED SIMULATED ANNEALING ALGORITHM
A. Applies the Approximation for Time Delayin the Near-Field Beam Pattern
Let us consider a planar array on the plane withsensors. The coordinates of the sensor identified by
CHEN et al.: OPTIMIZED SIMULATED ANNEALING ALGORITHM FOR THINNING AND WEIGHTING LARGE PLANAR ARRAYS 659
the indexes and the unit vector of the steering direction, can be presented as [17]
(1)
The absolute value for beam pattern (BP) of the steering direc-tion in a focusing distance can be expressed as [17], [18]
(2)where is the center frequency of the backscatter sonar signals,
is the array of the weights that are applied to control theSLP, and represents the time delays due to thepropagation from the sensor to the origin in the focusingdistance [19], [20]
(3)
where , is the speed of sound in the medium,the superscript “ ” indicates the transposition operator (bothand are row vectors), the subscript “ ” means the exact timedelays and is the Euclidean norm. The unit vector [17] canbe expressed as
(4)
where is the azimuth angle and is the elevation angle, andthe mapping is unique [17]. To enlarge thevalidity region of Fresnel approximation, this paper adopts thefollowing approximation that has been proposed in our previouswork [21]:
(5)
where the subscript “ ” denotes that the approximation is ap-plied to the time delays. , , and are three constants thatare applied to enlarge the system’s view scene. When ,
, and , (5) would become the time delay forfar-field planar arrays. As compared with the Fresnel approx-imation, (5) helps to enlarge the view scene and reduce thememory requirements for the delay time parameters storage.Use (5) in (2) as
(6)
denotes the vector of the sensor coordinate identified by theindexes presented as
(7)where , and is the distance betweentwo adjacent transducers along the -axis and along the -axis.Substituting (7) in (6) gives
(8)
where
(9)
Because of the symmetry properties shown in (9), the valueranges of and can be restricted to ,in analyzing the performance of BP.
B. Definition of the “Energy” Function for BothFar-Field and Near-Field Conditions
The aim of the optimization procedure is to determine theweight values that make the “energy” function freeze to thelowest state. The “energy” function is defined as
(10)
(11)
where is the maximum absolute value of BP in the focusingdistance ; is the number of focusing distances; is thenumber of active transducers; , , and are the weight fac-tors for each term; and denote the obtained and desiredCTR values respectively; denotes the desired SLP in bothnear-field and far-field; and is the set of values whichsatisfies
(12)
(13)
where the main lobe is excluded from the validity region,is the wavelength of the backscatter signals, and is the
660 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 36, NO. 4, OCTOBER 2011
TABLE ICOMPARISONS OF THE THINNING RESULTS AMONG THE METHOD OF TRUCCO ET AL. [5], CHEN ET AL. [16], AND OURS
� : number of active transducers; � : number of iterations; ��� : maximum sidelobe peak; CTR: current taper ratio.
Fig. 1. Normalized BPs with an SLP of �22 dB at focusing distance �: (a) � � 3.5 m; (b) � � 7.5 m; (c) � � 15 m; (d) � � 80 m.
diameter of the circular boundary for the received array.Assuming that there are beam signals in the firstquadrant of space, those beams need to be com-puted in the azimuth and the elevation directions, respec-tively. The , which is a set of discrete data, constrains the
to and. and are
two constants, because the maximum number of beams tobe formed is a constant for each particular 3-D underwaterimaging system.
C. The Procedure of the Optimized Simulated Annealing
In this section, the detailed procedure of the optimized SAalgorithm is illustrated. The optimized procedure is started by
considering a fully sampled array, where is the diameter ofthe received array, and the weight values which are out of thecircular boundary are always set to zero. The objective of theprocedure is to minimize the number of active transducers in thefully sampled array satisfying a desired CTR and SLP for bothfar-field and near-field conditions. The procedures are describedas follows.
Step 1. The weight values and the initial analog “tem-perature” for the first Markov chain are initialized. Theweight values are initialized with random values of 0 or 1. Theinitial “temperature” is chosen high enough so that thefirst configuration perturbation will be almost always accepted,even though it leads to a sharp increase in the “energy” function.
Step 2. Choose a random transducer weight value andassign to . At each iteration, all the transducers are vis-
CHEN et al.: OPTIMIZED SIMULATED ANNEALING ALGORITHM FOR THINNING AND WEIGHTING LARGE PLANAR ARRAYS 661
Fig. 2. (a) Optimized layout of the 401 active transducers. (b) Weight values of the 401 active transducers. (c) The number of active transducers versus the numberof iterations.
ited according to a random sequence that does not revisit thesame transducer before all others have been visited once.
1) If the chosen transducer is inactive, it can be activated bythe resurrection probability which is defined as follows:
(resurrection)
(14)
where is a weight factor, ,. If the resurrection probability is larger than a random
uniformly distributed number, a random weight value isassigned to the chosen transducer and the weight values
are updated. Otherwise, it goes to Step 2.2) If the chosen transducer is active, it always changes to in-
active and the weight values are updated. The inactivestate of the chosen transducer will be retained when the“energy” function is reduced by this variation; then, go toStep 3. Otherwise, the inactive transducer would be reacti-vated and the weight value would be changed as follows:
(15)
where is one of Matlab math func-tions that generate uniform random numbers between
.
3) If the new weight decreases the value of the “energy”function, it will be accepted as the next weight . Ifit increases this function, it will be accepted or restoredas the weight value with a probability that depends onthe “temperature” of the system; the probability can be ex-pressed as
if
otherwise(16)
where is the “energy” function in the th iteration,is new state “energy,” is the Boltzmann constant, andis the system “temperature.”
Step 3. If all transducers have been accessed, update the it-eration number and the system “temperature” function ,where
(17)
otherwise the optimization goes to Step 2.Step 4. If the iteration procedure satisfies the termination cri-
terion which is defined in (18) (the active transducers do notdecrease in the last iterations), the optimization will termi-nate; otherwise it goes to Step 2
(18)
662 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 36, NO. 4, OCTOBER 2011
Fig. 3. (a) Interelement spacing is equal to � with logarithmic scale in � when � � �. (b) Interelement spacing is equal to �� with logarithmic scale in �
when � � �.
Fig. 4. (a) Normalized BPs integration along � for a 100 � 100 rectangular grid planar array with all ones weight values (� � 80 m). (b) Normalized BPsintegration along � for a thinned 100 � 100 rectangular grid planar array with 401 transducers (� � 80 m).
III. EXPERIMENTAL RESULTS
To evaluate the efficiency of the method, we employed a 100100 rectangular grid planar array with a intertransducer
spacing. In [16], a maximum value of SLP equaled to 22 dBand a CTR of 2.75 with 403 transducers were achieved in thefar-field. Thus, the parameters in the optimized SA were set asfollows: ( 22 dB)and . The other parameters were fixed as follows:
, , , , ,, , 3.5 m, 7.5 m, 15 m, 80 m, and
the center frequency for the backscatter echoes was 300 kHz.The three focusing distances , , and are in the near-field,and the other distance is in the far-field. When 3.5 m, ,
, and equaled to 0.998, 0.500, and 0.505, respectively;when 7.5 m and 15 m, , , and equaled to1.000, 0.500, and 0.500, respectively; when 80 m, ,
, and equaled to 1.000, 0, and 0, respectively. The radiusof each pair excluding the main lobe was .
In the space , the number ofbeams’ intensity to be calculated was 200 200. After several
runs of the proposed SA, the best results were obtained. Table Igives the comparisons of the thinning results among the methodof Trucco et al. [5], Chen et al. [16], and ours.
As compared with the method of Trucco et al. [5], the op-timized planar array has fewer active transducers, lower CTR,and similar BP performance. As compared with the method ofChen et al. [16], the optimized planar array has fewer activetransducers, fewer iterations number, slightly larger CTR, andsimilar BP performance. The normalized BPs of the 401 activetransducers with an SLP of 22 dB at are shownin Fig. 1.
Fig. 2 shows the optimized layout of transducers, weightvalues of the optimized transducers, and the number of ac-tive transducers versus the number of iterations. The activetransducers’ positions are shown as dots in Fig. 2(a); the max-imum and minimum weight values for the active transducers[Fig. 2(b)] are about 1.75 and 0.58, respectively. The numberof active transducers in Fig. 2(c) shows a rapid descent slopeduring the first phase (from iteration 1 to about iteration 10)and is quasi-constant during the second phase (from aboutiteration 10 to iteration 128). The optimized array achieved a
CHEN et al.: OPTIMIZED SIMULATED ANNEALING ALGORITHM FOR THINNING AND WEIGHTING LARGE PLANAR ARRAYS 663
Fig. 5. (a) Normalized BPs integration along � for the sparse array with 403 transducers reported in [16] at the focusing distance � � 3.5 m. (b) NormalizedBPs integration along � for the sparse array with 403 transducers reported in [16] at the focusing distance � � 7.5 m.
similar angular resolution (measured at 3 dB) to the resolu-tion achieved by Trucco et al. [5] and Chen et al. [16], wherethe interelement spacing was equal to or . The angularresolutions are shown in Fig. 3, where 0.64 ,
0.32 , respectively.
IV. DISCUSSION
One of the main drawbacks in general thinning process isthe high level of SLP. But the optimized SA can constrain theSLP to a fixed level. Normalized BPs at focusing distance80 m are shown in Fig. 4 where normalized BPs of a 100100 rectangular grid planar array with all ones weight valuesare shown in Fig. 4(a), and normalized BPs of the thinned arraywith 401 transducers are shown in Fig. 4(b). As seen in Fig. 4,to reduce the number of active transducers, the SLPs that arefar from the main lobe are allowed to be increased during thethinning process. At the same time, the SLPs that are near themain lobe are constrained to a fixed level.
The SLPs are determined by the number of active trans-ducers, the locations of the active transducers, and the valuesof the transducers’ weights. The weighting parameter results in[16] were applied to the near-field with 3.5 m and7.5 m, and the normalized BPs are shown in Fig. 5, where theSLPs are increased and the maximum SLPs equal to 13.63and 18.76 dB, respectively.
In this paper’s thinning example, four sets of the delay timeparameters (for example 3.5 m, 7.5 m, 15 m,and ) are chosen for the focusing distances from 2 to5 m, from 5 to 10 m, from 10 to 20 m, from 20 to 140 m, re-spectively. The number of the focusing distances that participatein the thinning process can be varied according to each partic-ular 3-D sonar imaging system.
V. CONCLUSION
The optimized SA is proposed for thinning and weightinglarge planar arrays in both far-field and near-field of underwater3-D sonar imaging systems, and it achieves a similar BP per-formance with fewer active transducers as compared with other
published SAs. Because the optimized SA employs fewer activetransducers to obtain a fixed SLP and a fixed CTR, the costs ofthe thinned planar array and the signal computation load are re-duced. The underwater experimental data need to be analyzedto validate the optimized SA algorithm in our future work.
REFERENCES
[1] A. Davis and A. Lugsdin, “High speed underwater inspection forport and harbour security using Coda Echoscope 3D sonar,” in Proc.IEEE/MTS OCEANS Conf., 2005, pp. 2006–2011.
[2] R. K. Hansen and P. A. Andersen, “The application of real time 3Dacoustical imaging,” in Proc. OCEANS Conf., 1998, pp. 738–741.
[3] R. K. Hansen, U. Castellani, V. Murino, A. Fusiello, E. Puppo, L. Pa-paleo, M. Pittore, M. Gobbi, L. Bisone, K. Kleppe, and M. Hall, “Mo-saicing of 3D sonar data sets—Techniques and applications,” in Proc.IEEE/MTS OCEANS Conf., 2005, pp. 2326–2333.
[4] V. Murino and A. Trucco, “Three-dimensional image generation andprocessing in underwater acoustic vision,” Proc. IEEE, vol. 88, no. 12,pp. 1903–1948, Dec. 2000.
[5] A. Trucco, M. Palmese, and S. Repetto, “Devising an affordable sonarsystem for underwater 3-D vision,” IEEE Trans. Instrum. Meas., vol.57, no. 10, pp. 2348–2354, Oct. 2008.
[6] M. Palmese and A. Trucco, “Three-dimensional acoustic imagingby chirp zeta transform digital beamforming,” IEEE Trans. Instrum.Meas., vol. 58, no. 7, pp. 2080–2086, Jul. 2009.
[7] B. Maranda, “Efficient digital beamforming in the frequency domain,”J. Acoust. Soc. Amer., vol. 86, no. 5, pp. 1813–1819, 1989.
[8] B. O. Odelowo, “A fast beamforming algorithm for planar/volumetricarrays,” in Proc. 39th Asilomar Conf. Signals Syst. Comput., 2005, pp.1707–1710.
[9] B. E. Nelson, “Configurable computing and sonar processing—Archi-tectures and implementations,” in Proc. 39th Asilomar Conf. SignalsSyst. Comput., 2001, pp. 56–60.
[10] B. L. Hutchings and B. E. Nelson, “Gigaop DSP on FPGA,” in Proc.IEEE Int. Conf. Acoust. Speech Signal Process., 2001, pp. 885–888.
[11] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simu-lated annealing,” Science, vol. 220, no. 4598, pp. 671–680, May 1983.
[12] A. Trucco, “Thinning and weighting of large planar arrays by simulatedannealing,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 46,no. 2, pp. 347–355, Mar. 1999.
[13] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans.Antennas Propag., vol. 42, no. 7, pp. 993–999, Jul. 1994.
[14] R. L. Haupt, “Optimized weighting of uniform subarrays of unequalsizes,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1207–1210,Apr. 2007.
[15] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, andE. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys., vol. 21, no. 6, pp. 1087–1092, Jun. 1953.
664 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 36, NO. 4, OCTOBER 2011
[16] P. Chen, B. Shen, L. Zhou, and Y. Chen, “Optimized simulatedannealing algorithm for thinning and weighting large planar arrays,”J. Zhejiang Univ-Sci. C (Comput. & Electron.), vol. 11, no. 4, pp.261–269, Apr. 2010.
[17] H. L. Van Trees, Optimum Array Processing. Part IV of Detection,Estimation, and Modulation Theory. New York: Wiley, 2002, pp.235–237.
[18] R. O. Nielsen, Sonar Signal Processing. Boston, MA: Artech House,1991, pp. 55–91.
[19] L. J. Ziomek, “Three necessary conditions for the validity of the Fresnelphase approximation for the near-field beam pattern of an aperture,”IEEE J. Ocean. Eng., vol. 18, no. 1, pp. 73–75, Jan. 1993.
[20] A. Trucco, “A least-squares approximation for the delays used infocused beamforming,” J. Acoust. Soc. Amer., vol. 104, no. 1, pp.171–175, Jul. 1998.
[21] P. Chen, X. Tian, and Y. Chen, “Optimization of the digital near fieldbeamforming for underwater 3-D sonar imaging system,” IEEE Trans.Instrum. Meas., vol. 59, no. 2, pp. 415–424, Feb. 2010.
Peng Chen was born in Zhengjiang Province, China,in 1981. He received the B.Sc. and Ph.D. degrees inengineering instrument science and technology fromZhejiang University, Hangzhou, China, in 2003 and2009, respectively.
He is currently a Lecturer at Zhejiang Universityof Technology, Hangzhou, China. His major researchfields are sonar signal processing and parallel pro-cessing.
YaYu Zheng was born in Zhejiang Province, China,in 1978. He received the B.Sc. degree in powerelectronics and the Ph.D. degree in engineeringinstrument science and technology from ZhejiangUniversity, Hangzhou, China, in 2002 and 2008,respectively.
He is currently a Lecturer at Zhejiang Universityof Technology, Hangzhou, China. His major researchfields are signal processing and intelligent analysis.
Wei Zhu was born in Zhejiang Province, China, in1982. He received the B.Sc. and Ph.D. degrees inengineering instrument science and technology fromZhejiang University, Hangzhou, China, in 2004 and2010, respectively.
He is currently a Lecturer at Zhejiang Universityof Technology, Hangzhou, China. His major researchfields are signal processing and video coding.
