A STUDY OF SIDELOBE REDUCTION FOR LIMTCED DIFFRACTION BEAMS

Jim-yu Lu and James F. Greenleaf

Biodynamics Research Unit, Department of Physiology and Biophysics, Mayo Clinic and Foundation, Rochester. h4N 55905

ABSTRACT used in both !mumission and reception) at the expense of

~ i ~ i ~ ~ d diffraction beams have a lqe depth of field frame rate [101. The price paid for the sidelobe reduction

and are approximately depth-independent, even if they are of limited diffraction pulseecho systems is similar to that

produced with a finite nerefore, they may have for a conventional system in wbich a montage process is applications in medical imaging, tissue ~acterization. used to increase the depth of field while maintaining low nondeshctive evaluation of materials, as well as 0 t h ~ sidelobes and thus also lowing the frame rate. In this pa- physical branches such as optics and electromagnetics. per, we sh@ & ef6cac~ of & summation-subtraction However, limited diffraction beams have larger sidelobes method when stepwise weighting$ are used to than those of conventional beams at their focuses. A produce the beams. A transducer that could be used with

method has been proposed for reducing the sidelobes of the method is

limited difkacticm beams at the expense of frame rate. 11. THE SUMMATION-SUBTRACTION METHOD In this paper, we study the efficacy of the method when Two examples of limited difkaction beams. the stepwise aperture weighting functions are used to produce Bessel beam [I] and X wave [11,12], will be studied for the beams. Results show that the method works well their sidelobe reduction. Zeroth-order limited difkaction with a transducer of only 14 rings and 16 sectors. The beams have both a mainlobe and high sidelobes. and transducer has a 25 mm diameter and 3.5 MHz central second-order beamrj do not have the mainlobe but simi- frequency, and an produce a pulseeeho response that lar sidelobes. Therefore, subtraction of A-lines obtained maintains a -6-dB mainlobe width of about 1.83 mm with beams of different orders may result in a dramatic over a depth of field of about 150 mm. sidelobe reduction. If we assume that biol&cal soft tis-

L INTRODUCTION

Limited diftktion beams are a special class of soh- fions to the isotropic-homogene~u~ wave equation. They wonld propagate to infinite distance without spreading, provided that they were produced with an infinite apelture and energy [I]. Even if produced with a finite aperture. these beams have a large depth of field and approximate depth-independent propexty [I]. Because of these proper- ties, limited diffraction beams could have applications in medical imaging L2.31. tissue characterization [41. nonde- structive evaluation of materials [51, color Doppler imag- ing [61, and other wave related physical branches such as optics 171 and elecfmmagnetics 18-91.

Although limited difEraction beams have a large depth of field, they have higher sidelobes than those of conventional beams at their focuses. Sidelobes may lower contrast in medical imaging affecting the detection of low scattering objects such as m a l l cysts. Sidelobes also in- crease the effective sampling volume in tissue character- ization. In addition. sidelobes are a source of multiple scattering in nondestructive evalualion of materials.

We have developed a summation-subtraction method that reduces dramatically the sidelobes of l i i t e d diffrac- tion pulseecho systems (limited diffraction beams are

sues are composed of point scatterers that are distributed randomly in the half space, z > 0, and multiple scattering among the scatters is negligible, the A-line produced by a Bessel pulse-echo system after sidelobe reduction is given by [I01

4 2 ) = e ~ ~ ( f ) - [ e ~ , ( t ) l + ~ = o + ~ J , ( ~ ) I + ~ = ~ / = I ]

= 4?r2 7 r d r 1 d4 7 d z A ( r , 4 , z ) (1) -co -7, 0

[ J i ( a r ) - Jz (ar)] 3-' {T2 (w)e"Pa 1, where eJ2 (t) l + , , ~ and e ~ , ( t ) l+o=s/4 are A-lines obtained with m d - o r d e r Bessel beams that are rotated 4 4 rel- ative to each other, 40 is an initial angle of the second- order Bessel beams, eJ, (t) is an A-line obtained with a zeroth-order Bessel beam. A ( r , 4, z) denotes tbe Met- tion coefficients of scatterem and is a function of the spa- tial coordinates Cm cylindrical ccmdinates). r' = (r , 4, z ) . Jo(.) and Jz ( . ) are the zeroth- and second-order Bessel functions of the fist kind, F-' denotes the inverse Fourier transform with respect to the angular fresuency, w . T=(w) is a combined transmitting and receiving trans-

fer function of the transducer, and p = dTW/C)2-C12, where c is the speed of sound (assumed to be 1500 mls)

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in biological d t tissues, and a is a scabg factor of the Bessel functions. Similarly. the A-line obtained with an X wave pulseecho system after the sidelobe reduction is given by [lo1

. , where G(w,r,C) = ~,2(:rsin~) - J;($rsinC). ex,(t) and ex,@) are the A-lines obtained with the zroth- and second-order X waves [11,121, respectively,

H ( % ) = { is the ~eaviside step function

[131. ~ ' (wjc) = T2(w) is the a r m b i i transmitting and mxiving transfer function of the ixmsducer, no is a parameter that controls the high-frequency camponent of the X waves and C is the Axicon angle.

Frmn Eqs. (1) and (2) we see that I Ji(.) - J;( ) I is much d e r than J,Z(.) except in the mainlobe baause both J;() and Jz(.) have similar sidelobes but J;( . ) does not have a mainlobe. Therefore, the scatterm in the region where I J,Z( ) - J;( ) I is mall do not have significant contributions to the A-lines, e( t ) .

We now explain the sidelobe reduction in Fig. 1. If there is only one point scatterer in the space, 6, or A(r, 4, z ) = 6(?- 6). Fqs. (1) and (2) leprasent the point spread functions of pulseecho systems. The point spread functions in a hausverse plane of the zeroth-order. second-order before rotatim, and second-order after lr14 rotaticm beams are shown in P w l s (1). (2). and (3). respectively. The summation of Panels (2) and (3) is shown in Panel (4). and the subtraction of Panel (4) from Panel (1) results in a significant sidelobe reduction panel (5)). In Fig. 1, we assume that a = 12CQ.45 m-'. and the central wavelength is 0.6 mm (2.5 MHz central freclue~y).

The point spread functions in Kg. 1 are plotted along rhe'u diameters (Fig. 2). The parameters in Fig. 2 are as follows: a = 1217.51 m-I, and the central wavelength is 0.429 mm (3.5 MHz central fresuency).

m. STEPWISE APERTURE WEIGHTING IN ANGULAR DIRECTION Eqs. (1) and (2) are formulas for sidelobe redue

tion of limited diffractioo beams that are produced with a trmducer of an infinite a p e m . The fmulas for the sidelobe reduction with a finite aperhue transducer can be obtained with the Rayleigh-Sommerfeld diffrac-

tion formula [lo. 141. In this section. we study the influ- ence on the sidelobe reduction of limited diffraction beams produced with a 25 mm diameter transducer of stepwise aperture weightings in the angular direction. 4 , and exact aperture weightings in tbe radial direction. r. The trans- mitting or receiving transfer functions, T(w) and B(w/c) in Fqs. (1) and (2). of the transducer are assumed to be a Bladrman window function [I51 that peaked at 3.5 MHz with a relative bandwidth of about 81% (-6-dB band- width wer the central £requency). The speed of sound. c. is assumed to be 1500 m/s giving a central wavelength of about 0.429 mm.

Line plots along the diameters of the point spread functions of a Bessel pulseecho system before and after the sidelobe reduction are shown in Fig. 3. The trans- ducer is divided into sectors to apply the stepwise aperhue weigh* to produce the second-order beams. The side- lobes are reduced as the number of sectors is increased. The scaling factor of the Bessel beams is a = 1217.51 m-' and the depth of field is about 150.0 mm [I]. Line plots along the diameters of the point spread functions of an X wave pulseecho system before and after the side- lobe reduction are shown in Fig. 4. The parameters of the X wave are as follows: a. = 0.05 mm. C = 4.75". and the depth of field [Ill is about 150.4 mm. The maximum sidelobes in the 2?r angle of 4 are plotted versus the radial distance (Fig. 5), when a 3.5 MHz tone bmt is used to drive the Bessel pulseecho system in Fig. 3.

N. STEPWISE APERTURE WEIGHTING IN BOTH RADIAL AND ANGULAR DIRECTIONS

In this section. we study the sideIobe reduction when stepwise a p e m weightings are applied to both the radial and angular directions. Along the radial direction, T.

the transducer is divided into 14 rings. The positions of the rings are debmined by the lobes of the absolute values of the Bessel function. I Jo(rYr)l. With the abwe parameters (a = 1217.51 m-' and the diameter of the transducer of 25 mm), (Jo(ar)( has about five lobes (Fig. 2). The 6rst lobe is divided into 2 rings of equal distance and each of the rest is divided into 3 rings of equal distance. Results of the sidelobe reduction for the 14-ring transducer of different numbers of sectors are shown in Figs. 6 to 8 that correspond to Figs. 3 to 5, respx!ively. There is little difference between these two gmups of plots. Again. we see that sidelobes are reduced as the number of the sectors is increased. However, when the number of the sectors is greater than 16. the reduction of the sidelobes is diminished while number of the !mmducer elements and thus the system complexity may be increased dramatically.

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V. A PROPOSED TRANSDUCER

From the above studies, we may design a trans- ducer that can produce both the zeroth- and higher-der limited dift?action beams to reduce the sidelobes of lim- ited diffraction pulseecho systems with the summation- subtraction method (Fig. 10). The transducer is uunposed of 14 nngs as described in the last section and the rings are divided into 16 sectors. The total number of elements of the transducer is 209 (the center element is not divided into sectors and is only used for producing zeroth-order beams). Because the limited diffrachon beams are sym- me& about the center of the tmmducer in any opposite sectors, the number of w k s required to drive the trans- ducer is only 105 (the remaining elements are connected inside the transducm). To reduce acoustic cross talk, 1-3 ceramics/polymer composite material may be used to con- struct the transducer. The tramdux proposed in Fig. 10 has the same struchrre as that we cons!mcted and used with a wobbler for real-time. medical imagi i [la, ex- cept that the former is divided into 16 sectors.

VI. DISCUSSION

A. Frame Rate. The price paid for sidelobe reduc- tion of limited diffraction beams with the summation- subtraction method is that the frame rate is reduced to In. More higher-order limited diffraction beams can be used to further reduce the sidelobes [lo. 171, which lowers farther the frame rate. Low frame rate causes distortions when imaging fast moving obpCts such as the heart.

B. Dynamic Range. Because A-lines are subtracted in the summation-subtraction method, large conbibutions frona sidelobes may be subtracted. This redm the dy- namic range of the signals after the subtraction and thus lowers the signal--noise ratio.

C. Motion Artifmts Apparently. the smmnation- subtraction method is sensitive to the motion of objects a transducer. However, if the time between the adja- cent A-lines for the summation and subtraction is short, the hiluece of motion will be mall. Steering l i i t ed diffraction beams with a two-dimensional phased array [I81 may eliminate the motion of the transducer.

D. System Complexity. Like the montage process for M i g the depth of field while maintaining low side- lobes inconventional scanners, the summation-subtraction method for reducing the sidelobes of limited diffraction beams also incxeases the system complexity. However, limited dift?action beams continue to have an approxi- mate depth-independent property wer a large depth of field after the sidelobe reduction.

E OtherMethods. Because limited diffraction beams have an approximately depth-independent point spread

function w i e their deep depth of field, image restoration methods such as the W i ~ 6ltaing may be used to enhaace resolutiom as well as suppress the sidelobes [19. 201.

VIL CONCLUSION The summatiou-subkaction method for reducing

sidelobes of limited difflaction beams works not only with an exactly weighted finite aperture transducer [lo]. but with ! m m k s weighted approximately with a stepwise functim Results from the computer simulations show that with a 3.5 MHz central frequency (0.429 mm central wavelength), 14-ring and 16-sector transducer (Fig. 10). the sidelobes of a limited diffraction pulseecho system whase point spread function has a -6-dB mainlobe width of about 1.83 mm can be reduced dramatically over a lacge depth of field of about 150 mm, although the frame rate is reduced to 1/3.

wI. ACKNOWLEDGMENTS The authors appreciate the secretarial assistance of

Elaine C. Qume. This work was supported in part by grants CA 43920 and CA 54212 from the National Institutes of Health.

M. REFERENCES

1. Dumin. J. I. Miceli Jr., and J. H. Eberly, "DiRraction- free beams." Phys. Rev. Lett. 58(15):1499-1501, April 13. 1987. Jian-yu Lu and I. F. Greenleaf, "Ultrasonic nondiffracting Uansducer for medical imaging." IEEB Trans. Ultrason Ferroelec. Freq. Contr. 37(5):438-447, Sept. 1990. Jian-yu Lu and J. F. Greenleaf. "Pulse-echo imaging using a nondiffracting beam transducer," Ultrason. Med Biol. 17(3):265-281. May, 1991. Jian-yu Lu and J. F. Greenleaf, "Evaluation of a non- djffractiog transducer for tissue characterization," IEeE 1990 ultrasou. Symp. ~ m c . 9W938-9 2795-798, 1990. Jian-yu Lu and J. F. Greenleaf. "Producing deep depth of field and depth-independent resolution in NDE with limited diffraction beams." Ultrason. Imag. 15(2):134-149, April, 1993. P. A. Magnin. "A review of Doppler flow mapping tecb- niques." IEE 1987 Ultrason. Symp. PIOC. 87CHZ492-7. 2:969-977, 1987. J. Ojeda-Castaneda and A Noyola-lglesias, ''Nondiffmct- ing wavefields in grin and free-space," Microwave and Op- tical Tech. Lett. 3(12):430-433. Dec. 12, 1990. 1. N. Brittingham, "Pocus wave modes in homogeneous Maxwell's equations: transverse electric mode," J. Appl. Phys. 54(3):1179-1189, 1983. R. W. Ziolkowski. D. K. Lewis, and B. D. Codi, "Evi- dence of localized wave transmission," Phys. Rev. Lett. 62(2):147-150, Jan. 9, 1989.

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10. Jian-yu Lu and 1. F. Greenleaf, "Sidelobe reduction for limited diffraction pulseecho systems," IEE Trans. U1- trason Fen'uelec.. Freq. Contr. 4q6) (to be published in Nov., 1993).

11. Jian-yu Lu and J. F. Greenleaf, "Nondifhcting X waves-exact solutions to freeapace scalar wave equation and their finite apermre realizations,'' IEE Trans. Ultra- son. Ferroelec. Fmq. Contr. 39(1):19-31. Jan.. 1992.

12. Jian-yu Lu and 1. F. Greenleaf. "Bwperimental vai6cation of nondiffracting X waves," IEEE Trans Ultrason. Ferro- elec. Preq. Contr. 39(3):441-446, May, 1992.

13. R. Bracewell, The Fourier Transform and its Applications. New York, NY: McGraw-Hill Book Company, 1965, chs. 4 and 6.

14. J. W. Goodman, Intmduction toFourier Optics, New Yolk, NY: McGraw-Hiq 1968. chs. 2-4.

15. A. V. Oppenheim and R. W. Schafer. Digital Sigml Pm- cessing, Englewood Cliffs, NJ: PrenticeHall, Inc., 1975, ch. 5.

16. Jian-yu Lu, T. K. Song, R. R. Kinnick, and I . F. Greenleaf. "In vitro and in viva real-time imaging with ultrasonic limited dif6action beams," IEEE Trans. Med Imag. 12(4) (to be published in Dec., 1993).

17. I. P. Wild, "Anew methodof imageformation with annuhr ape.rtwes and application in radio astronomy." Pmc. Royal Soc. A 286:499-509. 1%5.

18. Jian-yu Lu and 1. F. Greenleaf, "Steering of limited diffrac- tion beams with a two-dimensional array transducer," IEEE 1991 Ultrason. Symp. Proc. 97.CH3118-7, 1:603-607, 1992.

19. Jian-yu Lu and 1. F. Greenleaf. "Diraction-limited beams and their applications for ultrasonic imaging and tissue chara@erization," in New Developmetus in Ultrasonic Transducers and Transducer Sysrems. F. L. Lizzi. Edi- tor. Pmc. SPIE 173392-119, 1992.

20. A. Rosenfeld and A. C. Kak, Digital Picture Processing, New Yok Academic Press, 1982.

Fig. 1 Point spread functions of a Bessel beam pulse - echo sysfem in a transverse plane that is parallel to the transducer surfaw (1) zeroth-order system, (2) second-order system before rotation, (3) s a d - o r d e r system after ~ 1 4 rotation around the beam axis. (4) summation of the two point spread functions of the second-order systems, and (5) absolute values of the subtraction between Panels (4) and (1). Repmduced with permission from Fig. 2 in Reference [1011.

Fig. - 2 Sidelobes of a Bessel pulse-echo system before (dotted lines for a moth-ordm beam, and dashed lines for second-order beams) and after (full lines) subtraction (absolute values).

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Sidelobe reduction for a Bessel pulseecho system when the Pansducer is welghted approximately with 14 nngs along the radial distance and is also weighted approximately along the angular duection The figure has the same format as that of Pig. 3.

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Sidelohe reduction for an X wave pulseecho sys- tem when the ~aosducer is weighted approximately wlth 14 rings along the radial &stance and is also weighted approxnnately along the angular &ection. The figure has the same format as that of Fig. 4.

Fig. 8 Sidelobe reduction for the same Beuel pulse- - echo system as that m Pig. 6, except that a long CW tone hwst (3.5 MHz) is used to drive the transducer and the vertical scale of the plots represents the m a ~ m u m of the envelope of the pulse-echo response along the angular direcnon from 0 to 2~ and plotted versus the radial distance. The figure has the same format as that of Pig. 5.

Fig. 9 A pmposed Pansducer for produnng both zeroth- - and second-order limited diffraction beams and perform- ing the summation-subtraction method to reduce the sidelobes. Apart from the 16 sectors, the Pansducer has the same structure as that we used in real-tnne medical imaging [161.

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